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Level set based robust shape and topology optimization under random field uncertainties

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Abstract

A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in loading and material properties is developed in this work. The proposed 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. The univariate dimension-reduction (UDR) method combined with Gauss-type quadrature sampling is then employed for calculating statistical moments of the design response. The combination of the above techniques greatly reduces the computational cost in evaluating the statistical moments and enables a semi-analytical approach that evaluates the shape sensitivity of the statistical moments using shape sensitivity at each quadrature node. The applications of our approach to structure and compliant mechanism designs show that the proposed RSTO method can lead to designs with completely different topologies and superior robustness.

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Abbreviations

C(x1, x2):

spatial covariance function

D :

spatial domain

E ijkl :

elastic tensor

a(x, ω):

random field

\(\bar {a}\left( x \right)\) :

mean function of a(x, ω)

a i (x) or ai:

ith eigenfunction of random field

g(z):

function of z

J:

objective functional

p(z):

joint probability density function

u :

state variable

V(x):

design velocity field

\(\emph{w}_{i }\) :

weight of the ith quadrature point

ϕ :

level set function

λ :

Lagrange multiplier

λ i :

ith eigenvalue of random field

ξ i (ω):

orthogonal random variables with zero mean and unit variance

μ :

mean performance

σ 2 :

performance variance

z :

vector of random variables

z i :

the i-th random variable of z

z ij :

the j-th quadrature node of z i

Θ:

sample space

ω :

an element of sample space Θ

Ω:

geometric shape of design

\(\partial \Omega \) :

boundary of Ω

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Acknowledgments

The grant support (CMMI-0522662) from National Science Foundation (NSF) and the support from the Center for Advanced Vehicular Systems at Mississippi State University via Department of Energy Contract No: DE-AC05-00OR22725 are greatly acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL, USA

    Shikui Chen & Wei Chen

  2. Korea Atomic Energy Research Institute, 1045 Daedeok-Daero, Yuseong-gu, Daejeon, Republic of Korea

    Sanghoon Lee

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  1. Shikui Chen
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  2. Wei Chen
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Correspondence to Wei Chen.

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Chen, S., Chen, W. & Lee, S. Level set based robust shape and topology optimization under random field uncertainties. Struct Multidisc Optim 41, 507–524 (2010). https://doi.org/10.1007/s00158-009-0449-2

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