1.

Introduction
1.1 What is Design Optimization?
Design Optimization = finding „best“ design within available means
▪ Criterion → objective (e.g. minimize mass)
▪ Available means → bounds, constraints (e.g. geometry, strength, manufacturing)
▪ Distinction → design variables (e.g. component thickness, exact shape, …)
Types of Optimization
▪ Size → Indirect change of geometry by design variables like A
▪ Shape → Direct change of geometry
▪ Topology → Change of geometry and connectivity
Examples:

1.2 What is Multidisciplinary Design Optimization?

MDO = Finding best design within available means, considering more than one discipline
Various disciplines involved in design task:
▪ Strength, durability, vibrations, acoustics (associated with structural mechanics/ materials)
▪ Dynamics and control (associated with rigid body dynamics)
▪ Aerodynamic performance (associated with fluid mechanics)
➔ MDO integrates disciplines of same formal structure to seek overall best design
Examples:

Complexity of design problems: MDO used to find good/acceptable solution

▪ Complex = system behavior dominated by interaction of parts
▪ Complexity increases with number of
o Design goals (from diff. disciplines)
o Design variables
o Teams with distributed responsibilities
o Variants
▪ Design variables from many components affect many disciplines → connections is complex

1.3 Definitions
Universal mathematical language for interdisciplinary design work
 ▪ Objective function and constraint functions are computed from quantities of interest
▪ h(x)=0 is called equality constraint; g(x) <= 0 is called inequality constraint

Compact Notation:

2. Fundamentals of Optimization
2.1 Problem Formulation
Examples see script.

2.2 Characteristics of Optimization Problem

2.2.1 Well-Posedeness
▪ Problem is well-posed when a unique solution exists

2.2.2 Convexity
▪ A function 𝑓(𝑥) is convex, if for every two points 𝑥1 and 𝑥2 the connecting line lies above the graph
▪ Convexity is useful because local minimum = global minimum

2.2.3 Linearity
In this course, a function 𝑓(𝒙) is called linear, if it can be written as: 𝑓(𝒙) = 𝛽0 + 𝛽1𝑥1 + ⋯ + 𝛽𝑑𝑥𝑑

➔ Distinguish between linear/nonlinear system equations and the corresponding optimization problem!

2.2.4 Monotonicity

Monotonicity ≠ Linearity, but linearity implies monotonicity

 2.2.5 Continuous vs. Discrete Optimization

→ Here in class only continuous optimization methods

2.2.6 Closed-form Solution vs. Numerical Methods

3. Design Space Exploration

Many different techniques to do optimization

Trouble makers:

▪ Black box
▪ Non-linearity
▪ Dimensionality

3.1 Formula Analysis

Graphical visualization of simple dependencies possible
→ Problem, when >2 design variables / complex connections → how to visualize/ make intuitive

3.2 Sampling
▪ Sample is a subset of whole solution, vector of design vectors (matrix) /w output values [xa,ya]
▪ Designs are chosen according to sampling method = Design of Experiment (DoE)
▪ Design = specified design vector
▪ Sampling used to probe design space → useful for black-box functions (only input/output needed)

3.2.1 Full Factorial Design

▪ Design variables xa assume values from discrete levels
▪ Levels are equally spaced and number of levels is equal for all design variables
▪ All combinations of variable values are evaluated/computed
▪ Number of sample points: N = levelsd → d = number of design variables

3.2.2 Monte-Carlo Design

▪ Relies on random sampling to explore Design Space
▪ Number of sample points can be chosen
→ useful, when function evaluation is expensive or sample data incomplete

3.2.3 Other Sampling Methods

▪ More advanced methods: smart distribution
of sample points → higher efficiency
▪ Thumb rule: Full Factorial very expensive,
Monte Carlo cheaper and simple. More
advanced techniques may improve results
→ Monte-Carlo recommended
 3.2.4 Effects in High Dimensions
▪ All sample points have similar distances to each other when d becomes large
→ difficult to classify good and bad designs

▪ Regions of interest become extremely small wrt design space

→ difficult to find regions of good designs when all variables are relevant & gradient info not available
→ loss of information when you zoom in

3.3 Sensitivities
▪ Measure variability of output, depending on input
▪ Measure importance of design variable
o Correlation coefficient
o Gradient
o Regression Coefficient

3.3.1 Correlation
How to quantifiy correlation?

Quantities:

▪ Mean
▪ Standard deviation
▪ Covariance
▪ Pearson correlation coefficient
→ quantifies degree of relation between two variables as linear function

3.3.2 Gradient
▪ For functions that can be expressed analytically, the gradient can also be expressed analytically as a function
of the design variables. This is not possible for black box functions
▪ For black box functions, the derivatives can be approximated numerically, e.g., by central finite differences

3.3.3 Linear Regression

▪ Regression coefficient and correlation coefficient are related by:

3.3.4 Numercial Example
▪ Several measures for sensitivity
▪ Distinguish between global and local sensitivities:
correlation and regression coefficients = global,
gradient = local
▪ In order to compare sensitivities of design variables,
you need to normalize somehow, typically by range
of interest.

4. Basic Mathematics
4.1 Standard Forms
▪ Negative Null Form

▪ Alternative Formulations

4.2 Optimality Conditions

▪ Interior Optimum
▪ Boundary Optimum

4.2.1 Interior Optima

▪ When 𝛻𝑓 𝒙* = 0, 𝒙∗ is called a stationary point
▪ Nature of stationary point:

▪ Criteria for positive definiteness:

o All eigenvalues positive
o All determinants of leading principal minors are positive

4.2.2 Boundary Optima

4.2.3 Lagrangian Function
- Lagrangian Function

- KKT Conditions

4.3 Basics of Calculus of Variations

- Consider problem of infinite number of design variables
 5. Optimization Algorithms

➔ Calculations see Excercises

6. Many Objectives and Disciplines

MOO = multiobjective optimization:

▪ 𝑚 > 1, more than one optimization objective

▪ Typically, optimization objectives are in conflict with each other.
▪ Challenge: how to quantify designer’s preference?

6.1 Pareto Optimization

▪ Pareto front in u-m-plane: set of non-dominated solutions
▪ A solution 𝒙∗ is non-dominated, if there is no other point 𝒙 with ∀𝑗: 𝑓𝑗 𝒙 ≤ 𝑓𝑗 𝒙∗ with 𝑗 = 1, … , 𝑚

▪ Also works for many design variables

▪ Advantage of Pareto optimization:
• Provides full information to designer
• No need for weighting
▪ Disadvantage:
• Not straightforward to compute
• Visualization limited to 2d (with more effort to 3d)
▪ Algorithms:
• Normal-Boundary Intersection (NBI)
• Nondominated sorting genetic algorithm II
 6.2 Objective Meta Functions

7. Solution Space Optimization

7.1 Robustness and Reliability
▪ Robustness = Ability to tolerate perturbations
▪ Reliability = Ability to perform function under perturbations
➔ How to treat design variables that vary on an interval U(x)?

7.1.1 Possibilistic View

▪ Optimize worst case → optimum moves away from non-robust location
▪ Replace objective and constraint functions
▪ May be too conservative
▪ One type of robust design optimization = RDO

7.1.2 Probabilistic View

▪ Optimize probabilities to satisfy constraints/requirements 𝑔 𝑥 ≤ 0 / 𝑓 (𝑥) ≤ 𝑓𝑐
▪ one type of reliability-based design optimization = RBDO

OR

▪ Optimize expectation and variance of objective function

▪ Weighting required to balance expectation and variance
▪ Another type of robust design optimization = RDO

7.1.3 Required Input for RDO & RBDO

▪ Often, in early phase design, only (1) is known.

▪ (2) is typically unknown or expensive/time-consuming to find out.
▪ (3) is subjective → tedious to specify and limited acceptance in design projects, robustness = € !
➔ New approach: use (1) as input for optimization and compute permissible variability or perturbation as
output: Solution Space Optimization
 7.2 Solution Spaces

7.3 Optimization Techniques

7.3.1 Stochastic Iteration

Good: Very robust, No gradient necessary, Applicable to high-dimensional problems with up to 100 design variables

Bad: Neighborhood search, no mechanism to identify global optima

Good & bad: Tends to overestimate box size

7.3.2 Corner Tacking

▪ For linear requirements/constraints
▪ Key idea:
(1) Every requirement is assigned one corner of the solution box.
(2) The optimization problem can then be reformulated as a standard non-linear optimization problem with
linear constraints

Good: Fast

Bad: Limited applicability (linear output functions, extendable to monotonous output

functions)

Good and bad: Conservative = 100% purity (no bad designs included) → small boxes
 7.3.3 Selective Design Space Projection

→
▪ Look at each parameter and project solution onto space to receive solution box

Good: Typically used in an interactive tool → trains intuition for high dimensional problems, Designer can directly
include not formalized knowledge about constraints

Bad: Expensive 1000-10000 function evaluations per diagram,Confusing when many design variables are relevant,
Cannot find good solution in high dimensions

Good and bad: Statistical assessment of purity (bad designs may be included) → large boxes boxes

8. Systems Design and Optimization

8.1 Challenges of System Design
▪ Complexity due to number of…
- Variants (e.g. vehicle design)
- Designers (e.g. flow of information between stakeholders)
- Design Goals
- Interacting Components/ design variables (may lead to counterintuitive system behavior)
- Uncertainty (makes dev. Process complex), e.g. conflicting requirements, difficult requirements,
aleatoric uncertainty, distributed engineering processes

8.2 Designing Systems

▪ systems engineering = a methodical, multi-disciplinary approach for the design, realization, technical
management, operations, and retirement of a system
▪ system = combination of elements that function together to produce the capability required to meet a need

8.2.1 V-Model
▪ Dev. Procedure for complex systems/ integration of disciplines
e.g. for automotive / aerospace

▪ Bottom-up difficult, when many engineers try to adjust many design

variables for many design goals
→ Systematic dev. of requirements according to V-Model
→ new issue: finding quantitative and realistic requirements due to uncertainty
 8.2.2 Target Cascading

▪ Quantitative implementation of V-Model: Break down top-level

requirements into sub-system requirements to enable parallel design activity
▪ Sub-system requirements ensure satisfaction of overall design goal
▪ Break down sub-system requirements further into sub-sub-system and then
into component requirements

No treatment of uncertainty → coordination necessary

▪ Between levels (vertical) to align feasibility (bottom-up view) with requirements (top-down view)
▪ Between branches (horizontal) for common variables

8.2.3 Solution Spaces

▪ Iterative development with one design causes conflict of goals / lack of robustness
▪ Solution spaces integrate requirements from different disciplines

8.3 Cooperative Design Framework

▪ Includes a collection of tools for top-down development of complex systems

8.3.1 Dependency Graphs

▪ Many paths of dependencies and cycles in complex systems
▪ Difficult to find causes to problems → avoid circular dependencies
▪ dependency graphs model physical dependencies
▪ helps to (1) avoid circular dependencies and (2) decompose systems
using top-down mappings
 8.3.2 Bottom-up Mappings
▪ Used to assess the performance of one design
▪ Depend on discipline
▪ May be a surrogate model that takes aggregated input instead of detailed input
→ great for concept development

8.3.3 Top-down Mappings

▪ Map permissible performance values onto regions of design variables = many designs
▪ Need to carefully balance (1) decoupling and (2) loss of solution space

8.4 Two views on Design

8.5 Current Research for Solution Space Improvement

Question: how to make solution spaces larger?
 8.5.1 2D Spaces
▪ Product of 2d spaces larger than product of intervals
▪ Volume increase by 1-2 orders of magnitude
▪ Stochastic iteration with box rotation:

→ Key ingredient: Design variables are pair-wise coupled and pairs are rotated

8.5.2 Solution-Compensation Spaces

▪ Early decision variables xa
▪ Late decision variables xb
▪

Powered by TCPDF (www.tcpdf.org)