Structural and Multidisciplinary Optimization

, Volume 41, Issue 4, pp 507–524 | Cite as

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

  • Shikui Chen
  • Wei ChenEmail author
  • Sanghoon Lee
Research Paper


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.


Robust design Topology optimization Shape optimization Level set methods Uncertainty Random field Dimension reduction 


C(x1, x2)

spatial covariance function


spatial domain


elastic tensor

a(x, ω)

random field

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

mean function of a(x, ω)

ai(x) or ai

ith eigenfunction of random field


function of z


objective functional


joint probability density function


state variable


design velocity field

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

weight of the ith quadrature point


level set function


Lagrange multiplier


ith eigenvalue of random field


orthogonal random variables with zero mean and unit variance


mean performance


performance variance


vector of random variables


the i-th random variable of z


the j-th quadrature node of z i


sample space


an element of sample space Θ


geometric shape of design

\(\partial \Omega \)

boundary of Ω



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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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  2. 2.Korea Atomic Energy Research InstituteDaejeonRepublic of Korea

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