Northwestern University

McCormick School of Engineering

Robust Shape &Topology Optimization under Uncertainty

http://ideal.mech.northwestern.edu/

Prof. Wei Chen Wilson-Cook Professor in Engineering Design

Integrated DEsign Automation Laboratory (IDEAL)

Department of Mechanical Engineering

Northwestern University

Based on Shikui Chen’s PhD Dissertation

Presentation Outline

1. Topology Optimization

Level set method

Challenge for topology optimization under uncertainty

2. Robust Shape and Topology Optimization (RSTO)

Framework for RSTO

RSTO under Load and Material Uncertainties

RSTO under Geometric Uncertainty

3. Examples

4. Conclusion

Topology Optimization

Size Shape

Topology

What is Topology Optimization?

Why do topology optimization?

• Able to achieve the optimal

design without depending on

designers’ a priori knowledge.

• More powerful than shape and

size optimization.

• A technique for optimum

material distribution in a given

design domain.

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~ 4 ~

Applications of Topology Optimization

Aircraft Structure Design

(Boeing, 2004)

Light Vehicle Frame Design

(Mercedes-Benz, 2008)

MEMS Design

Micro structure of

composite material

Most of the sate-of-the-

art work in TO is

focused on deterministic

and purely mechanical

problems.

~ 4 ~

pressure distribution on the

upper wing surface

Material Uncertainty

Load Uncertainty

Topology Optimization: State of The Art

Homogenization

(Bendsoe & Kikuchi, 1988) Ground Structure Method

SIMP

(Rozvany, Zhou and Birker,

1992)

E E0p

Solid Isotropic Material

with Penalisation (SIMP)

- power law that

interpolates the Young's

modulus to the scalar

selection field

( ) 0x

( ) 0x

( ) 0x

( ) 0x

( ) 0x

( ) 0x

D

Dynamic Geometric Model: Level Set Methods

Implicit representation

Benefits

– Precise representation of boundaries

– Simultaneous shape and topology opt.

– No chess-board patterns

– Accurate for geometric variations

( ) 0 \

( ) 0

( ) 0 \

x x

x x

x x D

( ) 0nV xt

Hamilton-Jacobi Equation

~ 6 ~

(M. Wang et al., 2003)

Formulation for Robust Shape and Topology Optimization

robust X

f (X)

[ , ]

. . 0

f f

g g

minimize

s t k

(Chen, 1996) Robust Design Model

* ( , , ) ( ( , , )) ( ( , , )

:

,

,

obj

Minimize

J u z J u z k J u z

Subject to

Volume constraint

Perimeter constraint on

( )

( )

D

N

div in

on

on

σ u f

u 0

σ u n g

Challenges in RSTO: • Modeling and propagation of high-dimensional

random-field uncertainty • Sensitivity analysis for probabilistic

performances

~ 7 ~

Random Variable and Random Field

A realization of a weakly correlated random field

A realization of a strongly correlated random field

X

k

A random variable

~ 8 ~

Framework for TO under Uncertainty

Update design using TO algorithm

Robust & reliable Design

Material uncertainty Loading uncertainty Geometric uncertainty

A

Characterization of correlation Dimension reduction in UQ Random field to random variables

Uncertainty Quantification (UQ)

Decomposition into deterministic TO sub-problems

C

Analytical sensitivity analysis for deterministic TO sub-problems

Sensitivity Analysis (SA) for probabilistic performances

Efficient sampling Dimension reduction in UP

Uncertainty propagation (UP)

B

Evaluation of probabilistic performances using Gauss

quadrature formula

Performance prediction using finite element simulations

~ 9 ~

Chen, S., Chen, W., and Lee, S., “Level Set Based Robust Shape and Topology

Optimization under Random Field Uncertainties”, Structural and Multidisciplinary

Optimization, 41(4), pg 507, 2010.

ith eigenvalue ith eigenvector

Module A: Uncertainty Representation

•Karhunen-Loeve Expansion •A spectral approach to represent a random field using eigenfunctions of the random field’s covariance function as expansion bases.

ξ: orthogonal random parameters

: mean function g

Ghanem and Spanos 1991; Haldar and Mahadevan 2000; Ghanem and Doostan 2006

Significance check

Select M when s is close to 1

•Truncated K-L Expansion

x - spatial

coordinate

- random

parameter

Random

Field

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Module B: Uncertainty Propagation

• Approximate a multivariate function by a sum of multiple univariate functions

• Accurate if interactions of random variables ξ are relatively small

• Greatly reduce sample points for calculating statistical moments

Univariate Dimension Reduction (UDR) Method (Raman and

Xu, 2004)

Approximate the integration of a function g(ξ) by a weighted sum of

function values at specified points

Numerical Integration with Gaussian Quadrature Formula

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iw weights

Single Dimension Gauss Quadrature Formulae

Provide the highest precision in terms of the integration order

Much cheaper than MCS

1

( )mk kk

i i

i

E g g p d w g l

The k-th order statistical moment of a function of a random variable

can be calculated by a quadrature formula as follows

il locations of nodes

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Tensor Product Quadrature vs. Univariate Dimension Reduction

x1

x2 1

36

1

9

1

36

1

9

4

9

1

9

1

36

1

9

1

36

1

36

1

9

1

36

1

9

4

9

1

9

1

36

1

9

1

36

x1

x2

1

1 _

1

122

1

, , ,

, , , ,

i

i n

i

i n i

m

y i j X i j X

j

m

y i j X i j X g

j

w g l

w g l

UDR

i jl

i jw

Tensor Product Quadrature

weights

location of

nodes

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Module C: Shape Sensitivity Analysis for Probabilistic Performances

*nJ u V ds

Using adjoint variable method and shape sensitivity analysis (Sokolowski, 1992),we can calculate (1) and (2), and further obtain

nV uSteepest Descent t | | 0nV

Expand the functions of mean and variance using UDR in an additive

format

_ _

_

*

2

1 12

1

( , , ) ( ( , , )) ( ( , , ))

1 ( , , )i i

i

n n

J Jn

i i

J

i

D J D J kD J

kD n D J Dz

u z u z u z

u μ

_

1

1 , , (1)i

n

J J

i

D D n D J zu μ

_

_

2

12

1

1(2)

i

i

n

J Jn

i

J

i

D D

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Example 1. Bridge Beam with A Random Load at Bottom

f

Domain size: 2 by 1 , 1f

(2) Deterministic Topology Optimization

Angle: Uniform distribution [-3pi/4, -pi/4],

magnitude: Gumbel distribution (1, 0.3)

(1) RSTO under loading uncertainty

f

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Example 1. RSTO (with A Random Load at Bottom) v.s. DSTO

Robust Deterministic

E(C) 25-point tensor-product quadrature 1410.70 1422.25

Monte Caro (10000 points) 1400.10 1424.99

Std(C) 25-point tensor-product quadrature 994.86 1030.93

Monte Caro (10000 points) 959.86 1042.93

Robust Design Deterministic Design

Example 2 A Micro Gripper under A Random Material Field

1

2

outf

outf

inf

Chen, S., Chen, W., and Lee, S., 2010, "Level set based robust shape and topology optimization

under random field uncertainties," Structural and Multidisciplinary Optimization, 41(4), pp. 507-

524. ~ 17 ~

Example 2 Robust Design vs. Deterministic Design

Parameters Volume Ratio Robust Design Deterministic

Design

Material Field 1 0.090 -0.065 -0.07

Material Field 2

0.098 -0.059 -0.055 1

0.3

0.5

E

E

d

1E

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Represents geometric

uncertainty by modeling the

normal velocity field as a

random field;

Naturally describes topological

changes in the boundary

perturbation process;

Can model not only uniformly too

thin (eroded) or too thick (dilated)

structures but also shape-

dependent geometric uncertainty

Geometric Uncertainty Modeling with A Level Set Model

( )( , ) ( ) 0n

dV

dt

XX z X

~ 19 ~

Eulerian

Description

Chen, S. and Chen, W., “A New Level-Set Based Approach to Shape and

Topology Optimization under Geometric Uncertainty”, Structural and

Multidisciplinary Optimization, 44, 1-18, April 2011

Module A: Geometric Uncertainty Quantification

Extracted boundary points from the

level set model

1

,N

i i i

i

a x a x a x

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Extending Boundary Velocity to The Whole Design Domain

Initial velocity on the boundary Extended velocity on the whole domain

( ) 0nVsign V

( )( , ) ( ) 0n

d XV X X

dtz

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Challenges in Shape Sensitivity Analysis under Geometric Uncertainty

Conventional SSA Problem : How to

change to minimize

J

Our problem: Need shape gradient

of and at the same

time

J J

JChallenge: Need shape gradient of

is with respect to

DJ

D

n

DJV

D

~ 22 ~

SSA under Geometric Uncertainties

The design velocity field should be mapped along the path

line from to

Deformed configuration (perturbed design), t = t

1x

2x

3x

1e2e

3e

1X

2X

3X

1E2E

3E

b

P

p

Underformedconfiguration (current design), t = 0

Path line

u(X) = U(x)

x =Ψ(X,t)

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Using Nanson’s relation

and Polar decomposition

theory, it was proved that

Based on large

deformation theory

( ) ( )n n

DJ DJV V

D DX x

Example: Cantilever Beam Problem

(0,1), 0.02nV N tX

Deterministic Design Robust Design

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Configurations of Robust and Deterministic Designs under Geometric Uncertainty

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Robust Design under Variations Deterministic Design under Variations

Comparison of Deterministic vs. Robust Design

( )Std C

B

B

A

A

A

A

C

C

D

D

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Robust Designs for Over-Etching and Under-Etching Situations

E

E F

F

Robust design for the

under-etching situation

Robust design for the

over-etching situation

Summary

Demonstrated the importance of considering

uncertainty in topology optimization

A unified, mathematically rigorous and

computationally efficient framework to

implement RSTO

First attempt of level-set based TO under

geometric uncertainty (TOGU)

Bridge the gap between TO and state-of-the

art techniques for design under uncertainty

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