Multi-material topology optimization with multiple volume constraints: a general approach applied to ground structures with material nonlinearity

Xiaojia Shelly Zhang, Glaucio H. Paulino, Adeildo S. Ramos

Research output: Contribution to journalArticlepeer-review


Multi-material topology optimization is a practical tool that allows for improved structural designs. However, most studies are presented in the context of continuum topology optimization – few studies focus on truss topology optimization. Moreover, most work in this field has been restricted to linear material behavior with limited volume constraint settings for multiple materials. To address these issues, we propose an efficient multi-material topology optimization formulation considering material nonlinearity. The proposed formulation handles an arbitrary number of candidate materials with flexible material properties, features freely specified material layers, and includes a generalized volume constraint setting. To efficiently handle such arbitrary volume constraints, we derive a design update scheme that performs robust updates of the design variables associated with each volume constraint independently. The derivation is based on the separable feature of the dual problem of the convex approximated primal subproblem with respect to the Lagrange multipliers, and thus the update of design variables in each volume constraint only depends on the corresponding Lagrange multiplier. Through examples in 2D and 3D, using combinations of Ogden-based, bilinear, and linear materials, we demonstrate that the proposed multi-material topology optimization framework with the presented update scheme leads to a design tool that not only finds the optimal topology but also selects the proper type and amount of material. The design update scheme is named ZPR (phonetically, zipper), after the initials of the authors’ last names (Zhang-Paulino-Ramos Jr.).

Original languageEnglish (US)
Pages (from-to)161-182
Number of pages22
JournalStructural and Multidisciplinary Optimization
Issue number1
StatePublished - Jan 1 2018
Externally publishedYes


  • Design update scheme
  • Discrete filter
  • Ground structure method
  • KKT conditions
  • Material nonlinearity
  • Multi-material topology optimization
  • Multiple load cases
  • Multiple volume constraints
  • Potential energy
  • ZPR update scheme

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Control and Optimization


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