ROBUST AND ACCURATE SIMULATION OF

ELASTODYNAMICS AND CONTACT

Minchen Li

A DISSERTATION

in

Computer and Information Science

Presented to the Faculties of the University of Pennsylvania

in
Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

2020

Chenfanfu Jiang, Assistant Professor

Computer and Information Science
Supervisor of Dissertation

Mayur Naik, Professor

Computer and Information Science
Graduate Group Chairperson

Dissertation Committee

Norman I. Badler, Professor Stephen H. Lane, Professor of Practice

Computer and Information Science Computer and Information Science
University of Pennsylvania University of Pennsylvania

Cynthia Sung, Assistant Professor Danny M. Kaufman, Senior Research Scientist

Mechanical Engineering Creative Intelligence Lab
University of Pennsylvania Adobe Research
ROBUST AND ACCURATE SIMULATION OF ELASTO-
DYNAMICS AND CONTACT

2020

Minchen Li
To my family and friends.

iii
Acknowledgements

First, I would like to deeply thank my parents who always understand and support me.
Thank Prof. Kun Zhou, Dr. Menglei Chai, and Prof. Jijun Li for introducing computer
graphics to me when I was an undergraduate student at Zhejiang University. Thank Prof.
Victor C.M. Leung and Prof. Wei Cai for providing me the opportunity to intern in a foreign
country at the first time of my life which shaped my decision on pursuing a PhD.
Thank Prof. Alla Sheffer, Prof. Eitan Grinspun, Dr. Danny M. Kaufman, Dr. Vladimir
G. Kim, Prof. Justin Solomon, Dr. Xinxin Zhang, and Prof. Robert Bridson. Having
the opportunity to work with you when I was a master student at University of British
Columbia provide me a solid research background, which contributes a lot to my PhD.
Thank Dr. Danny M. Kaufman for guiding me through my four internships at Adobe
Research and introducing elastodynamics and contact problems to me. Without you we
would never come up with the brilliant ideas.
Thank Dr. Ming Gao, Dr. Timothy Langlois, Yu Fang, Joshuah Wolper, Jiecong Lu,
Yupeng Jiang, Xuan Li, Yue Li, Dr. Yixin Zhu, Prof. Bo Zhu, Dr. Xinlei Wang, Yunuo
Chen, Ziyin Qu, Yuxing Qiu, Zachary Ferguson, Dr. Teseo Schneider, Prof. Denis Zorin,
Prof. Daniele Panozzo, Jessica McWilliams, Dr. Yufeng Zhu, Dr. Evan Peng for being
wonderful collaborators and consultants. Thank Prof. Norman I. Badler, Prof. Stephen H.
Lane, Prof. Cynthia Sung, and Dr Danny M. Kaufman for being on my committee.
Last but not least, I would like to express my greatest gratitude to my advisor Prof.
Chenfanfu Jiang who is both a best friend and a role model. It would be impossible for me
to complete all these great works without your support and guidance.

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 ABSTRACT
ROBUST AND ACCURATE SIMULATION OF ELASTODYNAMICS AND CONTACT
Minchen Li
Chenfanfu Jiang

Simulating elastodynamics and contact in a robust and accurate way not only benefits
designing realistic and intriguing animations and visual effects in computer graphics,
but is also essential for industrial design, robotics, mechanical engineering analysis, etc.
However, existing methods are constructed under strong assumptions that limit their
application scenarios to a relatively narrow range, which also make the methods sensitive
to algorithmic parameters such that extensive parameter tuning often needs to be performed
for nearly each different example to obtain consistent quality results. To tackle these
challenges, we propose a robust, accurate, and differentiable elastodynamics and contact
simulation framework that can always reliably produce consistent quality results for any
codimensional solids (volumes, shells, rods, and particles) in a wide range of material, time
step size, boundary condition, and resolution settings with interpenetration-free guarantees
but do not require algorithmic parameter tuning. Based on solid theoretical foundations,
our methods provide controllable trade-off between efficiency and accuracy for different
application scenarios. All the proposed features of our methods are thoroughly verified by
performing extensive experiments and analyses including comparisons to state-of-the-art
methods and ablation study on multiple design choices. Our framework frees designers
from extensive parameter tuning as when traditional methods are used, enables simulating
brand new phenomena that are never achieved before, and already demonstrate effectiveness
in a broader range of application scenarios like robotics design and engineering analysis.

v
Contents

Acknowledgements iv

Abstract v

1 Introduction 1

2 Background 8
2.1 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Implicit Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Optimization Time Integrator . . . . . . . . . . . . . . . . . . . 10
2.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Contact and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Decomposed Optimization Time Integrator 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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 3.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . . . . . 31
3.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1 Limited memory quasi-Newton updates . . . . . . . . . . . . . . 37
3.4.2 Initalizing L-BFGS . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.4 Penalty Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.5 Interface Hessians . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.7 Initializer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.8 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.9 Line Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.10 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5.1 Implementation and Testing . . . . . . . . . . . . . . . . . . . . 45
3.5.2 Termination Criteria . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.3 Iteration Growth with Domain Size . . . . . . . . . . . . . . . . 50
3.5.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.6 Varying Time Step, Material Parameters and Model . . . . . . . . 53
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Incremental Potential Contact 56

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Contact Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Trajectory accuracies . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.1 Constraints and constraint proxies . . . . . . . . . . . . . . . . . 63

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 4.3.2 Implicit Time Step Algorithms for Contact . . . . . . . . . . . . 66
4.3.3 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.4 Barrier Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Primal Barrier Contact Mechanics . . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Barrier-Augmented Incremental Potential . . . . . . . . . . . . . 70
4.4.2 Smoothly Clamped Barriers . . . . . . . . . . . . . . . . . . . . 72
4.4.3 Newton-Type Barrier Solver . . . . . . . . . . . . . . . . . . . . 73
4.4.4 Intersection-Aware Line Search . . . . . . . . . . . . . . . . . . 75
4.4.5 IPC Solution Accuracy . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.6 Constraint Set Update and CCD Acceleration . . . . . . . . . . . 77
4.5 Variational Friction Forces . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.1 Discrete Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.2 Challenges to Computation . . . . . . . . . . . . . . . . . . . . . 79
4.5.3 Smoothed Static Friction . . . . . . . . . . . . . . . . . . . . . . 80
4.5.4 Variationally Approximated Friction . . . . . . . . . . . . . . . . 81
4.5.5 Frictional Contact Accuracy . . . . . . . . . . . . . . . . . . . . 82
4.6 Distance Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6.1 Combinatorial Distance Computation . . . . . . . . . . . . . . . 85
4.6.2 Differentiabilty of d . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.7.1 Unit tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7.2 Stress tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7.3 Frictional contact tests . . . . . . . . . . . . . . . . . . . . . . . 97
4.7.4 Scaling, Performance, and Accuracy . . . . . . . . . . . . . . . . 99
4.8 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.8.1 Computer Graphics Comparisons . . . . . . . . . . . . . . . . . 105
4.8.2 Comparison with engineering codes . . . . . . . . . . . . . . . . 107

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 4.8.3 Large scale benchmark testing with SQP-type methods . . . . . . 108
4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Strain Limiting and Thickness Modeling 112

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.1 Cloth and Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.2 Strain Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.3 Thickness Modeling . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.4 Continuous Collision Detection (CCD) . . . . . . . . . . . . . . 118
5.2.5 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3 Formulation and Overview . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Constitutive Strain Limiting . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.1 Isotropic Constitutive Strain Limiting . . . . . . . . . . . . . . . 124
5.4.2 Anisotropic Constitutive Strain Limiting . . . . . . . . . . . . . . 127
5.5 Modeling Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5.1 Solving IPC with Thickened Boundaries . . . . . . . . . . . . . . 130
5.5.2 Challenges for CCD . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5.3 CCD Lower Bounding . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.4 Additive CCD . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.6 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.6.1 Cloth Material and Membrane Locking . . . . . . . . . . . . . . 137
5.6.2 Exact Strain Limits . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.6.3 Comparison with Splitting Models . . . . . . . . . . . . . . . . . 141
5.6.4 Strain Limiting Comparisons . . . . . . . . . . . . . . . . . . . . 144
5.6.5 Thickness Modeling . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6.6 Cloth Simulation Benchmark . . . . . . . . . . . . . . . . . . . . 150
5.6.7 Cloth Simulation Stress test . . . . . . . . . . . . . . . . . . . . 154

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 5.6.8 CCD Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.6.9 General Shells, Rods, Particles, and Coupling . . . . . . . . . . . 159
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 Conclusion and Future Work 167

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

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List of Figures

1.1 Severe deformation dynamics simulated with large steps. (a) The De-
composed Optimization Time Integrator (DOT) decomposes spatial do-
mains to generate high-quality simulations of nonlinear materials undergo-
ing large-deformation dynamics. In (b) we apply DOT to rapidly stretch
and pull an Armadillo backwards. We render in (c) a few frames of the
resulting slingshot motion right after release. DOT efficiently solves time
steps while achieving user-specified accuracies — even when stepping at
frame-rate size steps; here at 25 ms. In (d) we emphasize the large steps
taken by rendering all DOT-simulated time steps from the Armadillo’s
high-speed trajectory for the first few moments after release. . . . . . . . 4

1.2 Squeeze out: Incremental Potential Contact (IPC) enables high-rate time
stepping, here with h = 0.01s, of extreme nonlinear elastodynamics with
contact that is intersection- and inversion-free at all time steps, irrespective
of the degree of compression and contact. Here a plate compresses and then
forces a collection of complex soft elastic FE models (181K tetrahedra in
total, with a neo-Hookean material) through a thin, codimensional obstacle
tube. The models are then compressed entirely together forming a tight
mush to fit through the gap and then once through they cleanly separate
into a stable pile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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1.3 Our method facilitates robust, stable, intersection-free, and strictly strain-
limited large time step simulations of Lagrangian codimensional hypere-
lastic objects (volumetric bodies, shells, rods, and particles) in a unified,
frictional contact-enabled framework. . . . . . . . . . . . . . . . . . . . 7

2.1 Spatial discretization of deformable solids. . . . . . . . . . . . . . . . 9

2.2 Methods running with a fixed iteration cap can easily running into numeri-
cal explosion issues for different examples. . . . . . . . . . . . . . . . . 14

2.3 Traditional contact constraints definitions easily get invalid under large
displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Uniform deformation examples. . . . . . . . . . . . . . . . . . . . . . . 36

3.2 DOT’s decomposition. Layout and notation for a mesh (left) which is
decomposed into three subdomains (right). . . . . . . . . . . . . . . . . . 40

3.3 Deformation stress-tests. DOT simulation sequences of hollow cat (top)

and monkey (bottom) large-deformation, high-speed, stress-tests. . . . . . 41

3.4 Convergence and Timings. Per-example comparisons of iteration (top)

and timing (bottom) costs, per frame, achieved by each time step solver.
Note that the bars are overlaid, not stacked. . . . . . . . . . . . . . . . . 47

3.5 Subdomain to iteration growth. We plot iterations per DOT simulation

example as the size of the decomposition increases. We observe a trend of
sublinear-growth in iteration count with respect to the number of subdo-
mains, revealing promising opportunities for parallelization. . . . . . . . 48

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3.6 Convergence comparisons. Top: We compare the convergence of meth-
ods for a single time step midway through a stretch script with a 138K
vertex mesh; measuring error with the characteristic gradient norm (CN)
see Section 3.5.2. Middle: We observe DOT’s super-linear convergence
matches LBGHS-H and closely approaches Project Newton’s (PN), while
LBFGS-PD lags well behind and ADMM-PD does not converge. Bottom:
comparing timing, DOT pulls ahead of PN and LBFGS-H, with lower
per-iteration costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 Residual Visualization. With a decomposition (a), current deformation in

(b), and starting error in (c), we visualize DOT’s characteristic convergence
process. In the first few iterations error is concentrated at interfaces and is
then rapidly smoothed by the successive iterations. . . . . . . . . . . . . 54

4.1 Nonsmooth, codimensional collisions. Left: thin volumetric mat falls on

codimensional (triangle) obstacles. Right: a soft ball falls on a matrix of
point obstacles, front and bottom views. . . . . . . . . . . . . . . . . . . 65

4.2 Barriers. Left: log barrier function clamped with varying continuity. We
can augment the barrier to make clamping arbitrarily smooth (see our
appendix). We apply our C 2 variant for best tradeoff: smoother clamping
improves approximation of the discontinuous function while higher-order
continuity introduces more computational work. Right: our C 2 clamped
barrier improves approximation to the discontinuous function as we make
ˆ smaller. . . . . . . . . . . . . . . .
our geometric accuracy threshold, d, 74

4.3 Extreme stress test: rod twist for 100s. We simulate the twisting of a
bundle of thin volumetric rod models at both ends for 100s. IPC efficiently
captures the increasingly conforming contact and expected buckling while
maintaining an intersection- and inversion-free simulation throughout. Top:
at 5.5s, before buckling. Bottom: at 73.6s, after significant repeated
buckling is resolved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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4.4 Friction benchmark: Stiff card house. Left: we simulate a frictionally
stable “card” house with 0.5m × 0.5m × 4mm stiff boards (E = 0.1GP a).
Right: we impact the house at high-speed from above with two blocks;
elasticity is now highlighted as the thin boards rapidly bend and rebound. 79

4.5 Friction smoothing in 1D. Left: increasing orders of our polynomials bet-
ter approximate the friction-velocity relation with increasing smoothness.
Right: Our C 1 construction improves approximation to the exact relation
as we make our frictional accuracy threshold, v , and so the size of static
friction zone, smaller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6 Friction benchmark: Masonry arch. IPC captures the static stable equi-
librium of a 20m high cement (ρ = 2300kg/m3 , E = 20GP a, ν = 0.2)
arch with tight geometric, dˆ = 1µm, and friction, v = 10−5 m/s accuracy.
Decreasing µ then obtains the expected instability and the stable arch does
not form (see the supplemental videos Li et al. [2020b]). Inset: zoomed
100× (orange) highlights the minimal gaps with a geometric accuracy of
ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
small d. 83

4.7 Large deformation, frictional contact test. We drop a soft ball (E =

104 P a) on a roller (made transparent to highlight friction-driven deforma-
tion). Here IPC simulates the ball’s pull through the rollers with extreme
compression and large friction (µ = 0.5). . . . . . . . . . . . . . . . . . 84

4.8 Nonsmoothness of parallel edge-edge distance. When edge AB and CD

are parallel, the distance computation can be reduced to either (a) C − AB
point-edge or (b) D − AB point-edge. Then for the trajectory of C moving
down from above D, the distance gradient is not continuous at the parallel
point even though the distance is always continuously varying. . . . . . . 87

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4.9 Aligned, close and nonsmooth contact tests. Pairs of before and after
frames of deformable geometries initialized with exact alignment of corners
and/or asperities; dropped under gravity. We confirm both nonsmooth and
conforming collisions are accurately and stably resolved. . . . . . . . . . 90

4.10 Erleben tests Top: fundamental test cases to challenge mesh-based col-
lision handling algorithms proposed by Erleben [2018]. Bottom: IPC
robustly passes all these tests even when stepped at frame-rate size time steps. 91

4.11 Large Mass and Stiffness ratios. . . . . . . . . . . . . . . . . . . . . . 92

4.12 Funnel test. Top: the tip of a stiff neo-Hookean dolphin model is dragged
through a long, tight funnel (a codimensional mesh obstacle). Middle
top: due to material stiffness and tightness of fit the tip of the model is
elongated well before the tail pulls through. Middle bottom: extremity of
the deformation is highlighted as we zoom out immediately after the model
passes through the funnel. Bottom: finally, soon after pulling through, the
dolphin safely recovers its rest shape. We confirm that the simulation is
both intersection- and inversion-free throughout all time steps. . . . . . . 93

4.13 Stress test: extreme twisting of a volumetric mat for 100s. Left: IPC
simulation at 10s after 2 rounds of twisting at both ends. Right: at 40s after
8 rounds of twisting. This model, designed to stress IPC, has all of its 45K
simulation nodes lying on the mesh surface. . . . . . . . . . . . . . . . . 94

4.14 Trash Compactor. An octocat (left) and a collection of models (right) are
compressed to a small cube by 6 moving walls and then released. Here,
under extreme compression IPC remains able to preserve intersection- and
inversion- free trajectories solved to requested accuracies. . . . . . . . . . 95

4.15 Roller tests. Simulating the Armadillo roller from Verschoor and Jalba
[2019] (same material parameters) in IPC now captures the expected stick-
slip behavior for the high-friction, moderate stiffness conditions. . . . . . 96

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4.16 Codimensional collision objects: pin-cushions We drop a soft ball onto
pins composed of codimensional line-segments and then torture it further
by pressing down with another set of codimensional pins to compress from
above. IPC robustly simulates the ball to a “safe”, stable resting state under
compression against the pins. . . . . . . . . . . . . . . . . . . . . . . . . 97

4.17 Codimensional collision objects: rollers We modify the ball roller exam-
ple from Figure 4.7 by using only the edge segments (left) or even just
vertices (right) for the moving roller obstacle. For these extremely chal-
lenging tests IPC continues robust simulation exhibiting tight compliant
shapes in contact regions pressed by the sharp obstacles. . . . . . . . . . 98

4.18 High-speed impact test. Top: we show key frames from a high-speed
video capture of a foam practice ball fired at a fixed plate. Matching
reported material properties (0.04m diameter, E = 107 Pa, ν = 0.45,
ρ = 1150kg/m3 ) and firing speed (v0 = 67m/s), we apply IPC to simulate
the set-up with Newmark time stepping at h = 2 × 10−5 s to capture the
high-frequency behaviors. Middle and bottom: IPC-simulated frames at
times corresponding to the video frames showing respectively, visualization
of the simulated velocity magnitudes (middle) and geometry (bottom). . . 99

4.19 Stick-slip oscillations with friction simulated with IPC by dragging an

elastic rod along a surface. . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.20 Scaling tests. Top: applying increasing resolution meshes ranging from
3K to 219K nodes we examine the time (left) and memory (right) scaling
behavior of IPC on a range of resolutions of the twisting Armadillo and
twisting mat (Figure 4.13) examples. Bottom: frames from the highest-
resolution twisting Armadillo example (219K nodes, 928K tets). . . . . . 102

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4.21 Squishy ball test: Simulated by IPC an elastic squishy ball toy model
(688K nodes, 2.3M tets) is thrown at a glass wall. The left three frames
show side views before, at, and after the moment of maximal compression
during impact. The right-most frame then shows the view behind the glass
during the moment of maximal compression, highlighting how all of the
toy’s intricately intertwined tendrils remain intersection free. . . . . . . . 103

4.22 Simulation statistics for IPC on a subset of our benchmark examples.

Complete benchmark statistics are summarized in our supplemental doc-
ument Li et al. [2020b]. For each simulation we report geometry, time
ˆ d and v are generally set w.r.t. to
step, materials, accuracies solved to (d,
bounding box diagonal length l), number of contacts processed per time
step, machine, memory, as well as average timing and number of Newton
iterations per time step solve. When applicable, for friction we addition-
ally report number of lagged iterations, with number of iterations set to ∗
indicating lagged iterations are applied until convergence until (4.17) is
satisfied. We apply implicit Euler time stepping and the neo-Hookean ma-
terial by default unless specified in example name; i.e., “NM” for implicit
Newmark time stepping and “FCR” for the fixed-corotational material model.111

5.1 Strain limiting methods feature table. Here 2-step splitting refers to
solving elastodynamics before a post constraint projection which simul-
taneously handle contact and strain limiting, while 3-step splitting means
elastodynamics, contact, and strain limiting are all decoupled and solved
independently per time step. In the table only Chen and Tang [2010] solve
the 3 terms in a fully coupled way, but by using least squares the solution
is simply an L2 approximation to the original problem. . . . . . . . . . . 116

xvii
5.2 Membrane Locking. A cloth on sphere simulation (8K-node) with (a)
1× stiff membrane of cotton material (0.8M P a) suffers from membrane
locking where bending behavior is artificially stiffened and creased. Re-
ducing membrane stiffness to (b) 0.1× that of measured cotton values
removes artificial bending stiffness but we still observe overly sharp edge
creasing. Next an even softer membrane at (c), 0.01× cotton’s, then avoids
observable membrane locking artifacts altogether, but results in nonphys-
ical stretching inappropriate for cotton. However, (d) applying a softer
0.01× membrane together with our constitutive strain limiting model to
enforce measured cotton strain limits prevents membrane locking while
obtaining natural stretch limits for the material. . . . . . . . . . . . . . . 139

5.3 Extreme Strain Limits. Our constitutive strain limiting model stays ro-
bust and guarantees constraint satisfaction even with extreme (and even
nonphysical) strain limit constraints (1% in (a, c) and 0.1% in (b, d)). Here
the mesh only contains 8K (a, b) or 2K (c, d) nodes. Not in (d) that as
strain-limits tighten beyond those physically plausible and so more con-
straints become active w.r.t. to DoF count, a small amount of membrane
locking artifact becomes apparent; see our discussion below. . . . . . . . 140

5.4 Anisotropic strain limiting. Cloth on sphere drape with varying scales of
real-world membrane stiffness (a-c) and a cloth hang (d) all simulated with
our anisotropic strain limiting model at h = 0.04s. Here the shown frames
are the final static rest shape. We can see that the draped cloth exhibits
differing wrinkle patterns when compared to isotropy. . . . . . . . . . . . 142

xviii
5.5 Comparison of our fully coupled strain limiting solution with tradi-
tional splitting methods. Examples with both a cloth pinned at top and
pulled on bottom (a, b), and a cloth draped on sphere (c-e) can both demon-
strate the difference between fully coupled solutions (a and c) and the
artifacts generated by splitting strain limit solves from elastodynamics
(b,d and e). Here we show the final configuration of each simulation as
it comes to a rest. We can see that with splitting, large errors in elasticity
(membrane and bending) can generate incorrect wrinkling directions (b),
severe compression artifacts (d), or even numerical softening (d and e). . 143

5.6 Strain limiting comparisons. A cloth drape example where initially the
cloth is in xz plane (a, b), and a cloth on sphere example (c-f) showing
comparison of our fully coupled contact-aware constitutive strain limiting
method to ARCSim’s augmented Lagrangian based method under h =
0.04s. From the shown frames when the scenes become (nearly) static, we
can see ARCSim cannot guarantee strain limit satisfaction (b,e,f), and as
we decrease membrane stiffness in (e,f) to avoid membrane locking issue
in (d), the over- compression or stretching artifact gets more obvious. . . . 145

5.7 Strain limiting comparisons. Strain and runtime statistics of the cloth
drape example and the cloth on sphere example in comparison with ARC-
Sim. Here smax and smin are the maximum and minimum strain of all
triangles in a certain time step. As time step size decreases, ARCSim
results satisfy strain limits better, while as membrane stiffness decreaes,
which is necessary to avoid membrane locking, ARCSim results violates
strain limits more. On the other hand, our method always guarantee strain
limit satisfaction agnostic to simulation settings. . . . . . . . . . . . . . . 146

xix
5.8 Cloth stack comparisons. Ten 8K-node square cloth are dropped onto a
square board with friction µ = 0.1 (a). Here ARCSim Narain et al. [2012]
and Argus Li et al. [2018a] are tested with different cloth thickness settings
at h = 0.001s. ARCSim fails to resolve the contact at 5mm (b), and at
10mm there is no interpenetration but the dynamics is off (c). Argus can
successfully simulate cloth with 5mm and 10mm thickness (e,f), but at
1mm, it fails to resolve contact. See Figure 5.9 1st row for our results. . . 148

5.9 Sphere on cloth stack. Ten 8K-node square cloth are dropped onto a
square board with friction µ = 0.1 (1st row). For different IPC dˆ per
column, our method is able to provide controllable thickness modeling for
cloth (the stack heights are visually distinguishable). Then a soft elastic
sphere (E = 10KP a) is dropped onto to the cloth stack and compressed
to the extreme (2nd row), where with 10mm thick cloth there are also
intricate wrinkling behaviors. After the scenes become static (3rd row), we
can see the thickness effect is still robustly maintained. . . . . . . . . . . 150

5.10 Cloth on rotating sphere. A square cloth (85K-node 1st row, 246K-node
2nd row) is dropped onto a sphere and a ground with friction (µ = 0.4 for
both). Then the sphere starts to rotate, and the cloth follows. 8×104 P a and
8×103 P a bending Young’s modulus are used for the 85K- and 246K-node
cloth respectively to produce intricate wrinkling behaviors with different
frequency. Here from left to right we show the 25th (right before rotation),
50th, and 75th frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.11 Funnel. Three 26K-node square cloth are dropped onto a funnel with
friction (µ = 0.4). Then a tetrahedron moving collision object pushes the
cloth through the funnel. . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xx
5.12 Ribbon knot. A classic example in ACM Harmon et al. [2009] with 2
ribbons dragged apart to form a knot. Here our 2 ribbons contain 100K
nodes in total, and we drag them starting from rest shape (a) to nearly reach
the strain limit (b-d), where the friction coefficient is µ = 0.02. Our results
have no interpenetration or jittering artifacts. . . . . . . . . . . . . . . . . 153

5.13 Garments. A yellow dress with knife pleats produced by FoldSketch Li

et al. [2018c] is staged (a) and then draped on a mannequin to static equi-
librium (b) with our method that stitches the seams together. A multilayer
skirt is draped in a same way (c), and then used to simulate with an input
character animation sequence with large motion (d,e,f). As the mannequin
kicking with her leg, intricate garment details are captured by our method. 154

5.14 Pulling cloth on needles. A 26K-node square cloth is first dropped onto
a bed of needles formed directly by codimensional segments (a) and then
pulled to the left at 1m/s without snagging issue (b, c, d). Here from (a) to
(d) we show the 50th (right before pulling), 55th, 65th, and 85th frame. . . 155

5.15 Table cloth pull A 26K-node square cloth is put on a table board with a
cube, a tetrahedron, a cylinder and a torus solid (ρ = 2000kg/m3 , E =
0.1GP a) on it (a). The capability of our method on accurately resolving
controllable friction behaviors is shown here by pulling the cloth leftwards
with different speed (b-c). In (b) the pull is at 1m/s and nearly all objects
follow the cloth due to static friction. As the pull gets faster at 2m/s in (c),
only the cube is falling and the other objects stay on the table after some
slight move. Finally in (d) pulling the cloth at 4m/s left all objects on the
table because of the small duration of the action of dynamic friction. In (c)
and (d) there are also elastic waves on the cloth right after it is detached
from the objects and board. . . . . . . . . . . . . . . . . . . . . . . . . . 156

xxi
5.16 Twisting cylinder. A 1m-wide 88K-node cylinder cloth is twisted and
moved closer at 72◦ /s and 5mm/s respectively at both ends from rest
shape without gravity (a). Right after twisting, intricate folds start to form
(b). Here strain limiting is turned off to test simulation under extreme stress,
which however makes the center part a tiny knot (c). By setting a 1.5mm
IPC offset to model some inelastic thickness, the cloth looks more realistic
at the same frame (d). After 32.96s twisting with the offset setting, we get
lots of buckling behaviors (e) and our method stays robust with thickness
behaviors consistentely modeled. . . . . . . . . . . . . . . . . . . . . . . 157

5.17 CCD comparison statistics. We provide the per Newton iteration total
CCD querying time (excluding spatial hash) in milliseconds for different
simulation examples when we use different CCD methods (EV’s Li et al.
[2020b], MinSep Harmon et al. [2011], Lu et al. [2019], Root Parity Brochu
et al. [2012], BSC Tang et al. [2014], T-BSC Wang et al. [2015], and our
ACCD). The first 5 examples are with inelastic thickness, which is not
supported by Root Parity, BSC, and T-BSC methods. ”Infinite Loop” means
the method never reports ”no collision” even when querying a configuration
with tiny displacement that is interpenetration-free. . . . . . . . . . . . . 158

5.18 Cards shuffling. 54 poker cards are uniformly separated into 2 piles
initially and bent (a), and then one-by-one released in turns from bottom to
up to get shuffled (b). After all cards are released 2 handles are squashed
to make the pile align better (c), and finally we obtain a nicely aligned
shuffled cards pile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

xxii
5.19 Hairs. Left: two hair clusters (each with 650 60-segment rods) are twisted
to form braids and then the bottom end is released. Right: a typical hair
simulation example in McAdams [2009] where a hair cluster with one
end fixed is falling onto another hair cluster with both ends fixed (900
50-segment rods for each cluster). A sphere collision object is also added
below to add more interesting dynamics. . . . . . . . . . . . . . . . . . . 161

5.20 Noodles. 625 40cm-long noodles (each represented with a 200-segment

discrete rod, separating each other by 3.3mm initially) are dropped into a
13.85cm-wide bowl (fixed). Here we set IPC’s offset and dˆ to 1mm and
0.5mm respectively to model the noodles’ thickness at 1.5mm, which let
us reconstruct the surface mesh with 1.5mm-thick cylinders along the rods
for rendering. The controllable and high-quality modeling of thickness here
is also essential for the noodles to pile up and fill the bowl. Our method
robustly simulate this scene until static at around 23min per h = 0.02s
time step without any jittering artifacts. . . . . . . . . . . . . . . . . . . 162

5.21 Sprinkles. 25K 6mm-long and 2mm-thick sprinkles (each represented

with a single-segment discrete rod, separating each other by 3.3mm ini-
tially) are dropped into a 13.8cm-wide bowl (fixed). Here we set IPC’s
offset and dˆ to 1.5mm and 0.5mm respectively to model the sprinkle’s
thickness, which let us reconstruct the surface mesh with 2mm-thick cylin-
ders along the rods for rendering. Our method robustly simulate this scene
until static at around 6.5min per h = 0.02s time step without any jittering
artifacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.22 Sand on cloth. A 10 × 10 × 500 sand column is falling onto a square cloth
with its four corners fixed. Here each sand is represented with a single iso-
lated FEM node, with IPC offset and dˆ set to 2mm and 1mm respectively,
the sand-sand contacts are accurately captured which is essential for the
volumetric effects in this scene. . . . . . . . . . . . . . . . . . . . . . . . 164

xxiii
5.23 All-in. We simulate objects of all codimensions coupled together in a scene
with an Armadillo volume, a square cloth, and a sand column formed by
isolated particles released in sequence to fall onto a rod net. . . . . . . . . 165
1 Ablation study on barrier functions with different continuity. NC
means not converging after 10000 PN iterations when solving a certain
time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
2 CCD strategies ablation. . . . . . . . . . . . . . . . . . . . . . . . . . 174
3 Parallel-edge degeneracy handling. Left: Due to numerical rounding
errors for distances of edge-edge pairs decreases to 0 when the edges get
more and more parallel (see Figure 4.6 for an example); Middle: a zoom-in
view show clearly that the distance really starts decreasing when their
angle’s sine value is at around 10−9 ; Right: we here set a threshold to
force the use of point-point or point-edge formulas to compute the distance
of edge-edge pairs when their angle’s sine value is below 10−10 , which
introduces a negligible (for optimization) nonsmoothness. . . . . . . . . . 181
4 Ablation study on smoothed static friction. . . . . . . . . . . . . . . . 184

xxiv
Chapter 1

Introduction

Solid is one of the four fundamental states of matter, which can be seen everywhere in
the world, from food, furniture, and garments, to cars, plants, and animals. Being able to
simulate the dynamic behavior of solids with complex self-contact configurations under
various external conditions not only allow us to design realistic and intriguing animations
and visual effects in computer graphics, but also can benefit industrial design, robotics,
civil and mechanical engineering, material science, and bio-mechanics, etc.

Extensive researches Bergou et al. [2008], Gast et al. [2015], Grinspun et al. [2003],
Kane et al. [1999], Kaufman et al. [2008], Liu et al. [2017], Teran et al. [2005], Verschoor
and Jalba [2019] have been established for modeling, analyzing, and numerically simulating
elastodynamics and/or contact. However, robust and accurate simulation of these effects
is still challenging because mathematically, elasticity is highly nonlinear and non-convex,
and contact is non-smooth. Simulating elastodynamics with contact in a coupled system
then brings in more challenges. These pose significant difficulties to standard numerical
methods, making them inefficient and unpredictable under potentially extreme conditions
and sometimes even results in numerical explosions. In addition, many existing methods
are constructed under strong assumptions that limit their application scenarios to a relatively
narrow range and also make the methods sensitive to algorithmic parameters such that
extensive parameter tuning often needs to be performed for nearly each different example

1
to obtain consistent quality results. These make the methods inconvenient to use, and also
insufficient for a broader range of applications like robotics design, shape optimization, etc,
where the reliability of methods is essential.
In this dissertation, we aim at innovating robust, accurate, and differentiable elastody-
namics and contact simulation methods that can always reliably produce consistent quality
results for any codimensional solids (volumes, shells, rods, and particles) in a wide range of
material, time step size, boundary condition, and resolution settings with interpenetration-
free guarantees but do not require algorithmic parameter tuning. Based on solid theoretical
foundations, we design our methods to provide controllable trade-off between efficiency
and accuracy for different application scenarios. We also perform extensive experiments
and analyses to thoroughly verify all the above features. This work includes

• 7 ACM SIGGRAPH papers Fang et al. [2019, 2020], Li et al. [2019, 2020b], Wang
et al. [2020b], Wolper et al. [2019, 2020],

• 1 ACM Transactions on Graphics (ToG) article Wang et al. [2020a] presented at

SIGGRAPH 2020,

• 1 International Journal for Numerical Methods in Engineering (IJNME) article Jiang

et al. [2020], and

• 3 submissions Li et al. [2020c,d,e] to top tier conferences or journals under review.

Here we focus on Decomposed Optimization Time Integrator (DOT) Li et al. [2019],

Incremental Potential Contact (IPC) Li et al. [2020b], and Constitutive Strain Limiting and
Thickness Modeling Li et al. [2020c].

Decomposed Optimization Time Integrator Simulation methods are rapidly advancing

the accuracy, consistency and controllability of elastodynamic modeling and animation.
Critical to these advances, we require efficient time step solvers that reliably solve all
implicit time integration problems for elastica. While available time step solvers succeed

2
admirably in some regimes, they become impractically slow, inaccurate, unstable, or
even divergent in others. Towards addressing these needs, in Chapter 3 we present the
Decomposed Optimization Time Integrator (DOT), a new domain-decomposed optimization
method for solving the per time step, nonlinear problems of implicit numerical time
integration. DOT is especially suitable for large time step simulations of deformable bodies
with nonlinear materials and high-speed dynamics (Figure 1.1). It is efficient, automated,
and robust at large, fixed-size time steps, thus ensuring stable, continued progress of high-
quality simulation output. Across a broad range of extreme and mild deformation dynamics,
using frame-rate size time steps with widely varying object shapes and mesh resolutions,
we show that DOT always converges to user-set tolerances, generally well-exceeding and
always close to the best wall-clock times across all previous nonlinear time step solvers,
irrespective of the deformation applied.

Incremental Potential Contact Contacts weave through every aspect of our physical
world, from daily household chores to acts of nature. Modeling and predictive computation
of these phenomena for solid mechanics is important to every discipline concerned with
the motion of mechanical systems, including engineering and animation. Nevertheless,
efficiently time-stepping accurate and consistent simulations of real-world contacting
elastica remains an outstanding computational challenge. To model the complex interaction
of deforming solids in contact, in Chapter 4 we propose Incremental Potential Contact (IPC)
– a new model and algorithm for variationally solving implicitly time-stepped nonlinear
elastodynamics. IPC maintains an intersection- and inversion-free trajectory regardless of
material parameters, time step sizes, impact velocities, severity of deformation, or boundary
conditions enforced (Figure 1.2).

Constructed with a custom nonlinear solver, IPC enables efficient resolution of time-
stepping problems with separate, user-exposed accuracy tolerances that allow independent
specification of the physical accuracy of the dynamics and the geometric accuracy of
surface-to-surface conformation. This enables users to decouple, as needed per application,

3
(a) (d)

(b) (c)

Figure 1.1: Severe deformation dynamics simulated with large steps. (a) The De-
composed Optimization Time Integrator (DOT) decomposes spatial domains to generate
high-quality simulations of nonlinear materials undergoing large-deformation dynamics. In
(b) we apply DOT to rapidly stretch and pull an Armadillo backwards. We render in (c)
a few frames of the resulting slingshot motion right after release. DOT efficiently solves
time steps while achieving user-specified accuracies — even when stepping at frame-rate
size steps; here at 25 ms. In (d) we emphasize the large steps taken by rendering all
DOT-simulated time steps from the Armadillo’s high-speed trajectory for the first few
moments after release.

desired accuracies for a simulation’s dynamics and geometry.

The resulting time stepper solves contact problems that are intersection-free (and
thus robust), inversion-free, efficient (at speeds comparable to or faster than available
methods that lack both convergence and feasibility), and accurate (solved to user-specified
accuracies). To our knowledge this is the first implicit time-stepping method, across both
the engineering and graphics literature that can consistently enforce these guarantees as we
vary simulation parameters.

In an extensive comparison of available simulation methods, research libraries and

commercial codes we confirm that available engineering and computer graphics methods,

4
Figure 1.2: Squeeze out: Incremental Potential Contact (IPC) enables high-rate time
stepping, here with h = 0.01s, of extreme nonlinear elastodynamics with contact that is
intersection- and inversion-free at all time steps, irrespective of the degree of compression
and contact. Here a plate compresses and then forces a collection of complex soft elastic
FE models (181K tetrahedra in total, with a neo-Hookean material) through a thin, codi-
mensional obstacle tube. The models are then compressed entirely together forming a tight
mush to fit through the gap and then once through they cleanly separate into a stable pile.

while each succeeding admirably in custom-tuned regimes, often fail with instabilities,
egregious constraint violations and/or inaccurate and implausible solutions, as we vary
input materials, contact numbers and time step. We also exercise IPC across a wide
range of existing and new benchmark tests and demonstrate its accurate solution over a
broad sweep of reasonable time-step sizes and beyond (up to h = 2s) across challenging
large-deformation, large-contact stress-test scenarios with meshes composed of up to 2.3M
tetrahedra and processing up to 498K contacts per time step. For applications requiring
high-accuracy we demonstrate tight convergence on all measures. While, for applications
requiring lower accuracies, e.g. animation, we confirm IPC can ensure feasibility and
plausibility even when specified tolerances are lowered for efficiency.

5
Constitutive Strain Limiting and Thickness Modeling In Chapter 5 IPC is extended
to consider codimensional degree-of-freedoms. This enables a unified, interpenetration-
free, robust, and stable simulation framework that couples codimension-0,1,2,3 geometries
seamless through frictional contact (Figure 1.3). Extending the incremental potential
contact onto thin structures necessitates several new contributions. We introduce a C 2 -
continuous smooth barrier-based energetically-consistent strain limiting model for cloth
and show that a constitutive model perspective of such inequality constraints can be used to
strictly enforce strain limits without compromising any inconsistency with elastodynamics
and contact in the same implicit procedure. We formulate our approach for both isotropic
and anisotropic strain limiting requirements. To capture geometrically convincing finite
thickness effects of thin structures, we further devise a new offset barrier formulation for
the unsigned distance constraint in IPC with details that are key to robust and efficient
implementations. To fulfill the extreme accuracy requirement of our offset barrier that
fails any existing continuous collision detection (CCD) methods, we design our own
additive CCD (ACCD) method based on iteratively accumulating a rigorously derived
theoretical lower bound of time of impact. We perform extensive benchmark experiments
to validate the efficacy of our method in capturing intricate behaviors of thin structures
without suffering from artifacts or numerical difficulties even in extreme deformations and
large time steps.

Our research brings in significant contributions to computer graphics in both academia

and industry. We at the first time propose a robust and accurate self-contact handling
framework with friction and fully implicit nonlinear elastodynamics that can guarantee
interpenetration-free trajectories with codimension-0,1,2,3 objects modeled with geomet-
ric thickness and fully coupled exact strain limiting. In practice, these technical pieces
effectively turns simulation from an extensive tuning procedure to a plug-and-play process
where after setting up, the predictive results will come out in a finite amount of time, there
will be no need to worry about algorithm failures. What is more exciting is that our methods

6
Figure 1.3: Our method facilitates robust, stable, intersection-free, and strictly strain-limited
large time step simulations of Lagrangian codimensional hyperelastic objects (volumetric
bodies, shells, rods, and particles) in a unified, frictional contact-enabled framework.

also enable creating phenomena that are never being able to be produced by prior methods,
which opens up many new possibilities for the current computer graphics community. Like-
wise, our methods have also shown their effectiveness in helping mechanical engineering
analysis Jiang et al. [2020], topology optimization Li et al. [2020e], and robotices desgin
Li et al. [2020d], and we have multiple ongoing projects to keep pushing this further.
In what follows, we will first introduce the background knowledge of elastodynamics
and contact simulation in Chapter 2. Then we present our methods in detail from Chapter 3
to 5 for Decomposed Optimization Time Integrator (DOT), Incremental Potential Contact
(IPC), and Constitutive Strain Limiting and Thickness Modeling, respectively. Finally, we
conclude our research and discuss a wide range of future work to further extend our work
and spread its impact to more research and industry communities in Chapter 6.

7
Chapter 2

Background

2.1 Time Integration

The dynamic simulation of deformable solids is conducted by time integration, which

is usually discretized into a sequence of discrete time steps that compute internal (e.g.
elasticity) and external (e.g. gravity) forces on an object and integrate them over time to
obtain velocity and position updates to proceed.

2.1.1 Implicit Euler

Taking the most popular time integration scheme, implicit Euler, in computer graphics as
an example, in time step t when object has its nodal positions and velocities stacked up
into vectors xt and v t (Figure 2.1), we have the following update rules:

xt+1 = xt + hv t+1

(2.1)
v t+1 = v t + hM −1 (fint (xt+1 ) + fext )


, where h is the time step size in unit s, M is the nodal mass matrix computed using density
and volume of incident elements of the nodes, fext is the external force, and fint (xt+1 ) is
the internal force evaluated using nodal position xt+1 . As shown in Figure 2.1, usually a

8
volumetric object is represented by a tetrahedral mesh with both its internal and surface
nodes as the degree-of-freedoms in the simulation.

Figure 2.1: Spatial discretization of deformable solids.

Since the above update rules for xt+1 and v t+1 are implicitly defined upon each other,
the problem is often solved by first eliminating one of the variables (e.g. v t+1 ), solve
a (non)linear system of equations for the other one (e.g. xt+1 ), and then compute v t+1
using the solved xt+1 . One of the most intriguing features of implicit Euler is that it is
unconditionally stable even with arbitrarily large time step sizes. This means the simulation
with implicit Euler will never explode like explicit time integration schemes even when there
are challenging boundary conditions that triggers large deformation of objects. However,
to really achieve unconditionally stable simulation in practice without numerical issues, the
time integration problem posed by implicit Euler scheme is often reformulated as a smooth
minimization problem, and then solved via gradient-based methods with stabilization
techniques like line search to guarantee global convergence1 .

1
Global convergence: the method is guaranteed to converge to a local optimum starting from any initial
configuration.

9
2.1.2 Optimization Time Integrator

To reformulate the problem in Equation 2.1, we eliminate v t+1 , gather all terms on one side
of the equation and integrate this side w.r.t. xt+1 to obtain an incremental potential
1
E(x) = ||x − x̃t ||2M + h2 Ψ (x) (2.2)
2
where x̃t = xt +hv t +h2 M −1 fext is called the predictive position, and Ψ (x) = −
R
fint (x)dx
is the elasticity potential, which will be discussed in Section 2.2. Then minimizing this
E(x) effectively give us the updated position:

xt+1 = arg min E(x) (2.3)

x

. This is because by solving this minimization problem we obtain ∇E(xt+1 ) = 0, which is

equivalent to our update rule.
A standard way to robustly solve the reformulated minimization problem with global
convergence is to apply Newton-type method with line search. Newton-type method is
an iterative approach that approximate the nonlinear objective function using a quadratic
function and solve for a search direction to update the iterate in each iteration. Specifically,
in each Newton iteration i towards solving the minimization problem in time step t, with
the current iterate xi , the quadratic approximation to the problem given by Taylor expansion
is
1
min E(xi ) + pT ∇E(xi ) + pT H(xi )p (2.4)
p 2
where H(x) is a symmetric positive definite (SPD) proxy matrix containing 2nd-order
information of E(x), and the search direction p can be obtained by setting the gradient of
the objective in Equation 2.4 to 0 and solving the linear system

p = −H(xi )−1 ∇E(xi ) (2.5)

. After solving for p, xi+1 can be obtained as

xi+1 = xi + αp (2.6)

by line search that ensures E(xi+1 ) ≤ E(xi ), and then we proceed to the next iteration
until convergence (Algorithm 1).

10
Algorithm 1 Newton-type Method for Optimization Time Integrator
1: procedure N EWTON T YPE M ETHOD(xt , v t , h, xt+1 , v t+1 )

2: x ← xt
3: Eprev ← E(x), xprev ← x
4: do
5: H ← computeProxyMatrix(x)
6: p ← −H −1 ∇E(x)
7: α←1
8: do
9: x ← xprev + αp
10: α ← α/2
11: while E(x) > Eprev
12: Eprev ← E(x), xprev ← x
13: while not converged
14: xt+1 ← x, v t+1 ← (x − xt )/h

11
Proxy Matrix The proxy matrix H(xi ) needs to be SPD to ensure that the computed
search direction p is a descent direction (pT ∇E(xi ) < 0) that can guarantee line search
success. Here figuring out a way to efficiently compute a SPD H(xi ) per iteration that
can also be factorized efficiently and ensures fast convergence is essential to the practical
performance of Newton-type method. As we show in Li et al. [2019] (Section 3), accurate
2nd order information of E(xi ) is essential for fast convergence. The widely used Projected
Newton (PN) method directly computes ∇2 E(xi ) per iteration i and efficiently perform
positive semi-definite (PSD) projection per tetrahedron elasticity Hessian to assemble a
SPD proxy matrix. PN achieves fast convergence with its bottleneck at computing and
factorizing the Hessian in each iteration.

For elastodynamics simulation without contact (what we are discussing up to now), we

propose Decomposed Optimization Time Integrator (DOT) Li et al. [2019] to decompose
the simulation domain into smaller ones that could be easily factorized in parallel, and
utilize quasi-Newton L-BFGS update to perform cheap low-rank update to the proxy matrix
so that it does not need to be recomputed per Newton iteration and can still reach super
linear convergence. DOT has an overall best timing performance over all alternatives
on our test set containing challenging examples undergoing large deformation with large
frame-rate time step sizes. See Chapter 3 for more detail.

Linear solve To solve the SPD linear system Hp = −∇E(x) to obtain search direction,
Cholesky factorization combined with backward substitution can be applied. Eigen Guen-
nebaud et al. [2010] LDLT and CHOLMOD Chen et al. [2008] are two popular SPD linear
solvers based on Cholesky factorization, where the latter one supports multi-threading to
achieve even better performance. For large scale elastodynamics simulations with around
300K to 500K or a even larger number of nodes, these direct factorization methods may
easily run out of memory on nowadays personal desktop/laptop computers because the
factors often have much more nonzero entries than the original sparse proxy matrix. In
these situations, iterative linear solvers like conjugate gradient methods can be applied.

12
 Iterative linear solvers solve the linear systems based on applying matrix-vector multi-
plications iteratively, so it does not consume more memory than storing the matrix like in
direct factorization methods. In some situations like fluids simulation, the matrix-vector
multiplications can even be implemented in a matrix-free style, which directly computes the
matrix-vector multiplication product without explicitly computing and storing every entry
of the matrix, thus achieve even better memory efficiency. However, to achieve nice conver-
gence behavior for the iterative linear solve, suitable preconditioners that could scale down
the condition number of the system is essential for often ill-conditioned elastodynamics
problems, which is not an issue for direct factorization methods.

In Wang et al. Wang et al. [2020a], we propose Hierarchical Optimization Time In-
tegration (HOT) to solve elastodynamics problem in the material point method (MPM)
Stomakhin et al. [2013] setting. Similar to DOT, HOT builds the proxy matrix under
L-BFGS framework, and applies conjugate gradient method to solve the linear system with
a customized efficient Galerkin multigrid as the preconditioner. HOT achieves the best
timing performance compared to all existing implicit MPM solvers. Even for challenging
examples with ill-conditioned system and a large number of degree-of-freedoms where
applying direct factorization is impractical, HOT can still maintain efficient performance
and generate high quality results. See Wang et al. [2020a] for more details.

Line Search Line search is the key to ensure robustness of the time integration. By
making sure that each iteration will only decrease the incremental potential that we are
minimizing, numerical explosions can never happen within any time step when solved to
convergence. Then combined with the implicit Euler update rule which guarantees stability
across time steps, the entire simulation can be totally free from explosion. Since we have
made sure that the search direction p is always a descent direction, which means given
small enough α in each iteration there must be a point with lower energy, we do not need
and should not add any iteration cap to the backtracking/bisection line search loop because
inproper termination of line search can possibly bring back the numerical explosion issues.

13
With PN search directions, usually 0 to 2 halfing of α in each iteration would be enough for
elastodynamics simulation without contact.

Convergence Criteria How to properly determine convergence of the minimization is

in fact an essential component to ensure high quality results (no jittering, damping, or
softening artifacts) but it is often lack of attention in traditional methods. To reach better
timing performance, some methods in computer graphics often terminate the minimization
early by simply setting an iteration cap at e.g. 10. As we show in DOT Li et al. [2019],
this absolute number does not work generally for all examples, which could even lead
to explosion for challenging scenes (Figure 2.2). Other methods stop the minimization
by checking the relative energy decrease (e.g. |(E(xi ) − E(xi+1 ))/E(xi )|) per iteration.
This seemingly good measurement is not adaptive either since different scenes can have
very different energy value. In addition, a small relative energy decrease does not always
indicate convergence. For a nonconvergent method, it can have vanishing relative energy
decreases as it fails to make any significant progress at a certain point, but the iterate can
still be faraway from the optimum.

dragging
force

(a) initial frame (b) a middle frame

Figure 2.2: Methods running with a fixed iteration cap can easily running into numerical
explosion issues for different examples.

To come up with a termination criteria that appropriately measures the distance from
the current iterate to the optimal configuration, and to consider the different scaling in

14
different examples for generality at the same time, we start by extending the Characteristic
Norm (CN) Zhu et al. [2018] that provides this kind of measure for distortion minimization
into elastodynamics simulation in DOT Li et al. [2019] (Section 3). Then we observed that
CN only characterizes the scale of a single elasticity stiffness per scene, so in HOT Wang
et al. [2020a] we extend CN to support complex scenes composed of multiple objects with
different elasticity stiffness by applying different CN scaling per dimension on the gradien
vector. In IPC Li et al. [2020b] (Section 4), as contact and friction potentials are introduced
which have very different form than elasticity, instead of keep exploring CN we found that
the norm of search direction p is in fact a nice measure for the distance to convergence up
to 2nd order approximation accuracy when PN proxy matrix is used. These appropriate
convergence criteria not only allow us to easily set a general tolerance for very different
examples to obtain consistently high quality results, but also avoid performing too accurate
solves that does not make a big difference for visual applications. See our papers for more
details.

2.1.3 Discussion

With the optimization time integration framework, physical simulation of solids could
be formulated and realized in a consistent way by introducing new forces like contact
and friction either in a potential form or as constraints. In Section 2.2 we introduce
hyperelasticity constitutive models in linear finite element setting with a full spectrum of
codimension-0,1,2 elements for simulating volumetric objects, shells, and rods. There are
different spatial discretization choices for simulating solids that has different trade-offs
on multiple aspects of quality, timing performance, and accuracy such as MPM Jiang
et al. [2017a], Stomakhin et al. [2013], Wang et al. [2020a], Wolper et al. [2019] and
Eulerian-on-Lagrangian methods Fan et al. [2013], Weidner et al. [2018]. In this thesis we
focus on finite element method (FEM) and introduce the background knowledge of contact
and friction on points, edges, and triangle elements in Section 2.3. In certain cases Dirichlet
or Neumann boundary conditions will be needed for prescribing part of the physical system.

15
Please see the appendix for more details.

When time step size is large, the implicit Euler time integration scheme we focus on
here can have noticeable numerical damping. Although this is useful for quickly obtaining
static equilibrium with progressive dynamic time stepping that has much better conditioning
than direct static solve, improving implicit Euler with better energy conservation while
maintaining its stability is a meaningful future work. Other than Newton-type method, there
are alternative methods for solving the nonlinear optimization time integration problem
such as ADMM Boyd et al. [2011], Overby et al. [2017] that trade accuracy for better
efficiency. In this thesis we focus on Newton-type methods for its generality and fast
convergence, and encourage future works on accelerating Newton-type methods through
smart solving schemes, GPU optimizations, etc.

2.2 Hyperelasticity

Elasticity describes the ability of solids to retain rest shape under external forces. When
the object deforms, the elasticity force appears and points to the direction for object to
recover rest shape. The essential parts for computing elasticity force and also the force
Jacobian (energy Hessian) as needed by our optimization time integrator is to first figure
out how deformation is described using nodal positions (our simulation DoFs) and what
relation does deformation and elasiticity force has. This depends on the element we use to
discretize the continuum body, which split this section into 3 subsections, i.e. tetrahedral
elements for volumes (Section 2.2.1), triangle and hinge elements for shells (Section 2.2.2),
and segment and ”corner” elements for rods (Section 2.2.3).

2.2.1 Volume

Here we focus on a volume represented by a tetrahedral mesh.

16
Deformation Gradient The deformation can be described using a tensor field called
deformation gradient
∂x
F =
∂X
where x is the current nodal position (world space) and X is the rest shape nodal position
(material space). For tetrahedra meshes with piecewise linear displacement field per
tetrahedron, this tensor field is then piecewise constant, and it can be easily computed as
the affine transformation matrix of a tetrahedron from material space to world space:
h ih i−1
F = x2 − x1 , x3 − x1 , x4 − x1 X2 − X1 , X3 − X1 , X4 − X1 ∈ R3×3

Here x1 , x2 , x3 , x4 represent the current nodal position of the 4 nodes on a tetrahedron and
similarly X1 , X2 , X3 , X4 for rest shape. With the deformation gradient, we can now think
of elasticity energies as a measurement of how faraway our deformation gradient F on the
tetrahedra are from their closest rotation matrix since rotation only change orientation but
not shape. Note that by using F we already ignored the translational motion which does
not change shape either.

Neo-Hookean Elasticity Neo-Hookean is one of the most common nonlinear hyperelas-

ticity models for predicting large deformations of elastic mateirals. The energy density
function for this model is

µ λ
ψ(F ) = (tr(F T F ) − 3) − µ ln J + (ln J)2
2 2

for 3D problems where J = det F and µ and λ are lame parameters computed using
Young’s modulus E ∈ (0, +∞)P a and Poisson’s ratio ν ∈ [0, 0.5) of the material via

E Eν
µ= , λ=
2(1 + ν) (1 + ν)(1 − 2ν)

Here E describes how stiff the material is, and ν measures the ability of a material to
preserve volume when deforming. Then to compute the elasticity energy of the entire mesh
we integrate ψ(F ) over the whole domain which in discrete setting end up with a weighted

17
sum
X
Ψ (x) = Vt ψ(Ft (x))
t

where Vt is the volume of tetrahedra t in material space.

With the definition of the elasticity potential energy, elasticity force and force Jacobians
can be easily computed using the derivatives of the elasticity potential via chain rule:

∂Ψ ∂Ψ ∂F
−f = =
∂x ∂F ∂x (2.7)
∂F ∂ 2 Ψ ∂F
∇2 E(x) = ( )T
∂x ∂F 2 ∂x
Note that here we flatten F as a vector for simplicity, and since F is a linear function
∂2F
of x, we have ∂x2
= 0, which leads to a Hessian ∇2 E(x) with this simpler form after
∂2Ψ
applying the chain rule. To efficiently compute ∂F 2
∈ R9×9 , singular value decomposition
∂2Ψ
(SVD) could be utilized to take advantage of the inherent sparsity structure of ∂F 2
due to
its rotation invariant property Jiang et al. [2016].
One thing to be careful for FEM simulation with Neo-Hookean is that when element
gets degenerated or inverted (J ≤ 0) during the optimization iterates, the energy density
function with ln J term will be undefined, thus crash the program. To avoid this issue and
always ensure positive J for every element, line search filtering Smith and Schaefer [2015]
is needed. Before energy decrease line search, we compute a large feasible step size to start
from by solving a polynomial equation for each element t to find out the largest step size
αt that first brings its volume to 0:

Jt (xi + αt pi ) = 0

where xi and pi are nodal positions and search directions in iteration i, and the smallest
positive real root of the above equation is αt . If there is no smallest positive real root for
a certain tetrahedron t, it means that even taking a full step at 1 will not make Jt zero or
negative. Given all αt we can now compute a large feasible step size α0 as

α0 = (1 − s) min{αt }
t

18
where s is a slackness coefficient for numerical robustness usually set to 0.2, and then
the energy decrease line search can start from α0 . This line search filtering should really
be performed at any time when nodal position x gets changed. For example, before
moving Dirichlet nodes at the beginning of each time step, line search filtering should
also be performed when Neo-Hookean is used. If moving the Dirichlet nodes to target
position triggers element inversion, then augmented Lagrangian method can be applied to
progressively move the Dirichlet nodes during the optimization. Please see our appendix
for more details. One may be curious about whether using an elasticity energy that is
also defined for inverted element can avoid this issue. It does, but this kind of invertible
elasticity energy cannot always capture the correct elastic behavior, especially when there
are extreme compression as we show in Chapter 4.

2.2.2 Shell

Shells refer to thin objects like papers and cloth where their thickness is nearly negligible
compared to the other two dimensions. Therefore, for simplicity shells are usually repre-
sented by surfaces discretized as triangle meshes. However, to correctly model the elasticity
of shells, only applying elasticity on each triangle element is not enough. Because even
if shells are thin, they are still volumetric in practice. Thus there should also be elasticity
modeled for the thickness direction.
In computer graphics, shells are usually simulated with the discrete shell model Grin-
spun et al. [2003], where the elasticity is decomposed into two parts residing in the tangent
space (membrane) and the normal space (bending) of the shell surface.

Membrane The membrane part of shells’ elasticity models the resistence to shearing,
stretching, compression, and area changes in the tangent space, or on each triangle in the
discrete sense. It is simply the 2D version of the general hyperelastic model we talked about
for tetrahedral volume elements in Section 2.2.1. But there are several detailed differences
to pay attention.

19
 First the deformation gradient in the tangent space is represented using a 3 × 2 matrix
per triangle:
 −1
h i ||X2 − X1 || (X3 −X1 )·(X2 −X1 )
||X2 −X1 ||
F = x2 − x1 , x3 − x1   ∈ R3×2
||(X3 −X1 )×(X2 −X1 )||
0 ||X2 −X1 ||

Intuitively, the rest shape of the triangle is reparameterized in a 2D space by placing X1 at

the origin, and then aligning edge X2 − X1 to the x-axis.
The second difference is that the lame parameter λ is computed differently here for
shells as the thickness approaches zero Barber [2002]

Eν
λ=
1 − ν2

For integration over the domain, notice that the volume weighting Vt should be com-
puted by multiplying triangle area with thickness but not just using triangle area as Vt .
Finally, a triangle can never get inverted in 3D space because any configuration of the
triangle can be achieved by combining rotations with stretches or compressions. Therefore
even when Neo-Hookean is used, there is no need to perform line search filtering. The only
case that could make Neo-Hookean’s log term undefined is when a triangle degenerates to
a single line or point. But this rarely happens in floating point operations and can be easily
avoided by halfing the step size even when detected.
To model the complex behaviors of cloth with discrete shell model, anisotropic mem-
brane elasticity energy which provides different stiffnesses in different directions is also
widely applied. Even with anisotropic models that well capture the directionally dependent
strain-stress relation of real cloth, real-world cloth material parameters cannot be directly
used in cloth simulation like for volumetric bodies. This is because for piecewise linear
displacement field, the elasticity model on a mesh without infinitely large resolution can
have stiffer behavior due to discretization error. This behavior is called membrane locking
Chen et al. [2019]. In this situation a smaller than real material stiffness is used for cloth
simulation, together with a strain limiting constraint to avoid over stretchy artifact. In
Chapter 5 we propose a barrier-based constitutive strain limiting model which at the first

20
time guarantees an upper bound of stretch and perform a fully coupled implicit solve
together with accurately resolved contact and elastodynamics. Please see Chapter 5 for
more details.

Bending Essentially, bending resistence comes from the different scales of membrane
resistences at multiple layers in a shell. Since we represent shells using a single surface in
the middle of the volume and assumes that all layers have the same membrane deformation,
we need to model bending separately. Note that the normal direction shearing resistences
are ignored here as they do not contribute significantly to the dynamics of the shell when
thickness is tiny (below around 0.001× than tangent dimensions).
Grinspun et al. [2003] discretize bending energy on every interior edge (hinge) of
the triangle mesh. The dihedral angle θ between the two incident triangles of the edge
is computed, and a measurement on the difference of this angle between rest shape and
current configuration is used as the bending energy:
X 3||ēi ||2
Ψbend (x) = k (θi − θ̄i )2
i
Ā i

. Here ||ēi || is the rest length of edge i, Āi is the sum of the rest areas of the two triangles
incident to the hinge, θ̄i is the rest dihedral angle, and k is the bending modulus computed
as
Eξ 3
k=
24(1 − ν 2 )
where ξ is the thickness. Computing the dihedral angle and the derivatives are more
complicated than in standard hyperelasticity. We refer to Tamstorf and Grinspun [2013] for
detailed derivations.
There are several important points to the robustness of the discrete bending model.
First, when the incident triangles are degenerated, the dihedral angle can be undefined.
Thus barrier-type membrane elasticity like Neo-Hookean is recommended to prevent this
case. Using large enough membrane stiffness that does not trigger membrane locking
would often also avoid this issue. In addition, when the two incident triangles get coplanar,

21
the computation of dihedral angle and its derivatives can also experience degeneracies.
Luckily this issue could be prevented by handling contact and ensures interpenetration-free
trajectories for the simulated objects. With IPC (Chapter 4), this issue can be safely ignored
at the first time in cloth simulation.

2.2.3 Rod

Rods refer to another set of thin objects like hairs and threads where two of their dimensions
are much smaller than their length. Thus, rods are often represented simply by a polyline
located at their centerline, and simulted using discrete rod model Bergou et al. [2008]
in computer graphics. Similar to discrete shells, discrete rod model decomposes rods’
elasticity into stretching, bending, and twisting. Here we briefly introduce the stretching
and bending energy, while refer to Bergou et al. [2008] for the twisting energy.

Stretching The stretching energy of rods is often simply modeled as a cluster of springs
that share the change in length, which can be viewed as a one-dimensional hyperelasticity
model measuring the inline stress for rods to keep rest length:
X E ||x1 − x2 ||
Ψstretch (x) = Āi ¯li ( ¯ − 1)2
i
2 li

where Āi = πr̄i2 is the cross section area of segment i with r̄i being its radius in material
space, ¯li is the rest length of segment i, E is the Young’s modulus, and x1 and x2 are the
current nodal positions of the two end nodes of segment i. Here note that we are still
integrating over the volume of rods so the volume weighting writes Āi ¯li .

Bending Similar to discrete shell bending model, for rods the bending part of elasticity
is also based on angle measurement but at each interior node i:
X π (κbi )2
Ψbend (x) = Er4 (2.8)
i
4 l¯0 i

where E is the Young’s modulus, r is the radius, l¯0 i = ¯li + ¯li−1 is the sum of the rest length
of the two incident edges on point i, and κbi is a vector with magnitude 2 tan φ2 where φ is

22
the angle between the two incident edges:

2ei−1 × ei
κbi =
|ei−1 ||ei | + ei−1 · ei

Note that the rod bending energy we describe here are only for rods with a straignt rest
shape, and we refer to Bergou et al. [2008] for rod bending energy supporting curvy rest
shapes. From the expression of κbi , we can see that it becomes ill-defined when a segment
gets degenerated into a single point. Again this issue is also fine if we apply our IPC model
(Chapter 4) to guarantee an interpenetration-free simulation.

2.2.4 Discussion

So far we have covered hyperelasticity models from volumes to shells and rods. One may
note that there is an important idea throughout – even we use codimensional surface and
polyline meshes for geometric representation, we still treat them as continuum regions when
formulating the elasticity to correctly capture their elastic behaviors. Using codimensional
representations has multiple advantages. First, there are less DoFs on codimensional
meshes so the simulation can be more efficient. The other advantage is a little bit tricker,
which is that when using volumetric meshes to represent thin objects like cloth and hair, and
discretize the elasticity with piecewise linear displacement field, there will be shear locking
issues when the resolution is not infinitely high such that the object will be harder to bend
than reality due to discretization errors. Using codimensional representations and fully
ignore the shearing resistence effectively solve this issue and keeps the simulation efficient.
However, one difficulty comes out for codimensional objects – there is no geometric
thickness modeling. Even if thin objects has nearly negligible thickness, when they gather
together and form a pile, e.g. a deck of cards, thickness will matter for correctly modeling
the geometric behaviors. In Chapter 5 we discuss this issue in detail and propose an offset
version of IPC Li et al. [2020b] to consistently model thickness in a controllable way for
mixed dimensional simulations. This also brings up our next topic for the background
chapter – contact modeling.

23
2.3 Contact and Friction

Up to now we have discussed all the background knowledge about how to simulate elasto-
dynamics in a robust and accurate way with codimension-0,1,2 meshes ignoring contact.
In this section, we will briefly cover how to accurately model contact in an optimization
time integration framework while keep its unconditional stability.

2.3.1 Contact

Contact is a hard problem not only because of its nonsmooth nature that contact forces
only exist when objects collide, but also that describing contact between discrete surfaces
is challenging. In optimization time integrators, contact can be treated as an inequality
constraint which limits the trajectory of the simulated objects in an interpenetration-free
admissible set:

min E(x) s.t. g(x) ≥ 0

x

Here E(x) is our incremental potential discussed in Section 2.1.2, g(x) ≥ 0 defines the
admissible set, but the definition of g(x) has been challenging over decades.
Traditional methods Kane et al. [1999], Kaufman et al. [2008], Verschoor and Jalba
[2019] define contact constraints by first finding out all close primitive pairs (point-triangle
and edge-edge), and then either use the signed volume of the tetrahedron formed by each
primitive pair, or the signed distance between each primitive pair along an estimated normal
direction in the constraint function. However, these definitions are all localized, and can be
invalid when there are large displacement in a time step (Figure 2.3), which thus limits the
time step sizes that can be used for solvers based on these constraints.

Contact Constraints based on Precise Distances In Chapter 4, we propose to formulate

contact constraint between triangulated surfaces by computing the precise unsigned distance
between each primitive pair, and ensures that in every time step, these distances always

24
 Volume constraints [Kane Gap constraints [Harmon 2008,
1999, Müller 2015, etc]: Verschoor 2019, Otaduy 2009, etc]:

g(x) = V g(x) = d ⋅ n

Large
displacement

Constraint violated
without interpenetration!

Figure 2.3: Traditional contact constraints definitions easily get invalid under large
displacements.

stay positive at any point on the optimization update trajectory in each iteration i:

min E(x) s.t. ∀k ∈ C , a ∈ [0, 1], i Dk ((1 − a)xi + axi+1 ) > 0

x

where C contains all non-incident point-triangle and non-adjacent edge-edge pairs on the
surface of the simulated objects, and Dk (x) is defined as

DkPT =min||xP − (xT 1 + β1 (xT 2 − xT 1 ) + β2 (xT 3 − xT 1 ))||

β1 ,β2
(2.9)
s.t. β1 ≥ 0, β2 ≥ 0, β1 + β2 ≤ 1.

for point(xP )-triangle(xT 1 xT 2 xT 3 ) pairs, and

DkEE =min||x11 + γ1 (x12 − x11 ) − (x21 + γ2 (x22 − x21 ))||

γ1 ,γ2
(2.10)
s.t. 0 ≤ γ1 , γ2 ≤ 1.

for edge(x11 x12 )-edge(x21 x22 ) pairs. These small optimization problems can be written
as piecewise smooth functions depending on the relative positions of the primitives, see
Chapter 4 for more details. Since our Dk precisely measure the distance of each primitive
pair under any configuration, it is always consistent and valid to describe whether the
simulation has interpenetration. With continuous collision detection (CCD) to directly

25
ensure that in each optimization iteration, the update to xi is without any interpenetration
by taking a step size smaller than the ”time of impact”, our problem formulation can be
simplified to

min E(x) s.t. ∀k ∈ C Dk (x) > 0

x

together with a CCD line search filtering in each optimization iteration similar to the one
that prevents element inversion.

Barrier Method with Smooth Clamping Now with our rigorous mathematical defini-
tion of the constraints, next thing is how to solve it. A common way to tackle inequality
constrained optimization problems is to apply sequential quadratic programming (SQP),
where the objective and constraints are approximated and then a quadratic optimization
problem with linear constraints is solved in each iteration to progressively move towards
the local optimum. This is a direct extension to Newton-type methods without constraints,
but conducting line search in SQP is tricky, which makes it hard to maintain robustness.
What is worse, the linearized constraints can even form infeasible problems, making the
approximated subproblems unsolvable. Therefore, we apply barrier method to turn the con-
strained problem into an ”unconstrained” one, and then apply the same projected Newton
method to solve it:
X
min E(x) + κ b(Dk (x))
x
k

where b is barrier function that goes to infinity when the input distance approaches zero.
Apparently, considering all candidates from C in the barrier is too expensive as its size
grows quadratically with the number of DoFs. In addition, this is also unnecessary as
faraway primitive pairs may never collide even if we do not explicitly constrain them. Thus
we construct a special barrier function with only a local support but is everywhere at least
C 2 -smooth to keep our smooth optimization method (PN) effective. Please see Chapter 4
for more details.

26
2.3.2 Friction

Intuitively, friction forces exist in tangent space of each contact point, and they are in the
opposite direction of the relative sliding displacement or the moving tendency if there
is no relative sliding. The challenge of simulating friction lies in figuring out the static
friction force and accurately capturing the transition between static and dynamic modes.
Mathematically, friction forces are defined by Maximum Dissipation Principle (MDP)
Moreau [1973]. For each active contact primitive pair k, with uk = Tk (x)T (x − xt ) being
the local relative sliding displacement where Tk (x) ∈ R3n×2 is the tangent operator, friction
force Fk can be defined as

−µλk Tk uk
||uk || > 0

||uk ||
Fk = (2.11)
−µλk Tk f

||uk || = 0

. Here λk is the normal force magnitude, and f can take any vector with ||f || ≤ 1, for
which we clearly see the difficulties here. When ||uk || = 0, Fk can have non-smooth jumps,
and displacement alone is not enough to define Fk because for static friction, the force
magnitude and direction depends on other forces and the momentum balance that will be
reached. This means that static friction force and nodal displacement are implicitly coupled
and needs to be solved simultaneously with Fk also being an unknown variable. In addition,
unlike hyperelasticity models, there is no well-defined D(x) such that − ∂D∂x
k (x)
= Fk , so
friction cannot be easily involved in our optimization time integration framework.
In Chapter 4, we propose a mollified friction model that transit between static and
dynamic friction smoothly. This introduces the approximation error for static friction that
it can act on primitive pairs with a tiny nonzero sliding displacment. Our model expose
an upper bound of the velocity magnitude that experience static friction as a controllable
parameter to let users weigh between smoothness (solver efficiency) and accuracy of the
friction. Then we discretize friction temporally in a semi-implicit way, such that λk and Tk
are fixed as their values at the end of last time step. These make the rest of the terms in Fk
integrable, and thus we can come up with a potential energy for this approximated friction

27
model and easily incorporate it into our optimization time integrator. Please see Chapter 4
for more details.

28
Chapter 3

Decomposed Optimization Time

Integrator

3.1 Introduction

Simulation of deformable-body dynamics is a fundamental and critical task in animation,

physical modeling, and design. Algorithms for computing these simulations have rapidly
advanced the accuracy, consistency, and controllability of generated elastodynamic trajec-
tories. Key to these advances are a long-standing range of powerful implicit time stepping
models to numerically time-integrate semi-discretized PDEs Ascher [2008], Butcher [2016],
Hairer and Wanner [1996], Hairer et al. [2008]. With a few restrictions, incremental po-
tentials Ortiz and Stainier [1999] can be constructed for these models. These are energies
whose local minimizers give an implicit time step model’s forward map at each time step
Hairer et al. [2006], Kane et al. [2000], Kharevych et al. [2006]. Adopting this variational
perspective has enabled the application and development of powerful optimization methods
to minimize these potentials and efficiently forward step dynamic simulations Chao et al.
[2010], Liu et al. [2017], Martin et al. [2011], Overby et al. [2017].
For deformable nonlinear materials, the incremental potential is the sum of a convex,
quadratic discrete kinetic energy and a generally nonconvex, strongly nonlinear deformation

29
energy weighted by time step size; see e.g. (3.3) below. Then, as step size and/or simulated
distortions increase, the influence of this latter deformation energy term dominates. This, in
turn, requires the expensive modeling of nonlinearities. As we will soon see, too weak an
approximation of nonlinearity leads to large errors, artifacts, and/or instabilities, while too
large or frequent an update makes other methods impractically slow — reducing progress
and imposing significant memory costs.
To address these challenges we propose the Decomposed Optimization Time Integrator
(DOT), a new domain-decomposed optimization method for minimizing per time step
incremental potentials. DOT builds a novel quadratic matrix penalty decomposition to
couple non-overlapping subdomains with weights constructed from missing subdomain
Hessian information. We use this decomposition’s Hessian, evaluated once at start of time
step, as an inner-initializer to perform undecomposed L-BFGS time step solves on a domain
mesh with a single copy of the simulation vertices. Advantages of DOT are then:

No vertex mismatch. While the decomposed Hessian is built per subdomain it is eval-
uated with a consistent set of vertices taken from the full, undecomposed mesh. Penalty
weights thus add missing second-order Hessian data to subdomain vertices from neighbors
across decomposition boundaries. This gives an initializer to our global L-BFGS solve. We
use it to perform descent on the full mesh coordinates. This means vertices on interfaces
are never separated, by construction.

Convergence. DOT adds no penalty forces nor gradients. DOT penalty terms only
supplement our preconditioning matrix. Descent steps then precondition the undecomposed
and unaugmented incremental potential’s gradient and converge directly to the underlying
undecomposed system’s solution.

Efficiency. Subdomain Hessians are parallel evaluated and factorized once per time step.
We show that they can also be applied (via small backsolves) in parallel as initializer at
each iteration. Results then simply need to be blended together. This process is inserted

30
between first and second efficient, low-rank updates of each quasi-Newton step. This
couples individual, per-domain backsolves together for a global descent step. Resulting
iterations are then more effective than L-BFGS approaches and yet faster than Newton.

Contributions In summary, DOT converges at large, fixed-size time steps, ensuring

stable continued progress of high-quality simulation output. DOT is an automated and
robust optimization method especially suited for simulations with nonlinear materials,
large deformations, and/or high-speed dynamics. By automated we mean users need not
adjust algorithm parameters or tolerances to obtain good results when changing simulation
parameters, conditions, or mesh sizes. By robust we mean a method should solve every
reasonable time step problem to any requested accuracy given commensurate time, and
only report success when the accuracy has been achieved. To achieve these goals we

• construct a quadratic penalty decomposition to couple non-overlapping subdomains

with weights constructed from missing subdomain Hessian information;

• propose a resulting method using this domain-decomposition as inner initializer for

undecomposed, full mesh, quasi-Newton time step solves;

• develop a line-search initialization method for reduced evaluations;

• extend Zhu et al.’s [2018] characteristic norm for consistent elastodynamic simula-
tions; and

• perform extensive comparisons of recent performant nonlinear methods for solving

large-deformation time stepping.

3.2 Problem Statement and Preliminaries

We focus on solving one-step numerical time-integration models with variational methods.

Minimizing an incremental potential, E, we update state from time step t to t + 1 with a

31
local minimizer
xt+1 = argmin E(x, xt , v t ), (3.1)
x∈Rdn

for n vertex locations in d-dimensional space stored in vector x, with corresponding

velocities v. Here E(x, xt , v t ) is a combined local measure of deformation and discrete
kinetic energy, and x is potentially subject to boundary and collision constraints.
The deformation energy, W (x), is (e.g. with linear finite elements) expressed as a sum
over elements e in a triangulation T (triangles or tetrahedra depending on dimension),
X

W (x) = ve w Fe (x ) , (3.2)
e∈T

where ve > 0 is the area or volume of the rest shape of element e, w is an energy density
function taking the deformation gradient as its argument, and Fe computes the deformation
gradient for element e.
Concretely, as recent papers on solvers for time stepping in graphics Bouaziz et al.
[2014], Gast et al. [2015], Liu et al. [2013, 2017], Narain et al. [2016], Overby et al. [2017]
have almost exclusively focused on the implicit Euler time stepping model, our results in
the following sections will do so as well in order to provide side-by-side comparisons. For
implicit Euler the incremental potential is

1
E(x, xt , v t ) = xT M x − xT M xp + h2 W (x). (3.3)
2

Here xp = xt + hv t + h2 M −1 f , f collects external and body forces, M is the finite

element mass matrix, and velocity is updated by the implicit Euler finite difference stencil
xt+1 −xt
v t+1 = h
. In the next sections we restrict our attention to solving a single time step
and so, unless otherwise indicated, we simplify by specifying the incremental potential as
E(x) = E(x, xt , v t ).

Notation. Throughout we will continue to apply superscripts t to indicate time step, while
reserving subscripts incrementing i for inner solver iteration indices, and correspondingly
subscripts with increments of j for quantities associated with subdomains.

32
3.3 Related Work

Domain Decomposition Domain decomposition strategies have long been an effective

means of efficiently parallelizing large-scale linear problems Quarteroni et al. [1999]. Direct
connections with the Schur complement method and block-decomposed iterative solvers
for linear algebra are then easily established for faster solves. These methods offer exciting
opportunities for scalable performance that have also been applied in graphics Huang
et al. [2006], Kim and James [2012], Liu et al. [2016], Sellán et al. [2018], but can also
suffer from slower convergence or gapping between imperfectly joined interfaces Kim and
James [2012]. Domain decompositions, including the classic Schwartz methods, have also
been extended to the nonlinear regime Dolean et al. [2015], Xiao-Chuan and Maksymilian
[1994]. Here parallelization is easily obtained; however, methods generally offer linear
convergence rates Dolean et al. [2015]. While speeds can potentially be further improved
by incorporating ADMM-type strategies Parikh and Boyd [2012], overall application
is still hampered by ADMM’s underlying first-order convergence and the overhead of
working with additional dual variables Boyd et al. [2011]. For DOT, we design a domain
decomposition without dual variables that employs energy-aware coupling penalty terms in
the subdomain Hessian proxy that can be efficiently and easily parallelized for evaluation.
We then integrate this model as an inner component of a customized limited memory BFGS
to gain higher-order coupling across domains and so regain super-linear convergence with
a small, additional fixed linear overhead.

Optimization-based time integrators in graphics. Position-based dynamics methods

have become an increasingly attractive option in animation, being fast and efficient, but not
controllable nor consistent Müller et al. [2007]. Both Projective Dynamics (PD) Bouaziz
et al. [2014] and extended position-based dynamics Macklin et al. [2016] observe that with
small but critical modifications, iterated local and global solves can be brought more closely
into alignment with implicit Euler time stepping, although various limitations in terms of
consistency, controllability, and/or materials remain. Liu and colleagues [2017] observe

33
that, in the specific case of the ARAP energy density function, PD is exactly a Sobolev-
preconditioning, or in other words, an inverse-Laplacian-processed gradient descent of
the incremental potential for implicit Euler. Note that beyond ARAP this analogy breaks
down. Following on this observation they propose extending PD’s Sobolev-preconditioning
of the implicit Euler potential for models with a range of hyperelastic materials and thus
varying distortion energies. They further improve convergence of this implicit-Euler solver
by wrapping the Sobolev-preconditioner as an initializer for the limited-memory BFGS
algorithm (L-BFGS) to minimize the incremental potential. In the following we refer to
this algorithm as LBFGS-PD.

Concurrently, Overby et al. [2016, 2017] observe that PD can alternately be interpreted
as a particular variant of ADMM with local variables formulated in terms of the deformation
gradient. Overby and colleagues then show that an algorithm constructed in this way can
likewise be extended to minimize the implicit Euler potential over a range of hyperelastic
materials. In the following we will refer to this algorithm as ADMM-PD. Both ADMM-
PD and LBFGS-PD improve upon PD both in extending to a wide range of hyperelastic
materials and to increasing efficiency for ARAP-based deformation Liu et al. [2017],
Overby et al. [2017].

Optimization-based time integration In brief, albeit by a circuitous path, time stepping

solvers in computer graphics, starting from position-based methods, come full circle back to
minimizing the standard incremental potential for fully nonlinear materials. In this setting,
traditional time stepping in computational mechanics generally focuses on preconditioned
gradient-descent methods, often augmented with line-search Ascher [2008], Deuflhard
[2011]. These methods find local descent directions pi , at each iterate i, by preconditioning
the incremental potential’s gradient with a Hessian proxy Hi , so that pi = −Hi−1 ∇E(xi ).
For example, Sobolev-preconditioning Neuberger [1985] for implicit Euler gives a fixed
efficient preconditioner built with the Laplacian L so that Hi = M + h2 L for all time steps
Liu et al. [2017].

34
Newton-type methods Preconditioning with the Hessian Hi = ∇2 E(xi ), gives Newton-
stepping. To gain descent for large-time stepping this generally requires both line search
and a positive-definite fix of the stiffness matrix which otherwise can make the total Hessian
indefinite Nocedal and Wright [2006]. Many closely related positive-definite corrections
are suggested Nocedal and Wright [2006], Shtengel et al. [2017], Teran et al. [2005].
Here, following Liu and colleagues, we employ Teran et al.’s per-element projection
and refer to this method as Projected Newton (PN). Newton-type methods like these
have rapid convergence but solution of the Hessian, either directly or via Newton-Krylov
variations, are generally reported as too costly per-iteration to be competitive with less-
costly preconditioners and scale poorly Brown and Brune [2013], Liu et al. [2017], Overby
et al. [2017]. We revisit and analyze this assumption in Section 4.7 with a suite of well-
optimized solvers. We observe that in some ranges PN with a direct solve is actually
competitive and can even outperform LBFGS-PD and ADMM-PD, while in others the
situation is indeed often reversed. In order to retain Newton’s superlinear convergence with
improved efficiency, lagged updates of the Hessian preconditioner every few time steps
have long been proposed Deuflhard [2011], Hecht et al. [2012], Shamanskii [1967]. While
in some cases this can be quite effective, the necessary frequency of these updates can not
be pre-determined as it depends on local simulation state, e.g., transitions, collisions, etc,
and so generally leads to unsightly artifacts and inaccuracies, including ghost forces and
instabilities Brown and Brune [2013], Liu et al. [2017].

Quasi-Newton methods Alternately, quasi-Newton BFGS methods have long been ap-
plied for simulating large-deformation elastodynamics Deuflhard [2011]. L-BFGS can be
highly effective for minimizing potentials. However, an especially good choice of initializer
is required and makes an enormous difference in convergence and efficiency Nocedal and
Wright [2006]; e.g., Liu et al.’s [2017] Sobolev-initializer. Directly applying the Hessian
is of course clearly an ideal initializer for L-BFGS; unfortunately, it is generally a too
costly option. Lazy updates of the Hessian, for example at start of time step, have also been

35
considered Brown and Brune [2013], Liu et al. [2017]; but again have been discarded as too
costly and limiting in terms of scalability Liu et al. [2017]. In the following we will refer
to this latter method as LBFGS-H. Our development of DOT leverages these start of time
step Hessians at the beginning of each incremental potential solve, applying a new, inner
domain decomposition strategy to build an efficient, per-time step re-initialized method
that outperforms or closely matches best-per-example prior methods across all tested cases.

Trade-offs across time-integration solvers As both LBFGS-PD and ADMM-PD ap-

peared concurrently and, to our knowledge, have not previously been analyzed side-by-side,
we do so here for the first time along with PN and LBFGS-H, across a range of examples.
We show how all these methods alternately excel or fall-short in varying criteria and ex-
amples; see Section 4.7. ADMM-PD has fast initial convergence that rapidly trails off,
characteristic of ADMM methods, and thus is generally slowest, often failing to meet even
moderate accuracy tolerances. LBFGS-PD on the other hand performs admirably when a
rest shape stretch and squash

twist and stretch and squash

Figure 3.1: Uniform deformation examples.

global, uniform deformation is applied on a uniform mesh such as in bar stress-tests, see
e.g., Fig. 3.1. For these cases the Laplacian preconditioner effectively smoothes global error
generating rapidly converging simulations Zhu et al. [2018]. However, for everyday non-
uniform deformations on unstructured meshes common in many application, LBFGS-PD
will rapidly lose efficiency. Often, as deformation magnitude grows, LBFGS-PD becomes
slower than a direct application of PN and LBFGS-H, despite LBFGS-PD’s per-iteration
efficiency; see Section 4.7.
Based on our analysis of where these prior methods face difficulties we propose DOT

36
which efficiently, robustly, and automatically converges to user-designated accuracy toler-
ances with timings that outperform or closely match these previous methods across a wide
range of practical stress-test deformation scenes on unstructured meshes. DOT combines
the advantages of per-time step updates of second-order information with an efficient quasi-
Newton update of per-iteration curvature information. The two are integrated together
by a novel domain-decomposition we construct that avoids slow coupling convergence
challenges faced by traditional domain decomposition methods. Together in DOT these
components form the basis for a scalable, highly effective, domain-decomposed, limited
memory quasi-Newton minimizer of large time step incremental potentials with super-linear
convergence; see Section 4.7.

3.4 Method

We solve each time step by iterated descent of the incremental potential. At each inner
iteration i, we calculate a descent direction pi and a line search step, αi ∈ R+ . We then
update our current estimate of position at time t + 1 by xt+1 t+1
i+1 ← xi + αpi .

3.4.1 Limited memory quasi-Newton updates

We begin with a quasi-Newton update. At each iterate i, the standard BFGS approach No-
cedal and Wright [2006] exploits the secant approximation from the difference in successive
gradients, yi = ∇E(xi+1 )−∇E(xi ), compared to the difference in positions si = xi+1 −xi .
This approximation is applied as low-rank updates to an inverse proxy matrix Di = Hi−1
(roughly approximating ∇2 E −1 ) so that Di+1 yi = si . Limited memory BFGS (L-BFGS)
then stores for each iteration i just an initial, starting matrix, D1 and the last m {s, y}
vector pairs (we use m = 5). Joint update and application of Di+1 is then applied implicitly
with just a few, efficient vector dot-products and updates, with the application of the matrix
D1 sandwiched in between the first and second low-rank updates of each step.
Given iterate i’s implicitly stored proxy Di , and the initial proxy, D1 , we set Qi :

37
Rdn×dn × Rdn → Rdn to apply the limited-memory quasi-Newton update and application
of Di+1 . Then

pi = Qi D1 , xi = −Di+1 ∇E(xi ). (3.4)

L-BFGS’s performance depends greatly on choice of initializer D1 Nocedal and Wright

[2006]. Setting D1 to a weighted diagonal, for example, offers some improvement, as do
various inverses of block diagonals taken from the Hessian. Similarly we can, as discussed
above, apply Liu et al.’s [2017] inverse Laplacian or even, as proposed by Brown and
Bune [2013], could use lagged updates of the Hessian inverse itself — just once every few
time steps so as to not perform too many expensive updates and solves.

3.4.2 Initalizing L-BFGS

To consider the relative merits of potential initializers for time stepping we adopt a simple
perspective. We observe that each of the above initializers corresponds to the application
of a single iteration of a different, nonlinear method as an inner step in the L-BFGS loop
Zhu et al. [2018]. The better the convergence of the inner method generally the better
performance of the outer L-BFGS method. From this perspective Liu et al. [2017], for
example, can be seen to apply an inner iterate of inverse-Laplacian preconditioned descent
and so gain the well-appreciated smoothing of the parent Sobolev Gradient Descent method
Neuberger [1985]. Or alternately we could consider applying the start of time step inverse
Hessian D1 = ∇2 E(xt )−1 . The resulting optimization applies an iteration of the inexact
Newton method within L-BFGS that could potentially augment quasi-Newton curvature
with second-order information via the tangent stiffness matrix.
While this latter strategy could be highly effective to improve convergence Liu et al.
[2017], it is expensive both to factorize at every time step and to backsolve at every
iteration. Moreover, we observe that initializing with the full Hessian inverse may be an
unnecessarily global approach. In many cases large deformations are concentrated locally
and their effects are only communicated across the entire material domain over multiple

38
time steps. Likewise, initializing with the full inverse Hessian limits scalability as we must
work with its factors in every quasi-Newton iterate; see Section 4.7.
Motivated by these observations we instead design a method to update L-BFGS with
Hessian information using a decomposition that allows us to efficiently store and apply local
second-order information at each iterate. To do so we first construct a simple quadratic
penalty decomposition that connects non-overlapping subdomains with automatically
determined stiffness weights. Our decomposed optimization time integrator is then formed
by applying a single inexact-Newton descent iterate of this method as an inner initializer of
an undecomposed, full mesh L-BFGS step. The resulting method, as we see Section4.7,
then obtains well-improved, scalable convergence for large time step, large deformation
simulations.

3.4.3 Decomposition

An illustration of our decomposition is in Fig. 3.2. We decompose our full domain Ω into s
non-overlapping subdomains {Ω1 , ..., Ωs } partitioned by interfaces Γ = {γ1 , ..., γ|Γ | } along
element boundaries with duplicate copies of each interface vertex assigned to each subdo-
main participating at that interface. We construct decompositions that both approximately
minimize number of edges along interfaces (and so cross-subdomain communication) and
balance numbers of nodes per subdomain Karypis and Kumar [1998]. For each subdomain
j we then likewise store its interfaces in Γj ⊆ Γ .
Each subdomain j then has nj vertices stored in vector xj = (yjT , zjT ) ∈ Rdnj , formed
by the concatenation of a copy of its interface vertices zj , and its remaining subdomain
vertices, yj . Concatenation of all subdomain vertices x̂ = (xT1 , ..., xTs )T ∈ Rdn̂ then
includes unshared interior vertices and duplicated interface vertices so that n̂ > n. We
then collect interface vertices from the fully connected mesh as xΓ ⊂ x. Note that xΓ
are then distinct from their duplicated interface copies in the decomposition and can
serve as consensus variables for subdomain boundaries. Finally, per subdomain j, we
construct interface restriction matrices RΓj that extract vertices from xΓ participating in

39
that subdomain’s interfaces Γj .

Ω, 𝑥 = { , } Ω$ 𝛾$ Ω" 𝛾" Ω*

decompose
𝑧={ }
𝑥/ = { , }

y={ } 𝑥. = { } Γ$ = {𝛾$ } Γ" = {𝛾$ , 𝛾" } Γ* = {𝛾" }

Figure 3.2: DOT’s decomposition. Layout and notation for a mesh (left) which is
decomposed into three subdomains (right).

3.4.4 Penalty Potential

Building time stepping incremental potentials on this decomposed system we get a nicely
P
separable sum j∈[1,s] Ej (xj ), where Ej = E|Ωj is the restriction of the incremental
potential to subdomain j. This separable potential could be efficiently optimized as it allows
us to minimize each subdomain independently. Doing so, however, would erroneously
decouple subdomains.
One possibility for reconnecting subdomains would be to add explicit coupling con-
straints. However, this would also require dual variables in the form of Lagrange-multipliers.
As we are designing our decomposition for insertion within a quasi-Newton loop, addi-
tional dual variables are undesirable. Instead, we adopt a simple quadratic penalty for
each interface in Γ to pull subdomains together, with consensus variables xΓ as the bridge.
Augmenting the above separable potential with this penalty gives
X  1 2

L(x̂, xΓ ) = Ej (xj ) + kzj − RΓj xΓ kKj . (3.5)
2
j∈[1,s]

Here each Kj is a penalty stiffness matrix that pulls subdomain j’s interface vertices
towards globally shared consensus positions stored in xΓ . By driving |Kj | towards ∞
as we optimize L we could hypothetically remove interface mismatch. However, finding

40
 E = 1e5, n

E = 1e

Rest Shape Frame 200 Frame 282 Frame 291 Frame 335

Rest Shape Frame 200 Frame 282 Frame 291 Frame 335

Rest Shape

Frame 52
Rest Shape

Frame 52
Frame 106
Frame 76
Frame 106
Frame 76

Frame 162 Frame 239

Frame 162 Frame 239

Figure 3.3: Deformation stress-tests. DOT simulation sequences of hollow cat (top) and
monkey (bottom) large-deformation, high-speed, stress-tests.

practical values for Kj that sufficiently pull interface edges together, while avoiding ill-
conditioning, is generally challenging. For example, if we choose a penalty that is too
stiff it would pull interface domains together too tightly at the expense of overwhelming
elasticity and inertia terms in the potential. On the other hand, if we choose a penalty that is
too soft, subdomain potentials dominate and interface consensus would be underestimated.

3.4.5 Interface Hessians

We instead seek a penalty that automatically balances out the missing elastic and inertial
information along interfaces. In our setting we observe that this missing information is

41
directly available and forms a natural, automated weighting function for the augmented
potential in (3.5) above. Consider that each subdomain’s penalty term is effectively just a
proxy for the missing energies from neighboring subdomains. Specifically, the Hessian for
subdomain Ωj is  
2 ∂ 2 Ej (xj ) ∂ 2 Ej (xj )
∂ L  2∂yj 2 ∂yj ∂zj
=  (3.6)
∂xj 2 ∂ Ej (xj ) ∂ 2 Ej (xj )
+ Kj .
∂zj ∂yj ∂zj 2

Here, the upper and off-diagonal (symmetric) terms nicely match the unrestricted Hessian
on the full domain Ω,

∂ 2 Ej (xj ) ∂ 2 E(x) ∂ 2 Ej (xj ) ∂ 2 E(x)

= and = . (3.7)
∂yj 2 ∂yj 2 ∂yj ∂zj ∂yj ∂(RΓj xΓ )

However, our lower diagonal does not as

∂ 2 Ej (xj ) ∂ 2 E(x)
6
= , (3.8)
∂zj 2 ∂(RΓj xΓ )2
irrespective of whether or not we have agreement on subdomain interface vertex locations.
This is due to the absence of element stencils connecting interface nodes to neighboring
subdomains. While this missing information is not currently present in ∂ 2 L/∂xj 2 , it gives
us a natural definition for penalty weights. We assign weights Kj so that they generate the
missing interfacial Hessian information from adjacent elements bridging across neighboring
subdomains:
∂ 2 E(x) ∂ 2 Ej (xj )
Kj (x) = − . (3.9)
∂(RΓj xΓ )2 ∂zj 2
Our quadratic penalty then recovers otherwise missing components of kinetic and elastic
energy stencils across interface boundaries. Note that, as in PN, we project all computed
Hessian stencils.

3.4.6 Discussion

To minimize (3.5) we could apply alternating iterations solving first for optimal x̂ and then
for xΓ . Indeed, when we add a constraint Lagrangian term, j∈[1,s] λTj (zj − RΓj xΓ ) to
P

(3.5), this alternating process generates a custom ADMM Boyd et al. [2011] algorithm. We

42
initially considered this approach and find that it significantly outperforms classic ADMM.
We observe that automatic weighting with (3.9) nicely smooths error at boundaries; see e.g.
Fig. 3.7. However, we also find that this strategy is still not competitive with undecomposed
methods like PN, as too much effort is exerted to close gaps between subdomains. Instead,
we apply a single iteration of our penalty decomposition as an efficient, inner initializer with
second-order information. We then insert it within an outer, undecomposed quasi-Newton
step solved on the full mesh. This ensures unnecessary effort is not spent pulling interfaces
together, and updates our decomposition with global, full-mesh, curvature information.

3.4.7 Initializer

We now have all necessary ingredients to construct DOT. We initialize an L-BFGS update
with a single Newton iteration of our quadratic penalty in (3.5) as follows. At start of
time step t + 1 we define subdomain variables from current positions xt as x̂t = Sxt .
Here S ∈ Rdn̂×dn is a separation matrix that maps the n full-mesh vertices x to subdomain
coordinates with duplicated copies of interfacial vertices. Application of the inverse Hessian
is then

−1
D1t+1 = BS T ∂ 2 L(x̂t , xtΓ )/∂ x̂2 S ∈ Rdn×dn . (3.10)

Here B ∈ Rdn×dn is a diagonal averaging matrix with diagonal entries corresponding to

vertex v set to 1/nv where nv is the number of duplicate copies for the corresponding vertex
v in the decomposition. Iteration i of DOT is then applied by application of

pi = Qi (D1t+1 , xi ), (3.11)

followed by our custom line search and update detailed below.

43
3.4.8 Construction

Construction and application of the per time step DOT is efficient. At start of solve we first
compute and store factors of the augmented Hessians per subdomain,

Hj = ∇2 Ej (xtj ) + Kj (xtj ). (3.12)

We then observe that

D1t+1 = BS T diag(H1−1 , ..., Hs−1 )S, (3.13)

where diag(·) constructs a block diagonal Rdn̂×dn̂ matrix. Application of D1t+1 as initializer
in L-BFGS is then applied in parallel computation to any global vector q ∈ Rdn . We
first separate q to repeated subdomain coordinates (q1T , ..., qsT )T = Sq ∈ Rdn̂ . Then we
independently backsolve subdomains with their factors to obtain rj = Hj−1 qj . Finally, we
lift all rj back to full mesh coordinates by blending with r = BS T (r1T , ..., rsT )T to average
duplicate coordinates appropriately. See Algorithm 1 below for the full DOT pseudocode.

3.4.9 Line Search

After our quasi-Newton update we next perform backtracking line search on pi to ensure
sufficient descent. For quasi-Newton methods rule-of-thumb Nocedal and Wright [2006]
is to always initialize line search with unit step length, i.e. αstart = 1, to ensure large
steps will take advantage of rapid convergence near solutions. In the large time step, large
deformation setting however, we observe that the situation is nonstandard. We often start far
from solutions and so need to balance large, initial step estimates against costs of repeated
energy evaluations. For this purpose we apply an alternative initializer for line search. For
each search direction pi , DOT initializes with the optimal length of the one-dimensional
quadratic model,
 −pT ∇E(xi ) 
αstart = max 10−1 , T i 2 . (3.14)
pi ∇ E(xt )pi

Here the lower bound handles the rare instances where this local fit is too conservative.

44
3.4.10 Algorithm

Algorithm 1 contains the full DOT algorithm in pseudocode. The dominant costs for
runtime are energy costs: evaluations, gradients, Hessians, and SVDs; and subdomain
augmented Hessian costs: assembly, factorization and backsolves. Otherwise, costs for
our quasi-Newton loop itself are linear (dot products, vector updates, etc). Memory cost
is primarily the once per-timestep Cholesky factorization of the subdomain augmented
Hessians and their corresponding backsolves per iteration.

3.5 Evaluation

3.5.1 Implementation and Testing

We implemented a common test-harness code to enable the consistent evaluation of all

methods with the same optimizations for common tasks. In comparisons of our ADMM-PD
and LBFGS-PD implementations with their release codes Liu et al. [2017], Overby et al.
[2017] we observe an overall 2-3X speed-up across examples.
Our code is implemented in C++, parallelizing assembly and evaluations with Intel TBB,
applying CHOLMOD Chen et al. [2008] with AMD reordering for all linear system solves,
and METIS Karypis and Kumar [1998] for all decompositions. Note that for LBFGS-PD
and ADMM-PD we perform their global Laplacian backsolves in parallel, per-dimension,
and likewise factorize only the scalar Laplacian, one-time as a precompute. Following
Overby et al. [2017], ADMM-PD’s per-element energy minimizations are performed in
diagonal space for efficiency. Common energy evaluations and gradients are optimized
with AVX2 parallelization to achieve roughly 4× speedup. To do so we extended the
open-source SIMD SVD library McAdams et al. [2011] to support double precision for our
framework.
Unless otherwise indicated, all experiments below were performed on a six-core Intel
3.7GHz CPU and were simulated with times step sizes of either 10, 25 (majority), or 40

45
Algorithm 2 Decomposed Optimization Time Integrator (DOT)
Given: xt , E, S, B, 
Initialize and Precompute:
i=1

−1
Hj−1 ← ∇2 Ej (xtj ) + Kj (xt ) , ∀j ∈ [1, s] // get Cholesky factors
g1 ← ∇E(x1 )
// quasi-Newton loop to solve time step t + 1:
while kgi k > h2 hW ik`k do // termination criteria (Section3.5.2)

q ← −gi
for a = i − 1, i − 2, .., i − m
sa ← xa+1 − xa , ya ← ga+1 − ga , ρa ← 1/(yaT sa )
αa ← ρa sTa q
q ← q − αa y a
end for
(q1T , ..., qsT )T ← Sq
rj ← Hj−1 qj , ∀j ∈ [1, s]
r ← BS T (r1T , ..., rsT )T
for a = i − m, i − m + 1, .., i − 1
β ← ρa yaT r
r ← r + (αa − β)sa
end for
pi ← r


αstart ← max 10−1 , − pTi ∇E(xi ) / pTi ∇2 E(xt )pi

α ← LineSearch(xi , αstart , pi , E)
xi+1 ← xi + αpi
gi+1 ← ∇E(xi+1 )
i←i+1
end while

46
ms (as indicated). For consistent comparison with prior work we focus our analysis on
examples with the implicit Euler using the fixed co-rotational material Stomakhin et al.
[2012]. Below in Section 3.5.6 we also explore DOT with the Stable Neo-Hookean material
Smith et al. [2018].
We compile CHOLMOD with MKL LAPACK and BLAS, supporting multi-threaded
linear solves, and set the number of threads per linear solver to take full advantage of
multi-core architecture per method. Specifically, the single, full linear systems in each PN
and LBFGS-H iteration are solved with 12 threads per solver, while the multiple smaller
systems in each LBFGS-PD, ADMM-PD, and DOT iteration are solved simultaneously
with 1 thread per solver.
Finally, note that we summarize detailed statistics from all of our experiments in tables
in our supplemental document Li et al. [2019]. All tables referred to in the following are
found there.
350 3
10
300
LBFGS-PD
250 DOT (ours) 10 2
200 LBFGS-H
PN 10 1
150
100
10 0
50
0 10 -10
0 5 10 15 20 25 30 5 10 15 20 25 30

Figure 3.4: Convergence and Timings. Per-example comparisons of iteration (top) and
timing (bottom) costs, per frame, achieved by each time step solver. Note that the bars are
overlaid, not stacked.

3.5.2 Termination Criteria

We next focus on the important question of when to stop iterating an individual time step
solve. Clearly, for most simulation applications, it is not reasonable to manually monitor
the quality of each individual iterate, within every individual time step solve, in order
to decide when to stop. Analogously, applying the same fixed number of iterations for
all time steps, no matter the method, will not suffice as different steps in a simulation

47
 90

80
horse-7K(S)
70 horse-38K(S)
horse-79K(S)
Iteration Count
60 horse-7K(SS)
horse-38K(SS)
50 kingkong-18K(SS)
kingkong-48K(SS)
40 bunny-30K(SS)
kongkong-18K(TSS)
30 monkey-18K(TSS)
elf-23K(TSS)
20 hollowCat-24K(TSS)
horse-38K(TSS)
10
0 50 100 150 200 250 300
Number of Blocks

Figure 3.5: Subdomain to iteration growth. We plot iterations per DOT simulation
example as the size of the decomposition increases. We observe a trend of sublinear-
growth in iteration count with respect to the number of subdomains, revealing promising
opportunities for parallelization.

will have more or less nonlinearity involved and so will require correspondingly different
amounts of work to maintain consistent simulation quality. Without this simulations
can and will accrue inconsistencies, artifacts and instabilities. Similarly, relative error
measures for termination are not satisfactory because they are fulfilled when an algorithm
is simply unable to make further progress and so has stagnated far from a working solution.
On the other hand, the gradient norm provides an excellent measure of termination for
scientific computing problems where we seek high-accuracy. However, for animation and
many other applications we seek a way to reliably stop at stable, good-looking, consistent
simulations with reasonably low error, but not necessarily with vanishingly small gradients.
As observed by Zhu et al [2018], in these cases even gradient norm tolerances are not
suitable for choosing a termination criteria.

To address this problem for minimizing deformation energies in statics problems,

Zhu and colleagues propose a dimensional analysis of the deformation energy gradient.

48
They derive a characteristic value for the norm of the deformation gradient over the
mesh. Then, applying it as a scaling factor to the gradient norm obtains consistent quality
solutions across a wide range of problems, object shapes and energies. Here we observe
that a small modification of this analysis gives a corresponding scaling factor for the
incremental potential. We begin with the base case of the characteristic value for the
norm of the deformation gradient over the mesh as hW i||`||. Here hW i is the norm of the
deformation energy Hessian of a single element at rest, ` is a vector in Rn with each entry
the surface area of each simulation node’s one-ring stencil. For dynamic time stepping
we are then minimizing with the incremental potential (3.3) and so have a weighted
sum of discrete kinetic and deformation energies. At stationarity of (3.3) we then have
M (x − xt − xp ) + h2 ∇W (x) = 0. We then have proportional measures − h12 M (x − xt − xp )
1
at corresponding scale with ∇W (x), and similarly h2
M (x − xt − xp ) + ∇W . Then, for
consistent convergence checks across examples and methods at each iteration we simply
check the rescaled incremental potential

||∇E|| < h2 hW ik`k. (3.15)

In the following evaluations we refer to this measure as the characteristic gradient norm
(CN). After extensive experimentation across a wide range of examples, using Projected
Newton (PN) as a baseline, we find consistent quality solutions across methods using (3.15)
at increments of ; see the supplemental videos Li et al. [2019]. Moreover, we find that
solutions satisfying  = 10−5 are the first to avoid visual artifacts that we consistently
observe below this tolerance, including variable material softening, damping, jittering and
explosions. For all experiments in the remainder of this section, unless otherwise indicated,
we thus set our termination criteria with  = 10−5 using (3.15). All CN convergence
measures thus apply rescaling of the gradient norm. Convergence in CN per method then
confirms convergence to the same (reference) solution provided, e.g., by Newton’s method.

49
3.5.3 Iteration Growth with Domain Size

We next investigate the scalability of DOT as the number of subdomains in the decomposi-
tion grows. We apply DOT across thirteen simulation examples ranging in mesh resolution
from 7K to 136K vertices with a range of moderate to extreme deformation test scenes.
Each scene is solved to generate ten simulated seconds, time stepped at h = 25ms. For
each simulation example we create a sequence of decompositions by requesting target
subdomain sizes starting at 16K vertices (where possible) down by halves to 256 vertices.
We then simulate all of the resulting decompositions, spanning from 2 to 309 subdomains,
with DOT. In figure 3.5 we plot the the number of DOT iterations per simulation example
as the size of its decomposition increases. We observe a trend of sublinear-growth in itera-
tion count (per simulation) as the number of subdomains increases, revealing promising
opportunities for parallelization of DOT.

3.5.4 Performance

Fig. 3.5 suggests parallelization opportunities for DOT across a wide range of decom-
position sizes. Of course, how to best exploit these opportunities will vary greatly with
platform. Here we begin with two modest exercises starting with a six-core Intel 3.7GHz
CPU, 64GB memory. We script a set of increasingly challenging dynamic deformation
stress-test scenarios across a range of mesh shapes and resolutions. See, for example, Fig.s
1.1 and 3.3, our supplemental document and video Li et al. [2019] for example details. For
each simulation we target DOT to simply utilize available cores and so set the number of
subdomains in all simulations in this first exercise to six. In Fig. 3.4 we summarize runtime
statistics for these examples with DOT, PN, LBFGS-H, and LBFGS-PD across the full set
of these examples. We also test ADMM-PD on the full set of examples but find it unable
to converge on any in the set. See our convergence analysis below for more details on
ADMM-PD’s behavior here.

50
Timings Across this set we observe DOT has the fastest runtimes, for all but three
examples (see below for discussion of these), over the best timing for each example across
all converging methods: PN, LBFGS-H, and LBFGS-PD. In general DOT ranges from 10X
to 1.1X faster than PN, from 2.5X to 1X faster than LBFGS-H, and from 11.4X to 1.6X
faster than LBFGS-PD. The one exception we observe is in the three smallest meshes of
the horse stretch scalability example where the deformation is slow, so that the solves are
close to statics. Here we observe that LBFGS-H is, on average 1.3X faster than DOT on
smaller meshes up to 79K vertices. Then, as mesh size increases to 136K and beyond, here
too DOT becomes faster. Finally, for even larger meshes LBFGS-H can not fit in memory;
see Scaling below. Importantly, across examples, we observe that PN, LBFGS-PD and
LBFGS-H alternate as fastest as we change simulation example. Here PN ranges from 2X
slower to 4.6X faster than LBFGS-PD, while LBFGS-H often seems to be a best choice
among the three, but also can be slowest, i.e., ranging from 1.1X to 1.7X slower. Trends
here suggest that LBFGS-PD tends to do better for more moderate deformations while
PN and LBFGS-H often pull ahead for more extreme deformations but this is not entirely
consistent and it is challenging to know which will be the better method per example, a
priori. Finally, as we see below, both PN and LBFGS-H do not scale well to larger systems.

Scaling To examine scaling we successively increase mesh resolution for the horse TSS
example. On this machine (recall 64GB memory) both PN and LBFGS-H can not run
models beyond 308K vertices while DOT and LBFGS-PD can continue for examples up to
and including 754K vertices. We compare the performance among methods at these two
extremes and find that DOT is 1.9X faster than LBFGS-H at 308K nodes, while it is 2.7X
faster than LBFGS-PD at 754K nodes, where both PN and LBFGS-H can not run.

Changing Machines Next, in Table 4, we report statistics as we exercise DOT on both

our six-core machine and a sixteen-core Xeon 2.4GHz CPU. Here we correspondingly
set the number of subdomains in all simulations to six and sixteen respectively. Although
overall timings of course change, we see that DOT similarly maintains the fastest runtimes

51
 10 -2 10 -2
PN PN
ADMM-PD ADMM-PD
CN

LBFGS-PD

CN
LBFGS-PD
LBFGS-H LBFGS-H
DOT (ours) 10 -4 DOT (ours)
10 -4

0 100 200 300 400 500 600 0 50 100 150 200 250 300
iteration count time (s)

Figure 3.6: Convergence comparisons. Top: We compare the convergence of methods for
a single time step midway through a stretch script with a 138K vertex mesh; measuring error
with the characteristic gradient norm (CN) see Section 3.5.2. Middle: We observe DOT’s
super-linear convergence matches LBGHS-H and closely approaches Project Newton’s
(PN), while LBFGS-PD lags well behind and ADMM-PD does not converge. Bottom:
comparing timing, DOT pulls ahead of PN and LBFGS-H, with lower per-iteration costs.

across both machines, over the best timing for each example between PN, LBFGS-H and
LBFGS-PD. Here these three latter methods again swap one another in speeds per example.

Changing Decompositions Then, to confirm that there is a wide range of viable subdo-
mains settings for DOT, we examine performance as we vary subdomain sizes using the
same simulation example set from Section 3.5.3 above. In Table 5 we summarize statistics
for these simulations and observe that all simulations match or out-perform the best result
timing between PN, LBFGS-H and LBFGS-PD; the only exceptions being the smallest 7K
mesh horse simulations are slightly outperformed by PN and LBFGS-H.

3.5.5 Convergence

DOT balances efficient, local second-order updates with global curvature information from
gradient history. In Fig. 3.6 we compare convergence rates and timings across methods

52
for a single time step midway through the large stretch example of the 138K vertex horse
mesh. We observe super-linear convergence for DOT, matching LBFGS-H’s and closely
approaching PN’s, while LBFGS-PD lags well behind, and ADMM-PD characteristically
does not converge to even a much lower tolerance than the one requested. In turn, comparing
timings, DOT out-performs PN and LBFGS-H with lower per-iteration cost. In Fig. 3.7 we
visualize DOT’s characteristic process: in the first few iterations error is concentrated at
interfaces and is then quickly smoothed out by the successive iterations.

In addition to PN, LBFGS-PD, LBFGS-H and ADMM-PD, we also investigated stan-

dard Jacobi and Gauss-Seidel decomposition methods Quarteroni et al. [1999], and ex-
perimented with applying incomplete Cholesky as initializer for L-BFGS. Convergence
rates for the former two methods are slow, making them impractical compared to the above
methods. The latter method, interestingly, sometimes performs well, but is inconsistent as it
also can be slow and even fails to converge in other cases; see our supplemental document
Li et al. [2019] for details.

3.5.6 Varying Time Step, Material Parameters and Model

Here we compare behavior as we change material parameters and vary over a range of
frame-size time steps. We apply a twist, stretch and squash (TSS) script to an 18K vertex
monkey mesh. For the same example we apply time step sizes of 10, 25 and 40 ms
respectively and vary material parameters comparing across a range of Young’s modulus
and Poisson ratio. As summarized in Table 2, across all examples DOT obtains the fastest
runtimes with trends showing timings increasing for all methods as we increase time step
size and Poisson ratio. For varying stiffness, timings for DOT stay largely flat, while here
PN, LBFGS-H, and LBFGS-PD become slower with softer materials. Finally, we consider
changing the material model. We apply the Stable Neo-Hookean model to run the monkey
TSS example. Relative timings stays consistent as with the FCR model, where DOT ranges
from 1.5X to 2.4X faster than all alternatives.

53
3.6 Summary

10-3

10-4

10-5

(a) (b) (c)

10-6

iteration 1 iteration 9 iteration 17 iteration 25

Figure 3.7: Residual Visualization. With a decomposition (a), current deformation in (b),
and starting error in (c), we visualize DOT’s characteristic convergence process. In the
first few iterations error is concentrated at interfaces and is then rapidly smoothed by the
successive iterations.

In this work we developed DOT, a new time step solver that enables efficient, accurate
and consistent frame-size time stepping for challenging large and/or high-speed deforma-
tions with nonlinear materials. So far we have focused on CPU parallelization on moderate
commodity machines with medium-scale meshes ranging from 31K to 4.2M tetrahedra.
However, as we see in Section 3.5.3 above, DOT’s sub-linear scaling of iterations for more
decompositions makes extensions of DOT to large-scale systems exceedingly promising to
pursue. Concurrently for meshes at all scales we observe that while rule-of-thumb matching
domain count to available cores already exposes significant speed-up and robustness we
have also seen in Section 4.7 that across a wide range of decomposition sizes we maintain a
significant and consistent advantage. Thus we are also excited to explore DOT with recent
advances in batch-processed factorizations and solves on the GPU, e.g., with MAGMA

54
Abdelfattah et al. [2017], where L-BFGS low-rank updates can be efficiently performed via
map reduce. Likewise, while DOT offers speed and robust convergence at large time step,
decomposition also offers other promising opportunities. One exciting direction is applying
recent mesh-adaptation strategies Schneider et al. [2018] which can now be performed
on-the-fly, independently per subdomain.
When deformations are mild, uniformly distributed, at slow speeds and/or on small
meshes we see that the win for DOT is sometimes not as significant. If such cases are to be
expected then certainly alternatives such as PN, LBFGS-H, LBFGS-PD and ADMM-PD
may be reasonable choices as well. Our experience suggests that most scenarios are not
likely to limit the scope of a simulation tool to these cases. If so, then we propose DOT as
a one-size-fits-all method that improves or closely matches performance in these easier and
gentler cases and then shines for the challenging scenarios were previous methods become
stuck and/or exceedingly slow.
Our evaluation of DOT has so far focused on invertible energies. Efficiency with
noninverting energies, e.g., Neo-Hookean, may require custom-handling of the elasticity
barrier, e.g, by line search curing Zhu et al. [2018], and is an interesting future direction.
DOT solves elastodynamics for both rapidly moving and fixed boundary conditions
over a wide range of simulation conditions. As in ADMM-PD and LBFGS-PD, integrating
standard collision resolution methods using penalty or hard-constraints, can be directly
encorporated in DOT. However, custom-leveraging DOT’s structure for efficient contact
processing remains exciting future work.
Finally, while our decompositions from METIS have been effective they are certainly
not optimal for optimization-based time stepping. It would be interesting to explore custom
decompositions which specifically take advantage of the incremental potential’s structure
and even adaptive decompositions per-time step. Likewise, while the simple strategy
of matching subdomain count to cores already enables simple and easy-to-implement
advantages, it is of course in no way optimal. An exciting future investigation is exploring
per-task custom decompositions based on compute resources available.

55
Chapter 4

Incremental Potential Contact

4.1 Introduction

Contact is ubiquitous and often unavoidable and yet modeling contacting systems continues
to stretch the limits of available computational tools. In part this is due to the unique hurdles
posed by contact problems. There are several intricately intertwined physical and geometric
factors that make contact computations hard, especially in the presence of friction and
nonlinear elasticity.
Real-world contact and friction forces are effectively discontinuous, immediately mak-
ing the time-stepping problems very stiff, especially if the contact constraints are enforced
exactly. On the other hand, even small violations of exact contact constraints (which are
nonconvex) can lead to impossible to untangle geometric configurations, with a direct
impact on physical accuracy and stability. In addition, stiff contact forces often lead to
extreme deformations, resulting in element inversions for mesh-based discretization. Fric-
tion modeling then introduces further challenges with asymmetric force coupling and rapid
switching between sliding and sticking modes.
In this work, our goal is to achieve very high robustness (by which we mean the absence
of catastrophic failures or stagnation) for contact modeling even for the most challenging
elastodynamic contact problems with friction. Robustness should be obtained independent

56
of user-controllable accuracy in time-stepping, spatial discretization and contact resolution,
while maintaining efficiency required to solve large-scale problems. At the same time we
wish to also ensure that all accuracies – across the board – are efficiently attainable (of
course with additional cost) when required.

With these goals in mind, we reexamine the contact problem formulation, discretization
and numerical methods from scratch, building on numerous ideas and observations from
prior work.

Our Incremental Potential Contact (IPC) solver is constructed for mesh-based dis-
cretizations of nonlinear volumetric elastodynamic problems supporting large nonlinear
deformations, implicit time-stepping with contact, friction and boundaries of arbitrary codi-
mension (points, curves, surfaces, and volumes). A key principle we follow is that while
the physics and shape can be approximated arbitrarily coarsely, the geometric constraints
(absence of intersections of the approximate geometry and inversions of elements) are
maintained exactly at all times. We achieve this for essentially arbitrary target time steps
and spatial discretization resolution.

The key element of our approach is the formulation of the contact problem and the
customized numerical method to solve it. As a starting point, we use an exact contact
constraint formulation, described in terms of an unsigned distance function, and rate-based
maximal dissipation for friction.

For every time step, we solve the discrete nonlinear contact problem with a given toler-
ance using a smoothed barrier method, ensuring that the solution remains intersection-free
at all intermediate steps. We use a comparably smoothed, arbitrarily close, approximation
to static friction, also eliminating the need for an explicit Coulomb constraint, and cast
friction forces at every time step in a dissipative potential form, using an alternating lagged
formulation. All forces can then be solved by unconstrained minimization.

Our barrier formulation for contact has several important properties: 1) it is an almost
everywhere C 2 function of the unsigned distances between mesh elements, C 1 continuous
for a measure-zero set of configurations; 2) its support is restricted to a small part of the

57
configuration space close to configurations with contact. The former property makes it
possible to use rapidly converging Newton-type unconstrained optimization methods to
solve the barrier approximation of the problem, the latter ensures that additional contact
forces are applied highly locally and that only a small set of terms of the barrier function
need to be computed explicitly during optimization. Jointly they enable stable, conforming
contact between geometries.
To guarantee a collision-free state at every time step, feasibility is maintained throughout
all nonlinear solver iterations: the line search in our customized Newton-based solver is
augmented with efficient, filtered continuous collision detection (CCD) accelerated by a
conservative CFL-type contact bound on line search step sizes. Friction forces are resolved
directly in the same solver via our lagged potential with geometric accuracy improved by
alternating updates.

4.1.1 Contributions

In summary, IPC solves nonlinear elastodynamic trajectories that are intersection- and
inversion-free, efficient and accurate (solved to user-specified accuracies) while resolving
collisions with both nonsmooth and codimensional obstacles. To our knowledge, this is
the first implicit time-stepping method, across both the engineering and graphics literature,
with these properties.
We demonstrate the efficacy of IPC with stress tests containing large deformations, many
contact primitive pairs, large friction, tight collisions as well as sharp and codimensional
obstacles. Our technical contributions include

• A contact model based on the unsigned distance function;

• An almost everywhere C 2 , C 1 -continuous barrier formulation, approximating the

contact problem with arbitrary accuracy, with barrier support localized in the config-
uration space, enabling efficient time-stepping;

58
 • Contact-aware line search that continuously guarantees penetration-free descent
steps with CCD evaluations accelerated by a conservative-bound contact-specific
CFL-inspired filter;

• A new variational friction model with smoothed static friction, formulated as a lagged
dissipative potential, robustly resolving challenging frictional contact behaviors; and

• A new benchmark of simulation test sets with careful evaluation of constraint and
time stepping formulations along with an extensive evaluation of existing contact
solvers.

4.2 Contact Model

We focus on solving numerical time-integration for nonlinear volumetric elastodynamic

models with contact. These models can interact with fixed and moving obstacles which
can be of arbitrary dimension (surfaces, curves and points). The simulation domain is
discretized with finite elements. Given n nodal positions, x, finite-element mass matrix,
M , and a hyper-elastic deformation energy, Ψ (x), the contact problem extremizes the
extended-value action
Z T 1 
S(x) = ẋT M ẋ − Ψ (x) + xT (fe + fd ) dt.
0 2

on an admissible set of trajectories A , which we discuss below. Here fe are external forces
and fd are dissipative frictional forces. We assume, for simplicity, that all object geometry
is discretized with n-dimensional piecewise-linear elements, n = 1, 2, 3.

Admissible trajectories We construct a new definition of admissibility based on un-

signed distance functions that has a number of advantages. Most importantly, in the context
of our work, it naturally allows us to formulate exact contact constraints in terms of con-
straints on collisions between pairs of primitives (triangles, vertices and edges), and can be

59
defined in exactly the same way for objects of any dimensions (points, curves, surfaces and
volumes).

Specifically we define trajectories x(t), with x ∈ R3n as intersection-free, if for all

moments t, x(t) ensures that the distance d(p, q) between any distinct points p and q on the
boundaries of objects is positive. In the space of trajectories, the set of intersection-free
trajectories forms an open set AI , as it is defined by strict inequalities. Optimization
problems may not have solutions in this set; for this reason, we add the limit trajectories to
it, which involve contact. Specifically, we define the set of admissible trajectories A as the
closure of AI . In other words, a trajectory is admissible, if it is intersection-free, or there is
an intersection-free trajectory arbitrarily close.

Note that this closure is not equivalent to replacing the constraint d(p, q) > 0 with
d(p, q) ≥ 0; the latter is always satisfied for unsigned distances, so that all trajectories
would be admissible. This is not the case for our definition. Consider for example, a point
moving towards a plane. If its trajectory touches the plane and then turns back, an arbitrarily
small perturbation makes it intersection-free, and the trajectory is in A . However, if the
trajectory crosses the plane small perturbations do not make it intersection-free. This
highlights the need for our treatment even in the volumetric setting as the boundaries of
our mesh upon which we impose constraint are exactly surfaces whose potential collisions
include the point-face case above.

We can describe AI directly in terms of constraints on unsigned distances d between

surface primitives (vertices, edges, and faces in the simulation surface mesh and domain
boundaries). We denote this set of mesh primitives T . Equivalently to the more general
definition above, a piecewise-linear trajectory x(t) starting in an intersection-free state x0
is admissible, if for all times t, the configuration x(t) satisfies positive distance constraints
dij (x(t)) > 0 for all {i, j} ∈ B, where B ⊂ T × T is the set of all non-adjacent and
non-incident surface mesh primitive pairs.

We then observe that the distance between any pair of primitives is bounded from below
by the distance for triangle-vertex and edge-edge pairs, if there are no intersections. For

60
this reason, it is sufficient to enforce constraints dk (x(t)) > 0 continuously in time, for
all k ∈ C ⊂ B where C contains all non-incident point-triangle and all non-adjacent
edge-edge pairs in the surface mesh.

Time discretization Discretizing in time, we can directly construct discrete energies

whose stationary points give an unconstrained time step method’s update Ortiz and Stainier
[1999]. Concretely, given nodal positions xt and velocities v t , at time step t, we formulate
the time step update for new positions xt+1 as the minimization of an Incremental Potential
(IP) Kane et al. [2000], E(x, xt , v t ), over valid x ∈ R3n . For example the IP for implicit
Euler is then simply

E(x, xt , v t ) = 21 (x − x̂)T M (x − x̂) − h2 xT fd + h2 Ψ (x), (4.1)

where h is the time step size and x̂ = xt + hv t + h2 M −1 fe . IPs for implicit Newmark (see
Section 4.7) and many other integrators follow similarly by simply treating their update
rule as stationarity conditions of a potential with respect to variations of xt+1 .
Addition of contact constraints restricts minimization of the IP to admissible trajectories
Kane et al. [1999], Kaufman and Pai [2012] and so yields for our model the following time
step problem:
xt+1 = argmin E(x, xt , v t ), xτ ∈ A , (4.2)
x

where xτ , τ ∈ [t, t + 1], is the linear trajectory between xt and xt+1 .

Our goal is to define a numerical method for approximating the solution of this problem
in (4.2). Solving it is challenging due to the complex nonlinearity of the admissibility
constraint, especially in the context of large deformations.
In turn, when frictional forces in (4.2) include frictional contact, solving the time
step problem becomes all the more challenging as fd is now governed by the Maximal
Dissipation Principle Moreau [1973] and so must satisfy further challenging, asymmetric
and strongly nonlinear consistency conditions Simó and Hughes [1998]. We present a
friction formulation in Section 4.5 that is naturally integrated into this formulation via a
lagged dissipative potential.

61
 We further define a set of piecewise-linear surfaces as -separated, if the distance
between two boundary points of the set is at least , unless these are on the same element
of the boundary. An -separated trajectory is then a trajectory for which surfaces stay
-separated. We denote the set of such trajectories A .
To handle contact constraints, in our algorithm, we use the following overall approach:
(a) the IP function E is unmodified on A – the set of trajectories for which any -separated
trajectory extremizes the action are preserved; (b) we introduce a barrier term that vanishes
for trajectories in A and diverges as the distance between any two boundary points vanishes,
converting the problem to an unconstrained optimization problem. This barrier, together
with continuous collision detection within minimization steps, ensure that trajectories
remain in AI .
This algorithm then guarantees that the trajectories are modified, compared to the exact
solution, in an arbitrarily small, user-controlled (by ) region near object boundaries and, at
the same time, always remain admissible.

4.2.1 Trajectory accuracies

A discrete contacting trajectory is accurate if it satisfies 1) admissibility, 2) discrete mo-

mentum balance, 3) positivity, 4) injectivity and 5) complementarity.
In the discrete setting, momentum balance requires that the gradient of the incremental
potential, ∇x E(x), balance against the time-integrated contact forces. Its accuracy, is then
exactly measured by the residual error in the optimization of the constrained incremental
potential. We simply and directly control accuracy of momentum balance by setting
stopping tolerance in our nonlinear optimization; see Section 4.4.5.
In turn positivity means that contact forces’ signed magnitudes, λk , per contact pair
k ∈ C , are always non-negative and so push surfaces but do not pull. Our method
guarantees exact positivity.
Combined with admissibility, injectivity requires positive volumes for all tetrahedra
in the simulation mesh. This invariant is enforced when a non-inverting energy density

62
function, e.g. neo-Hookean, is modeled 1 .
Finally, the classic definition of complementarity in contact mechanics Kikuchi and
Oden [1988] is the requirement that contact forces enforcing admissibility can only be
exerted between surfaces if they are touching with no distance between them. We do
not allow dk (x) = 0, and so define a comparable measure of discrete -complementarity
requiring
λk max(0, dk (x) − ) = 0, ∀k ∈ C (4.3)

to measure how well contact accuracy is achieved. Discrete complementarity is then

satisfied whenever distances between all contact pairs, defined as surface pairs with nonzero
contact forces, are less than the  and converge to complementarity as we reduce .

4.3 Related Work

Computational contact modeling is a fundamental and long studied task in mechanics well
covered from diverse perspectives in engineering, robotics and computer graphics Brogliato
[1999], Kikuchi and Oden [1988], Stewart [2001], Wriggers [1995]. At its core the contact
problem combines enforcement of significant and challenging geometric non-intersection
constraints with the resolution of a deformable solid’s momentum balance. The latter task is
well-explored, often independent of contact Belytschko et al. [2000], Stuart and Humphries
[1996]. We focus below on related works in defining contact constraints, implicitly time
stepping with contact and friction, and barriers.

4.3.1 Constraints and constraint proxies

Contact simulation requires a computable model of admissibility and so a choice of contact

constraint representation. For volumetric models, admissibility generally begins with
description of a signed distance function. This allows a clean formulation of the continuous
1
When an invertible deformation model, e.g. fixed corotational, is modeled, injectivity need not be
preserved in computation. We primarily focus on non-inverting neo-Hookean but will also demonstrate the
weaker invertible case with fixed corotational.

63
model. However, when it comes to computing non-intersection on deformable meshes,
choices for representing non-intersection must be made and a diversity of constraint
representations exist. Contact constraints for deformable meshes, in both engineering
Belytschko et al. [2000], Wriggers [1995] and graphics Bridson et al. [2002], Daviet et al.
[2011], Harmon et al. [2008, 2009], Otaduy et al. [2009], Verschoor and Jalba [2019] are
most commonly defined pairwise between matched surface primitives.

Existing methods most often define a local, signed distance evaluation using a diverse
array of nonlinear proxy functions as well as their linearizations. These include linear
gap functions, linearized constraints built from geometric normals, as well as a number
of oriented volume constraints Kane et al. [1999], Sifakis et al. [2008]. These nonlinear
proxies, such as the tetrahedral volumes formed between surface point-face and edge-edge
pairs, are only locally valid. They can introduce artificial ghost contact forces when sheared,
false positives when rotated (e.g. for edge-edge tetrahedra), discontinuities when traversing
surface element boundaries, and, in many methods, must still be further linearized and so
introduce additional levels of approximation in order to solve a constrained time step.

Alternately gap functions and other related methods approximate signed distance
functions for pairs of primitives by locally projecting a linearized distance measure between
pairwise surface primitives onto a fixed geometric normal Otaduy et al. [2009], Wriggers
[1995]. As discussed in Erleben Erleben [2018] these “contact point and normal” based
constraint functions can be inconsistent over successive iterations and so are highly sensitive
to surface and meshing variations with well known failure modes if care is not taken. Indeed,
as we investigate in Section 4.8.3, even with iterative updates of these linear constraints
inside SQP-type methods, time stepping with gap functions and related representations
produces highly varied results whose success or failure is largely dependent on the scene
simulated. In turn all of these challenges are only further exacerbated when simulations
encounter the sharp, nonsmooth and even codimensional collisions imposed by meshed
obstacles Kane et al. [1999]; see e.g. Figure 4.1.

Recent fictitious domain methods Jiang et al. [2017b], Misztal and Bærentzen [2012],

64
Figure 4.1: Nonsmooth, codimensional collisions. Left: thin volumetric mat falls on
codimensional (triangle) obstacles. Right: a soft ball falls on a matrix of point obstacles,
front and bottom views.

Müller et al. [2015], Pagano and Alart [2008], Zhang et al. [2005] offer a promising
alternative. In these methods, motivated by global injectivity conditions Ball [1981]
negative space is separately discretized by a compatible discretization sometimes called
an air-mesh Müller et al. [2015]. Maintaining a non-negative volume on elements of this
mesh then guarantees non-inversion. However, as with locally defined proxy volumes, the
globally defined mesh introduces increasingly severe errors, e.g., shearing and locking
forces, as it distorts with the material mesh. In 2D this can be alleviated by local Müller
et al. [2015] or global Jiang et al. [2017b] remeshing, however this is highly inefficient in
3D, does not provide a continuous constraint representation for optimization, nor, even with
remeshing, can it resolve sliding and resting contact where air elements must necessarily
be degenerate Li et al. [2018b].
Alternately, discrete signed distance fields (SDF) representations can be constructed
via a number of approximation strategies over spatial meshes Jones et al. [2006]. However,
while state of the art adaptive SDF methods now gain high-resolution accuracy for sampling
against a fixed meshes Koschier et al. [2017], they can not yet be practically updated at rates
suitable for deformable time steps, much less at rates suitable for querying deformations
at every iterate within a single implicit time step solve Koschier et al. [2017]. At the
same time, discontinuities across element boundaries, while improved in recent works, still
preclude smooth optimization methods.

65
 We observe that while approximating signed distance pairwise between surface mesh
elements is problematic, unsigned distance is well defined. We then design a new contact
model for exact admissibility constraints in terms of unsigned distances between mesh-
element pairs. This model of constraint is constructed sufficiently smooth to enable efficient,
super-linear Newton-type optimization, maintains exact constraint satisfaction guarantees
throughout all steps (time steps and iterations) and requires evaluation of just mesh-surface
primitive pairs.

4.3.2 Implicit Time Step Algorithms for Contact

With choice of contact constraint proxy, g(x) ≥ 0, the solve for the implicit time-step
update is then the minimization of the contact-constrained IP Doyen et al. [2011], Kane
et al. [1999], Kaufman and Pai [2012],

min E(x, xt , v t ) s.t. g(x) ≥ 0. (4.4)

x

The variational problem (4.4), or its approximation is then minimized to compute the
configuration at each time step.
This is typically done with off-the-shelf Nocedal and Wright [2006] or customized
constrained optimization techniques. In engineering, commonly used methods include
sequential quadratic programming (SQP) Kane et al. [1999], augmented Lagrangian and
occasionally interior point methods Belytschko et al. [2000]. All such methods iteratively
linearize constraint functions and elasticity. However, both nonlinear constraint functions
and their linearizations are generally valid only in local regions of evaluation and so can lead
to intersections due to errors at larger time steps, faster speeds and/or larger deformations.
For example Kane et al.’s [1999] volumes are only valid under a strong assumption of the
relative position of contact primitive pairs.
In turn linearization of the full constraint set can also introduce additional error, result in
infeasible sub-problems, locking and/or constraint drift Erleben [2018]. This often requires
complex and challenging (re-)evaluations of constraints in inter-penetrating states. Even

66
when such obstructions are not present, iterated constraint linearization generally can not
guarantee interpenetration-free state except upon convergence and so often must resort to
small time steps and non-physical fail-safes in order to limit damage caused by missed
constraint enforcement.
Although SQP- Kane et al. [1999] and LCP/QP-based contact solvers Kaufman et al.
[2008] support and generally employ a variety of constraint-set culling and active-set update
strategies, e.g., incrementally adding newly detected collisions at each iteration Otaduy
et al. [2009], Verschoor and Jalba [2019], they also can become infeasible and generate
constraint drift when linearizing and filtering constraints.
Irrespective of how the contact-IP is solved and constraints are enforced, we then
remain faced with combinatorial explosion in the number of contact constraints to handle.
Determining the active set, i.e., finding which constraints are necessary for admissibility and
so can not be ignored, remains an outstanding computational challenges. At the same time,
to take large time steps or handle large deformation, we must resolve strongly nonlinear
deformation energies in balance with contact forces. This requires line search. However,
for constrained optimization methods, e.g., SQP, efficient line search in the presence of
large numbers of active constraints remains an open problem Bertsekas [2016], Nocedal
and Wright [2006]. For this reason, existing methods in graphics currently avoid line search
altogether and are, as a consequence, mostly restricted to quadratic energy models per time
step Otaduy et al. [2009], Verschoor and Jalba [2019] and, often, small time step sizes for
even moderate material stiffness Verschoor and Jalba [2019].

4.3.3 Friction

The addition of accurate friction with stiction only increases the computational challenge
for time stepping deformation Wriggers [1995]. Friction forces are tightly coupled to the
computation of both deformation and the contact forces that prevent intersection. These side
conditions are generally formulated by their own governing variational Maximal Dissipation
Principle (MDP) Goyal et al. [1991], Moreau [1973] and thus introduce additional nonlinear,

67
nonsmooth and asymmetric relationships to dynamics. In transitions between sticking and
sliding modes large, nonsmooth jumps in both magnitude and direction are made possible
by frictional contact model. Asymmetry, in turn, is a direct consequence of MDP: frictional
forces are not uniquely defined by the velocities they oppose, and are also determined
by additional consistency conditions and constraints, e.g., Coulomb’s law. One critical
consequence is that there is no well-defined potential that we can add to an IP to directly
produce contact friction via minimization.

To address these challenges, frictional contact is often solved by seeking a joint solution
to the optimality conditions of MDP together with the discretized equations of motion (the
latter are equivalent to optimality conditions for E). This requires, however, simultaneously
solving for primal velocity unknowns together with a large additional number of dual
contact and friction force unknowns. These latter variables then scale in the number of
active contacts and their number grows large for even moderately sized simulation meshes.

To solve these systems it is generally standard to apply iterative per-contact, nonlinear

Gauss-Seidel-type methods Alart and Curnier [1991], Bridson et al. [2002], Daviet et al.
[2011], Jean and Moreau [1992], Kaufman et al. [2014]. Here elasticity is again often, but
not always, linearized per time step, while contact and friction constraints are similarly
often approximated per iteration with a range of linear and nonlinear proxies. Alternate
iteration strategies Kaufman et al. [2008], Otaduy et al. [2009] have also been applied.
However, as in the frictionless setting, all such splittings remain challenging to solve with
guarantees for complex, real-world scenarios. Most recently, the same discrete formulation
has been solved with new custom-designed algorithms – both with nonsmooth Newton-type
strategies Bertails-Descoubes et al. [2011], Macklin et al. [2019] and an extension of
the Conjugate Residual method Verschoor and Jalba [2019] with improved accuracy and
efficiency.

68
4.3.4 Barrier Functions

Barrier functions are commonly applied in nonlinear optimization, especially in interior-

point methods Nocedal and Wright [2006]. Here primal-dual interior point methods
are generally favored with Lagrange multipliers as additional unknowns for improved
convergence. For contact problems, this impractically enlarges system sizes by orders-of-
magnitude. Here we focus on a primal solution suited for contact problems. Similarly,
the vast majority of the literature focuses on globally supported functions, which are
not viable for contact due to the quadratic set (collision primitive pairs) of constraints
that must be considered. Recently, a few works have begun exploiting locally supported
barriers Harmon et al. [2009], Schüller et al. [2013], Smith and Schaefer [2015]. Harmon
et al. [2009] propose a set of layered discrete penalty barriers that grow unbounded as
the configuration reaches toward contact. While well-suited for small time-step explicit
methods, the incremental construction of the barriers challenge application in implicit time
integration with Newton-type optimization. Most recently methods in geometry processing
Schüller et al. [2013], Smith and Schaefer [2015] propose locally supported barriers in
the context of 2D mesh parametrization to prevent element inversion and overlap. Our
formulation builds on a similar idea. Here we design smoothed, local barriers custom-
constructed for the challenges of resolving contact-response and preventing intersection
between 3D mesh-primitives.

4.3.5 Summary

In summary, state of the art methods for contact simulation are often highly effective per
example. However, in order to do so they generally require significant hand-tuning per
simulation set-up in order to obtain successful simulation output, i.e., stable, nonintersecting,
plausible, or predictive output. Currently many of the tuned parameters, as we discuss
in Section 4.8.3, are not physical but rather guided by expected intersection constraint
violation errors and stability needs, and so need to be experimentally determined by

69
many simulation test runs. Thus, to date, direct, fully automated simulation has not been
available for contact simulation – despite contact’s fundamental role in many design,
engineering, robotics, learning and animation tasks. Towards a direct, “plug-and-play”
contact simulation framework we propose IPC . Across a wide range of mesh resolutions,
time step sizes, physical conditions, material parameters and extreme deformations we
confirm IPC performs and completes simulations to requested accuracy without algorithm
parameter tuning.

4.4 Primal Barrier Contact Mechanics

In this section, we describe how we solve our time step problem (4.2) formulated in
Section 4.2. We defer consideration of friction to Section 4.5, focusing on handling contact
dynamics here. We solve the minimization problem (4.2), with primitive-pair admissibility
constraints using a carefully designed barrier-augmented incremental potential that can be
evaluated efficiently. In turn, to solve this potential we design a custom, contact-aware,
Newton-type solver, outlined in Algorithm 3, with constraint culling for efficient evaluation
of objectives, gradients and Hessians (Section 4.4.3).

4.4.1 Barrier-Augmented Incremental Potential

To enforce distance constraints dk (t) > 0, for all k ∈ C , we construct a continuous barrier
energy b (Section 4.4.2), that creates a highly localized repulsion force, affecting motion
only when primitives are close to collision, and vanishing if primitives are a small user-
specified distance apart. We then augment the time step potential E(x, xt , v t ) with a sum
of these barriers over all possible pairs in C . The barrier-augmented IP is then
X
Bt (x) = E(x, xt , v t ) + κ


b dk (x) , (4.5)
k∈C

with κ > 0 an adaptive conditioning parameter automatically controlling the barrier

stiffness (see Section 4.4.3 and our appendix for details.).

70
Algorithm 3 Barrier Aware Projected Newton
1: procedure BARRIER AWARE P ROJECTED N EWTON(xt , )

2: x ← xt
3: Cˆ ←ComputeConstraintSet(x, d)
ˆ . Section 4.4.6, 4.6.1
4: ˆ Cˆ)
Eprev ← Bt (x, d,
5: xprev ← x
6: do
7: ˆ Cˆ))
H ← SPDProject(∇2x Bt (x, d, . Section 4.4.3
8: ˆ Cˆ)
p ← −H −1 ∇x Bt (x, d,
9: // CCD line search: . Section 4.4.4
10: α ← min(1, StepSizeUpperBound(x, p, Cˆ))
11: do
12: x ← xprev + αp
13: Cˆ ←ComputeConstraintSet(x, d)
ˆ

14: α ← α/2
15: ˆ Cˆ) > Eprev
while Bt (x, d,
16: ˆ Cˆ)
Eprev ← Bt (x, d,
17: xprev ← x
18: Update κ, BCs and equality constraints . appendix
19: while h1 kpk∞ > d
20: return x

71
 Minimizing (4.5) enables the solution of contact-constrained dynamics with uncon-
strained optimization. Computing the energy naively, however, would require evaluation of
the barrier functions for all O(|T |2 ) pairs. To address similar challenges many simulation
methods simply remove constraints corresponding to distant primitives that are hoped to be
unnecessary for the current solution. However, this tempting operation is dangerous, as
significant errors and instabilities can be introduced when constraint sets are modified and
critical collisions can also be missed (see Sections 4.3 and 4.8). Instead, we design smooth
barrier functions that allow us to compute the barrier energy exactly and efficiently for all
constraints while evaluating distances only for a small subset of pairs of primitives that are
close and simultaneously ensuring that the rest smoothly evaluate to zero.

4.4.2 Smoothly Clamped Barriers

We begin by defining a smooth barrier function composed of terms that are local for every
primitive pair, that is each term is exactly zero if the two primitives are far away, enabling
reliable and efficient pruning of pairs in C without change to the solution.

We start by defining a computational distance accuracy target, dˆ > 0 (corresponding to 

in Section 4.2) that specifies the maximum distance at which contact repulsions can act. We
then construct a barrier potential that approaches infinity at zero distance, initiates contact
ˆ and applies no repulsion at distances
forces for pairs closer than the target distance, d,
ˆ
greater than d.

Considering the smooth log-barrier function commonly applied in optimization Boyd

ˆ where d is the unsigned distance evaluation be-
and Vandenberghe [2004] gives ln(d/d),
tween a primitive pair. However, simply truncating this function produces an unacceptably
non-smooth energy which cannot be efficiently optimized and is effectively no better than
simply discarding constraints. Some examples of problems this generates in optimiza-
tion are covered in the appendix. We thus propose a smoothly clamped barrier to regain

72
superlinear convergence for Newton-type methods

ˆ 2 ln d , 0 < d < dˆ


−(d − d)

ˆ
ˆ =
b(d, d) d
(4.6)
0
 ˆ
d ≥ d.
Our barrier function is now C 2 at the clamping threshold, and it is exactly zero for pairs
beyond the target accuracy (see Figure 4.2). Now, without harm, at any configuration x, we
only need to evaluate barrier terms for the culled constraint set
ˆ
Ĉ(x) = {k ∈ C : dk (x) ≤ d},

composed of barriers between close primitives. As we increase accuracy by specifying

smaller dˆ we then need to evaluate increasingly smaller numbers of contact barriers, albeit
with increased cost in nonlinearity.
ˆ itself is now C 2 , the distance function it evaluates
Next, while the barrier function b(d, d)
between primitives will be C 0 for certain unavoidable configurations; i.e., parallel edge-
edge collisions – see Figure 4.8. For this reason, we multiply the barrier terms for edge-edge
collisions by a mollifier that ensure our distance function is C 1 (and piecewise C 2 ) for all
primitive pair types. Distance evaluation and mollifier are discussed in detail in Section
4.6. Additional important considerations related to numerical stability and roundoff error
in distance evaluation are then detailed further in the appendix.

4.4.3 Newton-Type Barrier Solver

Projected Newton (PN) methods are second-order unconstrained optimization strategies

for minimizing nonlinear nonconvex functions where the Hessian may be indefinite. Here
we apply and customize PN to the barrier-augmented IP (4.5). At each iteration, we
project each local energy stencil’s Hessian to the cone of symmetric positive semi-definite
(PSD) matrices (see SPDProject function in Algorithm 3) prior to assembly. Specifically,
following Teran et al. [2005] we project per-element elasticity Hessians to PSD. We then
comparably project the Hessian of each barrier to PSD. Each barrier Hessian has the form
∂ 2b ∂b 2
∇x d(∇x d)T + ∇ d (4.7)
∂d 2 ∂d x
73
 3 3

Barrier energy
Barrier energy
2 2

1 1

0 0
0 0.5 1 1.5 0 0.5 1 1.5
Distance Distance

Figure 4.2: Barriers. Left: log barrier function clamped with varying continuity. We
can augment the barrier to make clamping arbitrarily smooth (see our appendix). We
apply our C 2 variant for best tradeoff: smoother clamping improves approximation of the
discontinuous function while higher-order continuity introduces more computational work.
Right: our C 2 clamped barrier improves approximation to the discontinuous function as
ˆ smaller.
we make our geometric accuracy threshold, d,

and so can be constructed as a small matrix restricted to the vertices in the stencil of the
barrier’s primitive pair. The addition of mass matrix terms then ensures that the assembled
total IP Hessian is symmetric positive definite (SPD). Originally we also investigated a
Gauss-Newton approximation to the above barrier Hessian, taking only the first, SPD term
in the sum. However, we find that resulting search directions are far less efficient than using
the full projected barrier Hessian.

Termination For termination of the solver we check the infinity norm of the Newton
search direction scaled by time step (but unscaled by line-search step size). Specifically
we solve each time step’s barrier IP to an accuracy satisfying h1 kH −1 ∇Bt (x)k∞ < d .
This provides affine invariance and a characteristic measure using the Hessian’s natural
scaling as metric. Accuracy is then directly defined by d in physical units of velocity
(and so is independent of time-step size applied) and consistently measures quadratically
approximated distance to local optima across examples with varying scales and conditions.

74
Barrier stiffness adaptation We automatically adapt our barrier stiffness to provide
repulsive scaling that balances necessary distances against conditioning from the barrier
stiffness. Our barrier-augmented potential, Bt , has two key parameters: dˆ and κ, that jointly
scale the effective stiffness of each contact barrier. The strength of our barriers’ contact
forces (equivalently Lagrange multipliers) are directly determined during minimization
by evaluating distances, dk , and stiffness, κ. When κ is too small, contact-pair distances
must become tiny to exert sufficient repulsion. On the other hand, when κ is too large,
contact-pair distances must be below dˆ in order to exert non-zero force, but at the same
time remain exceedingly close to dˆ so as to not exert too large a repulsion. Both cases thus
generate unnecessary ill-conditioning and nonsmoothness that challenge efficiency. As we
ˆ this frees κ to adaptively condition our
directly control geometric accuracy by setting d,
Newton-solver to improve convergence. While conceptually one could imagine finding
improved scalings of κ by hand, per example, this is unacceptable and inefficient for an
automated simulation pipeline. Instead, in our appendix, we derive our stiffness update
algorithm that automatically adapts barrier stiffness per iterate for improved conditioning.

Relation to homotopy solves While in IPC we directly set and solve for a desired target
ˆ a natural alternative is to solve with a homotopy as is typical in interior point
accuracy d,
methods. We initially experimented with this approach: solving for larger distances (and so
ˆ over successive nonlinear
less stiff systems) and then decreasing to the target distance, d,
solves. We find, however, that this is unnecessary for elastodynamics where the direct
barrier solves we employ are much more efficient. In part, this is because we typically have
a good warm start available from the prior time step.

4.4.4 Intersection-Aware Line Search

While our barrier energy is infinite for contact, this by itself does not guarantee that
constraints dk (t) > 0 are not violated by the solver. Standard line search Nocedal and
Wright [2006], e.g, back-tracking with Wolfe conditions, can find an energy decrease

75
in configurations that have passed through intersection, resulting in a step that takes the
configuration out of the admissible set.
Smith and Schaefer’s [2015] line-search filter computes the largest step size in 2D
per triangle and per boundary point-edge pair that first triggers inversion or overlap, and
then take the minimum as a step size upper bound for the current Newton iteration to
stay feasible. Taking inspiration from this line-search filter we propose a continuous,
intersection-aware line search filter for 3D meshes. In each line search we first apply CCD
to conservatively compute a large feasible step size along the descent step. We then apply
back-tracking line search from this step size upper bound to obtain energy decrease. CCD
then certifies that each step taken is always valid. When we apply barrier-based energy
densities (our default) for our elasticity potential, Ψ , i.e., neo-Hookean, we combine the
inversion-aware line search filter Smith and Schaefer [2015] with our intersection-aware
filter to obtain descent steps. In combination this guarantees that every step of position
update in our solver (and so simulation) maintains an inversion- and intersection-free
trajectory.

4.4.5 IPC Solution Accuracy

Revisiting accuracy we confirm momentum balance is directly satisfied by IPC after

convergence. For example, for implicit Euler we have

ˆ = 0 =⇒ M ( x − x̂ ) = −∇Ψ (x) +
X
∇x Bt (x, d) λk ∇dk (x), (4.8)
h2 k∈C

where our contact forces λk are given by barrier derivatives

κ ∂b
λk = − . (4.9)
h2 ∂dk

Comparable discrete momentum balance follows when we apply alternate time integration
methods, e.g. implicit Newmark. Positivity is then confirmed directly by (4.9) and observing
∂b
that our barrier function definition guarantees ∂dk
≤ 0. In turn our above line-search filters
guarantee admissibility and, when applicable for barrier-type elasticity energy densities,

76
injectivity. Finally, our barrier definition ensures discrete complimentarity is always satisfied
as contact forces can not be applied at distance more than  = dˆ away.

Figure 4.3: Extreme stress test: rod twist for 100s. We simulate the twisting of a
bundle of thin volumetric rod models at both ends for 100s. IPC efficiently captures the
increasingly conforming contact and expected buckling while maintaining an intersection-
and inversion-free simulation throughout. Top: at 5.5s, before buckling. Bottom: at 73.6s,
after significant repeated buckling is resolved.

4.4.6 Constraint Set Update and CCD Acceleration

Every line search, prior to backtracking, performs CCD to guarantee non-intersection,

while every evaluation of energies and their derivatives compute distances to update the
culled constraint set, Cˆ(x). To accelerate these computations, we construct a combined
spatial hash and distance filtering structure to efficiently reduce the number of primitive-pair
distance checks. Then, to further accelerate intersection-free stepping along each Newton
iterate’s descent direction, p, we derive an efficient conservative bound motivated by CFL
conditions Courant et al. [1967]. As in force evaluations we aim to avoid unnecessary
and expensive CCD computation on primitive pairs not in Cˆ. We leverage the fact that all
contact pairs not in Cˆ are at distances greater than d,
ˆ and use the maximal relative search

step in p of each such pair to obtain a conservative upper bound on step size. We then
need only perform the CCD tests on primitive pairs in Cˆ. This CCD culling generally
provides an average 50% speed-up for all CCD costs across our simulations, with negligible
increase in Newton iterations and an overall impact of 10% improvement in simulation
times. Details on these accelerations and our adaptive application of this bound (to avoid
taking overly conservative steps) are detailed in our appendix.

77
4.5 Variational Friction Forces

Frictional contact adds contact-dependent dissipative forcing to our system. At macroscale

these friction forces are modeled by the Maximal Dissipation Principle (MDP) Moreau
[1973]. MDP posits that frictional forces maximize rate of dissipation in relative motion
directions orthogonal to contact constraints up to a maximum magnitude imposed by limit
surfaces, e.g. as modeled by Coulomb’s constraint Goyal et al. [1991].

4.5.1 Discrete Friction

To include frictional contact in our time stepping, we add local friction forces Fk for
every active surface primitive pair, k ∈ Ĉ(x). For each such pair k, at current state x,
we construct a consistently oriented sliding basis Tk (x) ∈ R3n×2 . Each Tk is built so that
uk = Tk (x)T (x − xt ) ∈ R2 gives the local relative sliding displacement at contact k, in
the frame orthogonal to the distance vector between closest points on the two primitives
defining dk (x). See Section 4.6 and our appendix for details on construction of Tk (x). The
corresponding sliding velocity is then vk = uk /h ∈ R2 .

Maximizing dissipation rate subject to the Coulomb constraint defines friction forces
variationally

Fk (x, λ) = Tk (x) argmin β T vk s.t. kβk ≤ µλk (4.10)

β∈R2

where λk is the contact force magnitude and µ is the local friction coefficient.

Friction forces governed by (4.10) are bimodal. If kvk k > 0, there is sliding and the
corresponding friction force opposes it with Fk = −µλk Tk (x) kuukk k . If kvk k = 0, there is
sticking and the corresponding static friction force is Fk = −µλk Tk (x)f , where the friction
direction f can take any value in the closed 2D unit disk.

78
4.5.2 Challenges to Computation

Friction forces Fk are then challenging to solve for in three interconnected ways. First, Fk
is nonsmooth. In transitions between sticking and sliding modes, nonsmooth jumps in both
magnitude and direction are possible. Second, because of sticking modes, Fk in MDP is not
uniquely defined by displacements until we have found a solution satisfying stationarity:
X
∇Bt (x) − h2 Fk (x, λ) = 0. (4.11)
k∈C

Third, there is no well defined dissipation potential whose spatial gradient will generate fric-
tion forces. As a consequence, frictional contact forces do not naturally fit into variational
time-stepping frameworks.
To tackle these challenges, we first examine Fk as a nonsmooth function of uk . Next,
as in our barrier treatment of contact, we smooth the friction function with controlled and
bounded accuracy. Then, in order to apply friction as an energy potential in our variational
solve, we lag updates of the sliding bases Tk and contact forces λk over nonlinear solves
within each time step (or over time steps). This allows us to define a smooth dissipative
potential for friction that can be consistently integrated into our Newton-type solver.

Figure 4.4: Friction benchmark: Stiff card house. Left: we simulate a frictionally stable
“card” house with 0.5m × 0.5m × 4mm stiff boards (E = 0.1GP a). Right: we impact the
house at high-speed from above with two blocks; elasticity is now highlighted as the thin
boards rapidly bend and rebound.

79
4.5.3 Smoothed Static Friction

During each of our Newton iterations any transitions of sliding displacements to or from
sticking conditions will introduce large and sudden jumps in friction forces, Fk . These
discontinuities, if left unmollified, would severely slow and even break convergence of
gradient-based optimization; see Section 4.7. To enable efficient and stable optimization,
we smooth the friction-velocity relation in the transition to static friction.
We start with a useful and equivalent (re-)expression for friction forces:

Fk = −µλk Tk (x)f (kuk k) s(uk ), (4.12)

uk
with s(uk ) = kuk k
when kuk k > 0, while s(uk ) takes any 2D unit vector when kuk k = 0.
The friction magnitude function, f , is then correspondingly nonsmooth with f (kuk k) = 1
when kuk k > 0, and f (kuk k) ∈ [0, 1] when kuk k = 0.
To smooth f and so (4.12) with bounded approximation error, we first define a velocity
magnitude bound v (in units of m/s) below which sliding velocities vk = uk /h are
treated as static. Then, we define a smoothed approximation of f with f1 . We maintain
f1 (y) = 1 for all y > hv , (sliding) while for y ∈ [0, hv ], we require f1 (y) to smoothly and
monotonically transition from 1 to 0 over a finite range. This forms a bijective map from
velocity magnitudes to friction magnitudes for velocities below the v limit. For smoothing
we experiment with satisfying interpolating polynomials ranging from C 0 to C 2 . Increased
continuity order introduces greater smoothing and faster error reduction for decreasing v ,
at the cost of introducing greater nonlinearity into the IP solve. In the end, we find that our
C 1 interpolant

− y2 2y
+ , y ∈ (0, hv )

2v h2 v h
f1 (y) = (4.13)
1, y ≥ hv ,


provides best balance – yielding a continuous force Jacobian while introducing less nonlin-
earity and so fewer overall iterations in testing. See Figure 4.5 and our discussion in the
appendix.

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 1 1

0.5 0.5

Friction force
Friction force 0 0

-0.5 -0.5

-1 -1
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Tangent relative velocity Tangent relative velocity

Figure 4.5: Friction smoothing in 1D. Left: increasing orders of our polynomials better
approximate the friction-velocity relation with increasing smoothness. Right: Our C 1
construction improves approximation to the exact relation as we make our frictional
accuracy threshold, v , and so the size of static friction zone, smaller.

4.5.4 Variationally Approximated Friction

With a smooth and uniquely defined Fk for each uk , we are now able to define friction
forces solely based on nodal displacement unknowns. A next natural step would then be to
define a so-called dissipative potential Kane et al. [2000], Pandolfi et al. [2002] for inclusion
in our optimization. An ideal potential would be a scalar function with respect to x whose
gradient returns Fk . However, even with our smoothing, no well-defined displacement-
based potential for friction exists, and Fk cannot be approximated by a potential force
without introducing significant approximation errors. In other words, we do not have a
variational form of friction that we can yet minimize.
We start by making dependence of our friction on both Tk (x) and λk (x) explicit:

uk
Fk (x, λk , Tk ) = −µλk Tk f1 (||uk ||) . (4.14)
||uk ||

Now, if we set Tk = Tk (x) and λk = λk (x) this friction evaluation is exact. However, if we
decouple dependence of the evaluated sliding basis and contact force from x and instead
lag them to values, λn , T n , from a prior nonlinear solve (or previous time step) n, then all
remaining terms in the expression for friction are integrable. The lagged friction force is

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then Fk (x, λnk , Tkn ) and provides a simple and compact friction potential,

Dk (x) = µλnk f0 (||uk ||). (4.15)

Here f0 is given by the relations f00 = f1 and f0 (v h) = v h so that Fk (x) = −∇x Dk (x).
This potential provides easy-to-compute Hessian, ∇2x Dk (x), and energy contributions to
the barrier potential, described in detail in the appendix. Our full friction potential is then
D(x) = h2 k∈C Dk (x), and the frictional barrier-IP potential for the time step t + 1 is
P

ˆ + D(x).
Bt (x, d) (4.16)

Friction Hessian projection For our Newton method (Section 4.4.3), we again need to
project the friction potential Hessian to the space of PSD matrices. The friction Hessian
structure is similar to that of elasticity, in that it can be written as a product of the Tk
matrices. This allows us to apply the same strategy as used for elasticity Hessians, and so
we need only perform a 2 × 2 PSD projection for each friction term per primitive pair. This
is detailed in our appendix.

4.5.5 Frictional Contact Accuracy

Accuracy of friction forces generated by each solution of our IP (4.16) are defined by the
static threshold, sliding basis and contact force magnitudes.

Static friction threshold As we apply smaller v we decrease the range of sliding ve-
locities that we exert static friction upon and correspondingly sharpen the friction forces
towards the exact nonsmooth friction relation. Decreasing v thus reduces stiction error
while increasing compute times as we introduce a sharper nonlinearity in a tighter range;
see Figure 4.5. For accurate reproduction of dynamic behaviors with friction and for
visually plausible results, we observed that v = 10−3 `m/s, where ` is characteristic length
(i.e. bounding box size), works well as a default across a wide range of examples with
friction coefficients. See e.g., Figure 4.7. As static accuracy becomes important, we then

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find solutions with v = 10−5 m/s work well. We have further confirmed IPC convergence
down to v = 10−9 m/s. See, for example, our reproduction of the stable frictional contact
structures in the masonry arch and card house examples in Figures 4.4 and 4.6.

Figure 4.6: Friction benchmark: Masonry arch. IPC captures the static stable equilib-
rium of a 20m high cement (ρ = 2300kg/m3 , E = 20GP a, ν = 0.2) arch with tight
geometric, dˆ = 1µm, and friction, v = 10−5 m/s accuracy. Decreasing µ then obtains
the expected instability and the stable arch does not form (see the supplemental videos Li
et al. [2020b]). Inset: zoomed 100× (orange) highlights the minimal gaps with a geometric
ˆ
accuracy of small d.

Friction direction and magnitude We improve accuracy of the direction and magnitude
of the friction forces by solving successive minimizations of (4.16) within each time step.
For each solve we update the lagged T n and λn (warm-starting from the previous time
step) with results from the last nonlinear solve. Convergence of lagged iterations is then
achieved when we reach approximate momentum balance with
X
k∇Bt (xt+1 ) − h2 Fk (xt+1 , λt+1 , T t+1 )k ≤ d , (4.17)
k∈C

where d is the targeted dynamics accuracy.

We confirm lagged iterations rapidly converge over nonlinear solves with our FE models
for the well-known, standard frictional benchmarks, e.g., block-slopes, catenary arches and

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card houses. See Figures 4.6 and 4.4 and Section 4.7. However, we emphasize that we do
not have convergence guarantees for lagging. In particular, we have identified cases with
large deformation and/or high speed impacts where we do not reach convergence for T and
λ in the friction forces. Thus, in our large-deformation frictional examples we apply just
a single lagging iteration. In these cases, sliding directions and contact force magnitudes
in the friction force evaluation may not match. However, even in these cases, all other
guarantees, including non-intersection, momentum balance (as in the frictionless case)
and accurate stiction are maintained. More generally, we observe high-quality, predictive
frictional results for large deformation examples independent of the number of lagging
iterations applied; see e.g. Figure 4.15. We also emphasize that for frictionless models, IPC
continues to guarantee convergence for contact and elasticity with just a single nonlinear
solve per time step.

Figure 4.7: Large deformation, frictional contact test. We drop a soft ball (E = 104 P a)
on a roller (made transparent to highlight friction-driven deformation). Here IPC simulates
the ball’s pull through the rollers with extreme compression and large friction (µ = 0.5).

4.6 Distance Computation

Evaluating unsigned distance functions between point-triangle and edge-edge pairs requires
care as closed-form distance formulas change with relative position of surface primitives.

84
4.6.1 Combinatorial Distance Computation

Unsigned distances are given by the closest points on the two primitives evaluated.
Distance between a point vP and a triangle T = (vT 1 , vT 2 , vT 3 ) can be formulated as a
constrained optimization problem,

D PT =min||vP − (vT 1 + β1 (vT 2 − vT 1 ) + β2 (vT 3 − vT 1 ))||

β1 ,β2
(4.18)
s.t. β1 ≥ 0, β2 ≥ 0, β1 + β2 ≤ 1.

Similarly the distance between edges v11 -v12 and v21 -v22 is

D EE =min||v11 + γ1 (v12 − v11 ) − (v21 + γ2 (v22 − v21 ))||

γ1 ,γ2
(4.19)
s.t. 0 ≤ γ1 , γ2 ≤ 1.

Each possible active set of these two minimizations corresponds to a closed-form distance
formula. In each, at most two constraints can be active at the same time.

• When two constraints are active in either (4.18) or (4.19), the distance between
primitives is a point-point distance evaluation:

dP P = ||va − vb ||. (4.20)

Here va and vb correspond to vP and a vT i for (4.18), or to the two endpoints of the
edges in the edge-edge pair for (4.19).

• When a single constraint is active in either (4.18) or (4.19), the distance in both cases
becomes a point-edge distance evaluation:

||(va − vc ) × (vb − vc )||

dP E = . (4.21)
||va − vb ||

Here (va , vb ) corresponds to one of the triangle edges of T and vc = vP for (4.18), or
else (va , vb ) corresponds to one of the edges in the edge-edge pair and vc corresponds
to an endpoint of the other edge for (4.19).

85
 • When no constraints are active in either (4.18) or (4.19), distance computations are
simply parallel-plane distance evaluations. For the point-triangle pairing in (4.18)
this is
(vT 2 − vT 1 ) × (vT 3 − vT 1 )
dP T = |(vP − vT 1 ) · |, (4.22)
||(vT 2 − vT 1 ) × (vT 3 − vT 1 )||
while for the edge-edge pairing in (4.19) it is
(v12 − v11 ) × (v22 − v21 )
dEE = |(v11 − v21 ) · |. (4.23)
||(v12 − v11 ) × (v22 − v21 )||

For evaluations of d, ∇d, and ∇2 d, we apply the currently valid, closed-form distance
formula (either PP, PE, PT, or EE above) and its analytic derivatives. The formula to apply,
at each evaluation of a surface pair, is determined by the active constraint subset defined by
the current relative positions of the pair’s primitives. This information is computed and
stored together with our culled constraint set Cˆ data, and so is then available for direct
use whenever computing barrier energies and derivatives. This treatment is analogous to
storing and reusing singular value decompositions of deformation gradients for elasticity
computations. As in elasticity, our distance state and evaluations can efficiently be reused
for all energy and derivative evaluations at the same nodal positions. Correspondingly,
having now reduced general point-triangle and edge-edge distance evaluations to the above
closed-form formulas, we can directly compute and store our sliding bases, Tk (x), for
friction computation with respect to each case; please see our appendix for details.

4.6.2 Differentiabilty of d

In collision-resolution methods, close-to-parallel edge-edge contacts are notorious failure

modes – to the extent that existing methods often ignore this case by throwing out all
corresponding constraints Harmon et al. [2008]. However, despite the challenges imposed,
these constraint cases cannot be removed, as doing so would lead to intersection. The reason
for the difficulty in these cases is the (lack of) differentiability of the distance function for
some configurations. Each above analytic formula for distances corresponds to a subset of
the relative configuration space of a primitive pair. For example, for vertex-triangle pairs,

86
 ∇dC ∇dD = 0 ∇dC = 0 ∇dD
C D C D

A B A B
∇dA ∇dB ∇dA ∇dB
(a) (b)

Figure 4.8: Nonsmoothness of parallel edge-edge distance. When edge AB and CD

are parallel, the distance computation can be reduced to either (a) C − AB point-edge or
(b) D − AB point-edge. Then for the trajectory of C moving down from above D, the
distance gradient is not continuous at the parallel point even though the distance is always
continuously varying.

relative configurations are completely characterized by fixing the triangle and varying vP
positions. If the projection of vP to T is in the triangle interior, no constraints are active,
while if the projection lies on the interior of a triangle edge then one constraint is active.
Otherwise, two constraints are active.

Each of these geometric criteria defines a subset of R3 , where one of the three analytic
formulas is valid. The distance function is C ∞ inside each such domain, and, in general,
is C 1 at the boundaries between domains. However, the critical exception is in parallel
edge-edge configurations: at these points, the distance function is not differentiable (see
Figure 4.8). Configurations close to these parallel edge-edge conditions, when reached,
lead to unacceptably slow convergence of Newton iterations or even convergence failures
altogether. Numerically, the issue is similar to the C 0 -continuous friction problem we faced
in Section 4.5.2. To resolve this issue, we once again apply a local smoothing solution to
mollify the barrier corresponding to nearly parallel edge-edge contact conditions.

We smooth by multiplying all edge-edge barrier terms by a piecewise-polynomial

mollifier closely analogous to our static-friction smoother; recall Figure 4.5. Here, for
each edge-edge contact pair k, we define ek (x) to vanish when edges (v11 v12 − v21 v22 ) are

87
parallel and to smoothly grow to 1 as the edge-pair become far from parallel,

− 12 c2 + 2 c c < × ,

× ×
ek (x) = (4.24)
1 c ≥ × ,


where c = ||(v12 − v11 ) × (v22 − v21 )||2 and × = 10−3 ||v12

0 0
− v11 0
||2 ||v22 0
− v21 ||2 is defined
with respect to edge-edge vertex-pair rest positions v 0 .


Our mollified edge-edge barriers are then ek (x)b dk (x) and so now extend our barrier
potentials to a piecewise C ∞ , everywhere C 1 -continuous (for nonintersecting configura-
tions) barrier formulation. At the same time our barriers now remain sufficient to guarantee
that no collisions are missed: there are always point-triangle contact pairs at distance no
more than the parallel edge-edge distance; see our appendix for details on this. In turn,
our construction of the parallel-edge mollifier then minimizes its impact on edge-edge pair
barriers as they move away from degeneracy. While in principle increasing smoothness to
C 1 is sufficient to avoid most dramatic degeneracy failures, there are additional numerical
stability issues to be addressed related to nearly parallel edges. Please see our appendix for
details.
Now, with this third and last smoothing in place we have an overall time-stepping
potential for contact and friction that can leverage superlinear convergence and robustness
of Newton-type stepping. As we analyze in Sections 4.7 and 4.8 below (see especially
4.7.1) this gains robust simulation against failure – even when simulating challenging
conditions with unavoidable numbers of degenerate evaluations.

4.7 Evaluation
Our IPC code is implemented in C++, parallelizing assembly and evaluations with Intel
TBB, applying CHOLMOD Chen et al. [2008] with AMD reordering for linear system
solves in all examples (with the exception of the squishy ball example – see below) and
Eigen Guennebaud et al. [2010] for linear algebra routines. We run most experiments on a
4-core 3.5GHz Intel Core i7, a 4-core 2.9 GHz Intel Core i7, and a 8-core 3.0 GHz Intel

88
Xeon machine. Machine use per example is summarized along with performance statistics
and problem parameters in Figure 4.22 and in our supplemental document Li et al. [2020b].
The reference implementation, scripts used to generate these results and our benchmarks
are released as an open-source project.

Linear system computations and solves We compute elasticity and barrier Hessians
(with PSD projections) in parallel, and have designed and implemented a custom multi-
threaded, sparse matrix data structure construction routine that, given the connectivity
graph of nodes, efficiently builds the CSR format with index entries ready. While we utilize
efficient symbolic factorization and parallel numerical factorization routines in CHOLMOD
Chen et al. [2008] compiled with MKL LAPACK and BLAS, we also tested IPC with
AMGCL Demidov [2019] – a multigrid preconditioned solver. Here, we found behavior
is as might be expected, less memory overhead and faster linear solves by avoiding direct
factorization. However, for majority of examples the large deformations and many contacts
generate poorly conditioned systems. We then found AMGCL requires extensive parameter
tuning to perform well and still can not compete, in general, with the parallel direct solver.
All examples in the following then apply CHOLMOD for linear solves, with the exception
of our largest, squishy ball example (Figure 4.21), where we apply AMGCL.

Models and practical considerations We primarily employ the non-inverting, neo-

Hookean (NH) elasticity model and implicit Euler time stepping. In the following examples
we also apply and evaluate implicit Newmark time stepping, as well as the invertible fixed
corotational elasticity (FCR) elasticity model. While for clarity in the preceding we derive
IPC with unmodified distance evaluations, for numerical accuracy and efficiency our im-
plementation applies squared distances for evaluations of the barrier, we use b(d2 , dˆ2 ), and
related computations, thus avoiding squared roots. In turn expressions for contact forces,
λk , and related terms must be modified, from our direct exposition and derivations above.
To do so we rescale for consistent dimensions and units in our implementation; see our
appendix for details. Finally and importantly we note that IPC’s barrier formulation requires

89
nonzero separation distances to be strictly satisfied at initialization and then guarantees
it throughout simulation. Exact initialization at zero distance is neither possible (as the
barrier of course diverges) nor for that matter physically meaningful. Contact, including
resting contact, instead occurs around the specified geometric distance accuracy given by
the user. Here we demonstrate simulated configurations with distances down to 10−8 m
reached in simulation (e.g. arch in Figure 4.6) or initialized by users.

Evaluation and tests Below we first introduce a set of unit tests for seemingly simple
yet challenging scenarios with nonsmooth, aligned and close contacts (Section 4.7.1),
stress tests involving large deformation and high velocities (Section 4.7.2), and friction
(Section 4.7.3). We next study IPC ’s scaling, run time, and accuracy behavior as we
vary simulation problem parameters (Section 4.7.4). Finally, we present an extensive,
quantitative comparison with previous works in Section 4.8.

4.7.1 Unit tests

Figure 4.9: Aligned, close and nonsmooth contact tests. Pairs of before and after frames
of deformable geometries initialized with exact alignment of corners and/or asperities;
dropped under gravity. We confirm both nonsmooth and conforming collisions are accu-
rately and stably resolved.

Aligned, close and nonsmooth contact We apply a set of unit tests exercising closely
aligned, conforming and nonsmooth contact known to stress contact algorithms. We build
them with two simple models: a single tetrahedron and an 8-node unit cube; see Figure
4.9 and Figures 4.10. For contact handling, these seemingly simple tests are designed to

90
trigger degenerate edge cases that often cause failure in existing methods (see Section 4.8).
IPC resolves all cases including those in which we exercise exact parallel edge-edge (e.g.,
Figure 4.9 middle) and point-point (e.g., Figure 4.9, left) collisions. For unit tests like
Figure 4.9 right we drop objects into slotted obstacles so that they fit tightly with tiny gaps;
here IPC retrieves a tight conforming fit into a 1µm gap.

Figure 4.10: Erleben tests Top: fundamental test cases to challenge mesh-based collision
handling algorithms proposed by Erleben [2018]. Bottom: IPC robustly passes all these
tests even when stepped at frame-rate size time steps.

Erleben fundamental cases Erleben [2018] proposes unit tests (see Figure 4.10 top row)
for contact constraint failure testing. Here these tests are again simple but designed to
challenge mesh-based collision-handling algorithms. IPC again resolves all tests robustly
(see Figure 4.10, bottom row), even when stepped at frame-rate size time steps.

Tunneling Tunneling through obstacles when simulating high-speed velocities is a com-

mon failure mode in dynamic contact modeling. We thus add an example to our unit tests:
we fire an elastic ball (diameter 0.1m) at a fixed 0.02m thin board at successive speeds of
10m/s, 100m/s, and 1000m/s stepped at h = 0.02s. IPC accurately rebounds at large time
step without tunneling in all cases.

Large mass and stiffness ratio tests Contact resolution between objects with largely
varying scale, mass, and/or stiffness ratios has long-challenged time stepping methods due

91
 Figure 4.11: Large Mass and Stiffness ratios.

to ill-conditioning. In Figure 4.11, we simulate IPC dropping of a range of objects upon

each other with widely varying weight and stiffness. Here we apply E = 0.1GP a for the
sphere, board, and large cube, E = 1M P a for the small cube and the mat holding the
sphere, and E = 10KP a for the mat dropped on boards. For the stiff ball and large cube,
we set their respective densities to 2× and 10× that of softer objects (1000kg/m3 ) to add
large mass ratios to the challenge. Regardless of these different ill-conditioned settings, IPC
simulates all scenes robustly and efficiently without any artifacts; see also our supplemental
videos Li et al. [2020b].

Chains While resolving transient collisions exercises stability, large numbers of per-
sistent, coupled contacts, as in a long chain of elastic links, exercises contact constraint
accuracy. A small amount of constraint error integrated over time will cause such chains to
break. We simulate chains of 100 elastic links under gravity, observe stable oscillations
and shock-propagation while shorter chains stably bounce – all preserve constraints; see
our supplemental videos Li et al. [2020b].

4.7.2 Stress tests

We next consider IPC’s ability to resolve a range of extreme stress-test examples motivated
by well-known pre-existing challenges and previously proposed benchmarks.

Funnel To confirm contact resolution under strong boundary conditions, extreme com-
pression, and elongation, we pull a stiff NH material dolphin model through a codimensional
funnel mesh obstacle. We step IPC at large time steps of h = 0.04s with up to 32.3K

92
 funnel

Figure 4.12: Funnel test. Top: the tip of a stiff neo-Hookean dolphin model is dragged
through a long, tight funnel (a codimensional mesh obstacle). Middle top: due to material
stiffness and tightness of fit the tip of the model is elongated well before the tail pulls
through. Middle bottom: extremity of the deformation is highlighted as we zoom out
immediately after the model passes through the funnel. Bottom: finally, soon after pulling
through, the dolphin safely recovers its rest shape. We confirm that the simulation is both
intersection- and inversion-free throughout all time steps.

contacts per step. The resulting simulation is intersection- and inversion-free throughout
with the model regaining its rest shape once pulled through (Figure 4.12).

Thin volumetric meshes Thin geometries notoriously stress contact simulations. Like-
wise, as more simulation nodes are involved in collision stencils, simulation challenges
grow. Here we test IPC’s handling of extreme cases with both challenges, by simulating
single layer meshes of tetrahedra. Here IPC robustly handles the contacts with accurate
solutions at all time steps across a range of large deformation contact examples (Figures
4.4, 4.11 and 4.13).

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Figure 4.13: Stress test: extreme twisting of a volumetric mat for 100s. Left: IPC
simulation at 10s after 2 rounds of twisting at both ends. Right: at 40s after 8 rounds of
twisting. This model, designed to stress IPC, has all of its 45K simulation nodes lying on
the mesh surface.

Extreme and extended twisting As large deformation high-contact examples, we twist

thin mats (Figure 4.13), rods (Figure 4.3), and Armadillos (Figure 4.20, bottom) with
rotating speeds of 72◦ /s at both ends. We simulate the twist of both the rods and mats for
100s – efficiently capturing increasingly tight conforming contact and expected buckling in
all simulations.

Compactor test In Figure 4.14 we test “trash” compactor-type examples from Harmon
et al. [2009]. After releasing the compactor from the extreme compression point we clearly
see that the tentacles of the octocat model and correspondingly the sphere, mat, and bunnies
models are all cleanly separated.

Rollers compression and stick-slip instability To combine extreme deformation with

friction, we match the set-up of the kinematic roller test from Vershoor and Jalba [2019]
with the same originally applied, high friction coefficient µ = 0.5 (Figure 4.15). This scene
is highly challenging due to the competing large magnitude of the friction and the large
compression induced by the rollers. Here, with a moderately stiff material (E = 5 × 105 P a

94
Figure 4.14: Trash Compactor. An octocat (left) and a collection of models (right) are
compressed to a small cube by 6 moving walls and then released. Here, under extreme
compression IPC remains able to preserve intersection- and inversion- free trajectories
solved to requested accuracies.

Young’s modulus) we observe that IPC with our friction model obtains the expected stick-
slip instability effects that such competition should generate. In simulation we observe
deformation grows in opposition to static friction in the rollers until stress overcomes static
friction and we observe slip – this process is then repeated. This stick-slip effect is captured
by our Armadillo with moderate stiffness when tested with both the NH and FCR elasticity
models (see our supplemental videos Li et al. [2020b] for the motion). We also note, as
expected, when we subsequently test with softer material, i.e., E = 105 P a, we get smooth
rolling behavior for the Armadillo, as expected, without stick slip.

Codimensional collision obstacles Collision obstacles, especially in animation and gam-

ing, are often most easily expressed in their default form as triangle meshes or even un-
organized triangle soups. While highly desirable in applications, codimensional collision
types are not generally supported by available simulation methods, which often suffer
tunneling, snagging, and resulting instabilities when exposed to them. To our knowledge
IPC is the first algorithm to stably and accurately resolve collisions between volumes and
codimensional collision objects. We perform a set of tests dropping different objects on
planes, segments, and points, see e.g. Figures 4.1, 4.16 and 4.17. Collisions are stably
resolved and we see tight compliance to the sharp poking obstacles in contacting regions.

95
Figure 4.15: Roller tests. Simulating the Armadillo roller from Verschoor and Jalba [2019]
(same material parameters) in IPC now captures the expected stick-slip behavior for the
high-friction, moderate stiffness conditions.

Codimensional rollers What if we modify the roller test in Figure 4.7, leaving only the
edges or even only the points for the roller obstacles? This leads to our codimensional
roller tests (Figure 4.17). Here, with solely codimensional wire (just edges) and points (just
vertices) rollers, the big ball still pulls inwards, forming tightly pressed geometries in the
contact regions as it is compressed and pulled against and then out of the codimensional
rollers (Figure 4.17). For sharp point rollers we require negligible friction (for points we
apply µ = 10−3 ) to pull the ball inwards as the sharp points directly grab the deforming
surface.

Squeeze out stress test A plate compresses and then forces a collection of complex soft
material models into a tight conforming mush through a thin co-dimensional tube obstacle.
Once through they cleanly separate (Figure 1.2).

High speed impact test To examine IPC’s fidelity in capturing high-speed dynamics we
match the reported material properties and firing speed of an experiment of foam practice
ball fired at high-speed towards a fixed steel wall. In Figure 4.18 top, we show key frames
of a high-speed capture of the event. Middle: we visualize velocity magnitudes simulated
by IPC , stepped with implicit Newmark and the NH material, at the same corresponding

96
Figure 4.16: Codimensional collision objects: pin-cushions We drop a soft ball onto
pins composed of codimensional line-segments and then torture it further by pressing down
with another set of codimensional pins to compress from above. IPC robustly simulates the
ball to a “safe”, stable resting state under compression against the pins.

times in the simulation, and bottom the IPC-simulated geometry. Here we observe both
the expected shockwave propagating through the sphere during the finite-time collision
as well as the overall matching dynamics and shape across the simulation. Please see our
supplemental video Li et al. [2020b] for complete simulation moving through the phases of
inelastic collision impact: compression (first shockwave), restoration (second shockwave),
and release.

4.7.3 Frictional contact tests

To examine IPC’s frictional model we simulate a set of increasingly challenging frictional

benchmark tests. All utilize a tight accuracy of v = 10−5 m/s and apply lagged iterations
to update sliding bases and normal forces until the system is confirmed as fully converged
by satisfying (4.11).

Block tests We start by placing stiff elastic blocks on a slope with tangent at 0.5. Here
for µ = 0.5, IPC generates the expected result of frictional equilibrium – the block does not
slide. Switching to µ = 0.49, IPC then immediately sets the block sliding, again matching
the analytic solution.

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Figure 4.17: Codimensional collision objects: rollers We modify the ball roller example
from Figure 4.7 by using only the edge segments (left) or even just vertices (right) for
the moving roller obstacle. For these extremely challenging tests IPC continues robust
simulation exhibiting tight compliant shapes in contact regions pressed by the sharp
obstacles.

Frictionally dependent structures We test IPC on the challenging, frictionally depen-

dent stable structure tests from Kaufman et al. Kaufman et al. [2008]. We model both the
card house (Figure 4.4) and masonry arch (Figure 4.6) with stiff deformable materials. We
further extend the challenge of the arch with a precarious base balanced on sharp edges.
We obtain long-term stable structures with µ = 0.5 and µ = 0.2 respectively and confirm
that they fall apart as we reduce to µ = 0.2 and µ = 0.1 respectively (see our supplemental
document and video Li et al. [2020b] for statistics and animation).

Stick-slip instability Finally, we script the motion of the top of a thin, volumetric elastic
rod pushed slightly down towards, and then along a surface (µ = 0.35) to test stick-slip
oscillations. As in the Armadillo roller example, large static friction creates a buildup of
elastic energy in the rod which is released when the friction force, opposing sliding contact,
is exceeded by the tangential stiffness at the contact. This interaction between the friction
forces and the sliding velocities becomes periodic, and so induces self-excited oscillations
that buildup and dissipate energy; see Figure 4.19 and our supplemental video Li et al.
[2020b].

98
 video footage

67 m/s
velocity
magnitude (m/s)
0 m/s
0 67

simulation

Figure 4.18: High-speed impact test. Top: we show key frames from a high-speed video
capture of a foam practice ball fired at a fixed plate. Matching reported material properties
(0.04m diameter, E = 107 Pa, ν = 0.45, ρ = 1150kg/m3 ) and firing speed (v0 = 67m/s),
we apply IPC to simulate the set-up with Newmark time stepping at h = 2 × 10−5 s to
capture the high-frequency behaviors. Middle and bottom: IPC-simulated frames at times
corresponding to the video frames showing respectively, visualization of the simulated
velocity magnitudes (middle) and geometry (bottom).

4.7.4 Scaling, Performance, and Accuracy

Varying time step sizes Existing contact-resolution methods generally rely on small
time step sizes for simulation success. As demonstrated above, IPC is able to simulate
across a wide range of time step sizes h and so can capture a range of different frequency
effects. Choice of time step size for IPC is then simply a question of accuracy required
per application as balanced against efficiency needed, rather than a predicate required for
success. To investigate the effect of varying time step size, h, in IPC we simulate the tight

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Figure 4.19: Stick-slip oscillations with friction simulated with IPC by dragging an elastic
rod along a surface.

twisted rods example (Figure 4.3) for 6s. We range h from 0.002s to 2s. In Table 4.1 we
observe that transitioning from large to small time step sizes, our method improves its
per time-step performance – but not by orders of magnitude. This is because the costs of
intersection-free time stepping, distance computation and CCD do not change much. Since
we do not miss any contacts, the number of constraints we process decrease only sublinearly
as we decrease time step sizes. This is a key computational feature to ensure feasibility
and robustness. On the other hand, we happily observe that our method is robust even well
beyond standard time step sizes. While, in general, such excessively large step sizes beyond
frame-rate are not useful for dynamics, this offers a robust opportunity for quasi-statically
computing equilibria subject to challenging contact conditions. When we deploy IPC with
implicit Euler IP (taking advantage of numerical dissipation), these very-large time steps
rapidly compute equilibria with extreme contact conditions in just a few steps.

Scaling In Figure 4.20 top, we study scaling behavior of IPC , with twisting mat (Figure
4.13) and twisting Armadillo (Figure 4.20, bottom) simulations of increasing-resolution
meshes ranging from 3K to 219K nodes. Armadillo is a representative volumetric model
while the single-layer mat is an extreme example designed to especially stress IPC . The
mat meshes importantly have all simulation nodes on their surfaces and so, as contacts
tighten in the twisting mat, they can form arbitrarily dense Hessians. For the mat we

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Table 4.1: Increasing time step sizes to frame-rate and beyond. Here we demonstrate
tradeoffs in varying time step sizes for the same tight twisted rods example (2.5K nodes,
6.9K tets, 4.6K faces, E = 104 P a) with a 4-core 2.9GHz Intel Core i7 CPU, 16GB
memory. # iters is the number of Newton iterations per time step or in total for the
simulation sequences.
# constraints per time step total
h (s)
avg (max) t (s) # iters t (s) # iters
0.002 137 (430) 0.29 2.12 862 6351
0.005 194 (584) 0.36 2.37 435 2843
0.01 269 (707) 0.38 2.65 229 1591
0.025 435 (1.0K) 0.38 2.69 91 645
0.05 551 (1.2K) 0.46 3.06 56 368
0.1 597 (1.3K) 0.73 4.75 44 285
0.2 607 (1.2K) 1.79 14.37 54 431
0.5 653 (1.4K) 11.39 100.58 137 1207
1 708 (1.3K) 18.41 188.17 110 1129
2 843 (1.3K) 52.02 522.00 156 1566

observe iteration count, memory and contact counts increase linearly with resolution, while
timing increases in a slight superlinear trend. For the more standard volumetric Armadillo
model we observe iteration count remains flat as we increase resolution, while timing and
memory increase linearly.

In addition, when mesh sizes and contacts grow large, available memory can potentially
preclude application of direct linear solvers. To confirm IPC applicability in these settings
we simulate the firing of a 688K node, 2.3M tetrahedra, squishy ball model from Zheng and
James [2012] at a glass board using AMGCL’s Demidov [2019] multigrid-preconditioned
iterative linear solver. Here both the large element count and the large numbers of collisions
enabled by the toy’s many colliding tendrils introduce very large system solves during the

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 150 8000

memory (MB)
6000
100

time (s)
4000
50
2000

0 0
0 1 2 3 0 1 2 3
# nodes 10
5 # nodes 105

Figure 4.20: Scaling tests. Top: applying increasing resolution meshes ranging from 3K
to 219K nodes we examine the time (left) and memory (right) scaling behavior of IPC on a
range of resolutions of the twisting Armadillo and twisting mat (Figure 4.13) examples.
Bottom: frames from the highest-resolution twisting Armadillo example (219K nodes,
928K tets).

most contact-rich steps colliding against the glass (Figure 4.21).

Performance Comprehensive statistics on all simulations, models, parameters and per-

formance are reported in Table 4.22 and in our supplemental document Li et al. [2020b].
For reference dynamics please see our supplemental videos Li et al. [2020b].

Accuracy User-facing parameters in IPC have three accuracies that can be specified: 1)
dynamics accuracy (d ), defining how well dynamics are resolved; 2) geometric accuracy
ˆ defining how close objects can come to touching; and 3) stiction accuracy (v ), defining
(d),
how well static friction is resolved. All three provide users direct and intuitive control (with
meaningful physical units) of the trade-off between accuracy and compute cost.
In our extensive testing, IPC converges to satisfy these requested accuracies while

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Figure 4.21: Squishy ball test: Simulated by IPC an elastic squishy ball toy model (688K
nodes, 2.3M tets) is thrown at a glass wall. The left three frames show side views before, at,
and after the moment of maximal compression during impact. The right-most frame then
shows the view behind the glass during the moment of maximal compression, highlighting
how all of the toy’s intricately intertwined tendrils remain intersection free.

always maintaining an intersection- and inversion-free state. These guarantees (non-

intersection, non-inversion) hold even as we radically increase speed of collision at large
time step, apply extreme deformations, and model highly stiff materials. We have tested
this across a wide range of test examples with material stiffnesses up to E = 2 × 1011 P a
and have confirmed our IPC implementation’s ability to converge to tight tolerances for
all these measures when requested with d down to 10−7 m/s, v down to 10−8 m/s, and dˆ
down to 1µm.

As we discuss and demonstrate in Sections 4.3 and 4.8, all previously available methods
introduce computational error for these accuracy measures; to our knowledge, IPC is the
first to provide and expose direct and separable control of them. Our singular exception,
as detailed in Section 4.5 above, is the number of frictional lagging iterations applied.
When accurate friction is required, e.g., our arch, stick-slip and card house experiments,
we set no upper bound on this parameter. Then as discussed above, in these examples IPC
fully converges and is entirely parameter-free. However, (as detailed above) we do not
have convergence guarantees for lagging, and in our large-deformation frictional examples
we apply a single lagging iteration. In these cases, as discussed in Section 4.5.5, sliding

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directions and contact forces in the friction may not match. However, even in such cases
all other guarantees, including non-intersection are maintained. We observe high-quality
results regardless of number of lagging iterations applied or accuracies specified.
Finally, on the other end of the spectrum in many applications, e.g. animation, it
can be desirable to trade accuracy for efficiency. We confirm robust, plausible behavior
for IPC when we set very large, loose tolerances on all the above parameters, e.g. with
d = 10−1 m/s, while still maintaining feasible (non-inverting, non-intersecting) trajectory
guarantees.

Exact CCD admissibility check IPC’s collision aware line search ensures intersection-
free trajectories. Our implementation applies standard floating-point CCD2 combined with
the conservative advancement strategies detailed in Section 4.4 and our appendix to ensure
efficient, intersection-free stepping. Exact CCD then offers the possibility for aggressive
advancement of intersection-free steps and so improved efficiency. To this end we tested
the robust CCD methods from both Bridson et al. [2002] and Tang et al. [2014] but found
the reference implementations for each missed critical intersections in degeneracies. We
then reimplemented Bridson et al. [2002] with rationals. While this version now guarantees
exactness, it is much slower (∼30x) than our floating-point implementation. Currently we
apply this exact CCD just for re-analysis as a post step check after every Newton iterate to
test three of our challenging contact stress tests: octocat on codimensional “knives”, ball
roller and mat twist. We confirm that every step taken in every time step was intersection
free in these examples.

Varying material model A general expectation from unconstrained simulation is that

modeling with non-invertible materials like NH should be more costly than comparable
set-ups with invertible materials like FCR. However, when studying our large deformation
examples with contact we find that the picture is more complex. Here the larger bottleneck
is generally resolving contact barrier terms. In many examples we then observe that
2
https://github.com/evouga/collisiondetection

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simulations with NH and FCR have comparable cost. In a number of other simulations
with extreme contact conditions (e.g. pin-cushion and mat twist) element degeneracies
allowed by FCR actually increase overall cost of simulation well over the same simulations
with the NH material. Finally, in other cases where stress is most extreme (e.g. armadillo
roller and dolphin funnel), NH entails more cost than the comparable simulation with FCR.

4.8 Comparisons

We perform extensive quantitative comparisons with existing algorithms and commercial

codes used in both computer graphics (Section 4.8.1) and mechanical engineering (Section
4.8.2). Then, to more fairly compare across a large class of previous contacts algorithms
based on SQP-type methods, we implement their core contact resolution procedures in
a single framework, and perform a large scale comparison on our benchmark test set
(Section 4.8.3). While our implementations are not finely tuned as for the first two sets of
comparisons, this approach allows us to compare the core algorithmic components in a
common, objective and unbiased context.

4.8.1 Computer Graphics Comparisons

Contact algorithms in graphics often target performance with small compute budgets and
so admirably face many efficiency challenges in balancing fidelity against speed. We
investigate what happens if we push these methods’ settings to be most accurate without
regard to speed, e.g., max iteration caps of 1M per step and time steps down to 10−5 s. Here,
nevertheless, we still document failures, e.g., tunneling, non-convergence, instabilities and
ghost forces, even on very simple test examples.

Verschoor and Jalba [2019] We apply the reference implementation Verschoor and
Jalba [2019] to reproduce available scenes from Verschoor and Jalba with their default
and reported input parameters. Here we observe that small adjustments to time step

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sizes and material parameters lead to divergent simulations. Specifically, the Armadillo
roller example does not converge when applying the implementation’s default time step
of h = 10−3 for a range of stiffnesses of E = 5 × 104 , 5 × 105 , and 5 × 106 Pa, nor when
applying the default material setting E = 5 × 105 P a for a range of time step sizes of
h = 10−3 , 2 × 10−3 , 4 × 10−3 , and 10−2 s. In all these cases the implementation maxes out
at its default max-iteration cap of 1M.
We extract the Armadillo mesh, roller models and replicate the same example in IPC
with identical scene settings. Here it is noteworthy that IPC applies fully nonlinear NH and
FCR models with variational friction while the reference code (matching paper) linearizes
elasticity once per time step. As covered in Section 4.7.2, IPC obtains the stick-slip
oscillations expected in this setting (see also our video Li et al. [2020b]), when rolling
the Armadillo. This does not match the Verschoor and Jalba reference code nor paper
video. Artificial material softening due to the per-step linearization of Verschoor and Jalba’s
elasticity likely explains the difference. We confirm this in Section 4.7.2 where IPC’s fully
nonlinear simulation of the Armadillo roller, with a softer E = 105 P a (5× softer) does
not, as expected, stick-slip.

SOFA SOFA Faure et al. [2012] is an open-source simulation framework featuring a

range of physical models. These include deformable models via FEM. We modify a SOFA
demo scene to simulate the five-link chain example with the top link fixed and four free FE
links. We use the linear elasticity model (most robust) and found SOFA to provide a stable
solution for the chain with large time steps up to h = 10−2 s. We extend the chain to ten
links and are unable to find a converging time step size (tested down to h = 10−4 s).

Houdini Houdini SideFX [2020] is a widely used VFX tool that provides two performant
simulation methods for deformable volumes: 1) a FE solver with co-rotated linear and
neo-Hookean materials, and 2) Vellum, a state-of-the-art PBD solver. While capable of
producing impressive effects – especially for rapid collision denting and bouncing, we find
that both solvers suffer in different ways when enforcing contact constraints accurately is

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critical. As a simple demonstration we again apply the chain example.

Trying a simple, lower-stiffness, 5-link chain we aim Houdini’s FE solver towards

robustness over speed by finely tetrahedralizing the link rings (∼ 8000 tets per ring),
applying small time steps (we tried increasing solver substeps to h ≈ 1ms), and increasing
collision passes (up to 16). Up to and including these maximum settings we observe rings
tunneling through. We verify the same tunneling with both FE solvers provided in Houdini
18 (GNL,GSL), with both available materials. With similar stretchy material, IPC is able
to accurately resolve the chain collisions even with a much coarser mesh (∼ 500 tets per
ring), and frame-rate size time steps, e.g. h = 0.04s.

For the same 5-link scene, Houdini’s Vellum PBD system does better, avoiding tun-
neling. However, as we increase numbers of links different tradeoffs (expected of PBD)
are exposed. For example, a 35-link chain, requires collision passes and/or substeps to be
increased quite high to prevent tunneling. However, this unavoidably changes the material
(stiffer) and introduces biasing, in this case with sideways ghost forces. Careful experimen-
tation with substep, smoothing, and constraint iteration parameters do not help alleviate
these issues. For long chains (e.g. 100 links) we confirm IPC produces stable results, with
accurate physical effects (e.g. shockwaves). See our supplemental videos Li et al. [2020b].

4.8.2 Comparison with engineering codes

We compare IPC with two commercial engineering codes, COMSOL [2020] and ANSYS
[2020], and one open-source engineering simulation framework Krause and Zulian [2016].
For all three codes we set up exceedingly simple scenes involving small numbers of objects.
All three methods generate intersection during simulation and exhibit instabilities highly
dependent on parameters and tuning choices. In stark contrast to these three engineering
solutions, IPC resolves a range of contact problems, demonstrates robust output across
parameters, and ensures feasible trajectories.

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4.8.3 Large scale benchmark testing with SQP-type methods

We focus on frictionless contact to compare a wide range of recently developed, implicit

time-stepping algorithms. Removing the various and diverse treatments for friction allows
us to carefully consider behavior with contact for a broad set of recent methods Daviet
et al. [2011], Harmon et al. [2008], Jean and Moreau [1992], Kane et al. [1999], Kaufman
et al. [2008, 2014], Macklin et al. [2019], Otaduy et al. [2009], Verschoor and Jalba [2019]
in a common test-harness framework. This is because all these methods, once friction
is removed, follow a common iterated, Newton-type process to solve each time step as
follows: 1) To help reduce constraint violation heuristic distance offsets/thickenings are
applied to constraints; 2) at the start of each time step collision detection is performed to
update a running estimate of active constraints; 3) The currently determined active (and
possibly offset) constraint set and the IP energy are respectively approximated by first
and second-order expansions; 4) The resulting quadratic energy is minimized subject to
the linearized inequality constraints. This is a QP problem and so a bottleneck. A wide
range of algorithms thus focus particularly on the efficient solution of this QP with custom
approaches including QP, CR, LCP and nonsmooth-Newton strategies. Given the common
sequential QP structure, we will jointly refer to them going forward as SQP-type. 5) A
resulting displacement is then found and applied to the current iterate. This entire process
is then repeated until a termination criteria is reached.
The above methods then differ in amount of offset, choice of constraint function, active
set update strategy, IP approximations – most in graphics use just a fixed quadratic energy
approximation (and so linearized elasticity) per time step, and choice of QP solver.
Here we focus on the ability of these methods to achieve convergent and accurate solves
on a benchmark composed of our unit tests from Section 4.7.1 and a few additional low-
resolution examples. To eliminate uncertainty of errors from the wide range of QP methods,
we use the same state-of-the-art, albeit slow, QP solver Gurobi Gurobi Optimization [2019]
for all methods and test each simulation method across a grid of variations on an HPC
cluster.

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 We implement three common constraint types: the projected gap function, see e.g.,
Harmon et al. [2008]; the volume based proxy of Kane et al. [1999]; and the CCD-based gap
function, see e.g. Otaduy et al. [2009] and Vershoor and Jalba [2019]. For each constraint
type we test on a 3D sweep of (a) time steps (10−2 , 10−3 , 10−4 , 10−5 s), (b) constraint offsets
(10−2 , 10−3 , 10−4 , 10−5 ), and (c) both fully nonlinear SQP and the graphics-standard of per
time-step fixed quadratic approximation of the elastic energy with nonlinear constraints.
A general pattern appears in our results: for simulations to succeed all methods require
small time step and/or large constraint offset. With large time steps accuracy of the con-
straint linearization diminishes, thus larger constraint offsets are necessary to compensate
for constraint violations. A too large constraint offset leads to failures as the local QP may
become infeasible. Additionally, with large constraint offset, a constraint pair may initially
violate the constraints (a common-case for self-collision due to arbitrarily small distances
between elements). While it is possible to recover from such initial constraint violations,
this rarely happens in our experiments. In contrast, we (re-)confirm IPC is unconditionally
robust across all test cases and time steps in the benchmark.

4.9 Summary

In summary, IPC provides an exceedingly flexible, efficient, and unconditionally feasible

solution for volumetric, mesh-based nonlinear elasticity simulations with self or external,
volumetric or codimensional contacts. Guaranteeing intersection- and inversion-free output,
IPC allows both computer graphics and engineering applications to run simulations by
directly specifying just physically and geometrically meaningful parameters and tolerances
as required per application.
At the same time much more remains to be done. While we have enabled a first of its
kind “plug-and-play” contact simulation framework that provides convergent, intersection-
and inversion-free simulation, clearly costs rise as scene complexity (both in contacts
enforced and mesh resolutions) increase. There are thus many promising directions for

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future improvement that are exciting directions for exploration including further customized
Newton-type methods, practical speed exact CCD, extensions to higher-order elements and
improved convergence for frictional contact. We emphasize that we have no guarantee for
convergence of lagged friction for λ and T (although we do for stiction) and so another
meaningful avenue of future development is better exploration and understanding of its
behavior.
Our hope is to enable engineers, designers, and artists to utilize predicative, expressive,
and differentiable simulation, free from having to perform extra per-scene algorithmic
tuning or deviation from real-world physical parameters. We look forward to enabling
design, machine learning, robotics, and other processes reliant on automated and reliable
simulation output across parameter sweeps and iterations and hope to better enable artists
to use real-world materials and settings as useful design tools for creative exploration.

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Figure 4.22: Simulation statistics for IPC on a subset of our benchmark examples. Com-
plete benchmark statistics are summarized in our supplemental document Li et al. [2020b].
ˆ d
For each simulation we report geometry, time step, materials, accuracies solved to (d,
and v are generally set w.r.t. to bounding box diagonal length l), number of contacts
processed per time step, machine, memory, as well as average timing and number of
Newton iterations per time step solve. When applicable, for friction we additionally report
number of lagged iterations, with number of iterations set to ∗ indicating lagged iterations
are applied until convergence until (4.17) is satisfied. We apply implicit Euler time stepping
and the neo-Hookean material by default unless specified in example name; i.e., “NM” for
implicit Newmark time stepping and “FCR” for the fixed-corotational material model.
Example nodes, tets, faces h (s) ρ (kg/m3), d ̂ (m) μ, ϵv (m/s), ϵd (m/s)
contacts
avg. (max.) machine
memory
timing (s),
iterations
E (Pa), ν friction iterations (MB)
(per timestep) (per timestep)
1000, 4-core 2.9 GHz Intel Core i7,
Ball on points 7K, 28K, 10K 0.04 1e-3l N/A 1E-02l 126 (182) 229 2.8, 6.6
1e4, 0.4 16 GB memory
1000 4-core 2.9 GHz Intel Core i7,
Mat on knives 3.2K, 9.1K, 6.4K 0.04 1e-3l N/A 1E-02l 291 (472) 147 1.4, 5.5
2e4, 0.4 16 GB memory
500, 4-core 2.9 GHz Intel Core i7,
100 chains 20K, 49K, 40K 0.04 1e-3l N/A 1E-02l 40K (53K) 450 4.0, 2.4
1e7, 0.4 16 GB memory
1000 4-core 2.9 GHz Intel Core i7,
Dolphin funnel 8K, 36K, 10K 0.04 1e-3l N/A 1E-02l 7K (31K) 357 27.9, 39.7
1e4, 0.4 16 GB memory
1000 4-core 2.9 GHz Intel Core i7,
Pin-cushion compress 9K, 28K, 10K 0.04 1e-3l N/A 1E-02l 317 (496) 233 3.7, 9.5
1e4, 0.4 16 GB memory
1150, 4-core 2.9 GHz Intel Core i7,
Golf ball (NM) 29K, 118K, 38K 2E-05 1e-3l N/A 1E-02l 1K (4K) 861 12.1, 9.3
1e7, 0.45 16 GB memory
1000 8-core 3.0 GHz Intel Xeon,
Mat twist (100s) 45K, 133K, 90K 0.04 1e-3l N/A 1E-02l 264K (439K) 4,546 776.2, 34.5
2e4, 0.4 32GB memory
1000 8-core 3.0 GHz Intel Xeon,
Rods twist (100s) 53K, 202K, 80K 0.025 1e-3l N/A 1E-02l 243K (498K) 2,638 141.5 ,14.1
1e4, 0.4 32GB memory
Trash compactor: ball, 1000 8-core 3.0 GHz Intel Xeon,
15K, 56K, 22K 0.01 1e-3l N/A 1E-02l 6K (132K) 638 61.9, 29.4
mat and bunny 1e4, 0.4 32GB memory
1000, 8-core 3.0 GHz Intel Xeon,
Squeeze out 45K, 181K, 60K 0.01 1e-3l N/A 1E-02l 37K (277K) 1,700 252, 42.5
5e4, 0.4 32GB memory
1000, 4-core 2.9 GHz Intel Core i7,
Ball mesh roller 7K, 28K, 11K 0.01 1e-3l 0.5, 1e-3l, 1 1E-02l 2.3K (5.6K) 215 63.3, 58.6
1e4, 0.4 16 GB memory
1000, 4-core 2.9 GHz Intel Core i7,
Hit board house 6K, 15K, 11K 0.025 1e-4l 1.0, 1e-5l, 2 1E-02l 7K (13K) 186 10.0, 16.6
1e8, 0.4 16 GB memory
2300, 4-core 3.6 GHz Intel Core i7,
Cement Arch 216, 150, 324 0.01 1E-06 0.5, 1e-5l, * 1E-04l 101 (118) 54 0.05, 5.7
2e10, 0.2 32 GB memory
Stick-slip Armadillo roller 1000, 4-core 3.6 GHz Intel Core i7,
67K, 386K, 24K 0.025 1e-3l 0.5, 1e-3l, 1 1E-02l 8K (33K) 3,651 346, 66.8
(FCR) 5e5, 0.2 32 GB memory
1000, 8-core 3.6 GHz Intel Core i9,
Squishy ball (AMGCL) 688K, 2314K, 1064K 1E-03 1e-4l N/A 4E-02l 3.6K (105K) 19,463 328.3, 12.2
7e4, 0.4 64GB memory

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Chapter 5

Strain Limiting and Thickness

Modeling

5.1 Introduction

Cloth, papers, hairs, sands. These various kinds of solids have a common feature that one
or two or even all their dimensions are nearly negligible when compared to their other
dimensions or a cluster of them. To capture the intricate dynamic behaviors of these thin
structures with self-contact is challenging in many aspects.
First, codimensional shape representations like surface or segment meshes are often
used to avoid numerical stiffening on the shearing modes, that can make the material
artificially harder to bend. Even when codimensional shape representations are used, for
cloth there are still severe numerical stiffening on the bending mode unless the underlying
surface mesh has extremely high resolution. This happens for practical simulations of cloth
even when real-world anisotropic membrane stiffnesses are used. Using smaller membrane
stiffness avoid this issue, but it usually makes the cloth too stretchy. Therefore, modern
cloth simulation methods often apply a soft membrane energy to avoid locking but also add
a strain limit constraint to avoid stretchy behaviors. By limiting both the lower and upper
bound of the strain, some methods even go without membrane energies for simplicity and

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efficiency.

However, existing strain limiting method can hardly guarantee that the solved strains
are within the specified range, which potentially results in inconsistent material behaviors
in different time steps depending on how close the violation is to the specified bound. In
addition, all existing strain limiting methods handle the inequality strain limit constraint
via post projection separately from contact processing and elastodynamic solves. This not
only cannot guarantee simultaneous satisfaction of both contact and strain limit constraint,
but also can result in large error in elastodynamics (inertia and bending), which can lead to
severe artifacts.

On the other hand, with codimensional shape representation, the thickness of objects
are usually only considered when computing integral of kinetic and elasticity energies,
but ignored in contact handling. In fact, contact handling methods in state-of-the-art cloth
simulator even rely on an appropriately large thickness parameter as an offset to define and
handle the interpenetration-free constraints that could help avoid some interpenetration or
jittering artifacts. But as we show in Section 5.6.5, extensive parameter tuning, e.g. time
step size h, is required for this mechanism to work, making thickness control tricky for
thin materials, especially when simulating scenes where multiple objects gather together.
In addition, modeling thickness with constraint offset also poses significant difficulties to
existing state-of-the-art continuous collision detection (CCD) solvers on their robustness
and accuracy while maintaining efficiency.

CCD is an important component in contact simulation for avoiding interpenetration

when advancing nodal positions. It is often conducted by computing the smallest step sizes
(or times) that first bring the distances of any surface point-triangle or edge-edge pairs to 0
(first time of impact/collision), and then take the minimum of the computed values among
all primitive pairs. An inaccurate CCD return value can either lead to interpenetration
when it is larger than the true solution, or stall the simulation or optimization when it is
too tiny or even 0. Therefore, extensive researches have been conducted to explore various
trade-offs between efficiency and accuracy for different application scenarios. But none of

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them can fulfill our need on gauranteeing a finite separation as inelastic thickness ξ while at
the same time relying on the distances between offset surfaces that are orders-of-magnitude
smaller than ξ for computing contact forces.

5.1.1 Contribution

By posing the limit of strain as a constitutive behavior, we propose constitutive strain

limiting, a smooth barrier-based energetically-consistent strain limiting model supporting
both isotropic and anisotropic membrane resistance. With a barrier potential energy, our
strain limiting model can be accurately solved to satisfy any reasonable strain restriction
fully coupled with elastodynamics and contact, ensuring simultaneous strain limit and
interpenetration-free guarantees and consistent momentum balance even with frame-rate
time step sizes.
We thereby propose a unified framework for simulating mixed-dimensional elastody-
namics including elastic volumetric bodies, shells, rods, and particles all coupled together
through robust and accurate contact and friction. Unlike traditional mixed-dimensional elas-
todynamics simulation methods that mainly focus on designing the connections between
objects (mostly equality constraints), we here focus on accurately resolving the contact
and friction (the more challenging inequality constraints). By consistently maintaining a
constraint offset with our local CFL-based iterative CCD approach, our method at the first
time achieves controllable thickness effect under codimensional shape representation.

5.2 Related Work

5.2.1 Cloth and Rods

Cloth Baraff and Witkin [1998], Bridson et al. [2002], Harmon et al. [2009], Li et al.
[2020a], Narain et al. [2012], Volino and Thalmann [2000] and hair Choe et al. [2005],
McAdams et al. [2009], Müller et al. [2012], Selle et al. [2008], Ward and Lin [2003]

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simulation has been popular in computer graphics over decades. The most common cloth
simulation pipeline is based on implicit time integration with collision response computed
using a non-linear impact zone solver Baraff and Witkin [1998], Bridson et al. [2002],
Harmon et al. [2008], Li et al. [2020a, 2018a], Narain et al. [2012], Otaduy et al. [2009],
Tang et al. [2016, 2018]. As high resolution is essential for cloth simulation to capture
intricate behaviors, many existing works focus on improving the timing performance via
GPU acceleration Li et al. [2020a], Schmitt et al. [2013], Tang et al. [2013, 2016, 2018] or
smart solving schemes like multigrid Tamstorf et al. [2015] or splitting Goldenthal et al.
[2007]. Another challenge is to capture the complex anisotropic behaviors of cloth, which
triggered works that explores cloth material parameter estimation from real-world data
Bhat et al. [2003], Clyde et al. [2017], Miguel et al. [2012]. There are also works that
explore different spatial discretizations e.g. material point method Guo et al. [2018], Jiang
et al. [2017a], Eulerian-on-Lagrangian method Weidner et al. [2018], to obtain different
trade-offs among performance, quality, and accuracy in multiple aspects. Recently, Liang
et al. [2019] proposed an efficient differentiable cloth simulator which can be embeded
as a layer in deep neural networks, and Casafranca et al. [2020] and Sperl et al. [2020]
combined yarn-level simulation into continuum shell based cloth simulation to gain more
accuracy and details.
Observing that guaranteed contact resolution with controllable friction, strain limiting,
and thickness modeling of shells and rods is still an unsolved but essential problem, we
propose our method to tackle these challenges and make shell and rod simulation robust,
easy to control, and accurate.

5.2.2 Strain Limiting

Here we provide a literature review on general methods trying to maintain a strain limit or
range for cloth simulation, not only limited to standard strain limiting research.
Goldenthal et al. [2007] defines equality constraints on quad mesh edge length and
propose a fast projection method to efficiently post project predictive solutions towards

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 Constraint Splitting
Methods DoFs constrained Constraint solver
type choice
Goldenthal et al. Fast projection
Equality Quad edge length 3-step
[2007] (Implicit Euler)
English and Bridson Triangle edge midpoint
Equality 3-step Fast projection (BDF2)
[2008] distance
Thomaszewski et al. Lagged corotational
Equality 3-step Gauss-Seidel or Jacobi
[2009] small strain on triangles
Chen and Tang
Equality Triangle edge length 1-step Least squares
[2010]
Wang et al.
Range SVD of triangle strain 2 or 3-step Gauss-Seidel
[2010]
Narain et al.
Range SVD of triangle strain 3-step Augmented Lagrangian
[2012]
Hernandez et al. SVD of triangle strain Projected Gauss-Seidel
Range 2-step
[2013] with angle dependency for LCP
Wang et al.
Range SVD of triangle strain 2-step SQP
[2016]
Jin et al. Complementary (similar
Inequality Triangle edge length 3-step
[2017] to fast projection)
Chen et al. Green-Lagrangian strain Fast projection
Equality 3-step
[2019] on nodes (MLS) (Implicit Euler)

Figure 5.1: Strain limiting methods feature table. Here 2-step splitting refers to solving
elastodynamics before a post constraint projection which simultaneously handle contact
and strain limiting, while 3-step splitting means elastodynamics, contact, and strain limiting
are all decoupled and solved independently per time step. In the table only Chen and Tang
[2010] solve the 3 terms in a fully coupled way, but by using least squares the solution is
simply an L2 approximation to the original problem.

satisfying edge length constraints. English and Bridson [2008] applies non-conforming
mesh strategy and define equality constraint on triangle edge midpoint distances. They
also solve the constraint via post projection by extending the fast projection method in
Goldenthal et al. [2007] to support BDF-2 integrator. Thomaszewski et al. [2009] uses
lagged corotational small strain on triangles to define equality constraints and apply post
projection via Gauss-Seidel or Jacobi solver. Chen and Tang [2010] defines equality
constraints on triangle edge length. With equality constraints per edges or triangles, these
methods still suffer from some degree of membrane locking when the constraints are
accurately satisfied except for English and Bridson [2008] but there the non-conforming
mesh issues introduce extra challenges.

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 Thus, there are methods constraining the singular values of deformation gradient per
triangle to a finite range to avoid membrane locking Hernandez et al. [2013], Narain et al.
[2012], Wang et al. [2010, 2016]. Moreover, Jin et al. [2017] only defines inequality
constraints on triangle edge length. Similar to English and Bridson [2008] that reduce the
constraint number by investigating different choice of discretization, Chen et al. [2019]
novelly defines equality constraints on the Green-Lagrange strain per node computed by
Moving Least Squares (MLS).
Here only Thomaszewski et al. [2009] and Hernandez et al. [2013] explicitly dealt
with anisotropic strain limiting, while the latter one is customized for anisotropic but not
isotropic cases. Goldenthal et al. [2007], Narain et al. [2012], and Chen et al. [2019]
mentioned possible ways that could extend the proposed methods to support anisotropy.
Except for Chen and Tang [2010] that applies Least Squares approximation to the entire
system without friction and only for elastostatics, most existing methods solves strain
limiting, elastodynamics, and contact separately, where strain limit and contact constraints
cannot be simultaneously satisfied. Hernandez et al. [2013], Wang et al. [2016], and Wang
et al. [2010] conduct post projection for strain limiting and contact simultaneously, but the
errors from elasticity and inertia introduced by splitting the elastodynamics and constraint
projection can still potentially lead to artifacts.

5.2.3 Thickness Modeling

To simulate shells or rods using codimensional shape representations, although the thickness
of the objects are orders-of-magnitude smaller than other dimensions and it usually can be
ignored, in situations where multiple objects gather together through contact, e.g. a pile of
cards (Figure 5.18), a bowl of noodles (Figure 5.20), etc, correctly modeling thickness is
still important to obtain realistic behaviors.
Directly simulating shells or rods using volumetric meshes easily solve the thickness
modeling issue geometrically, where elasticity can also be succinctly modeled with pure
translational DoFs Hauptmann and Schweizerhof [1998]. However, obtaining a correct

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elastic behavior is challenging in this situation. Linear element volumetric shell has the
well-known shear locking issue. Although higher-order elements are much more accurate,
they are too expensive. Thus, traditional solid shell methods use either reduced integration
with hourglass mode stabilization Reese [2007] or assumed natural strain methods Cardoso
et al. [2008] to achieve locking free linear element that maintains the efficiency, which
unfortunately brings back the complexity into the model.

Getting back to the discrete shell or rod regime, contact methods usually apply an offset
in the constraint definition for modeling thickness to a certain degree Li et al. [2018a],
Narain et al. [2012]. However, as shown in Li et al. [2020b], this offset is in fact a
relaxation to the contact handling problem that try to balance contact constraint errors
against time step size and collision speed for avoiding interpenetration or explosion issues.
Relying on it to consistently model thickness for codimensional objects is not practical
as we show in Section 5.6.5. In our work, we extend IPC with an offset barrier that can
guarantee a minimal separation between the mid-surface/line geometries to consistently
model thickness. This new formulation also extends the capability of simulation with
codimensional objects to a lot more phenonmena than before (e.g. Figure 5.20, 5.21, etc).

5.2.4 Continuous Collision Detection (CCD)

Here we provide a brief literature review of CCD methods focusing on piece-wise linear
surfaces with linear trajectory. Provot [1997] formulated computing first time of impact
for point-triangle or edge-edge pairs as first solving a cubic equation for coplanarity and
then performing overlap checks to see whether collision is happening. This later becomes a
standard formulation of CCD and followed up by many works Harmon et al. [2009], Tang
et al. [2011]. However, based on floating-point root-finding, although the computation
can be quite efficient, large numerical errors can result in both false positive or false
negative output especially when distances are tiny and the configuration is degenerate (e.g.
parallel edge-edge, etc). Although performing the root-finding and overlap checks with

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rational numbers can avoid any numerical issue, the computational cost becomes orders-of-
magnitude more expensive. Therefore, there are methods Brochu et al. [2012], Tang et al.
[2014], Wang et al. [2015] applying exact arithmetic for improved robustness while still
maintaining efficiency. Some other methods Harmon et al. [2011], Lu et al. [2019] perform
CCD by requesting a small separation for better avoiding interpenetration, which is also the
strategy applied in IPC Li et al. [2020b] on top of a floating-point root-finding CCD solver1 .
But when we try to maintain a finite separation ξ between midsurface/midlines for inelastic
thickness modeling and relying on the orders-of-magnitude smaller distances between the
offset surface for computing contact forces, we observed that all existing CCD methods
fail hardly, either leading to interpenetration or totally stall our optimization (see Section
5.6.8). Therefore, by deriving a theoretical lower bound of the time of impact, we propose
our additive CCD (ACCD) method that stay in the floating-point regime but robustly
approximate time of impact monotonically without the error-prone direct root-finding
computations and thus works with high accuracy even for extreme cases.

5.2.5 Coupling

Finally we review simulation methods that couple codimensional DoFs. Martin et al.
[2010] derive a higher-order integration rule, or elaston, measuring stretching, shearing,
bending, and twisting along any axis without distinguishing between forms of different
dimension (solids, shells, rods). Their single code accurately models a diverse range of
elastoplastic behaviors where collision is simply point-wise penalty model since the objects
are represented by a set of spheres. Chang et al. [2019] present a unified method to simulate
deformable elastic bodies consisting of mixed-dimensional components by defining all
types of connections between objects of different codimensions via equality constraints.
They employed collision detection and resolution strategy in Bridson et al. [2002]. On
the other hand, the material point method Jiang et al. [2017a] discretizes elasticity on
Lagrangian particles while solve the momentum balance on the DoFs of a background
1
https://github.com/evouga/collisiondetection

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Eulerian grid. The contact between any codimensional objects are easily handled via the
Eulerian grid in a unified way, but sticky and gap artifacts can be visible for not high enough
grid resolution. Our method demonstrate that the recently proposed Incremental Potential
Contact (IPC) Li et al. [2020b] model can naturally couple all different codimensional
objects together through robust and accurate contact and friction without any artifacts or
special treatment. In addition, our framework also at the first time conduct a fully implicit
coupled solve of elastodynamics, contact, and strain limiting.

5.3 Formulation and Overview

We focus on the combined solution of meshed, codimensional models simulated jointly and
so coupled arbitrarily via contact. To do so we generalize the IPC model to codimensional
DoF and so must address the challenges introduced by thin codimensional models both
coupled by and stressed via contact and largely imposed boundary conditions. Here we
first cover our generalization of IPC to mixed codimensional models and then introduce the
key components of our method that enable its simulation.

Contact-aware elastodynamics. For elastodynamics, we perform implicit time stepping

with optimization time integration Gast et al. [2015], Kaufman and Pai [2012], Li et al.
[2019], Liu et al. [2017], Overby et al. [2017], Wang et al. [2020a] to minimize an In-
cremental Potential (IP) Kane et al. [2000] with line search, ensuring stability and global
convergence. For a wide range of time steppers and assuming hyperelasticity, the IP to
update from time step n to n + 1 is defined as
1
E(x) = ||x − x̂n ||2M + αh2 Ψ (βx + γxn ), (5.1)
2
with the timestep update given by xn+1 = argminx E(x). Here h is time step size, M
is the mass matrix, and Ψ is an elastic energy potential. Scaling factors α, β, γ ∈ [0, 1]
and explicit predictor x̂n then determine the time step method applied. For example, here
we focus primarily on the graphics-standard implicit Euler with α, β = 1, γ = 0 and

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x̂n = xn + hv n + h2 M −1 fext . Alternate implicit time stepping, e.g., with implicit midpoint
or Newmark, are similarly matched by applying alternate scalings and predictors Kaufman
and Pai [2012].
Incremental Potential Contact (IPC) then augments the IP with contact and dissipative
potentials Li et al. [2020b]:

1
E(x) = ||x − x̂n ||2M + αh2 Ψ (βx + γxn ) + B(x) + D(x). (5.2)
2

Here B(x) and D(x) are respectively the IPC contact barrier and friction potentials. The
contact barrier B(x) enforces strictly positive distances between all primitive pairs. Fol-
lowing Li et al. [2020b], a custom Newton-type solver is applied to solve the IP with
Continuous Collision Detection (CCD) filtering used in each line search, at every Newton
iteration to ensure intersection-free trajectories throughout. Please see Li et al. [2020b] for
details on the IPC algorithm and solver implementation.

Mixed-dimensional hyperelasticity. To enable a unified simulation framework for cou-

pled volumetric bodies, shells, rods and particles, we compute mass and volume for all
codimensional elements by treating them as continuum regions with respect to standard
discretizations (see Section 5.2 and below), and so construct the total elasticity energy Ψ (x)
for the IP as
Ψ (x) = Ψvol (x) + Ψshell (x) + Ψrod (x). (5.3)

Here, as representative examples, we apply fixed Corotated elasticity Stomakhin et al.

[2012] for volumes (Ψvol ); discrete shells hinge bending Grinspun et al. [2003], Tamstorf
and Grinspun [2013], combined with isotropic/orthotropic StVK Chen et al. [2018], Clyde
et al. [2017] and neo-Hookean membrane models, for shells (Ψshell ); and the quadratic spring
energy for stretching, combined with the discrete rods bending model Bergou et al. [2008]
for rods (Ψrod ). Here the generalized IPC framework is agnostic to these elasticity and time
stepper choices. The above are currently selected as best for comparison and evaluation
given the models standard in existing available codes for comparison. For discrete models,

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along with properly integrating all lower-dimensional energies with an accurate volume
weighting per element, we further parameterize rod bending moduli via Kirchhoff rod
theory for direct material setting. As, to our knowledge, this is not previously covered in
the literature Bergou et al. [2008], we cover the necessary details for completeness in our
appendix. Now, with per-domain elasticity summed, our IPC model couples objects of
arbitrary codimensions directly, without splitting, via frictional contact. Specifically all
codimensional DoF are now associated with a potential energy and so are free to move
by time stepping. In turn they are coupled by IPC barriers including all point-triangle and
edge-edge pairs from surfaces (both volumes’ and shells’), rods, and particles; point-edge
pairs from all rod nodes and particles to rod segments; and finally point-point pairs between
particles.

Constitutive strain limiting When it comes to codimensional shell models all but the
highest-resolution meshes suffer from locking artifacts for cloth materials. To mitigate these
effects strain-limiting has long been critical for simulating cloth materials with realism and
accuracy. Inspired by IPC’s intersection-free and inversion-free barrier formulation Li et al.
[2020b], we construct a new constitutive barrier model that directly enforces strain limiting
while maintaining rest-state consistency. To do so we designed a barrier energy that 1)
diverges at the strain limit, and 2) has vanishing gradient and Hessian (and so neither force
nor preconditioning contribution) at rest shape. This keeps whichever underlying membrane
elasticity model we choose to apply energetically consistent. With this barrier construction,
we apply our IPC solver and guarantee strain-limit-satisfaction for all practical (and well
below) reasonable limits (verified down to 0.1%) for all iterations and so for all time
steps in each IPC simulation. See Section 5.6. We show how our formulation augments
existing membrane energies as an added strain limiting potential for general isotropic cases
(Sec. 5.4.1), and also demonstrate its application as a new extension for orthotropic STVK
membranes augmented to include anisotropic strain limits (Sec. 5.4.2). Because of its
potential-based model, our isotropic (respectively anisotropic) constitutive strain limiting is

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then easily and directly applied in IPC (Eq. 5.2), as just a simple addition to (respectively
modification of) the elasticity potential, Ψ , in the IP. This gives a mesh-based simulation
framework that produces trajectories simultaneously ensuring intersection-free and strain
limit satisfying steps. In turn the resulting dynamics satisfy accurate momentum balance
fully coupling frictional contact forces, strain-limiting and elasticity. Thus, as demonstrated
in Section 5.6.3, our method then avoids artifacts commonly generated by force-splitting
errors in traditional projection-based strain-limiting methods.

Thickness modeling. Thin structures typically have small relative thicknesses whose
elastic behavior can be indirectly captured by codimensional DoF models. However, cor-
rectly and directly modeling thickness is crucial for capturing correct geometric behaviors.
Consider, for example, a deck of cards. Each card has a nearly imperceptible thickness
when considered individually. However, when stacked, their combined thickness is large
and clear, and can not be ignored. While contact-processing strategies often introduce
thickness parameters they are generally applied to mitigate inaccuracies. As analyzed in
Li et al. [2020b], these thicknesses can not be consistently enforced and moreover must
be changed heuristically per scene and example (e.g., based on collision speeds) to avoid
simulation failures. To consistently enable thickness modeling for codimensional simula-
tions we extend the IPC formulation Li et al. [2020b] to capture the geometric thickness of
codimensional structures modeled by directly enforcing distance offsets. Our treatment
imposes a strict guarantee that mid-surfaces (respectively lines) of shells (respectively rods)
will not move closer than the applied thickness values even as these thicknesses become
characteristically small; see Sec. 5.5. This enables us to account for thickness in the
contacting behavior of codimensional structures and so capture effects that are challenging
and in some cases have not previously been modeled nor demonstrated.

Additive CCD Contact modeling between thin codimensional models with small but
finite thicknesses challenges all collision-detection routines. Here then, most importantly,
we see unacceptable failures and/or inefficiencies in all available existing CCD methods

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and codes. We see this both for standard floating point root-finding CCD methods as
well as more recent developments in exact CCD methods. These issues are attributable to
the well-known numerical sensitivity of the challenging underlying root-finding problems
posed. Here, most specifically, we see that IPC is reliant on a bounded guarantee of
non-intersection for all CCD evaluations. Large enough conservative bounds can help
Li et al. [2020b] account for these inaccuracies, ensuring non-intersection, but they do
so at the cost of progress and so can even stall convergence altogether when it comes
to codimensional models. To address these issues we derive a new, simple to compute,
numerically robust, additive CCD (ACCD) for approaching requested CCD bounds that is
stable for the exceedingly small evaluation distances required. While we most immediately
focus here on ACCD’s application within our IPC framework, as we show in the following
sections, ACCD is exceedingly simple and so easy to implement when compared to all
prior CCD routines (exact and floating point) with both improved performance, robustness
and guarantees, and so is suitable for easy replacement in all other applications where CCD
modules are employed.

5.4 Constitutive Strain Limiting

Here we begin by constructing our constitutive strain-limiting model. We first start with
the general, isotropic regime with a strain-limiting potential (Sec. 5.4.1) that augments
arbitrary, existing membrane energies. Then we demonstrate extension to the anisotropic
regime with an orthotropic StVK membrane that directly applies anisotropic strain limits
(Sec. 5.4.2).

5.4.1 Isotropic Constitutive Strain Limiting

We define isotropic strain-limiting constraints per element t as

σit < s, ∀i, (5.4)

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where the singular value decomposition of each triangle t’s deformation gradient, F t =
T
U t Σ t V t , gives Σ t = diag(σ1t , σ2t ). The imposed constraint bound s is then the requested
strain limit with practical bounds generally selected for cloth with s ∈ [1.01, 1.1] Bridson
et al. [2002], Provot et al. [1995].

Inspired by IPC’s contact barrier formulation, we then construct a local energy that
enforces this unilateral strain-limit. We can begin with a typical log-barrier strategy. This
gives energies
X
κs − ln(s − σit ), (5.5)
i

with κs defining the barrier stiffness for limit s. While ensuring strain-limits this bar-
rier is problematic. First, from the modeling perspective, it applies forcing everywhere.
Specifically, constitutive behavior is unnecessarily changed far from strain limits where the
underlying membrane model should be untouched and, worse yet, applies ghost forces at
rest. Second, from the computational perspective, because this barrier is globally non-zero,
evaluating it (and its derivatives) in optimization is prohibitively expensive.

From these issues we then observe that strain-limit forcing for triangles far from the
strain limit is then both unacceptable and unnecessary. Instead, we want local support for
nonzero forces just close to where these limits are achieved. One simple and seemingly
plausible remedy, to achieve this localized support region, is to clamp the barrier in Eq. 5.5
to zero beyond a small strain threshold ŝ (e.g. ŝ = 1) giving barrier energies


t
− ln( s−σi )

σit > ŝ
s−ŝ
X
κs bst
i (x), bst
i (x) = . (5.6)
0 σit ≤ ŝ

i

Unfortunately, while providing local support these energies are only C 0 continuous and so
do not allow for practical gradient-based optimization necessary for the efficient solution
of the IP.

To smooth our strain-limiting while providing local support we then propose a C 2

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smoothly clamped barrier for strain-limiting

t t
−( ŝ−σi )2 ln( s−σi ) σit > ŝ

s−ŝ s−ŝ
bst
i (x) = (5.7)
0 σit ≤ ŝ


.
Now, with barriers in hand, we could potentially next consider treating these barriers
directly as constraints (as in primal barrier and interior point methods). However, doing
so would then simply sum these barriers over elements and so would obtain inconsistent
behavior as we change meshes. Instead, to provide consistent behavior we impose strain
limiting constitutively as an energy density integrated over the volume of cloth to obtain
the potential Z
X
ΨSL (x) = κs bst
i (x)dV
i Ω
X (5.8)
t st
≈ κs V bi (x).
t,i

For our approximation, we use V t = At ξ t , with At and ξ t respectively the area and
thickness of triangle t. Our final isotropic strain-limiting potential, ΨSL , is then C 2 with
local support and can simply be added to our total potential in Equation 5.3. In turn, strain
limiting is then directly handled at each time step by optimization of the IPC Equation
5.2, while ensuring that strain-limit barriers are not violated in each line-search step.
During steps when most triangles remain under the strain limit threshold, little additional
computation is then required. While, when stretch increases, we can adjust the necessary
nonzero terms from newly activated barriers and so, as we show in Section 5.6, strictly
enforce strain limits while balancing all applied forces.

Strain-Limit Stiffness With our barrier potential defined we now specify setting its
stiffness. The mollified clamping strategy we apply for our strain-limit model is inspired
by IPC’s smoothing of contact barrier energies Li et al. [2020b]. Here notice an extra
ingredient for strain-limiting is the applied 1/(s − ŝ)2 factor. This scaling enables us to
apply our barrier directly to a limit-normalized strain, measured by y = (ŝ − σit )/(s − ŝ)

126
with 
−y 2 ln(1 + y)

y < 0,
bst
i (y) = (5.9)
0 y ≥ 0.


In turn, as our applied strain-limit (s) and/or clamping threshold (ŝ) is varied with applica-
tion, the barrier function w.r.t. y remains unchanged. Only the gradient of the linear map
from σit to y varies. This allows us to apply a single, consistent initial barrier stiffness κs
(we use 1KP a for all examples) across all choices of differing strain limits and clamping
thresholds. Here the barrier potential curve is then simply linearly rescaled each time, to a
different strain range, and so provides consistent conditioning to the system. Finally, to
avoid numerical issues introduced by tiny gaps between a current strain and the imposed
strain limit (e.g., when extreme boundary conditions are imposed as in Figures 5.14 and
5.15) we adapt barrier stiffness when needed. Starting with our initial κs , we increase it by
2× (up to max bound of κs = 0.1M P a) whenever the strain gap s − σit of a triangles t is
less than 10−4 (s − ŝ) over two consecutive iterations.

5.4.2 Anisotropic Constitutive Strain Limiting

Above we have constructed strain-limiting as an isotropic constitutive model. We formed a

barrier energy that can be integrated and so directly added to potential energy in order to
augment existing membrane models with hard strain limits. The key to making this work is
the application of our C 2 continuous clamping so that the application of strain-limits does
not alter rest-shape gradients nor rest-shape Hessians.
Alternately, we can directly apply a similar constitutive strain-limiting strategy to
modify membrane elasticity to now include strain limits. To do so we simply substitute an
anisotropic membrane model with a barrier energy that prevents violation of strain limits,
while matching the original membrane energy gradient and Hessian at rest.
This second strategy is particularly motivated by the need to incorporate strain-limiting
into available data-driven models that have been constructed to carefully fit measured cloth
data. Here we demonstrate the specific application to the anisotropic, data-driven model of

127
Clyde and colleaguesClyde [2017], Clyde et al. [2017].
Their constitutive model energy is
a11 2 a22 2
ψ̃(Ẽ11 , Ẽ12 , Ẽ22 ) = η1 (Ẽ11 )+ η3 (Ẽ22 )
2 2 (5.10)
2
+ a12 η2 (Ẽ11 Ẽ22 ) + G12 η4 (Ẽ12 ),

where Ẽ = DT ED is the reduced and aligned Green-Lagrangian strain, Ẽ i3 = Ẽ 3i = 0,

and column vectors of D are the tangent and normal bases of the shell in material space.
Functions η are then a sum of polynomial functions with real-valued exponents αji that
satisfy η(0) = 0 and η 0 (0) = 1. Here the first constraint enforces zero-energy, zero-stress
rest configurations, while the second constraint allows a natural correspondence between
the parameters aαβ and G12 and linear elasticity at infinitesimal strain. Clyde et al. use
dj
X µji
ηj (x) = ((x + 1)αji − 1).
i=1
αji

Their measured data is then restricted to deformations within the fracture limit or elasticity
range of the cloth. Beyond this range meaningful extrapolation is then unlikely, due to
possible overfitting inside the range (exponents αji from the fitting can range up to 105 ).
To address this limitation Clyde et al. propose a quadratic extrapolation for simulation.
However such extrapolation is not physically meaningful. To usefully apply data-driven
modeling either fracture should be captured beyond this regime or else a stable and
controllable strain limit imposed to stay in bounds. Here, we focus on controllably imposing
strain limit while respecting the underlying model fitting.

Barrier Formulation Starting from the same constitutive model we redefine the η basis
with barriers
η(E) = −E max log((E max − E)/E max ).

For consistency note this basis again satisfies both η(0) = 0 and η 0 (0) = 1 and so is valid
for elasticity. This barrier also diverges at E max and so ensures that E never exceeds
measured strain limit bounds. It thus captures data-driven strain limits and at the same time

128
avoids extrapolation. Here, as the underlying model is anisotropic, warp, weft, and shearing
directions all can impose the have different, measured E max bounds enabling preservation
of measured anisotropy for all strain limits.
Free model parameters aαβ and G12 , are then matched at ∇2 ψ(0, 0, 0) with the underly-
ing model’s fit; see Appendix for details. Note this formulation frees us from computing
the sensitive (and potentially expensive) polynomials with large real-valued exponents. See
Section 5.6.2 for experiments on the proposed model. Finally we note that, in contrast to
the strategy we demonstrated in our isotropic case from the last section (where we worked
directly on bounds w.r.t. the deformation gradient), here this second energy is symmetric
for stretch and compression and so is not inequality-based.

5.5 Modeling Thickness

In its original form IPC maintains intersection-free paths for volumetric models by en-
forcing positivity of unsigned distances dk , as an invariant between all non-adjacent and
non-incident surface primitive pairs k Li et al. [2020b]. This is suitable for volumetric
contact where this constraint permits arbitrarily close, but never intersecting, surfaces. For
codimensional models, however, this constraint is no longer sufficient.
When the 3D deformation of thin materials is reduced to the deformation of a 2D
surface or 1D curve, elasticity can be well-resolved on surfaces and curves, but contact
can not. Neglecting to account for finite thickness in codimensional contact generates
unacceptable artifacts (see e.g. Figure 5.16) and clearly fails to capture geometries formed
by thin structure interactions (see e.g. Figures 5.9 and 5.18).
To model thickness in contact for codimensional objects we equip each midsurface
element i with a finite thickness ξi . The distance constraint for primitive pairs k on the
midsurface, formed between element primitives i and j, is then

ξi + ξj
dk (x) > ξk = .
2

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Boundaries of codimensional materials are then approximated
with a rounded cross-section, while for interaction between zero-
thickness materials our distance constraints reduce to that of the
original IPC constraint (e.g. for volume-to-volume contact). Fi-
nally for volume-to-codimensional contact, volumes can continue
to maintain zero-thickness boundaries while interacting with finite
thickness codimensional boundaries.

5.5.1 Solving IPC with Thickened Boundaries

For a numerically robust and efficient implementation, IPC applies squared distances (with
appropriate rescalings) to compute with equivalent barrier functions. This change swaps
ˆ to an equivalent rescaled implmentation
the model’s derived contact barrier b(dk (x), d)
using b(d2k (x), dˆ2 ) Li et al. [2020b]. Here we directly apply our thickened contact barrier
via squared distances

ˆ = b(d2 (x) − ξ 2 , 2ξk dˆ + dˆ2 ),

bξ (dk (x), d) k k

so that contact forces correctly diverge at dk (x) = ξk , and nonzero contact forces are only
applied between mid-surface pairs closer than dˆ + ξk .
Here then the barrier function, gradient, and Hessian only need to be evaluated with the
modified input distance, while distance gradient and Hessian remain unchanged as

d(d2k (x) − ξk2 ) dd2 (x)

= k .
dx dx

Evaluation of contact force magnitudes for computing friction forces as well as adaptive
stiffness (κ) updates for contact barriers follow similarly.
Most constraint set computations then require simple and comparable simple modifica-
tions. For spatial hashing, bounding boxes of all mid-surface primitive i are extended by
ξi /2 in all dimensions on left-bottom and top-right corners before locating voxels. This
enables hash queries with dˆ to remain unchanged. Next, for broad-phase contact pair

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detection, query distances dˆ are now updated to dˆ + ξk to check for bounding box overlap.
Or else, equivalently, bounding box primitives are extended as in the spatial hash. Finally,
to accelerate continuous collision detection (CCD) queries, spatial hash construction also
requires similar bounding box extension, while for its broad phase, applied gaps are ξk
(rather than 0).

5.5.2 Challenges for CCD

While the above initial modifications for thickness are straightforward, finite thickness and
codimensional DoF introduce new computational challenges for CCD queries. Here we
analyze these challenges and develop a new CCD method to address them.
Standard-form IPC applies position updates in each iteration to obtain minimal sepa-
ration distances of s× the current separation distances, dcur
k , at time of CCD evaluations.

Here s ∈ [0.1, 1) is a conservative rescaling factor (generally set to s = 0.2) that allows
CCD queries to avoid intersection, even when surfaces are very close Li et al. [2020b]. To
similarly resolve finite thicknesses, our conservative distance bound is now s(dcur −ξk )+ξk .
Concretely, the barrier evaluated parameters, d2k (x) − ξk2 , must always remain positive.
In turn we require all CCD evaluations, for each displacement p, to provide us with
sufficiently accurate times t ∈ [0, 1] to satisfy positivity. If impact would occur for a pair
along p, we require a time t so that the scaled displacement, tp, ensures distance remains
greater than ξk and is as close as possible to the target separation distance ξk + s(dcur
k − ξk ).

Doing this with thickness is much more challenging than in volumetric IPC where
updated distances are targeting sdcur (and must be greater than 0). With thin materials
dcur − ξk can be as small as 10−8 m, while in practice ξk is regularly at the scale of
10−4 m (e.g. thickness values of ∼ 3 × 10−4 m for cloth and ∼ 1 × 10−4 m for hairs are
standard). Consider then that without thickness (ξk = 0) updating distances to sdcur within
100% relative error are then acceptable. On the other hand for standard values of finite
ξk , distance updates would require relative errors from the CCD evaluation of around
0.1% = 10−8 m/10−4 m in order to avoid interpenetration.

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 Obtaining CCD evaluations to this accuracy is extremely challenging for available
methods. As a starting example we tested the floating point CCD solver from Li et
al. [2020b], requesting a 0.2(dcur − ξk ) + ξk minimal separation for two challenging
codimensional examples with thickness (see Figures 5.20 and 5.21) with ξk > 0. Here
the CCD solver returns t = 0 time of impact (toi) erroneously even if we remove the
conservative scaling factor (i.e. set s = 0) altogether (Section 5.6.8). Note that in doing so
this error stops simulation progress altogether.

5.5.3 CCD Lower Bounding

Next we derive a useful lower bound value for CCD queries that is numerically robust and
can be efficiently evaluated in floating point. This lower bound provides a conservative,
guaranteed safe step size and also a clear measure to test a CCD query’s validity: any
CCD-type evaluation with smaller toi has clearly failed. In Section 5.5.4 we then apply this
bound to derive a simple, effective and numerically accurate explicit CCD solver that is
robust and efficient for progress even in the challenging CCD evaluations we require for
thin material simulations.
Here, without loss of generality, we will focus on the edge-edge case between edge
pairs (x0 , x1 ) and (x2 , x3 ) with corresponding displacements p0 , p1 , p2 and p3 . The distance
function between arbitrary points parameterized respectively by γ and β on each edge at
any time t is then

f (t, γ, β) = ||d(t, γ, β)||, with

d(t, γ, β) = (1 − γ)x0 (t) + γx1 (t) − ((1 − β)x2 (t) + βx3 (t)),
(5.11)
xi (t) = xi (0) + tpi , and

t, γ, β ∈ [0, 1].

A CCD evaluation then seeks the smallest positive real t satisfying

f1 (t) = min f (t, γ, β) = 0.

γ,β

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If such t exists we call it ta . Parameters (γa , βa ) = argminγ,β f (ta , γ, β) in turn give the
corresponding points colliding at time ta on each both edges.
We can then express ta as
f (0, γa , βa )
ta = .
||(1 − γa )p0 + γa p1 − ((1 − βa )p2 + βa p3 )||
Challenges to CCD evaluations then occur in determining γa and βa , when degeneracies
and numerical error make floating-point methods both prone to false positives and false
negatives.
We can however, directly and accurately perform a distance query w.r.t. the primitives
at start (t=0) of evaluation, to find the distance, f1 (0), that certainly satisfies f1 (0) ≤
f (0, γa , βa ). Then triangle inequality gives ||(1 − γa )p0 + γa p1 − ((1 − βa )p2 + βa p3 )|| ≤
||(1 − γa )p0 + γa p1 || + ||(1 − βa )p2 + βa p3 ||, and since γa , βa ∈ [0, 1] we get ||(1 − γa )p0 +
γa p1 || ≤ max(||p0 ||, ||p1 ||) and ||(1 − βa )p2 + βa p3 || ≤ max(||p2 ||, ||p3 ||). Put together
we then have the practical and directly computable bound on toi
f1 (0)
ta ≥ . (5.12)
max(||p0 ||, ||p1 ||) + max(||p2 ||, ||p3 ||)
We then note that even when there is no smallest positive time satisfying f1 (t) = 0 (and so
no impact on the interval), our bound remains well-defined as a conservative step size.
We next observe that, perhaps surprisingly, state-of-the-art floating-point CCD solvers
can and will return toi results smaller than our lower bound, and so are clearly in error
(Section 5.6.8).
Finally, to improve our bound, we observe that it holds independently of the choice
of the reference frame. Thus we can further tighten the bound on each CCD query
independently by picking frames that reduce the norm of displacement vectors pi . For
example by subtracting each pi by the average 41 i pi .
P

5.5.4 Additive CCD

We now apply our lower bound to build a new CCD algorithm that iteratively updates
and adds our lower bound over successive conservative steps towards toi. The resulting

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additive CCD (ACCD) method then robustly solves for a bounded toi, monotonically
approaching each CCD solution, while only requiring explicit calls to evaluate distances
between updated primitive positions.

At the start of each CCD query, to initialize the ACCD algorithm (see Algorithm 4), we
first center the collision stencil’s displacement at origin to reduce our bound’s denominator,
lp = max(||p0 ||, ||p1 ||) + max(||p2 ||, ||p3 ||), and so increase the step size we can safely
take. If there is no relative motion (lp = 0), we of course simply return no collision and so a
p
full unit step is valid. We then compute the requested minimal separation g = s( dsqr − ξ)
to the offset surface based on the current squared distance dsqr and the scaling factor s. For
this we use a formula that is more robust to cancellation error; see lines 8-9 in Algorithm 4.

Starting with a most conservative time of impact t = 0 (line 10) we create a local
scratch-pad copy of nodal positions, xi , and initialize the current lower bound step, tl , with
Equation 5.12 (line 11).

We then enter our iterative refinement loop (lines 12-21) to monotonically improve
our toi estimate t. At each iteration, we update our local copy of nodal positions xi with
the current step tl (lines 13-14). If this new position achieves our target distance to the
offset surface (becomes smaller than g) we have converged and the previous t is the time
of impact that brings distance just up to g (line 17). If not, we update our toi estimate by
adding the current tl to t (line 18). Note that we always add our first lower bound step to
t (line 16) as it is guaranteed to not bring distance closer than g. If our toi is now larger
than tc , the current minimum first time of impact (or can be simply set to 1), we can return
no collision (lines 19-20). Otherwise we compute a new local, lower bound, tl , from the
updated configuration (line 21) and begin the next iteration.

ACCD thus provides an exceedingly simple-to-implement, numerically robust CCD

evaluation. It requires only explicit calls to distance evaluations and so no numerically
challenged root-finding operations. In turn, ACCD is able to support very small thickness
offsetting and also controllable accuracy and so flexible tuning for performance vs accuracy

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Algorithm 4 Additive CCD
1: procedure A DDITIVE CCD(xi , pi , ξ, s, tc , t)
P
2: p̄ ← i pi /4
3: for i in {0, 1, 2, 3} do
4: pi ← pi − p̄

5: lp ← max(||p0 ||, ||p1 ||) + max(||p2 ||, ||p3 ||)

6: if lp = 0 then
7: return false

8: dsqr ← computeSquaredDistance(x0 , x1 , x2 , x3 )
p
9: g ← s(dsqr − ξ 2 )/( dsqr + ξ)

10: t←0
p
11: tl ← (1 − s)(dsqr − ξ 2 )/(( dsqr + ξ)lp )
12: while true do
13: for i in {0, 1, 2, 3} do
14: xi ← xi + tl pi

15: dsqr ← computeSquaredDistance(x0 , x1 , x2 , x3 )

p
16: if t > 0 and (dsqr − ξ 2 )/( dsqr + ξ) < g then
17: break

18: t ← t + tl
19: if t > tc then
20: return false
p
21: tl ← 0.9(dsqr − ξ 2 )/(( dsqr + ξ)lp )

22: return true

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trade-offs in standard CCD applications. In Section 5.6.8 we compare ACCD with state-
of-the-art CCD solvers based on exact arithmetics and conservative strategies and show
that ACCD succeeds in all cases with the best timing performance while other methods
frequently fail.
Note that our stopping criteria requires s > 0 to provide finite termination, meaning
that ACCD is never aimed at computing the exact toi, which is also not really needed by
contact simulation methods. In our simulation we use s = 0.1 for every example.

5.6 Evaluation
We implement our methods in C++, applying Eigen for linear algebra Guennebaud et al.
[2010] and CHOLMOD Chen et al. [2008], compiled with Intel MKL LAPACK and BLAS
as our linear solver. Details of our computed derivatives and algebraic simplifications
for numerical robustness and efficiency are detailed in our appendix. All evaluations are
executed on either an Intel 16-Core i9-9900K CPU 3.6GHz × (32GB memory), an Intel
Core i7-8700K CPU @ 3.7GHz × 12 (64GB memory), or an Intel Core i7-9700K CPU @
3.6GHz × 8 (32GB memory) as detailed per experiments below.

Line Search As discussed above, CCD-based filtering is critical in line-search for col-
lision. For strain-limiting we must now additionally ensure line search will not violate
any imposed strain limits. In the isotropic case, although 2 × 2 SVD has a closed form
solution, it does not provide a polynomial equation for easy feasible step size computation.
Therefore, in line search, after we obtain an interpenetration-free step size, we simply
half step sizes whenever we detect a strain limit violation during backtracking until we
find a step size satisfying both strain limit and decreases energy. In practice, because our
strain-limit barriers are included in the Hessian and gradient computation, we find that
Newton search directions are effective at avoiding strain-limits and, in cases when needed,
at most 3 bisections for strain-limiting are required. We apply the same strategy here for
the anisotropic case. However, we do note that here in our the anisotropic case, it would

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be reasonable to formulate quadratic equations amenable to directly computing largest
feasible strain-limit satisfying step sizes. We leave this for future work.

Experiments Below we begin with a study evidencing membrane locking behaviors for
standard cloth materials and simulation meshes (Section 5.6.1). We then demonstrate our
method’s ability to strictly satisfy strain-limits while fully coupling all physical forces
in Section 5.6.2. To our knowledge our method is the first to enable strict satisfaction
of strain-limits and also is the first to fully couple strain-limiting, elasticity and contact
forces. To verify this we next consider comparison against prior methods. As discussed
in Section 5.2 existing methods in strain-limiting generate artifacts including locking,
jittering, and interpenetration. These issues result from two sources: 1) inaccuracies from
applied splitting models and 2) inability to satisfy strain-limits in computation. Here, for
the first time, we first separately analyze the problems that are created by splitting models
(Sec. 5.6.3), and then in Sec. 5.6.4 consider the artifacts and inaccuracies (generated by
state-of-the art cloth code) that are then jointly resulting from both splitting errors and
inability to enforce requested strain-limits. In Section 5.6.5 and 5.6.8 we then analyze
our thickness modeling comparing both with existing cloth simulation codes and prior
methods for CCD evaluation. Finally with all pieces in place, we consider our method
applied to prior cloth simulation benchmarks (Section 5.6.6); and stress tests demonstrating
new challenges for thin materials (Section 5.6.7) and fully coupled systems of arbitrary
codimension with consistent thickness modeling (Section 5.6.9).

5.6.1 Cloth Material and Membrane Locking

Here we first examine the impact of membrane locking on real-world materials and strain-
limiting’s ability to mitigate them. In Penava et al. [2014], density, thickness, and di-
rectionally dependent membrane strain-stress curves for multiple varieties of cloth are
measured and validated. However, directly applying these real-world cloth parameters
for simulation with standard-resolution meshes can produce severe membrane locking.

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This is unavoidable unless exceedingly high and so often impractically large mesh sizes
are used. Here we examine this locking behavior, first examining our IPC model without
strain-limiting. We also note that such locking behaviors are independent of algorithm
and, for example, are also easy to demonstrate in ARCSim Narain et al. [2012] with their
real-world captured material parameters Wang et al. [2011]; see Figure 5.6d.

We start by considering the behavior of a simple 1m-by-1m unstructured mesh dropped

on a fixed sphere and ground plane (with friction of µ = 0.4 for all). We apply the measured
cotton cloth density (472.6kg/m3 ) and thickness (0.318mm) from Penava et al. [2014]
and then consider varying membrane stiffness while keeping bending Young’s modulus at
0.8M P a and Poisson’s ratios for both membrane and bending to the measured value for
warp direction as 0.243. Specifically Penava et al. [2014], find membrane Young’s moduli
ranging from 0.8M P a to 32.6M P a for varying in-plane directions.

For our example, we then find that even when applying an isotropic membrane model,
with the smallest determined membrane stiffness (0.8M P a), we observe severe locking
artifacts (see e.g. Figure 5.2a) where bending is artificially stiffened. If we next lower
the smallest measured membrane stiffness by 0.1×, we then see that artificial bending
artifacts are largely eliminated but we still obtain sharp creasing artifacts forced by the
dominating membrane energy (see e.g. Figure 5.2b). Next, if we try an even smaller 0.01×
scaling of membrane stiffness, observable membrane locking effects are now gone, but as
expected the resulting material is much too stretched and so not even close to the desired
material behavior (Figure 5.2c). This simple example nicely demonstrates the challenges
of membrane locking when simulating stiff real-world cloth materials. We then note that
these artifacts are only exacerbated in more challenging simulations with, for example,
moving boundary conditions and tight contact.

Next we apply a strict strain-limiting, here via our constitutive model, to constrain
strain within the elasticity range measured by Penava et al. [2014]. The broadest range
in all directions for cotton allows a stretch factor up to 6.08%. Here, applying this bound,
even with 0.01× scaling of membrane stiffness, we now regain a simulation free from

138
membrane locking and, since restricted to measured strain limits, also free of unnatural
stretching artifacts (Figure 5.2d).

(a) 1x stiffness (b) 0.1x stiffness

(c) 0.01x stiffness (d) 0.01x stiffness + SL

Figure 5.2: Membrane Locking. A cloth on sphere simulation (8K-node) with (a) 1× stiff
membrane of cotton material (0.8M P a) suffers from membrane locking where bending
behavior is artificially stiffened and creased. Reducing membrane stiffness to (b) 0.1× that
of measured cotton values removes artificial bending stiffness but we still observe overly
sharp edge creasing. Next an even softer membrane at (c), 0.01× cotton’s, then avoids
observable membrane locking artifacts altogether, but results in nonphysical stretching
inappropriate for cotton. However, (d) applying a softer 0.01× membrane together with
our constitutive strain limiting model to enforce measured cotton strain limits prevents
membrane locking while obtaining natural stretch limits for the material.

5.6.2 Exact Strain Limits

Above we have demonstrated the well-known importance and impact of applying strain-
limiting for simulating cloth materials. Here we investigate our method’s ability to tightly
and accurately enforce strain limits. As we will demonstrate this holds both for extreme
(e.g. at 0.1%) and anisotropic limits.

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Accuracy across decreasing strain limits To confirm accuracy, controllability and ro-
bustness of our constitutive strain limiting we apply increasingly severe strain limits of 1%
and then 0.1% on both the sphere drape example from above and a standard membrane
locking test of a two-corner pinned cloth Chen et al. [2019], Jin et al. [2017]. Here, see
Figure 5.3, we observe stable simulation output with stretch on all triangles satisfying the
prescribed limit constraints. Closer examination of the draped sphere tests in Figure 5.3

(a) 1% (b) 0.1%

(c) 1% (d) 0.1%

Figure 5.3: Extreme Strain Limits. Our constitutive strain limiting model stays robust
and guarantees constraint satisfaction even with extreme (and even nonphysical) strain
limit constraints (1% in (a, c) and 0.1% in (b, d)). Here the mesh only contains 8K (a, b) or
2K (c, d) nodes. Not in (d) that as strain-limits tighten beyond those physically plausible
and so more constraints become active w.r.t. to DoF count, a small amount of membrane
locking artifact becomes apparent; see our discussion below.

also confirms that both are free from artificial bending stiffness. For contrast, compare this
result with the artificial stiffening in Figure 5.2a, where the maximum stretch exceeds 4.5%
despite applying the measured membrane stiffness directly. Here, as we are applying our
strain-limits unilaterally, this agrees with Jin et al.’s [2017] argument that unilateral strain-
limits can allow compression to avoid membrane locking better than bilateral enforcement

140
– at least to a certain degree.
However, even unilateral enforcement is not a perfect panacea if limits are very tight.
For an extreme example as we push strain-limits to a very small 0.1% strain limit we
can finally observe some slight but distinct sharp edge-creasing artifacts in Figure 5.3b
and similarly locking in Figure 5.3d. Although such an extreme limit is unlikely and
generally not encountered for cloth, this experiment does highlight an important point: if
most unilateral stain-limit constraints are active then they behave similalrly to the bilateral
case. Thus, as discussed in Chen et al. [2019], when most strain limits are at the bound,
active constraint numbers can again exceed the number of simulated DOFs, and so locking
can once again be encountered. Here we focus on formulating a controllable and robust
constitutive model for strain limiting. We then hope to further enable increasing the range
of locking-free configurations by exploring alternate discretization for the strain constraints
as in Chen et al. [2019] but note that this is only be possible when the underlying method,
as proposed here, can accurately guarantee constraint satisfaction.

Anisotropic strain limiting For anisotropy, the story is similar. In Figure 5.4 we demon-
strate a cloth hang (two top corners fixed) and our same cloth drape on sphere example.
Again applying measured stiffness values, here from Clyde [2017], generates membrane
locking (Figure 5.4a). Then scaling down membrane stiffness 0.1× and 0.01× while
preserving measured strain limits again provides the expected improved results (Figure
5.4b,c) while in Figure 5.4 (again with 0.01× membrane stiffness), we see both har strain
limits and anisotropic wrinkling behavior.

5.6.3 Comparison with Splitting Models

Existing strain-limiting methods introduce errors from two primary sources. First, at
the modeling level, they split strain-limiting from the resolution of elasticity. Second,
irrespective of the splitting model applied, algorithms applied to solve them are inaccurate
and often unable to satisfy even moderate requested strain-limits. All prior methods then

141
 (a) 1x stiffness with SL (b) 0.1x stiffness with SL

(c) 0.01x stiffness with SL (d) 0.01x stiffness with SL

Figure 5.4: Anisotropic strain limiting. Cloth on sphere drape with varying scales of
real-world membrane stiffness (a-c) and a cloth hang (d) all simulated with our anisotropic
strain limiting model at h = 0.04s. Here the shown frames are the final static rest shape.
We can see that the draped cloth exhibits differing wrinkle patterns when compared to
isotropy.

introduce errors from both of these sources and so it has remained unclear what problems
originate from each. Here, we first separately analyze the problems that are created by
splitting models, by solving each step of the splitting to tight accuracy. This allows us
to show that the splitting itself introduces errors that are unavoidable irrespective of how
accurately the constraints could be enforced. Then, in the next section, we will examine
solution accuracy and see that errors in the strain-limit solve itself then produce inconsistent
and so often uncontrollable results for simulation.

To examine splitting errors we follow the standard strain-limiting, splitting strategy and
so divide each time step solve into two sequential steps. The first solves a predictor step
that includes all forces for the whole system, with the exception of the strain-limit, to obtain
an intermediate configuration x̂. Next, the second step projects the predictor to satisfy the
strain-limit. To do so we minimize our constitutive strain limiting energy summed with the

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mass-weighted L2 distance from the final timestep solution to x̂.

In the simplest case we can consider the effect of this splitting when there are no contact
forces. We start with a pinned and pulled square cloth. We fix its two top corners and apply
weights to pull its two bottom corners downwards; see Figure 5.5a and b. For comparison,
without strain limiting, the cloth is stretched well over 10× (not shown) while with our
constitutive strain limits, strains are restricted to the measured elasticity range, and, in this
fully coupled solution, we see the expected vertically aligned wrinkles from stretching at
the two bottom pulled corners; see Figure 5.5a. On the other hand in Figure 5.5b, we see
that splitting strain limiting from the elasticity solve produces obvious biasing artifacts at
the two bottom corners where the wrinkles are aligned in non-physical directions. These
errors from the decoupled forces are then increasingly severe as time step increases (here
we show with h = 0.04s).

(a) our method (b) split (c) our method (d) split (e) split plus lower bound

Figure 5.5: Comparison of our fully coupled strain limiting solution with traditional
splitting methods. Examples with both a cloth pinned at top and pulled on bottom (a,
b), and a cloth draped on sphere (c-e) can both demonstrate the difference between fully
coupled solutions (a and c) and the artifacts generated by splitting strain limit solves from
elastodynamics (b,d and e). Here we show the final configuration of each simulation as it
comes to a rest. We can see that with splitting, large errors in elasticity (membrane and
bending) can generate incorrect wrinkling directions (b), severe compression artifacts (d),
or even numerical softening (d and e).

Next, to consider splitting errors when subject to contact, we again consider the cloth
on sphere example. Here we apply a neo-Hookean membrane elasticity to help the splitting
method avoid possible triangle degeneracies. Note that with our fully coupled model we

143
do not require the neo-hookean model as triangle degeneracies are prevented by point-
triangle constraints between neighbors. Here in Figure 5.5d we then see that splitting with
contact now suffers from severe compression artifacts, which again comes from splitting
between membrane and bending energies. For comparison, consider our fully coupled
strain-limited solution in Figure 5.5c. In turn the error in the split solution suggests that an
additional lower bound on strain (e.g. as applied in ARCSim) might be helpful to avoid
these compression errors. However, if we additionally add a strain-limit lower bound (e.g.
at 0.7) then the splitting method is indeed now free from severe compression artifacts, but
now due to the errors in membrane and bending the cloth still remains unnaturally flat
against the floor; see Figure 5.5e.
Of course as with all time-step splitting methods, errors can be reduced by applying
increasingly smaller time step sizes. Here, for example, we find that visual errors between
the split model and our fully coupled solve disappear at h = 0.004s. However, as expected,
the resulting large and often unacceptable increase in compute time wipes out any expected
gains from splitting (we will see this theme again in the next sections’ comparisons with
existing cloth codes). Likewise the necessary decrease in time step size then varies with
example and scene so that robust, controllable time-stepping with splitting, does not appear
possible.

5.6.4 Strain Limiting Comparisons

As we just demonstrated how severe the artifact a two-step splitting solve can result in,
what if contact and strain limiting projection is further split that results in 3 substeps, and
that the strain limit cannot always be satisfied exactly? This leads to our comparison to
ARCSim Narain et al. [2012].
Without contact, we consider the cloth drape example again. Instead of making the
two bottom corners heavier, this time we place the initial configuration of the cloth in xz
plane, so the swing motion will make the drape challenging in certain frames. In Figure
5.6(a, b) we show the frames at 4s as the cloth nearly becomes static. We use the default

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strain limit, [0.9, 1.1], in ARCSim for the comparison, and in ARCSim we use their default
cotton material. Comparing to our method, even with this wide range of strain limit the
result obtained by ARCSim under h = 0.04s has obvious over-stretched triangles near the
fixed corners, which forms a larger arc. In Figure 5.7 we can see that ARCSim violate the
strain limit a lot both in stretch and compression. Even if we decrease the time step size to
h = 0.01s to make the problem easier, ARCSim still cannot exactly satisfy the strain limit
constraints, although the violation does get smaller. With h = 0.001s, ARCSim finally can
closely satisfy the constraints, but it needs 87min for the 4s simulation sequence, which is
more than 10× that of our method running with h = 0.04s. In addition, it is unclear how
time step size needs to be decreased in different simulations to ensure ARCSim results to
satisfy the strain limit, while only our method guarantees constraint satisfaction agnostic to
simulation settings.

In fact for the augmented Lagrangian strain limiting solver in ARCSim source code
there is an iteration cap set to 100. Even if we increase this number up to 1000, the strain
limit satisfaction still cannot be guaranteed.

(a) ours (b) ARCSim (c) ours (d) ARCSim 1x stiff (e) ARCSim 0.1x stiff (f) ARCSim 0.01x stiff

Figure 5.6: Strain limiting comparisons. A cloth drape example where initially the cloth
is in xz plane (a, b), and a cloth on sphere example (c-f) showing comparison of our
fully coupled contact-aware constitutive strain limiting method to ARCSim’s augmented
Lagrangian based method under h = 0.04s. From the shown frames when the scenes
become (nearly) static, we can see ARCSim cannot guarantee strain limit satisfaction
(b,e,f), and as we decrease membrane stiffness in (e,f) to avoid membrane locking issue in
(d), the over- compression or stretching artifact gets more obvious.

With contact, we consider the 8K-node cloth on sphere example again. For ARCSim,

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 h=0.04s if not Max smax Avg smax Min smin Avg smin Total runtime
specified over time steps over time steps over time steps over time steps (min)
Cloth drape
1.0988 1.0884 0.9801 0.9824 6.5
ours
Cloth drape
2.1281 1.4836 0.7686 0.8760 2.0
ARCSim
Cloth drape
1.8959 1.3197 0.8127 0.8879 8.0
ARCSim h=0.01s
Cloth drape
1.1110 1.1009 0.8959 0.9093 87.0
ARCSim h=0.001s
Cloth on sphere
1.0617 1.0535 0.9548 0.9652 7.5
ours
Cloth on sphere
1.4176 1.0875 0.2397 0.9220 1.0
ARCSim
Cloth on sphere
1.3796 1.1587 0.0568 0.8599 1.5
ARCSim 0.1x stiff
Cloth on sphere
1.7944 1.3216 0.6714 0.8445 1.0
ARCSim 0.01x stiff

Figure 5.7: Strain limiting comparisons. Strain and runtime statistics of the cloth drape
example and the cloth on sphere example in comparison with ARCSim. Here smax and smin
are the maximum and minimum strain of all triangles in a certain time step. As time step
size decreases, ARCSim results satisfy strain limits better, while as membrane stiffness
decreaes, which is necessary to avoid membrane locking, ARCSim results violates strain
limits more. On the other hand, our method always guarantee strain limit satisfaction
agnostic to simulation settings.

we observed very similar membrane locking issue as in Figure 5.2a when we use ARC-
Sim’s default cotton material (Figure 5.6d). But with this stiff membrane, the strain limit
constraints are well satisfied except for several challenging time steps in the beginning.
When we multiply the stretching stiffness by 0.1× (Figure 5.6e), the membrane locking
issue gets much better, but now there are more triangles violating the strain limit in the
whole simulation. This shows that the performance of ARCSim’s strain limiting method is
very sensitive to membrane stiffness. Now if we multiply the stiffness by 0.01× as usually
used in our method (e.g. Figure 5.2d), the result of ARCSim shown in Figure 5.6f violates
the strain limits even more, and it is suffering from the extreme compression issue similar
to our splitting ablation study in Figure 5.5e. In addition, with 0.01× membrane stiffness,
the animation generated by ARCSim is suffering from jittering issue, which is because

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their contact and strain limiting are further split into separate projection steps, making
the constraints hard to be simultaneously satisfied. This issue has been discussed in their
limitation section.

5.6.5 Thickness Modeling

Here we conduct an experiment on a cloth stack example to demonstrate the thickness

modeling challenge in existing cloth simulator, and then show the controllability of our
method on intricately modeling the thickness of objects with robustness guarantees (more
examples in Section 5.6.9).
We construct a scene with 10 8K-node cloth in xz-plane falling simultaneously at
different height and orientation onto a square board (Figure 5.8a). This should form a
cloth pile where correctly modeling the thickness of each cloth matters a lot on obtaining a
reasonable pile height. This seemingly simple example already exposes challenges with
existing shell simulators in handling geometric thickness with contact. Here we compare
against two widely used state-of-the-art cloth simulators, ARCSim Narain et al. [2012] and
Argus Li et al. [2018a], of which the source codes are available online.

ARCSim ARCSim Narain et al. [2012] computes2 collision response applying Harmon
et al’s [2008] non-rigid impact zones and standard ”repulsion thickness” parameters in
order to both model cloth thickness and to help stabilize collision processing constraints.
ARCSim’s default thickness setting is 5mm and can be changed by users. This parameter
is then associated in code with additional thickness parameters directly applied for collision
projection, analysis and contact force computations. For the default 5mm thickness setting,
even decreasing time step size down to 0.001s, ARCSim still reports failures for collision
handling and we see severe artifact in the simulation results; see Figure 5.8b. We then try
with 10mm thickness, and here ARCSim succeeds in resolving contacts but still generates
large artifacts in geometry and dynamics; see Figure 5.8c. We did not push ARCSim further
2
We use the most recent ARCSim v0.3.1 in our testing.

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 (a) initial frame (b) ARCSim 5mm (c) ARCSim 10mm

(d) Argus 1mm (e) Argus 5mm (f) Argus 10mm

Figure 5.8: Cloth stack comparisons. Ten 8K-node square cloth are dropped onto a square
board with friction µ = 0.1 (a). Here ARCSim Narain et al. [2012] and Argus Li et al.
[2018a] are tested with different cloth thickness settings at h = 0.001s. ARCSim fails to
resolve the contact at 5mm (b), and at 10mm there is no interpenetration but the dynamics
is off (c). Argus can successfully simulate cloth with 5mm and 10mm thickness (e,f), but
at 1mm, it fails to resolve contact. See Figure 5.9 1st row for our results.

with even smaller time step sizes as generating this simulation sequence with h = 0.001s
already makes it excessively slow for computation when compared to our method (details
later).

Argus Argus Li et al. [2018a] combined ARCSim and a new contact solver Daviet et al.
[2011] to provide improved contact and friction processing for shell simulations. It also
applies and exposes the same thickness parameter from the underlying ArcSim code to
users. Using Argus to simulate our cloth on stack example at h = 0.001s with default
5mm thickness (Figure 5.8e) and the thicker 10mm (Figure 5.8f) both work, and we can
observe the height difference of the pile is also captured. But as we decrease thickness
towards thinner cloth materials sizes. e.g. setting thickness to 1mm (which is our default dˆ
for IPC cloth simulations), Argus produces severe artifacts; see Figure 5.8d. Again, for all
3 different thickness settings, Argus’ timing is worse than our method (details later).
In summary what we see here is that for state-of-the-art methods, they require small
enough time step size to avoid failures. With smaller thickness, the required time step size

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is probably even smaller, which makes the simulation expensive and it is unclear what time
step size will work for a specific thickness. This makes parameter setting challenging as
not only time step size needs to be tuned to ensure success, but there are also the damping
parameters to be tuned to obtain realistic dynamics.

For our method, by setting our dˆ (the distance we start exerting contact forces) to
different values, i.e. 1mm, 5mm, and 10mm here, we successfully model different
thickness for the cloth on stack example (Figure 5.9 1st row) without any artifact. For this
1s animation ARCSim and Argus both took around 260 minutes to finish under h = 0.001s,
while our method can apply h = 0.04s and only took around 100 minutes for 1mm and
5mm, and 156 minutes for 10mm, which are much faster. For dˆ = 10mm, the maximum
number of active contact constraints (not including friction) reaches 4.7M due to the large
activation distance. Although this shows that our method is robust even with a large number
of contacts, it would still be interesting to explore constraint culling strategies where
uncolliding, e.g. 2-ring neighbor contact pairs, can be appropriately ignored to gain more
efficiency in large dˆ situations.

Since our contact model exert larger forces as distance gets smaller, it naturally provides
a constitutive model between the distance (thickness) and the contact force in the normal
direction where indentation effect can also be naturally captured if we drop a elastic ball
onto our cloth stack (Figure 5.9 middle and bottom rows). In the middle row we show
the frame when the ball gets the most compression, and we see that it creates a valley on
the cloth stack and also generate wrinkles in the 10mm case. For the frame where the
scene becomes nearly static (Figure 5.9 last row), we still clearly see the different height
of the piles. This simulation also shows that IPC naturally couples elastic body and shells
together, which is further explored later in Section 5.6.9.

Another way of modeling thickness in our method is to set an offset to the constraint
definition. This is different from using different dˆ that for a nonzero offset ξ, objects are
guaranteed to be separated from each other at ξ, and contact force starts to exert when
object distance is below ξ + dˆ and goes to infinity when distance is ξ. For simulating

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 1mm 5mm 10mm

Figure 5.9: Sphere on cloth stack. Ten 8K-node square cloth are dropped onto a square
board with friction µ = 0.1 (1st row). For different IPC dˆ per column, our method is
able to provide controllable thickness modeling for cloth (the stack heights are visually
distinguishable). Then a soft elastic sphere (E = 10KP a) is dropped onto to the cloth
stack and compressed to the extreme (2nd row), where with 10mm thick cloth there are
also intricate wrinkling behaviors. After the scenes become static (3rd row), we can see the
thickness effect is still robustly maintained.

codimensional objects, these 2 mechanisms could be combined together, where larger

dˆ introduces more elastic response while a nonzero ξ provides guarantees for minimum
thickness/volume even under extreme compression (e.g. Figure 5.16). For scenes where
consistently modeling thickness matters (e.g. Figure 5.16, 5.19, 5.20, 5.21, 5.22, 5.23), we
set ξ to the true thickness value of the object or slightly smaller, and then set dˆ around ξ
depending on how much elastic response we need.

5.6.6 Cloth Simulation Benchmark

We next, in this section, confirm our method’s performance on challenging benchmark tests
from prior work.

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Cloth on rotating sphere Here we run a classic example in Bridson et al. [2002] where
after dropping a square cloth onto a sphere on the ground, the sphere starts rotating. With
friction on the sphere and ground at µ = 0.4 and between cloth at µ = 0.01, we see that
the cloth follows the sphere to rotate together with friction accurately resolved, and fine
wrinkling details are captured without any locking issue or stretchy artifacts (Figure 5.10)
thanks to our robust and accurate strain limiting model. With higher resolution (246K
nodes) and smaller bending stiffness, the wrinkles become more intricate with higher
frequency.
85K

246K

Figure 5.10: Cloth on rotating sphere. A square cloth (85K-node 1st row, 246K-node
2nd row) is dropped onto a sphere and a ground with friction (µ = 0.4 for both). Then the
sphere starts to rotate, and the cloth follows. 8 × 104 P a and 8 × 103 P a bending Young’s
modulus are used for the 85K- and 246K-node cloth respectively to produce intricate
wrinkling behaviors with different frequency. Here from left to right we show the 25th
(right before rotation), 50th, and 75th frame.

Funnel Inspired by Tang et al. [2018] and Harmon et al. [2008], we run a funnel example
where multiple square cloth are dropped onto a funnel with friction, and then pushed
through by a scripted collision object (Figure 5.11). Here we set µ = 0.4 to hold the cloth
in earlier frames and to make the problem more challenging, we use a 4-node tetrahedron,

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with sharp corners and a very different resolution, as the scripted collision object. Our
method robustly simulate this challenging scene with multiple layers of cloth and the sharp
moving obstacle without jittering nor intersection artifacts.

Figure 5.11: Funnel. Three 26K-node square cloth are dropped onto a funnel with friction
(µ = 0.4). Then a tetrahedron moving collision object pushes the cloth through the funnel.

Ribbon knot Here we look at cloth contact under large stress from a classic example in
Harmon et al. [2009]. 2 ribbons are initially placed intertwind with each other and then
stretched to form a knot (Figure 5.12). In our simulation the cloth contains 100K nodes
(10× that of the original example) and is nearly stretched to its strain limit, resulting in
extreme stress in the center and form a much smaller knot than in Harmon et al. [2009].
We also applied a µ = 0.02 friction here to realistically capture the intricate dynamics. Our
simulation has no interpenetration or jittering issue.

Garments Garment is an important application of cloth simulation where 2 main chal-

lenges lie in multilayered garment and garment simulation with moving characters. To test
the robustness of our method we first simulate the yellow dress (15K nodes) produced by
FoldSketch Li et al. [2018c] with knife pleats where cloth are folded and then placed on top
of each other and stitched, which is very challenging to simulate. We start the simulation
from flat patterns staged around the 13K-node mannequin (Figure 5.13a), and by efficiently
time stepping at h = 0.04s with the spring forces to stitch seams together, we quickly
obtain the static equilibrium of the draped garment with all knife pleats simulated with
correct layering free from any intersection (Figure 5.13b).

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 (a) (b)

(c) (d)

Figure 5.12: Ribbon knot. A classic example in ACM Harmon et al. [2009] with 2 ribbons
dragged apart to form a knot. Here our 2 ribbons contain 100K nodes in total, and we drag
them starting from rest shape (a) to nearly reach the strain limit (b-d), where the friction
coefficient is µ = 0.02. Our results have no interpenetration or jittering artifacts.

Then we create a multilayer skirt (30K nodes) using Sensitive Couture Umetani et al.
[2011], and obtain a realistic drape in the same way (Figure 5.13c). Next we will use this
looser garment to simulate with a character animation sequence that has large movement.
To setup the simulation we download a skeleton sequence from Adobe Mixamo3 , generate
skinning animation with our mannequin, and then use the animated mannequin sequence
as kinetic object to simulate with garment. In Figure 5.13(d,e,f) we show 3 frames of the
simulation featuring the kicking motion. With these large movement, our method stays
stable and efficient while capturing intricate garment details.

The multilayer garment simulation with the input character animation is efficiently
simulated at 2min per h = 0.04s time step, where it would be interesting to explore
whether replacing the stiff stitch springs with change of variable strategy will further
accelerate our convergence.

3
https://www.mixamo.com/

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 (a) (b) (c)

(d) (e) (f)

Figure 5.13: Garments. A yellow dress with knife pleats produced by FoldSketch Li et al.
[2018c] is staged (a) and then draped on a mannequin to static equilibrium (b) with our
method that stitches the seams together. A multilayer skirt is draped in a same way (c),
and then used to simulate with an input character animation sequence with large motion
(d,e,f). As the mannequin kicking with her leg, intricate garment details are captured by
our method.

5.6.7 Cloth Simulation Stress test

Now we perform stress tests on our method with challenging examples that require a
high level of robustness and accuracy. To our knowledge we are the first to achieve these
simulations without any artifacts and algorithmic parameter tuning.

Pulling cloth on needles When simulating cloth with sharp contacts, snagging artifact
is a notorious issue for existing methods. Now we show that our method is snagging-free
with sharp contact. We form a needle bed directly using codimensional segments and drop

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a square cloth onto it (Figure 5.14). With a large time step size at h = 0.02s, we see that
our cloth is safely becoming static on the bed of needles without any jittering. Then we pull
the left side of the cloth with 1m/s speed moving Dirichlet boundary condition, and we
see that the cloth can be pulled through on top of the bed of needles without snagging. In
this example as we have extreme stretching with sharp contact, our adaptive strain limiting
stiffness adjustment strategy is effective to make sure the simulation stays numerically
feasible and efficient.

(a) (b)

(c) (d)

Figure 5.14: Pulling cloth on needles. A 26K-node square cloth is first dropped onto a
bed of needles formed directly by codimensional segments (a) and then pulled to the left at
1m/s without snagging issue (b, c, d). Here from (a) to (d) we show the 50th (right before
pulling), 55th, 65th, and 85th frame.

Table cloth pull Pulling a table cloth out from under a set of objects with the objects
staying on the table while not following the cloth is a classic magic trick4 . To simulate
it, requires an accurate friction model and a method that can robustly handle the large
stresses and normal forces from pulling, as well as the contact with sharp edges. Here, we
successfully simulate this trick with both heavy and stiff objects on table by pulling the
cloth at high (4m/s) speed; Figure 5.15d. Similarly, we correctly capture the trick failing
if we pull at too small a speed where, due to stiction, some objects follow the table cloth
4
https://www.juggle.org/table-cloth-pull-trick/

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pull and so fall off; and finnally confirm that with an even slower pull even more objects
fall; see Figure 5.15b and c. Here IPC naturally resolves stiff elastic bodies with cloth
contact by coupling all directly in an accuarte solve. In Section 5.6.9 we demonstrate more
coupling simulations. This is also a nice example that demonstrates the robustness of our
method on handling very different tessellations and stiffnesses (the cube and tetrahedron
only have 8 and 4 nodes), and extreme boundary conditions that pulls the cloth at a high
speed under large friction forces given by the heavy objects.

(a) before pulling (b) pulling at 1m/s (c) pulling at 2m/s (d) pulling at 4m/s

Figure 5.15: Table cloth pull A 26K-node square cloth is put on a table board with a cube,
a tetrahedron, a cylinder and a torus solid (ρ = 2000kg/m3 , E = 0.1GP a) on it (a). The
capability of our method on accurately resolving controllable friction behaviors is shown
here by pulling the cloth leftwards with different speed (b-c). In (b) the pull is at 1m/s and
nearly all objects follow the cloth due to static friction. As the pull gets faster at 2m/s in
(c), only the cube is falling and the other objects stay on the table after some slight move.
Finally in (d) pulling the cloth at 4m/s left all objects on the table because of the small
duration of the action of dynamic friction. In (c) and (d) there are also elastic waves on the
cloth right after it is detached from the objects and board.

Cloth cylinder twist Now we test cloth simulation under extreme stress. We made a
1m-wide cylinder cloth with 88K nodes, and twist and move the two sides together at
72◦ /s and 5mm/s respectively without gravity and strain limiting (Figure 5.16). Right
after twisting we get interesting folds patterns (Figure 5.16b), but due to the extreme stress
and the codimensional shape representation the center part quickly becomes a tiny knot.
Therefore, we add an offset of 1.5mm to the IPC to model an inelastic thickness for the
cloth, and then the same frame looks more realistic with thickness supporting the structure

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(Figure 5.16c). This is a nice example demonstrating the need for thickness modeling when
simulating with codimensional shape representations, which will be further analyzed in
Section 5.6.5. After 32.96s twisting, our cylinder cloth experience lots of buckling (Figure
5.16e) and our method stays robust with thickness behaviors consistentely modeled.

(b) (c) no offset

(a) rest
shape
(d) (e)

Figure 5.16: Twisting cylinder. A 1m-wide 88K-node cylinder cloth is twisted and moved
closer at 72◦ /s and 5mm/s respectively at both ends from rest shape without gravity (a).
Right after twisting, intricate folds start to form (b). Here strain limiting is turned off to
test simulation under extreme stress, which however makes the center part a tiny knot (c).
By setting a 1.5mm IPC offset to model some inelastic thickness, the cloth looks more
realistic at the same frame (d). After 32.96s twisting with the offset setting, we get lots of
buckling behaviors (e) and our method stays robust with thickness behaviors consistentely
modeled.

5.6.8 CCD Benchmark

Up to this point all our examples can be nicely simulated with the conservative CCD
strategy in IPC Li et al. [2020b]. But when we start exploring more complex examples in
the next Section 5.6.9 with much more degenerate contact pairs and also higher demand on
CCD accuracy for inelastic thickness (constraint offset), we found that the CCD can simply
return 0 toi, completely stalling our optimization incorrectly. Multiple alternative CCD
methods also turned out failing. This triggered our innovation of ACCD, which stays robust

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and accurate even for our complex examples where all alternative methods fail. Then by
running all our examples in this chapter using ACCD we found that for easier cases ACCD
still works nicely and can provide comparable or even faster timing performance. Here we
provide some of our comparison statistics in Figure 5.17.
ACCD
EV’s MinSep Root Parity BSC T-BSC
(ours)
Sprinkles Tiny toi Tiny toi N/A N/A N/A 72.8
Noodles Tiny toi Tiny toi N/A N/A N/A 98
Aleka Tiny toi Tiny toi N/A N/A N/A 855.5
Braids 1012.8 Tiny toi N/A N/A N/A 965.4
Card Shuffle Tiny toi Tiny toi N/A N/A N/A 165.2
Runtime Infinite
Needle bed 55.6 Tiny toi Intersection 43.4
error loop
Runtime
Garment 262.3 Tiny toi Intersection Tiny toi 229.6
error
Cloth stack Runtime Infinite
498.3 Tiny toi 448.5 441.5
(1mm) error loop
Cloth stack Runtime Infinite
236.8 Tiny toi 191.9 189.6
(5mm) error loop
Cloth stack Runtime Infinite
135 Tiny toi 124.4 123.3
(10mm) error loop

Figure 5.17: CCD comparison statistics. We provide the per Newton iteration total CCD
querying time (excluding spatial hash) in milliseconds for different simulation examples
when we use different CCD methods (EV’s Li et al. [2020b], MinSep Harmon et al. [2011],
Lu et al. [2019], Root Parity Brochu et al. [2012], BSC Tang et al. [2014], T-BSC Wang
et al. [2015], and our ACCD). The first 5 examples are with inelastic thickness, which is
not supported by Root Parity, BSC, and T-BSC methods. ”Infinite Loop” means the method
never reports ”no collision” even when querying a configuration with tiny displacement
that is interpenetration-free.

We tested minimal separation (MinSep) CCD Harmon et al. [2011], Lu et al. [2019]
by wrapping it with the same conservative CCD strategy in IPC Li et al. [2020b] that the
minimal separation is set to sdcur where s is the slackness and dcur is the current distance,
and if the returned toi is smaller than 10−6 the query will be performed again without any

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slackness (s = 0) to make it easier. For Root Parity Brochu et al. [2012], BSC Tang et al.
[2014], and T-BSC Wang et al. [2015] that only return ”has collision” or ”no collision”
without computing toi, we apply a monotonic bisection for each query until finding a step
size without collision and return it after multiplying by 1 − s. Since the bisection still
cannot directly extend the methods to compute the time or step size that first brings the
distance of a contact pair to ξ even if we combine it with distance computation, we do
not test these 3 methods on our examples with inelastic thickness (the first 5 examples in
Figure 5.17).
For IPC’s conservative strategy with EV’s CCD, except for the complex scenes, it
works well with slightly slower timing than our ACCD. One interesting thing to note is that
EV’s CCD handles degenerate cases like parallel edge-edge via querying extra point-edge
and point-point pairs, while our robust ACCD only needs to query the general pairs (e.g.
point-triangle and edge-edge pairs for surface only scenes). On the other hand, MinSep
always return tiny values, sometimes even 0, for all the listed examples. For examples
without inelastic thickness (constraint offset), Root Parity performs well with nice efficiency
but there are missed collisions leading to interpenetration in the needle bed and garment
example; BSC always report runtime error with an internal message ”Inflection points are
not handled exactly in BSC”; and T-BSC returns tiny value for the garment example, and
trapped in the loop for finding a step size without collision in all the other four examples.

5.6.9 General Shells, Rods, Particles, and Coupling

In this section we demonstrate IPC simulation across all codomains. IPC simulation is thus
extended to includes the full spectrum from volumetric materials to codimensional shells,
rods and even, if desired, particles. Because these domains then interact via IPC barriers
we also are able to seamlessly and directly couple all such mesh-based models in the
same common simulation. Here we then demonstrate the application of this coupling and
further highlight the advantages and suitability of our consistent and controllable thickness
modeling for this range of thin material interactions.

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Card shuffle First, let’s looks at stiff shells by modeling playing cards. Traditional card
simulation examples mainly focus on accurately resolving the friction. Here we not only
model friction, but also show that we can accurately capture the bending and thickness
effects of the card by simulating card shuffling. We uniformly separate 54 poker cards into
2 piles and apply Dirichlet boundary condition on their sides to bend them in preparation for
shuffling (Figure 5.18a). Then we release the Dirichlet boundary conditions from bottom
up card-by-card alternatingly from the left and right pile (Figure 5.18b). The released
cards quickly retain their rest shape, falling onto the ground, and form another shuffled
pile without intersection throughout. After all cards are released we use 2 handles to make
the pile align better (Figure 5.18c). By setting the IPC offset and dˆ to 0.1mm and 0.2mm
respectively to accurately resolving the thickness of every card, we ensure that the height
of the final pile is visually the same with reality (Figure 5.18d).

(a) (b)

(c) (d)

Figure 5.18: Cards shuffling. 54 poker cards are uniformly separated into 2 piles initially
and bent (a), and then one-by-one released in turns from bottom to up to get shuffled (b).
After all cards are released 2 handles are squashed to make the pile align better (c), and
finally we obtain a nicely aligned shuffled cards pile.

Hairs Next we consider rod models. Here we test our method on two hair simulation
examples. First, we show twisting of two hair clusters that forms braids and then the bottom
handle is released so that the braids untie (Figure 5.19a-e). Then we test a typical hair
simulation example in McAdams et al. [2009] where one end of a hair cluster is fixed, and

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the other end falls onto another hair cluster with two ends fixed. Here we also added a
sphere below the falling hair cluster to make the dynamics more interesting. Our method
robustly simulate both scenes without any intersection at 2 to 3 minutes per h = 0.01s time
step, and accurately capture the intricate hair dynamics. Although hair thickness is nearly
negligible, we still set an offset of 0.08mm (the thickness of a hair strand) and dˆ = 0.1mm
for IPC to make sure that thickness behaviors are always consistently captured even with
large stress as in the braids example.

(f)

(g)

(a) (b) (c) (d) (e) (h) (i) (j) (k)

Figure 5.19: Hairs. Left: two hair clusters (each with 650 60-segment rods) are twisted to
form braids and then the bottom end is released. Right: a typical hair simulation example
in McAdams [2009] where a hair cluster with one end fixed is falling onto another hair
cluster with both ends fixed (900 50-segment rods for each cluster). A sphere collision
object is also added below to add more interesting dynamics.

Noodles Beyond hairs which are extremely thin, our contact-driven thickness modeling
naturally allows us to also capture the contacting geometries of even thicker materials
directly with discrete rods. Here we first show a simulation of noodles falling into a bowl
(Figure 5.20). We setup the scene by arranging 625 vertically placed 40cm-long noodles
(each represented with a 200-segment discrete rod) into a xz-plane matrix separating each
other by 3.3mm and place them on top of a 13.85cm-wide fixed bowl (Figure 5.20a). We
set IPC’s offset and dˆ to 1mm and 0.5mm respectively to model the noodles’ thickness
at 1.5mm, which let us reconstruct the surface mesh with 1.5mm-thick cylinders along

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the rods for rendering. After dropping the noodles, they quickly fill up the bowl (Figure
5.20bc) thanks to our controllable and high-quality modeling of thickness. Our method
robustly simulate this scene until static at around 23min per h = 0.02s time step without
any jittering artifacts. If we do the calculation, the total volume of the noodles is about
4.42 × 10−4 m3 , which is right below the hemisphere bowl’s volume 6.96 × 10−4 m3 . These
numbers further verified that our static equilibrium is consistent with reality.

(a) initial (b) static frame (c) static frame (lower view, bowl hidden)

Figure 5.20: Noodles. 625 40cm-long noodles (each represented with a 200-segment
discrete rod, separating each other by 3.3mm initially) are dropped into a 13.85cm-wide
bowl (fixed). Here we set IPC’s offset and dˆ to 1mm and 0.5mm respectively to model the
noodles’ thickness at 1.5mm, which let us reconstruct the surface mesh with 1.5mm-thick
cylinders along the rods for rendering. The controllable and high-quality modeling of
thickness here is also essential for the noodles to pile up and fill the bowl. Our method
robustly simulate this scene until static at around 23min per h = 0.02s time step without
any jittering artifacts.

Sprinkles Now let’s do something creative by representing each sprinkle using a single
ˆ Here we place 25 × 25 × 40
segment with its thickness modeled by IPC offset and d.
6mm-long and 2mm-thick sprinkle separating each other by 3.3mm above a 13.8cm-wide
fixed bowl (Figure 5.21a). We set IPC’s offset and dˆ to 1.5mm and 0.5mm respectively to
model the sprinkle’s thickness, which let us reconstruct the surface mesh with 2mm-thick
cylinders along the rods for rendering. Since sprinkle often does not deform, here we
make offset 3× large as dˆ so that thickness behaviors are more inelastic. Then we drop

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the sprinkles, and they quickly fill up the bowl (Figure 5.21bc). Our method robustly
simulate this scene until static at around 6.5min per h = 0.02s time step without any
jittering artifacts. If we calculate the total volume of the sprinkles by summing up each
capsule volume 25000(lπr2 + 34 πr3 ) ≈ 5.76 × 10−4 m3 , and calculate the hemisphere
bowl’s volume 23 πrb3 = 6.88 × 10−4 m3 , the close match of these numbers again nicely
verified our simulation results.

(a) initial (b) static frame (c) static frame (lower view, bowl hidden)

Figure 5.21: Sprinkles. 25K 6mm-long and 2mm-thick sprinkles (each represented with
a single-segment discrete rod, separating each other by 3.3mm initially) are dropped into a
13.8cm-wide bowl (fixed). Here we set IPC’s offset and dˆ to 1.5mm and 0.5mm respectively
to model the sprinkle’s thickness, which let us reconstruct the surface mesh with 2mm-thick
cylinders along the rods for rendering. Our method robustly simulate this scene until static
at around 6.5min per h = 0.02s time step without any jittering artifacts.

Sand on cloth Here we show that our method can even handle contacts between single
particles, which has codimension 3. We arrange single particles into a 10 × 10 × 500 cuboid
separating each other by 4mm and let the high sand column fall onto a cloth with its four
corners fixed (Figure 5.22). We compute the mass of each sand using ρ = 1600kg/m3 and
V = 4/3πr3 ≈ 4.2 × 10−9 m3 . With IPC offset and dˆ set to 2mm and 1mm respectively,
the volume of each sand is consistently modeled during contact, which is essential to
connecting every sand together through interaction and let them move as a flow. Here we
rendered each sand as a 2mm-wide sphere. This simulation highly resembles the Discrete
Element Method (DEM) Cundall and Strack [1979], Yue et al. [2018] where near-rigid

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bodies, or particles, are treated as perfectly rigid, and an interaction zone (analogously
ξ < d < ξ + dˆ in our situation) is coated outside each of them for contact forces to be
calculated based on the depth, area, or volume of the intersections among these zones
Jiang et al. [2020]. DEM is popular in geomechanics communities since its reasonable
simplification ignores the deformation for the particles while accurately simulates the
momentum and contact forces through contact laws. What is different here for our method
is that we relate our contact forces directly to distances with a barrier function, which
guarantees interpenetration-free while for DEM the geometric contact accuracy would need
to be controled by carefully choosing time step size and contact model stiffness.

(b) (c)

(a) initial

(d) (e)

Figure 5.22: Sand on cloth. A 10 × 10 × 500 sand column is falling onto a square cloth
with its four corners fixed. Here each sand is represented with a single isolated FEM
node, with IPC offset and dˆ set to 2mm and 1mm respectively, the sand-sand contacts are
accurately captured which is essential for the volumetric effects in this scene.

All in Finally, we show a scene that couples objects of every codimension together. Here
an Armadillo volume, a square cloth, and a sand column formed by isolated particles are
falling onto a rod net (Figure 5.23a), where we release them in sequence at 0s, 2s, and
4s. The rod net is constructed by alternating the y-coordinates of the rods in x direction
up and down by 1mm, so that adjacent x-direction rods are on the different sides of the
y-direction rods. We can see that this net structure provides interesting interactions with

164
the Armadillo (Figure 5.23b). Then after the cloth falls and covers the Armadillo on the net,
we see cloth-rod contacts are also accurately captured (Figure 5.23c). When sand colume
hit the cloth, the cloth slightly deforms and the sand particles flow into the 2 valleys of
cloth supported by the Armadillo and the rod net (Figure 5.23d,e). Here IPC offset and dˆ
are 0.5mm and 1mm everywhere.

(b) (c)

(a) initial
(d) (e)

Figure 5.23: All-in. We simulate objects of all codimensions coupled together in a scene
with an Armadillo volume, a square cloth, and a sand column formed by isolated particles
released in sequence to fall onto a rod net.

5.7 Summary

In summary, this chapter extends the incremental potential contact (IPC) framework from
volumetric deformables Li et al. [2020b] to thickness and strain limit-aware modeling
of codimensional/mixed-dimensional structures. Independent from the elasticity model
or the time step size, the resulting framework guarantees out-of-the-box intersection-free
output with a strict satisfaction of strain limits (down to 0.1%) and an accurate capture of
geometrically meaningful thickness. Furthermore, the proposed additive CCD (ACCD)
approach shows extremely stable behaviours for challenging first-time-of-impact tasks. It
can also be used as an extremely easy-to-implement and efficient alternative CCD algorithm

165
for other graphics applications.

166
Chapter 6

Conclusion and Future Work

6.1 Conclusion

In this dissertaion, we at the first time proposed a robust and accurate self-contact handling
framework with friction and fully implicit nonlinear elastodynamics that can guarantee
interpenetration-free trajectories, supporting a full spectrum of codimension-0,1,2,3 objects
modeled with geometric thickness and fully coupled exact strain limiting. Specifically, this
includes

• a decomposed optimization time integrator for large-step elastodynamics,

• a rigorous formulation on a globally consistent contact constraint definition between

triangulated surfaces,

• a scalable barrier method for inequality constrained optimization,

• a mollified friction model with controllable accuracy and consistent transition be-
tween static and dynamic modes,

• a semi-implicit friction discretization with a potential energy that can be seamlessly

incorporated into optimization time integration framework,

167
 • a constitutive strain limiting method that guarantees constraint satsifaction and can
be solved simultaneously with accurate contact and elastodynamics,

• a consistent thickness model for simulation with codimensional shape representation,

and

• a simple but robust and efficient additive continuous collision detection approach
that outperforms all existing alternatives.

Based on solid theoretical foundations, our methods provide controllable trade-off between
efficiency and accuracy for different application scenarios. We also performed extensive
experiments and analyses to thoroughly test and verify all the proposed features. Our
framework frees designers and engineers from extensive algorithmic parameter tuning as
when traditional methods are used, enables brand new phenomena to be easily simulated,
and can be applied in a wide range of applications from other communities like mechanical
engineering, robotics design, etc.

6.2 Future Work

We consider four main aspects that futher extend our work and spread its impact into more
research and industry communities.

More Physical Models With our explorations in elastoplasticity and fracture simulation
Fang et al. [2019], Wolper et al. [2019, 2020], and solid-fluid coupling Fang et al. [2020],
it would be meaningful to incorporate all these different physics into our IPC based
optimization time integration framework. Thus multiphsics phenomena like a bullet fired
towards a glass window, origami structrue design for quadroptor wings, a robot swims
under water, etc, could also be accurately simulated with robostness and predictiveness
to further benefits more complex application scenarios. The challenges would lie in how
to couple fracture dynamics, fluid dynamics, and plasticity, into our optimization time

168
integration framework, where most of these effects have their own evolving rules and do not
have a well-defined potential form. The idea of our semi-implicit temporal discretization
of friction is certainly promising to be considered, and a recently advanced Hyperplasticity
theory Houlsby and Puzrin [2007] is also of interest.

Easier Calibration to Reality In Li et al. [2020d] we found that matching simulation

results to real experiments requires several iterations of calibration on the physical param-
eters of the system. Although we matched every physical setting of the quadroptor that
can be refered to like dimensions and material stiffness, the damping coefficient of the
material is still hard to match. This issue will be magnified for even more complex systems
where there will be more physical factors that are hard to quantify. Given that our methods
are differentiable, it would be great if a physical parameter calibration system could be
designed to combine computer vision techniques to directly extract information from video
footage of real experiments, and then apply some optimization or searching algorithm to
automatically figure out the best physical parameters that matches reality. This will be very
meaningful for simulation-in-the-loop material or robotics design applications.

Better Efficiency Then it would be critical to improve the timing of our methods. Cur-
rently we still require several minutes per frame to obtain high quality results with 50K to
100K DoFs. Although this timing matches or is even faster than existing accuracy targetted
methods, and it is acceptable for engineering and robotics applications, it still needs much
faster speed to scale up to even higher resolutions, and to be applied in a wider range of
scenarios like gaming and virtual reality (VR) where real-time or interactive-rate responses
are required. Therefore, it would be meaningful to explore implementation improvements
on advanced computing architectures like what we explored on GPU for MPM Wang et al.
[2020b]. Devising new mathematical or numerical approaches like higher-order basis with
a smaller number of DoFs, and hierarchical optimization schemes Wang et al. [2020a]
would also be helpful. In addition, applying machine learning techniques to accelerate
certain steps of the simulation by assuming a specific application target would also be

169
interesting.

Differentiable Simulation Framework Finally, it would be impactful to design a differ-

entiable simulation framework based on our optimization time integrators. Although our
methods are differentiable by construction, computing sensitivity information in an efficient
and robust way is still challenging especially for large scale dynamic problems. In addition,
traditional simulation-in-the-loop design optimizations often only rely on 1st-order sensi-
tivity information, which can make the convergence slow and would require algorithmic
parameter tuning to obtain reasonable results. It would be interesting to explore 2nd-order
optimization methods for design optimizations with consistent and scalable approximation
to the 2nd-order sensitivity information. Providing this differentiable simulation framework
that are directly applicable to various design tasks would greatly benefit many research and
industry fields.

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Appendix A: Smoothing

Let f (x) be a function we wish to smooth. It is C 1 continuous everywhere except at x = a

where it is only C 0 continuous. Applying a function g(x) that is C 1 continuous everywhere
with g(a) = 0, we have a smoothed function f (x)g(x) that is C 1 continuous everywhere.
For x 6= a we have (f (x)g(x))0 = f 0 (x)g(x) + f (x)g 0 (x) is C 0 continuous everywhere.
At x = a, the left and right derivatives of f (x)g(x) are then

lim (f (x)g(x))0 = lim− f 0 (x)g(a) + f (a)g 0 (a),

x→a− x→a
(1)
0 0 0
lim (f (x)g(x)) = lim+ f (x)g(a) + f (a)g (a).
x→a+ x→a

As f (x) is C 0 continuous at x = a, limx→a− f 0 (x) and limx→a+ f 0 (x) are both bounded.
Then with g(a) = 0 we then have left and right derivatives of f (x)g(x) both equal f (a)g 0 (a)
at x = a,
lim (f (x)g(x))0 = lim+ (f (x)g(x))0 = f (a)g 0 (a) (2)
x→a− x→a

Thus (f (x)g(x))0 is likewise C 0 continuous at x = a and f (x)g(x) is correspondingly C 1

continuous.

Appendix B: Barrier Continuity and Testing

ˆ as
The continuity of our C 2 barrier function is confirmed when d < d, ∂b ˆ
= (d−d)(2 ln ddˆ −
∂d
dˆ ∂2b ˆ
d
+ 1) and ∂d2
= −2 ln ddˆ + (dˆ − d) d+3d
d2
ˆ Thus the left and right
both vanish as d → d.
derivatives of b at d = dˆ are both equal at zero up to 2nd order.
Our motivation for applying a C 2 clamped barrier rather than a less nonlinear, but still
smooth, C 1 barrier is to provide 2nd-order derivatives suitable for our Newton-type solver.
Thus it is genralluy better to have a continuous Hessian for improved convergence Nocedal
and Wright [2006]. Nevertheless, here we also provide an ablation study applying all both
ˆ and C 1 (b = (d − d)
C 0 (b = − ln(d/d)) ˆ ln(d/d)),
ˆ along with our final choice of our C 2
ˆ 2 ln(d/d))
(b = −(d − d) ˆ barrier in IPC on a set of examples in Figure 1.

171
 Barrier ablation

C0 C1 C2 (IPC)
Examples
# iters, t (s) / time # iters, t (s) / time # iters, t (s) / time
step step step
mat on knives 5.51, 2.16 5.59, 1.43 5.47, 1.40

2 mats40x40 fall NC 26.83, 10.64 26.17, 10.48

octocat on knives NC 9.29, 4.21 7.73, 3.76

sphere1K roller NC 47.64, 9.41 45.22, 9.02

mat40x40 twist (10s) 7.74, 1.93 8.04, 2.07 7.56, 1.89

sphere1K pin-cushion 12.82, 1.27 9.22, 0.75 8.93, 0.69

rods630 twist (10s) 6.65, 1.05 3.05, 0.38 3.07, 0.40

Figure 1: Ablation study on barrier functions with different continuity. NC means not
converging after 10000 PN iterations when solving a certain time step.

From the results we see that for the C 0 barrier, optimization can be non-convergent.
Here this can be a result if the local minima is right inside the clamped region of the barrier,
where the gradient does not change smoothly – here intermediate values may not be found
to balance terms so decrease the total gradient. While, in comparison to C 1 , our C 2 barrier
is generally 5%–10% faster due to the continuity of the Hessian – this is reflected in less
iteration counts.

Appendix C: CFL-inspired Culling of CCD

As IPC requires performing CCD for every Newton iteration, CCD clearly becomes a
bottleneck. Therefore we propose a novel CFL-inspired culling strategy to accelerate CCD.
Recall that our culled constraint set contains all surface primitive pairs with distances
ˆ Thus all remaining primitives outside the culled constraint set have farther
smaller than d.

172
 ˆ We place all vertices participating in these primitives in a
distances lower-bounded by d.
set F . At each iteration i with a search direction pi , we find the index ` of the simulation
node with the largest component in p (i.e., its effective search-direction “velocity”) among
all remaining node pairs, ` = argmaxκ∈F ||pik ||. We then compute a conservative bound on
the largest-feasible step size for surface primitives not in Cˆ as
dˆ
αF = . (3)
2||pi` ||
We then apply conservative CCD (see below) to the remaining primitive pairs in Cˆ(xi ),
obtaining a large feasible step size, αCˆ, for that set. Then α0 = min(αF , αCˆ) is our
maximum feasible step size for iteration i. We then apply α0 as starting step size to
bootstrap backtracking for Newton iteration i and so ensure decrease with feasibility.
Computing large, feasible step sizes in this way effectively reduces CCD cost by
two-orders of magnitude. However, for scenes that contain high speed motions, αF can be
overly conservative (smaller than needed) which then would increase iteration counts and
so cause energy related computations to increase overall cost unnecessarily.
Thus we adapt by balancing between applying full CCD for all candidate pairs provided
by spatial hash and applying our CFL-like strategy. Here we will designate the exact
feasible bound computed from applying CCD to all primitive pairs in the spatial hash as
αS .
In each step we first compute αF and αCˆ – they are both efficient and inexpensive to
find. Next we observe that the culled bound αCˆ is often very close to the exact bound
αS , while computing this exact bound is generally one-third of the total timing cost in a
single iteration. We thus proceed by computing αS if αF < 21 αCˆ. Otherwise we apply our
CFL-type bound and apply α0 = min(αF , αCˆ) .
We thus avoid overly restrictive step sizes when αF and αCˆ are already quite close –
meaning their minimum should also be quite close to αS . In practice we observe that our
CFL-like assisted CCD culling strategy provides a 50% speed-up for all CCD related costs
with nearly the same iteration counts. Ovxerall this results in an average 10% speedup for
IPC; see Figure 2.

173
 CCD ablation

full CCD CFL combined (IPC) full CCD CFL combined (IPC)
Examples
# iters, t (s) / time step # iters, t (s) / time step CCD related Timing (s) CCD related Timing (s)

mat on knives 5.34, 1.66 5.47, 1.40 97.50 47.25

2 mats40x40 fall 23.22, 10.21 26.17, 10.48 273.38 162.10

octocat on knives 6.99, 3.92 7.73, 3.76 206.61 129.93

sphere1K roller 49.38, 9.89 45.22, 9.02 482.13 394.01

mat40x40 twist (10s) 7.37, 2.13 7.56, 1.89 147.60 62.91

sphere1K pin-cushion 8.35, 0.77 8.93, 0.69 37.25 19.61

rods630 twist (10s) 3.04, 0.47 3.07, 0.40 23.89 9.13

Figure 2: CCD strategies ablation.

Appendix D: Conservative CCD

CCD is generally applied to compute a time of impact corresponding to a step size that
would bring distances between primitives to 0. In our barrier setting this ”largest feasible
step size” needs to be made conservative by backing away from an exact zero distance. A
simple strategy would be a conservative rescaling with a factor c ∈ (0, 1); e.g., by starting
the line search at 0.5 or 0.9 of the total step along the descent direction. However, for the
CCD computations rounding error can be severe for the tiny contacting distances we allow
and so even small naive scaling factors (e.g., 0.1) can allow unacceptable intersections
in such cases while in others is a much too conservative bound unnecessarily slowing
convergence.
Rather than directly finding and then conservatively rescaling a CCD-computed step
length that takes us to intersection we directly compute via CCD a step size along pi
that will bring primitives to a distance of (1 − c)dc > 0. Here dc is the current distance
between the primitives. Standard CCD libraries 1 directly provide this option, e.g., exposed
as an η parameter. In turn this modifies coefficients of the polynomial equations solved
1
we use https://github.com/evouga/collisiondetection

174
during CCD, and effectively reduces numerical rounding errors that cause issues for direct
scaling. In our implementation we apply c = 0.8 between mesh primitives, c = 0.9 for
mesh-to-plane, and similarly c = 0.8 for computing the large feasible step size to avoid
element inversion when barrier elasticity energies, e.g., neo-Hookean, are applied.
Finally, to ensure we remain intersection- and inversion- free at each step. We apply
a post-step check whenever nodal positions are displaced (e.g. line search and initial
movements of boundary conditions and collision objects. These include the edge-triangle
intersection check filtered by our spatial hash and a volume check for every tetrahedral
element. In exceedingly rare cases when an edge-triangle intersection or negative volume
are detected, we half the final step size bound.

Appendix E: Equality Constraints for Moving Collision Ob-

jects and Time-Varying Boundary Conditions

In many scenarios, e.g., scripting animations, kinematic objects, and engineering tasks,
scripted kinematic collision objects (CO) and/or moving positional (Dirichlet-like) bound-
ary conditions (BC) are required. Contact algorithms generally handle these functionalities
by either directly prescribing and updating nodal positions of CO/BC nodes at start of each
time step, or interpolating them in substeps across a step. Remainder of simulation DOF
are then solved w.r.t. the prescribed nodes being fixed at “current” positions. However,
such strategies are extremely limiting, with simulations generally restricted to small time
step sizes, speeds, and/or deformations as the BO and/or CO become faster and more chal-
lenging. For example, directly prescribing CO or BC nodes can often generate tunneling
artifacts, and for moving BC, simulations can often fail simply by inverting elements when
barrier energies like neo-Hookean energy are applied. To address these issues we formulate
scripted dynamic BC and CO as equality constraints in IPC. This simultaneously ensures
that intersections (and inversions) are avoided even while scripted motions are applied at
large time step.

175
 We start by computing the prescribed nodal positions at the beginning of each time
step, and then apply CCD and element inversion detection to find a large feasible step size
towards the prescribed position from the current one (see above).
If it is safe to apply them fully without causing intersection or inversion, we simply
update prescribed nodal positions, solve the remainder of simulation DOF and are done.
However, if our feasible step size is smaller than taking a full step (we use a criterion of
0.999), we first move the prescribed nodes as far as we can conservatively (see Section 6.2)
and then add new equality constraints for each of these prescribed nodes provided by either
CO or BC scripting.
As in our treatment of intersection-free inequality constraints, we build an unconstrained
form for each by applying an augmented Lagrangian. For every such prescribed node, we
add a Lagrangian and a penalty potential. We initialize each Lagrange multiplier to 0 and
each penalty stiffness to 106 . Concretely, we add an energy

√ κA
− mk λTA,k (xk − x̂k ) + mk ||xk − x̂k ||2 (4)
2

to our barrier-form incremental potential for each prescribed node k with corresponding
destination x̂k in the current time step, if any of the prescribed nodes could not reach its
destination during our start-of-time-step test.
We measure satisfaction of BC/CO node constraints at each Newton iteration i by
calculating their total in-time-step progress as
sP
||x̂t+1
k − xik ||2
ηA = 1 − Pk t+1 , (5)
k ||x̂k − xtk ||2

where x̂t+1 is the prescribed BC/CO positions for time step t + 1. Then, whenever a current
iterate is both close to satisfying stationarity, via the stopping criteria, and progress, with
ηA < 0.999, we either increase the BC/CO penalty stiffness or else update the Lagrange
multipliers, via the first-order update rule, see Algorithm 1 below. Finally, whenever
ηA ≥ 0.999, we fix all prescribed nodes at current position and solve for remaining DOF in
order to avoid unnecessary slow down of convergence due to added stiffness.

176
Algorithm 5 Augmented Lagrangian Update Rule
1: for each PN iteration i do

2: ...
3: if h1 ||pi ||∞ < max(10−2 l, d ) and ηAL < 0.999 then
4: if ηAL < 0.99 and κAL < 108 then
5: κAL ← 2κAL
6: else
7: for each constrained node k do
√
8: λAL,k ← λAL,k − κAL mk (xik − x̂t+1
k )

For codimensional surface and segment collision objects their nodal mass is computed
by estimating their nodal volume as the half sphere with diameter being the average length
of the incident edges. For point collision objects we set their mass to be the average nodal
mass of the simulated objects. Alternatively, simply setting all codimensional nodal masses
to be the average of the simulated objects should also be fine. These estimated masses are
only used for moving codimensional collision objects and they do not affect any physical
accuracy.

Appendix F: Adaptive Barrier Stiffness

Stiffness, and so difficulty in solution of the barrier comes from two sources: dˆ and likewise
ˆ this frees κ to adaptively condition
κ. As we can improve accuracy by directly decreasing d,
our solves for improved convergence.
A feature of our barrier framework is that the estimation of Lagrange multipliers are
efficiently self-adjusted by the constraint values, in our case, the dk ’s. However, if κ is
not set appropriately, the contact primitives would either need to get extremely close to
get enough repulsion (barrier gradient) when κ is too small, or need to get a distance right
below dˆ for a small repulsion if κ is set too large, both resulting in slow convergence
because of ill-conditioning and nonsmoothness. Thus adaptively setting and/or adjusting κ

177
is essential.
Intuitively, the smaller dk is, the better the complimentarity condition is satisfied. This
is certainly true from optimization theory, but is troublesome in numerical computation.
In numerical optimization, since double precision floating point number is used, when dk
gets tiny, the rounding errors will significantly slow down convergence and even result in
incorrect intermediate computations. Therefore, we must also prevent dk from being too
tiny.
If we view dˆ as a control on the upper bound acting distance of our contact forces,
κ can be seen as an indirect control on the lower bound of the acting distance as larger
κ can provide the same amount of “repulsion” at a larger distance, avoiding the need to
push distances to become too tiny. Tiny distances not only make CCD less robust and
make the optimization less efficient in our numerical simulation, but also are not physically
reasonable in science. Generally speaking, the space between the nucleus of two bonded
atoms is around 10−10 meters. There is no way for two macroscopic touching objects to
get to that close. On the other hand, recall that the energy gradient of our optimization is

X ∂b
ˆ = ∇E(x) + κ
g(x, κ, d) ˆ
∇dk (x, d), (6)
k
∂d k

where at subproblem convergence, the IP gradient ∇E(x) balances with the barrier gradient
P ∂b
κ k ∂d ˆ In addition, the barrier stiffness κ also influences the condition of
∇dk (x, d).
k

the energy Hessian. Thus we adapt barrier stiffness strategy based on balancing the two
gradients, iteratively increasing κ when needed, and applying lower and upper bounds
obtained from conditioning analysis on the Hessian. Here dk can then avoid being too
small or too close to dˆ to provide improved convergence regardless of d,
ˆ material, h, and

our other input settings change.

The idea of balancing gradients can be traced back in optimization literature No-
cedal and Wright [2006] for estimating appropriate initial stiffnesses of barriers. In
ˆ 2 which gives us −gc · ∇E(x)/||gc ||2 (where
our case, we solve argminκ ||g(xj , κ, d)||
P ∂b
gc = k ∂d ˆ at start of each time step to obtain an estimate of κ that seeks to
∇dk (x, d))
k

178
balance the two gradients at xj . However, we observe that the effectiveness of this balanc-
ing term is highly dependent on the xj applied. It can be quote far from the configuration at
solution and so potentially can leads to poorly scaled or even negative values for κ. Thus,
we extend our analysis from the balancing gradients to additionally include conditioning of
the Hessian. This, in turn, obtains an effective estimate to provide a lower bound of κ in
support of the gradient balancing strategy.
Our analysis seeks to keep the scaling of the diagonal entries of the Hessian, at small
distances, close to the mass. Specifically, a scaling characterization of the Hessian diagonal
2
in ∇2 b at d = 10−8 l is easily estimated by taking its first term ∇dT ∂d
∂ b
2 ∇d in point-point

formula as
∂ 2b 2
−16 2 ∂ b
c∇2 b = (||∇dT ||2 )| −8
d=10 l = 4 × 10 l (10−8 l).
∂d2 ∂d2
We then set the lower bound of κ to provide at least 1011 times of the average lumped nodal
mass m̄ on the diagonal entries when d = 10−8 l and so enable production of sufficient
repulsion at larger distances, that is

κmin = 1011 m̄/c∇2 b

ˆ effectly capturing the curvature

Note that our κ lower bound will be different for different d,
change of the barrier. With this lower bound, we now can safely use the gradient scheme
with bounded κ value.
Distance can still become small if resultant stress or applied compression (e.g, BCs) in
the scene are extreme. We thus add an additional, final κ adjustment that doubles κ, when
needed, in between Newton iterations. After every Newton iteration, if we detect that there
are contact pairs having a characteristic distance smaller than minimum dˆ = 10−9 l both
before and after this iteration, and the distance is decreasing in this Newton iteration, we
double the κ value. Although we do not observe divergence of the adapted κ values we
apply a fixed upper bound of κmax = 100κmin .
To summarize, our adaptive barrier stiffness strategy is:

1. At start of each time step, compute κg giving smallest gradient, and set κ ←

179
 min(κmax , max(κmin , κg )).

2. After each Newton iteration, if any contact pair has distance smaller than dˆ both
before and after this iteration, and the distance is decreasing, set κ ← min(κmax , 2κ).

Appendix G: Distance Computation Implementation

Point-point and point-edge constraint duplications

As we discuss in Chapter 4 many point-triangle and edge-edge distances can and will
reduce to point-point or point-edge distance in computation. Thus there can be multiples of
exactly the same point-point and/or point-edge stencils in our constraint set. While it is
tempting to simply either ignore or remove these duplicates, neither strategy is effective.
Ignoring duplication in code can lead to significant redundant computation of the same
force and Hessian. On the other hand removing them introduces inconsistency into our
objective energy, leading to poor convergence or even divergence over iterations. Instead,
we track duplicate stencils, computing their energy, gradient, and Hessian evaluations only
once for each distinct stencil and then multiply their entries appropriately so that all terms
are correctly applied but still avoiding redundant and expensive computation.

Nearly parallel edge-edge distance

When computing distances between two nearly parallel edges using the edge-edge plane
distance formula (4.23), numerical rounding errors will generate huge gradient and Hessian
values, and even results in wrong distances (Figure 3) because of the ill-defined normal,
making our optimization intractable with double precision floating point numbers. There-
fore, we check the angle between edges, forcing each case to reduce to the most appropriate
point-point or point-edge constraints if the sine value is smaller than 10−10 . This clearly
makes the distance function C 0 continuous again at the threshold. However, the nonsmooth-
ness is nearly negligible (Figure 3), while our multiplying energy smoother ek,l (x) is also

180
 0.2

EE distance (numerical)
0.15

0.1

0.05

0
0 0.5 1
EE angle sine value 10 -14
0.1695137880219 0.1695137880219
EE distance (numerical)
EE distance (numerical)

0.1695137880218 0.1695137880218

0.1695137880217 0.1695137880217

0.1695137880216 0.1695137880216

0.1695137880215 0.1695137880215

0.1695137880214 0.1695137880214
0 0.5 10 0.5 1
EE angle sine value 10 -9 EE angle sine value 10
-9

Figure 3: Parallel-edge degeneracy handling. Left: Due to numerical rounding errors

for distances of edge-edge pairs decreases to 0 when the edges get more and more parallel
(see Figure 4.6 for an example); Middle: a zoom-in view show clearly that the distance
really starts decreasing when their angle’s sine value is at around 10−9 ; Right: we here set a
threshold to force the use of point-point or point-edge formulas to compute the distance of
edge-edge pairs when their angle’s sine value is below 10−10 , which introduces a negligible
(for optimization) nonsmoothness.

extremely small when edges are nearly parallel. In practice we find this robustly avoids
numerical issues and converges well for all benchmark tests; see Section 4.7.

181
Appendix H: Tangent and Sliding Modes

After reducing the general point-triangle, and edge-edge distances to one of the closed form
formulas, see Section 4.6, we can directly compute the sliding basis operators for each of
the four types of contact pairs required for computing friction.
We start by defining the basis, Pk (x) ∈ R3×2 , formed by the two orthogonal 3D unit-
length column vectors spanning the tangent space of the contact pair k, and a selection
matrix Γk ∈ R3×3n which computes relative velocity vk = Γk v of each contact pair k.
Then we can define the sliding basis Tk (x) = ΓkT Pk (x) that maps tangent space relative
velocity or displacement to the stacked global vector. Here we then list the construction for
P and Γ for each contact distance type:

Point (x0 ) – Triangle (x1 x2 x3 )

h i
Pk (x) = x2 −x1 x2 −x1 (7)
||x2 −x1 ||
,n × ||x2 −x1 ||

(x2 −x1 )×(x3 −x1 )

where n = ||(x2 −x1 )×(x3 −x1 )||
. Each row of Γk is

[..., 1, ..., (−1 + β1 + β2 ), ..., −β1 , ..., −β2 , ...] (8)

where β’s are those defined in D P −T (4.18).

Edge (x0 x1 ) – Edge (x2 x3 )

h i
Pk (x) = x1 −x0 x1 −x0 (9)
||x1 −x0 ||
,n × ||x1 −x0 ||

(x1 −x0 )×(x3 −x2 )

where n = ||(x1 −x0 )×(x3 −x2 )||
. Each row of Γk is

[..., 1 − γ1 , ..., γ1 , ..., γ2 − 1, ..., −γ2 , ...] (10)

where γ’s are those defined in D E−E (4.19).

182
Point (x0 ) – Edge (x1 x2 )
h i
Pk (x) = x2 −x1
, (x2 −x1 )×(x0 −x1 )
||x2 −x1 || ||(x2 −x1 )×(x0 −x1 )||
(11)

Each row of Γk is
[..., 1, ..., η − 1, ..., −η, ...] (12)

where (1 − η)x1 + ηx2 is the closest point to x0 on edge x1 x2 .

Point x0 – Point x1
h i
Pk (x) = t, x1 −x0 (13)
||x1 −x0 ||
×t
e×(x1 −x0 )
where t = ||e×(x1 −x0 )||
and e is (1, 0, 0) if (x1 − x0 ) is not colinear with (1, 0, 0), or e is
(0, 1, 0). Each row of Γk is [..., 1, ..., −1, ...].

Appendix I: Friction Implementation

Since we lag the sliding basis in friction computations to T n = T (xn ) and normal forces to
λn in (see Section 4.5) from either the last time step or the last friction update iteration n,
all other terms are integrable. The lagged friction is then

uk
Fk (x, λn , T n , µ) = −µλn Tkn f1 (||uk ||) (14)
||uk ||

and gives us a simple and compact friction potential

Dk (x) = µλnk f0 (||uk ||). (15)

Here f0 is given by f00 = f1 and f0 (v h) = v h so that Fk (x) = −∇Dk (x). In turn this
likewise provides a simple-to-compute Hessian contribution
 f 0 (||u ||)||u || − f (||u ||)
1 k k 1 k
∇2 Dk (x) = µλnk Tkn 3
uk uTk
||uk ||
(16)
f1 (||uk ||)  n T
+ I2 Tk .
||uk ||

183
  
1 0
where I2 =  . Projecting this Hessian to PSD then simply requires projecting the
0 1
2 × 2 matrix
f10 (||uk ||)||uk || − f1 (||uk ||) f1 (||uk ||)
3
uk uTk + I2 (17)
||uk || ||uk ||
to SPD as Tkn is symmetrically multiplied on both of its two sides.
The above model is general so that f0 and f10 are both easy to define for a range of f1
choices:

x2
1. C 0 Ff : f0 (x) = 2v h
+ v h
2
, f1 (x) = x
v h
, and f10 (x) = 1
v h
;

3 2 2
2. C 1 Ff : f0 (x) = − 3x2 h2 + xv h + v3h , f1 (x) = − 2xh2 + 2x
vh
, and f10 (x) = − 22xh2 + v2h ;
v v v

x4 x3 3x2 x3 3x2
3. C 2 Ff : f0 (x) = 43v h3
− 2v h2
+ 2v h
+ v h
4
, f1 (x) = 3v h3
− 2v h2
+ 3x
v h
, and f10 (x) =
3x2 6x 3
3v h3
− 2v h2
+ v h
.

Importantly this also emphasizes that there are never any divisions by ||uk || in all of our
energy, gradient, and Hessian computations for friction as they are always cancelled out.
This ensures that so that the implemented computation can be robust and accurate.
Our friction model applies our C 1 – (2) in the above.. This design choice again provides
a continuous Hessian for better convergence in our Newton-type method. Here we also
provide a comparison of behavior of the C 1 model w.r.t. to the different orders of smoothed
friction model on our arch and ball roller example (Figure 4).
C0 C1 (IPC) C2
Examples
# iters, t (s) / time step # iters, t (s) / time step # iters, t (s) / time step

sphere1K roller 1.37, 0.01 1.24, 0.01 1.26, 0.02

1m-height arch (static) 53.60, 12.21 45.22, 9.79 52.83, 11.42

Figure 4: Ablation study on smoothed static friction.

184
 We observe that C 1 friction model provides a “sweet-spot”: improvement over C 0 due
to the C 1 model’s continuous hessian, while it also improves over the C 2 model with less
additional nonlinearity.

To compute friction potential, f0 is well defined without any division by 0.

For friction gradient/force, a robust implementation requires algebraically deriving

f1 (||uk ||)
||uk ||
, which does not contain division by 0 by construction of f1 .

For friction Hessian, the inner 2 × 2 matrix is

f10 (||uk ||)||uk || − f1 (||uk ||) f1 (||uk ||)

3
uk uTk + I2
||uk || ||uk ||

f10 (||uk ||)||uk ||−f1 (||uk ||)

, one needs to first algebraically derive ||uk ||2
which does not contain division
uk uT
by 0 by construction, and note that k
||uk ||
is always bounded and it equals 0 when ||uk || = 0,
f1 (||uk ||)
in this case since ||uk ||
> 0 is always true there’s no need for SPD projection.

Let uk = (x, y),

f10 (||uk ||)||uk || − f1 (||uk ||) f1 (||uk ||)

uk uTk + I2
||uk ||3 ||uk ||
 
0
f (||uk ||) −f1 (||uk ||)  x2 xy
= 1 u u
k k
T
+ 
||uk ||2 ||uk ||3 xy y2
 
2 2
f1 (||uk ||) x + y 0
+ 3
 (18)
||uk || 0 2
x +y 2
 
0
f1 (||uk ||) T f1 (||uk ||)  y 2 −xy
= uk uk + 
||uk ||2 ||uk ||3 −xy x 2

f10 (||uk ||) f1 (||uk ||)

= 2
uk uTk + 3
ūk ūTk
||uk || ||uk ||

where ūk = (−y, x). So for ||uk || ≥ v h when we have f10 (·) = 0 and f1 (·) = 1, the matrix
is always SPD (must compute using ūk , the original formula can still be not SPD due to
numerical error of subtraction!); otherwise we project the 2x2 matrix.

185
Appendix J: Squared Terms
In our implementation, we apply squared distances in our evaluations to avoid numerical
errors and inefficiencies that can be introduced by taking squared roots – especially in
gradient and Hessian computations. Concretely, our barrier terms are applied as b(d2 , dˆ2 )
throughout our implementation. This manipulation leaves our problem formulation with
unsigned distances unchanged as have d > 0 ⇔ d2 > 0. However, we must be careful with
units in order to preserve appropriate scaling of dimensions – especially in terms of frictio.
Fortunately, we can sort this out directly via the chain rule.
To ensure that units of normal force remain correct we now observe that directly
plugging in d2 and dˆ2 into Equation (4.9) in our Chapter 4 no longer computes the contact
2 ˆ2
∂b
force magnitudes λk defined there. Here we now have − hκ2 ∂(d2 ) (dk , dk ). Applying the
k

squared formulation we rewrite the stationarity of the barrier as

x − x̂ κ X ∂b ∂(d2k )
M ( 2 ) = −∇Ψ (x) − 2 ∇dk (x) (19)
h h k∈C ∂(d2k ) ∂(dk )

which is M ( x−x̂ ) = −∇Ψ (x) − hκ2 ∂b

P
h2 k∈C ∂(d2k ) 2dk ∇dk (x). In turn this allows us to extract
the correct contact force magnitudes (when using squared distances) as λk = − hκ2 ∂(d
∂b
2 ) 2dk .
k

Appendix K: Isotropic Strain Limiting Derivative Deriva-

tions
Will need derivatives of singular values:
∂Sii /∂F = ui viT
∂ 2 Sii ∂(ui viT ) T
(∂F )2
= ∂F
= ∂u
∂F i
v + ui ∂v
i T
∂F
i

∂ui ∂vi
where ∂F
and ∂F
can be found in Xu et al. [2015].
Then if we call the barrier bt,i for triangle t, i-th singular value, we have
∂bt,i ∂bt,i ∂St,ii ∂Ft
∂x
= ∂St,ii ∂Ft ∂x
∂ 2 bt,i ∂St,ii T ∂ 2 bt,i ∂St,ii ∂bt,i ∂ 2 St,ii
∂x 2 = ∂x 2
∂St,ii ∂x
+ ∂St,ii ∂x2

186
 where
∂Ft
- ∂x
can be found in anisotropic strain limiting implementation detail section.
∂bt,i κs s−St,ii (ŝ−St,ii )2
- ∂St,ii
= (s−ŝ)2 (2(ŝ − St,ii ) ln( s−ŝ ) + s−St,ii
)
2
∂ bt,i κs s−St,ii ŝ−St,ii ŝ−S 2
- 2
∂St,ii
= (s−ŝ)2 (−2 ln( s−ŝ ) − 4 s−S
t,ii
+ s−St,ii
t,ii
)
∂St,ii ∂St,ii ∂Ft
- ∂x
= ∂Ft ∂x
2
∂ St,ii T ∂ 2 St,ii ∂Ft
- ∂x2
= ∂F
∂x
t
∂Ft2 ∂x

For efficiency we can compute

∂bt
P ∂bt,i ∂St,ii ∂Ft
∂x
= i ( ∂St,ii ∂Ft
) ∂x
∂ 2 bt,i T ∂ 2 bt,i ∂Ft
= ∂F
P
i ( ∂F 2 ) ∂x
t
∂x2 ∂x t
∂ 2 bt,i ∂S T ∂ 2 bt,i ∂St,ii ∂bt,i ∂ 2 St,ii
- where ∂Ft2
= ∂Ft,ii
t
2
∂St,ii ∂Ft
+ ∂St,ii ∂Ft2
P ∂ 2 bt,i
- for SPD projection we can process the 6x6 matrix i ( ∂Ft2 )

Appendix L: Anisotropic Strain Limiting Derivations

Quadratically Approximating Data-Driven Model

- 1st order derivatives

∂ψ/∂ Ẽ11 =a11 η10 (Ẽ11
2
)Ẽ11 +a12 η20 (Ẽ11 Ẽ22 )Ẽ22
∂ψ/∂ Ẽ22 =a22 η30 (Ẽ22
2
)Ẽ22 +a12 η20 (Ẽ11 Ẽ22 )Ẽ11
∂ψ/∂ Ẽ12 =2G12 η40 (Ẽ12
2
)Ẽ12
when Ẽ = 0, element is at rest shape, all gradients are equal to 0.
- 2nd order derivatives
∂ 2 ψ/∂ Ẽ11
2
=2a11 η100 (Ẽ11
2 2
)Ẽ11 +a11 η10 (Ẽ11
2
)+a12 η200 (Ẽ11 Ẽ22 )Ẽ22
2

∂ 2 ψ/∂ Ẽ11 ∂ Ẽ22 =∂ 2 ψ/∂ Ẽ22 ∂ Ẽ11 =a12 η200 (Ẽ11 Ẽ22 )Ẽ22 Ẽ11 +a12 η20 (Ẽ11 Ẽ22 )
∂ 2 ψ/∂ Ẽ22
2
=2a22 η300 (Ẽ22
2 2
)Ẽ22 +a22 η30 (Ẽ22
2
)+a12 η200 (Ẽ11 Ẽ22 )Ẽ11
2

∂ 2 ψ/∂ Ẽ12
2
=4G12 η400 (Ẽ12
2 2
)Ẽ12 +2G12 η40 (Ẽ12
2
)
other terms are 0
when Ẽ = 0, ∂ 2 ψ/∂ Ẽ11
2
= a11 ∂ 2 ψ/∂ Ẽ11 ∂ Ẽ22 =∂ 2 ψ/∂ Ẽ22 ∂ Ẽ11 = a12 ∂ 2 ψ/∂ Ẽ22
2
=
a22 ∂ 2 ψ/∂ Ẽ12
2
= 2G12 so to quadratically approximate the data-driven model at Ẽ = 0

187
using our barrier model, we just need to use the same stiffnesses. This is very simple, but
also suggest that our approximation might not work very well to match intricate behaviors
as the data-driven model.

Implementation Details

For simplicity, if we align the cloth weft and warp directions to coordinate axis, we can eas-
ily compute Ẽ from 3x2 deformation gradient F = [x2 −x1 , x3 −x1 ][X20 −X10 , X30 −X10 ]−1 :
Ẽij = 12 (Fmi Fmj −δij ) so ∂ Ẽij /∂Fkl = 21 (δil Fkj +Fki δjl ) ∂ 2 Ẽij /∂Fkl ∂Fmn = 12 δmk (δil δnj +
δni δjl )
   
−I h i
Then together with ∂ψ above and dF/dx = {  I  } : B where B = [X20 −
−I I
X10 , X30 − X10 ]−1 so

dF/dx
     
−(B11 + B21 ) B11 B21
     
∗ ∗ ∗ 
     
    
     

 ∗ 
 
∗ 
 
∗
 (20)
= , , 
−(B12 + B22 )
     
 B12  B22 
     
∗   ∗   ∗ 
     

     
∗ ∗ ∗

we are able to implement the constitutive model: ∂ψ/∂x= ∂ψ/∂ Ẽ * ∂ Ẽ/∂F * ∂F/∂x

∂ 2 ψ/∂x2 =(∂F/∂x)T * ∂ 2 ψ/∂F 2 * ∂F/∂x where ∂ 2 ψ/∂F 2 =(∂ Ẽ/∂F )T * ∂ 2 ψ/∂ Ẽ 2

* ∂ Ẽ/∂F +∂ψ/∂ Ẽ * ∂ 2 Ẽ/∂F 2

188
Appendix M: Rod Bending Modulus
In discrete rods model Bergou et al. [2008], the bending energy is directly integrated over
the length of the rod as
n
X α(κbi )2
Ebend (x) = ¯li (21)
i=1

Here κbi is the curvature binormal at a vertex, ¯li = |ēi−1 | + |ēi | where |ēi | is the rest length
of edge i, and α is an adjustable parameter to control the stiffness of bending. As we
know when a rod becomes thicker, it is harder to bend, which means we need to manually
increase α accordingly and this makes seting up examples inconvenient.
According to Kirchoff rod theory Dill [1992], the bending energy of rod at a unit length
is defined as
1 4π 2
Er κ
2 4
κb
where E is the bending modulus in P a unit, r is the radius of the rod, and κ = l̄/2
is the
point-wise curvature Bergou et al. [2008]. Integrating this quantity over rod length then
gives us Z
1 4π 2
Ebend (x) = Er κ dl
Ω 2 4
X ¯li 1 π κbi
≈ Er4 ( ¯ )2 (22)
i
22 4 li /2
X π (κbi )2
= Er4
i
4 ¯li
Combining with Equation 21 we finally know that the α in Bergou et al. [2008] is associated
with the rod’s radius as
π
α = Er4
4
so now the change of rod thickness under a certain bending modulus E can be automatically
reflected in the bending stiffness. This enables us to easily setup rod examples starting
from using real material thickness and Young’s modulus of e.g. hairs, iron, etc, without
having to tune the unintuitive bending stiffness α.

189
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