Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2461–2476 | Cite as

Robust multiphase topology optimization accounting for manufacturing uncertainty via stochastic collocation

  • Vahid KeshavarzzadehEmail author
  • Kai A. James
Research Paper


This paper presents a computational framework for multimaterial topology optimization under uncertainty. We combine stochastic collocation with design sensitivity analysis to facilitate robust design optimization. The presence of uncertainty is motivated by the induced scatter in the mechanical properties of candidate materials in the additive manufacturing process. The effective elastic modulus in each finite element is obtained by an interpolation scheme which is parameterized with three distinct elastic moduli corresponding to the available design materials. The parametrization enables the SIMP-style penalization of intermediate material properties, thus ensuring convergence to a discrete manufacturable design. We consider independent random variables for the elastic modulus of different materials and generate designs that minimize the variability in the performance, namely structural compliance. We use a newly developed quadrature rule, designed quadrature, to compute statistical moments with reduced computational cost. We show our approach on numerical benchmark problems of linear elastic continua where we demonstrate the improved performance of robust designs compared with deterministic designs. We provide the MATLAB implementation of our approach.


Multimaterial topology optimization Additive manufacturing Robust design optimization Stochastic collocation 


Funding information

The authors wish to acknowledge the support of the National Science Foundation, which funded this research through grant number CMMI-1663566.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignChampaignUSA
  2. 2.Department of Aerospace EngineeringUniversity of Illinois at Urbana-ChampaignChampaignUSA

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