An Overview on Finite Element Analysis

and Structural Optimization

Dr János Plocher, Ph.D. | Academic Simulation Program Lead

©2025 ANSYS, Inc.

Outline

Dr János Plocher, PhD

Academic Simulation Program Lead
The Ansys Suite
Ansys Germany GmbH
janos.plocher@ansys.com
An Introduction to FEA
Real-World Applications

Structural Optimization
Theory | Objectives & Constraints | General
Considerations | Application Examples

Ansys Innovation Space

Curated Knowledge and Learning Material

Q&A

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The Ansys Suite

An Overview

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Electronics 3D Design Embedded software & Safety Fluids

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Process integration & optimization
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for Simulation (MDS) Kit (TETK) Ansys Motion Ansys Forte
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Developer Ecosystem* Ansys Twin Builder® Ansys Discovery Ansys AI+ Ansys Lumerical
INTERCONNECT Ansys Speos®

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alone product Ansys Lumerical CML Compiler

Academic Products Academia Products upon request

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Structural Analysis Solutions
3D Design Structures

Ansys’ up-front simulation tool allowing you to design Ansys’ flagship tool for detailed structural analysis to
(CAD) and explore ideas, iterate and innovate quickly solve complex structural engineering problems utilizing
across multiple physics using fast GPU-based solvers. implicit, explicit solvers and advanced meshing tech.

Mechanical LS-DYNA

Provides in-depth Integrates into Ansys

analysis of structural Mechanical for
and coupled-field powerful explicit
behaviors for broad simulations. A large
structural analysis array of capabilities
needs through a suite and material models
of finite element enable complex
analysis (FEA) models with great
solutions. scalability.

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Structural Analysis Solutions
3D Design Structures

GPU-accelerated simplified topology optimization,

Back-2-CAD reverse engineering & validation in one Advanced structural optimization
tools

Mechanical LS-DYNA

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Ansys Workbench
• Ansys Workbench allows setting up and
performing all simulations within one
powerful platform
• Material data and geometry is shared
through workflows to perform all kinds
of analyses (static, dynamic, thermal,
fatigue, etc.)
• Analyses can be derived from one
another for submodels or further
detailed investigation

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An Introduction to FEA

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 Why Do We Need Engineering Simulation?
Traditional Design
Change
Material Physical Change of
Concept CAD Model Testing of Production
Selection Prototype design
prototype

Physical Prototyping
Manufacturing & Processing → Expensive and time-consuming

Validation
Simulation Driven Design
Material Virtual Physical
Concept CAD Model Testing Production
Selection Prototyping Prototype
Cost and time
Virtual savings
Prototyping Physical Prototyping
Manufacturing & Processing Verification
→ Cheaper and
faster process

FEA-simulations enables you

to…
drive eliminate
innovation risk

manage increase
complexity quality

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Virtual Prototyping Using Numerical Models

1 2 3
“Finite Elements” used as
CAD model imported Geometry divided into variables in physics equations,
“Finite Elements” Software computes results

Telescoping Truss Model “Mesh” created for Truss Model Deformation Results

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Numerical Analysis – Finite Element Analysis (FEA) and Method (FEM)

• FEM exploits computational power to calculate solutions for a range of

different engineering problems, offering good approximate solution 1
to complex problems in the field of e.g. solid mechanics (e.g., statics
and dynamics).

• At the core of FEM lies the discretization, which means a model is

divided into a finite number of smaller bodies or elements that are
connected at common points, the so-called nodes.
2
• Each node has a limited number of degrees of freedom (DoFs), so e.g.
node
a continuum solid becomes discretized into a finite number of DoFs,
based on the number and type of elements.

• For each of those elements a set of algebraic equations are solved in

an iterative fashion. Element type: Tetrahedron

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 Numerical Analysis – From Physical Problem to Final Product

Physical Problem Numerical Model Physical Prototype Final Product

Solid Mechanics
e.g. bracket
Physical principles System of algebraic
equations
Governing equations
FEA approximate
SM FM TA
Fluid Mechanics
Euler-Bernoulli beam Navier-Stokes motion Fourier’s law of heat SM
e.g. pipe deflection of viscous fluids transfer
2 2𝑤
d2
𝐸𝐼
d2 𝑢
=𝑞
d
𝜕u d
⋅ ∇ u − 𝜈∇2 u ==
+ u 𝐸𝐼
1
− 𝑞∇𝑝 + 𝑔 𝑞𝑥 = −𝑘
𝑑𝑇 𝑲 𝐮 = 𝐅
d𝑥 2 d𝑥 2 𝜕𝑡 𝜌 𝑑𝑥
d𝑥 2 d𝑥 2
Stiffness Nodal Force
matrix vector vector
Thermal Analysis Boundary conditions (disp.)
e.g. heat sink
SM FM TA

⇒
Fixed end Parallel flow Insulation at x=0
𝜕𝑢 2
d 𝑢 𝑑𝑇
𝐮 = ሾ𝐊ሿ−1 𝐅
θ 0 = ቤ =0 = −1; 𝑢 0 = 𝑢 1 = 0 −𝑘 ቤ =0
𝜕𝑥 𝑥=0 d𝑦 2 𝑑𝑥 𝑥=0
…

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 Numerical Analysis – From Physical Problem to Final Product

Physical Problem Numerical Model Physical Prototype Final Product

Solid Mechanics
e.g. bracket
Physical principles System of algebraic
equations
Governing equations Boundary conditions
Euler-Bernoulli beam Fixed end Free end FEA approximate
deflection
d2 d2 𝑢 𝜕𝑢 d2 𝑢 𝐹
SM
Simplification 𝐸𝐼 =𝑞 θ 0 = ቤ =0 𝜏 0 = 2อ =−
d𝑥 2 d𝑥 2 𝜕𝑥 𝑥=0 d𝑥 𝐸𝐼
𝑥=𝐿
𝑲 𝐮 = 𝐅
Discretization
𝑞
1 Element 1 2 Element 2 3 Element 3 4 Stiffness Nodal Force
matrix vector vector
(disp.)
𝑢1 𝐹1 𝑁𝑜𝑑𝑒 1: 𝑢1−3 = 0

⇒
𝑢2 = 𝐹2
𝑁𝑜𝑑𝑒 4: 𝐹1 ≠ 0
𝑢3 𝐹3
𝐮 = ሾ𝐊ሿ−1 𝐅
𝑲 𝐮 = 𝐅
Analytical

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 Numerical Analysis – From Physical Problem to Final Product

Physical Problem Numerical Model Physical Prototype Final Product

Solid Mechanics
e.g. bracket
Physical principles System of algebraic
equations
Governing equations Boundary conditions
FEA approximate

• The PDE is also referred to as the strong from SM

Simplification whereas it integral is called the weak form.
• Weak form has no need𝑞for higher-order derivatives 𝑲 𝐮 = 𝐅
Discretization and is thus more suitable for FEA:
Stiffness Nodal Force
• Common methods for deriving and solving the weak matrix vector vector
form of the governing equations are: (disp.)
• Principle of Minimum Potential Energy

⇒
• Galerkin Method of Weighted Residuals
𝐮 = ሾ𝐊ሿ−1 𝐅
Analytical

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 FEA Workflow
Physical Problem Numerical Model Physical Prototype Final Product
Physical principles

CAD-geometry (e.g. shape, Boundary conditions (e.g.

cross-sections, assemblies) supports, forces, etc.) and
and material (e.g. steel, analysis type (e.g. structural
rubber, wood, etc.) Geometry & vs fluid)
Pre-processing
Material

Numerical solution (i.e.

Validation Mathematical
Sanity check (i.e. model approximation) to the system of
do the results match the equations at selected points obtained
expectation, are the results Post- using FEA
processing
plausible?)

Evaluation of outputs/state variables (e.g.

displacement, velocity)

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Structural Analysis – FEA Basic Equations
Equilibrium Constitutive Laws Compatibility

Internal stresses and must be The constitutive material Apart from cracks in a structure,
in equilibrium everywhere behavior (stress-strain and displacements/strains must
which presupposes a local strain-displacement continuous if the structure is
equilibrium (i.e. the sum of all relationship) must be satisfied. continuous.
forces is zero).

Body and
surface Displacements
forces
*
Stresses Strains

* Lecture Unit: Stresses and Strains with Ansys Discovery | Ansys

Note: Capital letters generally refer to the global and
lowercase letters to the element matrix.
𝐅 = 𝑲 𝐔 † Intro to Material Elasticity - ANSYS Innovation Courses
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Structural Analysis – Stresses & Strains

• Engineering (nominal) data is commonly used when

FS

𝑆𝑡𝑟𝑒𝑠𝑠 𝝈
designers and engineers discuss structural problems Plastic
Elastic
regime
× true
or test results; however, these are not always regime
accurately reflecting the stresses and strains a part × UTS
FS
experiences. × engineering
𝜎
• Engineering stresses and strains are accurate if a part 𝐸=
is only elastically deformed. 𝜀

• The differences become apparent once a part is

deformed plastically, because it starts to deform
significantly with drastic changes in the cross-section
(e.g., typical necking behavior of metals after
necking
ultimate tensile strength is reached).
0 𝑆𝑡𝑟𝑎𝑖𝑛 𝜺
• Taking the instantaneous change in cross-section
area and length into account provides the true stress-
strain data, which give insights into the fundamental 𝐹 Δ𝐿
Stresses 𝜎= 𝜀= Strains
properties of materials. 𝐴0 𝐿0

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Structural Analysis – Engineering vs. True Stress
In contrast to the engineering stresses, which is determined by the applied force F
by the original cross-section area A0, the true stress is expressed as the force applied
on the current cross-section area A.

F A
Conservation of volume:
𝑆𝑡𝑟𝑒𝑠𝑠 𝝈

true
𝐴0 × 𝐿0 = 𝐴 × 𝐿
𝐹 L
𝜎𝑡𝑟𝑢𝑒 = Strain:
𝐴 ∆𝐿 𝐿−𝐿0 𝐿
× 𝜀𝑒𝑛𝑔 = = = −1
𝐿0 𝐿0 𝐿0
F A0
×
engineering L0 𝐹 𝐹×𝐿
𝜎𝑡𝑟𝑢𝑒 = =
𝐹 𝐴 𝐴0 × 𝐿0
𝜎𝑒𝑛𝑔 =
𝐴0

𝐹
𝑆𝑡𝑟𝑎𝑖𝑛 𝜺 𝜎𝑡𝑟𝑢𝑒 = × (1 + 𝜀𝑒𝑛𝑔 ) = 𝜎𝑒𝑛𝑔 × (1 + 𝜀𝑒𝑛𝑔 )
𝐴0

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Structural Analysis – Engineering vs. True Strain
• Step-wise vs. continuous deformation

o Engineering strain o True strain ∆𝐿 ∆𝐿2 𝑑𝐿

∆𝐿1
𝐿 ∆𝑳𝒏
∆𝐿 𝐿−𝐿0 𝑑𝐿 𝐿
𝜀𝑒𝑛𝑔 = = 𝜀𝑡𝑟𝑢𝑒 =න = ln
𝐿0 𝐿0 𝐿0 𝐿 0 𝐿0 ∆𝑳𝟐
𝐿 ∆𝑳𝟏 𝑑𝐿
𝜀𝑒𝑛𝑔 = −1 𝜀𝑡𝑟𝑢𝑒 = ln(1 + 𝜀𝑒𝑛𝑔 )
𝐿0
𝑇𝑖𝑚𝑒 𝑡

Tension Compression

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21
Structural Analysis – Equivalent (von Mises) Stress
• The state of stress in 3D space can be described by a 3 × 3 tensor. Every stress component could be visualised
and analysed individually, however, sometimes it might be more convenient to investigate just one plot,
encompassing a single scalar value, the so called equivalent (von Mises) stress, which serves as an indicator for
failure:
𝜎𝑥𝑥 𝜎𝑥𝑦 𝜎𝑥𝑧 2 2 2 2 2 2
𝜎𝑥𝑥 − 𝜎𝑦𝑦 + 𝜎𝑦𝑦 − 𝜎𝑧𝑧 + 𝜎𝑧𝑧 − 𝜎𝑥𝑥 + 6 𝜎𝑥𝑦 + 𝜎𝑦𝑧 + 𝜎𝑧𝑥
𝜎𝑦𝑥 𝜎𝑦𝑦 𝜎𝑦𝑧 ⇒ 𝜎𝑣 =
𝜎𝑧𝑥 𝜎𝑧𝑦 𝜎𝑧𝑧 2
𝝈𝒛𝒛
• This so called von Mises yield criterion constitutes a means for determining the 𝝈𝒛𝒚
onset of material yielding and is the most widely used criterion. Other popular
criteria are for yielding in isotropic material are the Tresca criteria as well as the 𝝈𝒛𝒙
𝝈𝒚𝒛 𝝈𝒙𝒛
Hill criteria for anisotropic behaviour.
𝝈𝒙𝒚
• The von Mises stress can also be calculated from the principal stresses:
𝝈𝒙𝒙
𝜎1 0 0 𝝈𝒚𝒙
𝜎1 − 𝜎2 2 + 𝜎2 − 𝜎3 2 + 𝜎3 − 𝜎1 2 𝝈𝒚𝒚
0 𝜎2 0 ⇒ 𝜎𝑣 =
0 0 𝜎3 2

Note: Recall that the two expressions (original and principal coordinate axis) provide the same equivalent stress value for a given stress condition.

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Structural Analysis – Equivalent (von Mises) Stress
• In the principal coordinate system, the von Mises stress can be visualised as a cylindrical (yield) surface. On
every point (locus) on the surface, von Mises stress values are identical.
𝜎1
𝜎1
Von Mises hydrostatic axis
yield surface

1 2 2 2
𝜎𝑣 = 𝜎 − 𝜎2 + 𝜎2 − 𝜎3 + 𝜎3 − 𝜎1
2 1
𝜎3

• The length of the cylinder can extend 𝜎2 𝜎2 𝜎3

infinitely, which means that the von Mises
stress is equal as long as the ratio between 𝜎1 𝜎1
the principal stresses remain equal.
• The radius of the cylinder can grow or shrink,
meaning the von Mises stress increases or
decreases, respectively.
𝜎3 𝜎3

𝜎2 𝜎2
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Structural Analysis – Linear and Nonlinear FEM
• All phenomena we see in nature have some degree of nonlinearity. To simplify various physical problems, it is common to
assume linear relationships and present a solution that is sufficiently accurate, but still an approximation.

• In the underlying system of equations in the linear analyses, it is possible to observe the relation between the force and the
displacement, in which the stiffness matrix is constant. However, outside the material’s linear elastic region, for large
deformations, the stiffness matrix is no longer constant, and the linear assumption cannot be applied.
𝑲 𝐮 = 𝐅
• Since the stiffness matrix is a function of its geometry and material properties, it is possible to
Stiffness Nodal Force
highlight three main sources of non-linearity.
matrix vector vector
(disp.)

Large deformation Contact nonlinearity Nonlinear material

In the stress-strain curve, when the
Large deformations can affect load Contacts between parts of an assembly structure is stressed beyond its elastic limit
distribution, introducing a nonlinear can introduce nonlinearity in a model, for (linear comportment), plasticity introduces
displacement response. example by relative motion or contact loss. nonlinearities to the model.

• It is possible that the nonlinearities present in the model are a result of a combination of factors, not just one alone. Thus, it
is important to analyse what is the main source of nonlinearity and think carefully, according to the wanted application, if it is
possible to simplify the model to a linear comportment.
Introduction to nonlinearities – Ansys Innovation Course

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Structural Analysis – Linear and Nonlinear FEM

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 Element order
Meshing – Overview Linear (lower order element)
Quadratic (higher order element)

Linear approximation

Number of 1D (Beam elements) Actual curve Mesh method

2D (Shell elements) Algorithm used by the
dimensions 3D (Solid elements) software to create
Quadratic approximation the mesh

Element shape Mesh structure Mesh metrics

How predictable and well defined The quality of the mesh is
the location of the nodes are defined by mathematical
Quadrilateral Triangular
parameters (metrics)

Structured x Unstructured
Hexahedral Tetrahedral

The Fundamentals of FEA Meshing for Structural Analysis I Ansys Blog

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Meshing – Mesh Independence & Convergence
Geometric representation Mesh Independence Mesh Metrics Mesh Refinement Good
The mesh doesn’t alter the Coarse or fine mesh
Mesh
All the geometric details are The quality criteria are
well captured solution (convergence study) within a correct range depending on the complexity

Element Size
0.005m
What is convergence? The process of convergence consists in decreasing the element size and analyzing its impact in the
solution of the underlying partial differential equations.

Smaller More accurate Larger Longer Optimal

mesh size solution simulations runtimes balance
Element Size
0.003m

…when it is possible to obtain the converged solution after which mesh refinement does not
A mesh is independent…
change the result anymore.

Refine the mesh in the Compare the results for the

Choose a result to study,
area of interest by various runs and plot a
such as stress values, plastic
changing the mesh size and convergence curve in a chart
strain, fatigue life, etc.
maintaining element type number of nodes vs stress
and order. values, for example.

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Meshing – Converged Solution

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Static vs. Dynamic Analysis
• Drop test of all forms
• Impacts
• Product misuse / severe loadings
• Product failure / fragmentation • Large Deformations
• Containment safety and • Complex Contact
penetration mechanics
• Nonlinear Material
• Large plasticity in mechanisms
• Sports equipment design
• Manufacturing processes like
machining / cutting / drawing

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Application of Implicit vs. Explicit Methods

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 Garbage-In-Garbage-Out-Principle

Garbage-In Garbage-Out
Think critically about what
assumptions you are making
and why!

Simulation Tool

Garbage can Nicer-looking can, but still garbage!

© 31 Gal. Metal Trash Can | 3D CAD Model Library | GrabCAD

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Real-World Applications

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Aerospace & Defense

Drone Strike Parachute

Containment
Image source CADFEM
Deployment

Shuttle Water Landing

Fan Blade Out Safety Barrier Crash
Image source AWG
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Automotive

Truck Water Wading Simulation Wall impacting a battery pack Hydroplaning with Flexible Tires
(Current density + temperature)

Fuel Tank Integrity Coupling w. DEM to simulate mud Drive Side Airbag Folding and Deploy
or snow deposition
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Manufacturing

Hot Forming with Roller Hemming Multiphysics – Braiding (Courtesy Shrinkwrap

Lancing Adhesive Hemming Dynamore)

Sheet Winding Roll Forming Sheet Metal Press Hemming Rubber Forming
Forming

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35
Bio-Medical

Balloon Inflation in an Artery

Heart Modeling EndoVascular Surgical

ElectroPhysiology + Mechanics Process Simulation
+ Flow

Syringe Drop test

Helmet Design
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Beating Heart Automatic generation of the Purkinje network (top)
and muscle fibers (bottom)
LS-DYNA model
Challenge
• Provide a high-fidelity multi-physics evaluation tool for beating human heart simulation. User input:
• Accelerate device or clinical app development by testing implanted device performance, • geometry
e.g. pacemakers, stent meshes, prosthetic valves, etc. • physics
• database
• Develop coupled physics-based tool for patient-specific risk assessment.
• Reduce animal testing, time for device development and failed clinical interventions.
Unified pre-processing, execution, and post-processing for electro-physiology,
Solution fluids and structures, incl. relevant physical coupling.
• A complete ecosystem for fast pre- and post-processing, optimization and a
full suite of design and analysis tools. (SpaceClaim, Discovery, Workbench Meshing)
• Accuracy: Implicit time integration solution for nonlinear material models for valves, Transmembrane
Bundle of
implants and surrounding soft tissues. Potential
Bachmann
• Robustness: Strongly coupled fluid-structure interaction (FSI) provides stable
solutions for coupled mechanical and hemodynamic simulations. Aortic valve

Benefits
• Strongly coupled multi-physics capability on a single code platform
enabling the solution of complex problems in a single run.
• Predict physiological response to device or intervention in silico. Mitral valve
Action
• Reduce the cost of physical testing.
Potential
Bundle of His

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Structural Design Optimization

An Introduction

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 Traditional Method of Design and Manufacturing

• Traditional way of manufacture has influenced the way products are

designed
• Experience drives design (a good thing)
• Traditional design made from Boolean operations such as subtraction,
considers the tooling needs and processes
• Leads to limitations in design and optimization benefits

[1] Vehicle upright – manufactured via CNC Milling

A process that
uses a physics
driven approach

Traditional Design Topology Optimized Design

[1] Muzzupappa, Maurizio & Barbieri, Loris & Bruno, Fabio. (2011). Integration of topology optimization tools and knowledge management into Best COUB OIC/YouTube
the virtual Product Development Process of automotive components. Int. J. of Product Development. 14. 14 - 33. 10.1504/IJPD.2011.042291.
39 ©2025 ANSYS, Inc.
"The art of structure is where to put the holes“
Robert LeRicolais (1894-1977)

Where would you first remove material from the beam

to make it lighter?

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"The art of structure is where to put the holes“
Robert LeRicolais (1894-1977)

Where would you first remove material from the beam

to make it lighter?

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"The art of structure is where to put the holes“
Robert LeRicolais (1894-1977)

42 ©2025 ANSYS, Inc.

Structural Optimization

The Pursuit for Lightweight, Cost-Efficient and High-Performing Products

/ Benefits / Specific Use Cases

• Improved performance • Additive Manufacturing Synergy
• Cost efficiency • Robotic Arms
• Weight reduction • Surface Texture of Body Panels
• Safety & Reliability • Ribs and Spars Dimensioning
• Sustainability • Aerodynamic Wing Profiles
• Design flexibility • Electronic Housings

/ Industries / Objectives & Constraints

• Aerospace • Minimize or maximize e.g. compliance,
• Automotive stress for a specific volume or weight
• Biomedical Engineering • Consideration of additional
• Civil Engineering manufacturing (milling, printing,
• Consumer Goods extrusion, etc.) and design (symmetry,
repetition, etc.) constraints.
/ “Evolution is to nature what optimization
processes are to the designer.“
- Neri Oxman
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Structural Optimization – Material Distribution Methods

/ Size Optimization
• Domain of the design is known a priori and
remains fixed
• Typical parametric optimization

/ Shape Optimization
• Goal: find the optimal shape of the domain
• Boundary manipulation

/ Topology Optimization
• The number, location, shape and size the of holes
and the connectivity of the domain are unknown
a priori
• Improves specific performance

44 ©2025 ANSYS, Inc.

At a Glance – Structural Optimization in Ansys

/ Shape Opt.

/ Topology Opt.
• Density-Based (SIMP)
• Level Set
• Mixable-Density

/ Topography Opt.

/ Lattice Opt.

45 ©2025 ANSYS, Inc.

General Information on Optimization
Characteristics of an optimization algorithm? Search mechanisms
i) Iterative, Numerical Guided random Enumerative
techniques search techniques techniques
ii) tries to improve the
Direct Indirect Heuristic
solution, approximatio
Evolutionary
algorithms Dynamic Branch-
method method n
iii) has a search mechanism and program and-
Newton Shooting Simulated Genetic ming Bound
iv) solves equilibrium condition method method annealing algorithm

Design Space:
i) Convex Convex Non-Convex
ii) non-convex

Design variable formulation:

i) Discrete vs Continuous material
distribution vs
ii) boundary variation Discrete Continuous Boundary variation

Search mechanism:
i) Deterministic vs Stochastic
ii) Gradient vs Non-gradient (e.g. Det. Gradient Descent Stoch. Gradient Descent Non-gradient*
heuristic) Particle swarm optimization - Wikipedia
46 ©2025 ANSYS, Inc.
Topology Optimization – General Formulation
Finding the “optimal” design within the
• The purpose of topology optimization is available scope
to find the optimal lay-out of a structure 1. What is our objective for “optimal” design?
within a specified region. 2. What are the available options & constraints?
3. How do we describe different designs?
• The only known quantities in the
problem are the applied loads, the
possible support conditions, the volume
of the structure to be constructed and Design Variable State Variable
possibly some additional design Objective function min → 𝑓 𝐬, 𝐮(𝐬)
𝐬∈ℝn
restrictions such as the location and size
Inequality constraints s. d. ℎ𝑖 𝐬, 𝐮(𝐬) ≤ 0 𝑤𝑖𝑡ℎ 𝑖 = 1, … , 𝑛ℎ
of prescribed holes or solid areas.
Equality constraints 𝑔𝑗 𝐬, 𝐮(𝐬) = 0 with 𝑗 = 1, … , 𝑛 𝑔
• In this problem the physical size and the Equality constraints 𝐬𝑚𝑖𝑛 ≤ 𝐬 ≤ 𝐬max
shape and connectivity of the structure
are unknown.
Martin P. Bendsøe, Ole Sigmund, Topology Optimization Theory, Methods, and Applications (2004).

47 ©2025 ANSYS, Inc.

What's behind? Explanation of the optimization methods
Optimization Problem

Unconstraints Constraints
12 12

10 10
𝑓 𝐬, 𝐮(𝐬)
8 𝑓 𝐬, 𝐮(𝐬) 8 ℎ 𝐬, 𝐮(𝐬)
6 6

4 4

2 2 𝑔 𝐬, 𝐮(𝐬)
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6

min → 𝑓 𝐬, 𝐮(𝐬) min → 𝑓 𝐬, 𝐮(𝐬)

𝐬∈ℝn
𝐬∈ℝn
s. d. ℎ𝑖 𝐬, 𝐮(𝐬) ≤ 0 𝑤𝑖𝑡ℎ 𝑖 = 1, … , 𝑛ℎ

𝑔𝑗 𝐬, 𝐮(𝐬) = 0 with 𝑗 = 1, … , 𝑛 𝑔

𝐬𝑚𝑖𝑛 ≤ 𝐬 ≤ 𝐬max

48 ©2025 ANSYS, Inc.

48
Topology Optimization – Objectives & Constraints
• Objective function can be defined as
‐ Minimizing compliance (maximizing the stiffness)
‐ Maximizing natural frequencies
‐ Minimizing stress
‐ …
Topology Optimization
• Design constraints could be:
‐ Maximum weight, Maximum volume
‐ Maximum stress, maximum deflection/rotation of a
point
‐ Maximum cost
‐ Manufacturing constraints (e.g. support angle for AM)
‐ …
• Design variables & state variables could be:
‐ Density of each element of the structure
‐ Thickness of each element of the structure
‐ Elemental strain energy of each element
‐ …

49 ©2025 ANSYS, Inc.

SIMP vs Level-Set Method
SIMP Level-Set
• Mature, 30-year-old technology • Relatively new, 10-year-old technology
• For each finite element, a pseudo-density is
introduced as design variable. A value of 1 describes • This method directly deals with the boundary of the
that an element is active, 0 refers to inactive shapes during the optimization process.
elements.
• Optimized geometry is obtained by the shape sensitivity
• Intermediate densities are penalized using power law and topology sensitivity of the compliance.
to achieve a clear near 0-1 solution.
• Level set function describing structure in domain
• Weighting of the pseudo densities with a global (evolution equation = Hamilton-Jacobi).
penalty parameter transforms the discrete
optimization problem into a continuous problem
→Smooth and watertight unambiguous
→Boundaries/contours are not well defined boundaries/contours, facilitating interpretation of
optimized shapes
→Inertial and surface loads difficult to apply
→Mass, modal or response constraints are easy to apply
→Mass is ambiguous and response and modal analysis
require workarounds →Surface loads and manufacturing constraints (size &
thickness) are easy to handle
→Manufacturing constraints are hard to handle

50 ©2025 ANSYS, Inc.

SIMP vs Level-Set Method
SIMP Level-Set
• Mature, 30-year-old technology • Relatively new, 10-year-old technology
• For each finite element, a pseudo-density is
introduced as design variable. A value of 1 describes • This method directly deals with the boundary of the
that an element is active, 0 refers to inactive shapes during the optimization process.
elements.
• Optimized geometry is obtained by the shape sensitivity
• Intermediate densities are penalized using power law and topology sensitivity of the compliance.
to achieve a clear near 0-1 solution.
• Level set function describing structure in domain
• Weighting of the pseudo densities with a global (evolution equation = Hamilton-Jacobi).
penalty parameter transforms the discrete
optimization problem into a continuous problem
→Smooth and watertight unambiguous
→Boundaries/contours are not well defined boundaries/contours, facilitating interpretation of
optimized shapes
→Inertial and surface loads difficult to apply
→Mass, modal or response constraints are easy to apply
→Mass is ambiguous and response and modal analysis
require workarounds →Surface loads and manufacturing constraints (size &
thickness) are easy to handle
→Manufacturing constraints are hard to handle

51 ©2025 ANSYS, Inc.

Topology Optimization Methodology – SIMP
‐ Assumes continuous elemental densities for the structure –
Design variables are elements’ relative densities.
‐ Intermediate densities are penalized using power law to
achieve a clear near 0-1 solution.
‐ Objective is to minimise Compliance (total strain energy), i.e.
maximise stiffness
SIMP (penalised)
Penalization
1
0,8
0,6
0,4
0,2
0
0 0,2 0,4 0,6 0,8 1
𝐸(𝜌) = 𝐸 × 𝜌𝑃𝑒𝑛𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛
SIMP (unpenalised)
Note: The power-law approach is physically permissible as long as simple conditions on the power are satisfied (e.g. p ≥ 3 5
for Poisson’s ratio v=0.3) → H-S bounds → you will get an error in Discovery if you choose a material like rubber with v=0.49. 2
52 ©2025 ANSYS, Inc.
 Topology Optimization Methodology – Level Set Method
This method directly deals with the boundary of the shapes during the optimization
process.
Optimised geometry is obtained by the shape
sensitivity and topology sensitivity of the compliance

Level set function describing structure in domain

(omega)
Cartesian Grid

Surface
Level set function
Evolution equation
discretised with grid-
(Hamilton-Jacobi)
points (centre of element)
53 ©2025 ANSYS, Inc. Structural and Multidisciplinary Optimization (2014), 51(5): 1159-1172
Lightweighting Strategies in Additive Manufacturing

Topology Optimization Latticing

Goal: Stiffness/strength-optimal Goal: Lightweight +

designs (lightweight in the ‘strict’ multifunctional properties
sense) (lightweight in the ‘loose’ sense)

→ strength-to-weight ratio, energy

→ high specific stiffness/strength absorption, dissipation of heat and
→ Reduced energy and material usage vibration
→ Reduced energy and material usage
→ Improved warpage in metal-AM

54 ©2025 ANSYS, Inc.

Topology Optimization vs Generative Design
Goal
Topology Optimal design for specific Generative
requirement
Optimization Design

• Simply removal of material until most • Optimization of material lay-out (TO is

efficient design for objective and generally underlying GE) under consideration
(volume/mass) constraint is achieved but not limited to manufacturing constraints,
→ Single design material properties and other design
requirements
→ Portfolio of designs
Method for structural
optimization
• Often automated process with a dedicated GE
design engine (e.g. linked with ML/AI)
→ up-front design exploration & feedback

Enabler for digitalization

CAD/product design

55 ©2025 ANSYS, Inc.

At a Glance – Generative Design Optimization in Ansys

Generative
Design

Real-time
decisions

Explore
possibilities

57 ©2025 ANSYS, Inc.

Jet Engine Bracket Generative Design
Diverse Training Data

prediction confidence
95%
Challenge
• Coming up with new jet engine bracket designs as quickly as Displacement
possible that meet structure requirements and constraints.
Von Mises stress
• Moreover, it is important to make use of existing designs that have
accumulated over years, retaining the knowledge from previous
50 100 250
projects.
training samples
Discovery SimAI

Solution
• Up to 250 training samples, with different and topologically diverse bracket
designs, were run to build a global AI model. (SimAI, optiSLang, Discovery)
• Based on knowledge from previous projects and simulations, optiSLang can Unseen Shape Verification
Displacement
predict a new shape, while SimAI shows you its physical behavior in less than
a few seconds. The AI model can be extended and retrained with new designs
at any time to capture additional physical behavior. (optiSLang, SimAI) Discovery SimAI

Benefits
• 0 scraps, use all the designs you have from the past to move forward,
even a bad design is a source of knowledge.
• Reduce your current workflow by 90%, go directly from CAD to physics.
• Use 100% of your team and let AI guide them with new ideas, no deep
knowledge of physics required.
Von Mises stress

58 ©2025 ANSYS, Inc.

Topology Optimization in Ansys Discovery
Level-Set Topology Optimization
Objectives
Stiffness | Strength | Natural Frequency | Balance
Stiffness and Frequency | Excess Material
Manufacturing Constraints
Min./Max. thickness | Pull Direction| Table Direction|
Overhang Prevention
Protected Depth
Exclusion regions
Multiple Loading Conditions
Concurrent design optimization

59 ©2025 ANSYS, Inc.

Topology Optimization in Ansys Mechanical
LS & SIMP Topology Optimization
Response Constraints
Volume | Mass | Centre of Gravity | Moment of Inertia
| Compliance | Displacement | Reaction Force | Global
Stress | Local von Mises Stress | Natural Frequency |
Dynamic Compliance | Thermal Compliance |
Temperature | Custom Criterion
Manufacturing Constraints
Member Size | Pull Direction | Extrusion | Overhang
Prevention | Housing
SIMP Level-Set Design Constraints
Cyclic Repetition | Symmetry| Uniform | Pattern
Repetition
Protected Depth
Exclusion regions
Multiple Loading Conditions
Concurrent design optimization

60 ©2025 ANSYS, Inc.

 Topology Optimization in Ansys Mechanical
Symmetry and Cyclic Minimize Mass under Frequency Constraint Pull-Direction (no Hole)

Pattern repetition

Housing

61 ©2025 ANSYS, Inc.

 GE-Bracket Challenge – Structural Optimization
Aircraft Engine Bracket Challenge
• Minimization of weight
• 4 load cases
• Ti-6Al-4V

[1]

[1] - Carter, W.T., Erno, D.J., Abbott, D.H., Bruck, C.E., Wilson, G.H., Wolfe, J.B., Finkhousen, D.M., Tepper, A., Stevens, R.G.: The GE Aircraft Engine Bracket Challenge:
An Experiment in Crowdsourcing for Mechanical Design Concepts
63 ©2025 ANSYS, Inc.
GE-Bracket Challenge – Structural Optimization

Load Case 4 Load Case 1-4

Load Case 3 Load Case 1

Load Case 2

64 ©2025 ANSYS, Inc.

GE-Bracket Challenge – Structural Optimization
Problem Definition Topology Optimisation Smooth boundary representation Convert to CAD geometry Validation

CAD
…

65 ©2025 ANSYS, Inc.

GE-Bracket Challenge – Structural Optimization

Load Case 4 Load Case 1-4

Load Case 3 Load Case 1

Load Case 2

66 ©2025 ANSYS, Inc.

Topology Optimization Workflow in Mechanical

67 ©2025 ANSYS, Inc.

Topology Optimization Workflow in Mechanical

69 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints

70 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints

/ Static / Thermal / Modal / Harmonic

Compliance & Environmental Modal | max. Force |
Stress Temperature & 1st Frequency 0-5000
Heating through Hz
movement

71 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints

Combine different Assign from Give weight to Assign from different

type of objectives different analysis each objective results in each analysis

72 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints

What it the best design regarding mass-savings, CO2-emission or Cost-reduction?

Optimization driven by…

CO2-EMISSION
 min ( alu Massalu +  steel Masssteel )
( alu , steel )
( P1 )  uB  1.5e−5m
+plane symetry

( alu , steel ) = 1
COST

MASS

The systems contains two parts

made of Aluminum and Steel

74 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints

75 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints
Pull-Out
Housing

No Manufacturing Constraints

No Housing Constraint

✓ enables you to deliver a

✓ enables you to create design that is compatible
watertight designs that with the casting process
enclose a given set of by preventing the
faces formation of undercuts
and internal holes

With Housing Constraint Pull Out Direction in Both Directions

77 ©2025 ANSYS, Inc.

Topology Optimization Objectives & Constraints
Complexity Index
AM Overhang

No AM Overhang Constraint no Complexity Index

With 45 degree AM Overhang Constraint

With Complexity Index value 2.0

✓ aims to create self-supporting designs that prevent the

✓ enables you to minimize the creation of overly complex
formation of these overhanging regions.
structures
78 ©2025 ANSYS, Inc.
Topology Optimization Objectives & Constraints
Member Size
Extrusion

Member Size Minimum Member Size Maximum Member Size Gap Size
No Extrusion Constraint

✓ Minimum member size prevents the formation of thin features in

the optimized shape
✓ Maximum member size property prevents the formation of thick
features in the optimized shape
With Extrusion Constraint
✓ Gap Size constraint prevents the formation of closely spaced
✓ aims to create uniform cross-section design. features in the optimized shape

79 ©2025 ANSYS, Inc.

Topology Optimization – General
Considerations

©2025 ANSYS, Inc.

Topology Optimization – General Considerations

Tech Tips
L-Bracket Example

Will the topology be (a) the same

or (b) different for a low and high
mesh size?
vs

81 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

L-Bracket Example

vs

82 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

L-Bracket Example

Plastic Steel

Will the topology optimized solution for the two

different materials (a) change or (b) remain the
same?
vs

83 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

L-Bracket Example

Plastic Steel

vs

84 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations
Different materials … 1 𝜈 𝜈
− − 0 0 0
𝐸 𝐸 𝐸
→ different Young’s Modulus E and Poisson’s Ratio 𝝂 𝜈 1 𝜈
𝜀𝑥𝑥 − − 0 0 0 𝜎
→ different 4th order elasticity tensor 𝐸 𝐸 𝐸 𝑥𝑥
→ different elastic behavior 𝜀𝑦𝑦 𝜈 𝜈 1 𝜎𝑦𝑦
𝜀𝑧𝑧 − − 0 0 0 𝜎
→ different compliance = 𝐸 𝐸 𝐸 𝑧𝑥
𝛾𝑦𝑧 1 𝜎𝑦𝑧
𝛾𝑧𝑥 0 0 0 0 0 𝜎𝑧𝑥𝑥
𝐺
𝛾𝑥𝑦 1 𝜎𝑥𝑦
0 0 0 0 0
𝐺
1
0 0 0 0 0

min von Mises Stress max

𝐺

𝑬
𝑮=
𝟐(𝟏 + 𝝂)

0 𝝂 0.5

85 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

L-Bracket Example

vs vs

86 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

L-Bracket Example

87 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

Manufacturing
Constraint:
Table Direction (Y)

Ensure realistic
constraints are
applied → Sanity
Check

Y-Direction

88 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

Brake Pedal Example

Will a (a) larger or (b)

smaller design space
yield a higher
performance (e.g. lower
compliance)?

vs

89 ©2025 ANSYS, Inc.

Topology Optimization – General Considerations

Brake Pedal Example

vs
2 1

90 ©2025 ANSYS, Inc.

Structural Optimization in Action

©2025 ANSYS, Inc.

At a Glance – Structural Optimization Application

Unsp
rung
& Rot
atio
nal
Ma
ss


1.6 kg Equal Stiffness

-45% Weight
<1 kg
Developing a Hybrid CFRP Wheel With Ansys Software

92 ©2025 ANSYS, Inc.

At a Glance – Structural Optimization Application

Harmonic analysis to increase the

“dynamic stiffness”

-71% Stress
From plastic to elastic

12x Stiffness
93 ©2025 ANSYS, Inc.
At a Glance – Structural Optimization Application

289 MPa 195 MPa

Manifold Original Shape

Max. Stress 1 [MPa] 289 195
229 MPa
262 MPa Max. Stress 2 [MPa] 262 229
Improvement [%] 32 / 12

/ Shape Optimization
• Easy setup
• Only meshing parts
• Program Controlled Settings
• Best practice post-processing of
TO-Solution -32% Stress
From plastic to
elastic

< 15 min

94 ©2025 ANSYS, Inc.

At a Glance – Structural Optimization Application

Alcoa Bracket Original Lattice

Weight [g] 859 448
Max. Stress [MPa] 855 821
Max. Disp. [mm] 1.17 2.15
Saving [%] 52

/ Lattice Optimization
• Easy & fast validation
• Automatic generated & applied variables on
initial geometry
• Solid material is substitute by beams
• Outer shape doesn’t change -52% Weight
• Knockdown factors are a fast approach for Maintain Max
validation, properties of lattices are mapped Displacement
onto solid elements
< 2 min

95 ©2025 ANSYS, Inc.

Simulation-Driven Bell Crank Design Optimization for LPBF
RUNNING SNAILS RACING TEAM & ACONITY 3D GMBH

Structural Optimization Validation/Fatigue AM Process Simulation LPBF Print

“GPU-based topology optimization
turned out to be a game changer,
accelerating our virtual prototyping
time by 3x over conventional CPU-
based FEA, helping us to build
confidence in choosing the best
possible design and setup prior to
sending the parts to production.”

— Markus Hofmann and Luis Atzenhofer,

Mechanical Engineers, Running Snail Racing Team

-40% Weight* +75% Stiffer*

* Compared to milled design

96 ©2025 ANSYS, Inc.

Simulation-Driven Bell Crank Design Optimization for LPBF
RUNNING SNAILS RACING TEAM & ACONITY 3D GMBH

Simulation-driven Design Optimization of a Formula Student Bellcrank for Additive Manufacturing

97 ©2025 ANSYS, Inc.

 Combining additional Ansys solutions
Lattice

Structural Optimization

Strength assessment - nCode

Additive Suite
Ansys Motion

98 ©2025 ANSYS, Inc.

 Ansys Innovation Space

Curated knowledge materials, FAQs, tutorials and

more

©2025 ANSYS, Inc.

 Ansys Innovation Space

Learning and community platform designed for students, educators and engineering professionals to enhance their
engineering knowledge with free learning resources

/ Ansys Innovation Courses / Ansys Marketplace

• Hundreds of free, award-winning • Certify your knowledge by investing in
courses on engineering and product Course Completion Badges and Ansys
topics spanning many physics Certifications on various topics and
categories. levels.

/ Ansys Learning Forum / Ansys Educator Hub

• The go-to place for engineering and • The first stop for educators seeking
simulation questions and answers. resources that support teaching
Engage one to one with Ansys experts with simulation. Resources are
and peers. organized by physics areas and
specialized topics.

/ And more!

100 ©2025 ANSYS, Inc.

Free Learning Resources at your Fingertips

• Free, online physics and engineering courses

• Include video lectures, accompanying handouts,
simulation exercises, quizzes
• Perfect for use with our free student download

Visit ansys.com/courses to learn more

101 ©2025 ANSYS, Inc.

Ansys Certifications and Course Completion Badges
Learning products design with the power of engineering simulation
made for students & professionals

/ Certifications / Course Completion Badges

• Offered in fluids, structures, and electronics • Offered for all Ansys Innovation Courses (AIC)
• Two levels: Associate and Professional Certification • Proof of successful completion of AIC and Tracks
• Successful completion comes with digital certification
• Links to description of credentials earned
• Easily sharable on professional networks and resume
• Easily sharable on professional networks and resume

102 ©2025 ANSYS, Inc.

Ansys Resources on Structural Optimization

Setup

Fine-Tuning Efficiency

103 ©2025 ANSYS, Inc.

Ansys Innovation Space – Educator Hub (AERs)

Ansys Education Resources – Teaching

Materials

104 ©2025 ANSYS, Inc.

Education Resources (AERs) Related to FEA Fundamentals

105 ©2025 ANSYS, Inc.

 Industrial Example of DfAM with Ansys
Repairing Bone Loss with Simulation-Generated, Patient- Real-Time Generative Design Drives Innovation | Ansys
Specific Implants | Ansys

Predator Cycling Optimizes the Cycling Experience With Ansys Topology Optimization | Lightweighting & Shape
Simulation (ansys.com) Optimization

106 ©2025 ANSYS, Inc.

Q&A

©2025 ANSYS, Inc.