Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach

Xiaojia Shelly Zhang, Heng Chi, Glaucio Paulino

Research output: Contribution to journalArticlepeer-review


We introduce a general multi-material topology optimization framework for large deformation problems that effectively handles an arbitrary number of candidate hyperelastic materials and addresses three major associated challenges: material interpolation, excessive distortion of low-density elements, and computational efficiency. To account for many nonlinear elastic materials, we propose a material interpolation scheme that, instead of interpolating multiple material parameters (such as Young's modulus), interpolates multiple nonlinear stored-energy functions. To circumvent convergence difficulties caused by excessive distortions of low-density elements under large deformations, an energy interpolation scheme is revisited to account for multiple candidate hyperelastic materials. Computational efficiency is addressed from both structural analysis and optimization perspectives. To solve the nonlinear state equations efficiently, we employ the lower-order Virtual Element Method in conjunction with tailored adaptive mesh refinement and coarsening strategies. To efficiently update the design variables of the multi-material system, we exploit the separable nature and improve the ZPR (Zhang–Paulino–Ramos) update scheme to account for positive sensitivities and update the design variables associated with each volume constraint in parallel. Four design examples with three types of nonlinear material models demonstrate the efficiency and effectiveness of the proposed framework.

Original languageEnglish (US)
Article number112976
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Oct 1 2020
Externally publishedYes


  • Adaptive refinement and coarsening
  • Hyperelastic materials
  • Large deformations
  • Multi-material topology optimization
  • Virtual Element Method (VEM)
  • ZPR update scheme

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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