Metamaterial topology optimization of nonpneumatic tires with stress and buckling constraints

Yeshern Maharaj, Kai A. James

Research output: Contribution to journalArticlepeer-review


This article presents the design of a metamaterial for the shear layer of a nonpneumatic tire using topology optimization, under stress and buckling constraints. These constraints are implemented for a smooth maximum function using global aggregation. A linear elastic finite element model is used, implementing solid isotropic material with penalization. Design sensitivities are determined by the adjoint method. The method of moving asymptotes is used to solve the numerical optimization problem. Two different optimization statements are used. Each requires a compliance limit and some aspect of continuation. The buckling analysis is linear, considering the generalized eigenvalue problem of the conventional and stress stiffness matrices. Various symmetries, base materials, and starting geometries are considered. This leads to novel topologies that all achieve the target effective shear modulus of 10 MPa, while staying within the stress constraint. The stress-only designs generally were susceptible to buckling failure. A family of designs (columnar, noninterconnected representative unit cells) that emerge in this study appears to exhibit favorable properties for this application.

Original languageEnglish (US)
Pages (from-to)1410-1439
Number of pages30
JournalInternational Journal for Numerical Methods in Engineering
Issue number7
Early online dateNov 14 2019
StatePublished - Apr 15 2020


  • buckling constraints
  • metamaterial design
  • nonpneumatic tire
  • stress-constrained optimization
  • topology optimization

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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