design optimization

1. Design Optimization under
Uncertainty: Status & Promises
Dr. Wei Chen
Integrated Design Automation Laboratory (IDEAL)
Department of Mechanical Engineering
Northwestern University
Panel Presentation
7th World Congress on Structural and Multidisciplinary Optimization
May 24th, 2007

2. ( )
1 2
, , , n
y g x x x
= L
1
x
2
x
n
x
M
Moments
(mean, variance,…)
Reliability
Distribution
Uncertainty
quantification,
representation
Analysis model
Uncertainty
propagation and
estimation
Design scenarios
Robust design
Reliability
Based Design
(RBDO)
Different Aspects of Design under Uncertainty

3. Design Scenario
Robustness
0
Probability
Density (pdf)
Performance y
Target M
Bias
Minimizing the effect of variations
without eliminating the causes
μy
σy σy
Performance g
R=Area = Prob{g(x)≥c}
Reliability
pdf
To assure proper levels of
“safety” for the system designed
C
24% 76%

4. • Probabilistic (distribution function, random field)
• Non-probabilistic
– Possibility-based (fuzzy set)
– Interval analysis
– Evidence theory
• Aleatory (stochastic, irreducible) vs. Epistemic
(reducible)
– reliability interval, Bayesian reliability
Uncertainty Representation & Quantification
89%
3%
3%
5%

5. Uncertainty Propagation & Estimation
• Local expansion based methods
– Taylor series method
• Simulation based method
– Monte Carlo Simulation
– Importance sampling, stratified sampling, adaptive sampling,…
• MPP (Most probable point based methods)
– FORM (first order reliability method)
– SORM (second order reliability method)
• Numerical integration based method
– Full factorial numerical integration (tensor product quadrature)
– Dimension reduction method
• Functional expansion based method
– Polynomial chaos expansion method
• Metamodeling based method (Kriging, Radial Basis Function,
moving least square, etc.)
6%
3%
19%
37%
3%
30%
One paper on comparative study 3%

6. Applications
• Structure
• Simultaneous structure & aerodynamics
• Manufacturing process
• Medical
• Automotive & aerospace
• Military (undersea vehicle, penetration
warhead
7%
7%
83%
3%

7. Research Trends/Emerging Topics
• Hybrid method (MPP+DRM, metamodeling+DRM)
• Nonlinear system (multi-critical points, convergence control)
• Asymmetric, non-Guassian, correlated random variables
• System reliability using hierarchical multi-level
optimization/probabilistic target cascading
• Confidence and prediction interval of using RSM for
RBDO/Uncertainty of using multiple fidelity models
• Multiobjective optimization for making tradeoffs (mean vs.
variance, multiple performance, reliability vs. others)
• Introduction of parallel computing

8. Future Opportunities
•Nano-, bio-engineering
•Incorporating model, data, algorithmic
uncertainties
•From reliability analysis to reliability-
based design
•MDO under uncertainty
•Integration with rigorous decision
making framework

9. Optimization under Uncertainty in Nanoengineering
(Olson 1997)
Atomic
scale
Submicro
scale
Micro
scale
Macro scale
Component
Assembly
Product
meter
10 -10 10 -9 10 -6 10 -3 10 -2 10 -1 1.0 10.0
Material
How to propagate the effect of
stochastic random material
microstructure on product
performance (fatigue, fracture,
creep etc.)?

10. Incorporating Model, Data, Algorithmic Uncertainty
( )
1 2
, , , n
y g x x x
= L
1
x
2
x
n
x
M
Moments
(mean, variance,…)
Reliability
Distribution
Uncertainty
quantification,
representation Analysis model
Uncertainty
propagation and
estimation
( ) ( , ) ( ) ( ) ( )
e m
y y ε δ ε
= + + +
x x θ x x x
ｈ
( )
ε x
( )
δ x - Bias (Modeling error) Validation
( )
h
ε x - Numerical error Verification
- Experimental error Validation
experiment computation
θ - Uncertainty in model parameter Validation/Calibration

11. From Reliability Analysis to RBDO
RBDO Methods
Using FORM
Double Loop
RBDO
Serial Loop
RBDO
Single Loop
RBDO
PMA, RIA SORA SLSV, MVFOSM

12. MDO under Uncertainty
Targets
Design of Top-Level Subsystem
Design of Subsystem A Design of Subsystem B
X
A
X A
Y B
Y B
x
R0
RA,YA RB,YB
…
…
In decomposition, how should we match probabilistic characteristics
between different subsystems?
Non-hierarchical
Hierarchical

13. • Without uncertainty
– objective function f = V(Y) = V(Y(X))
V - value function, e.g. profit
• With uncertainty
– objective function f =
E(U) - expected utility. The preferred choice is the alternative
(lottery) that has the higher expected utility.
Integration with Rigorous Decision Making Framework
pdf (V)
V (e.g. profit)
A B
U (V)
V
1
0
worst best
Risk neutral
Risk averse
Risk prone
∫
= dV
)
V
(
pdf
)
V
(
U
)
U
E(
What is the cost
of failure?