Multiscale design of nonlinear materials using a Eulerian shape optimization scheme

Ahmad R. Najafi, Masoud Safdari, Daniel A. Tortorelli, Philippe H. Geubelle

Research output: Contribution to journalArticlepeer-review


Motivated by recent advances in manufacturing, the design of materials is the focal point of interest in the material research community. One of the critical challenges in this field is finding optimal material microstructure for a desired macroscopic response. This work presents a computational method for the mesoscale-level design of particulate composites for an optimal macroscale-level response. The method relies on a custom shape optimization scheme to find the extrema of a nonlinear cost function subject to a set of constraints. Three key “modules” constitute the method: multiscale modeling, sensitivity analysis, and optimization. Multiscale modeling relies on a classical homogenization method and a nonlinear NURBS-based generalized finite element scheme to efficiently and accurately compute the structural response of particulate composites using a nonconformal discretization. A three-parameter isotropic damage law is used to model microstructure-level failure. An analytical sensitivity method is developed to compute the derivatives of the cost/constraint functions with respect to the design variables that control the microstructure's geometry. The derivation uncovers subtle but essential new terms contributing to the sensitivity of finite element shape functions and their spatial derivatives. Several structural problems are solved to demonstrate the applicability, performance, and accuracy of the method for the design of particulate composites with a desired macroscopic nonlinear stress-strain response.

Original languageEnglish (US)
Pages (from-to)2981-3014
Number of pages34
JournalInternational Journal for Numerical Methods in Engineering
Issue number12
StatePublished - Jun 30 2021


  • analytical sensitivity analysis
  • fixed grid
  • gradient-based shape optimization
  • nonlinear GFEM

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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