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A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in various sources such as loading, material properties, and geometry has been developed. The approach integrates the state-of-the-art level set methods for shape and topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional random-field uncertainty with a reduced set of random variables, the Karhunen-Loeve expansion is employed.
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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.
~ 3 ~
~ 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
~ 10 ~
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
~ 11 ~
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
~ 12 ~
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
~ 13 ~
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
~ 14 ~
~ 15 ~
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
~ 16 ~
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
~ 18 ~
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
~ 20 ~
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
~ 21 ~
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)
~ 23 ~
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
~ 24 ~
Configurations of Robust and Deterministic Designs under Geometric Uncertainty
~ 25 ~
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
~ 26 ~
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
~ 28 ~