Multiphase topology design with optimal material selection using an inverse p-norm function

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We present an original mathematical formulation for optimizing structural topology while simultaneously identifying an optimal set of design materials that are selected from a larger set of candidate materials. This design task is analogous to that, which is commonly encountered in additive manufacturing applications in which the 3D printer can print parts containing up to 3 distinct materials that can be selected from a larger suite of usable materials. The material distribution is parameterized via the shape functions with penalization formulation in which a set of activation functions, which are derived from a partition of the unit hypercube, is used to determine the effective local elasticity modulus within a single finite element. Additionally, we introduce an inverse p-norm function, which is used to ensure that the optimized material properties converge to a set of discrete values corresponding to the available candidate materials. The algorithm has been implemented on a set of 2D benchmark problems. Numerical results show that the formulation combining the inverse p-norm function with the activation functions successfully produces optimized multimaterial solutions containing no more than the prescribed number of distinct materials.

Original languageEnglish (US)
Pages (from-to)999-1017
Number of pages19
JournalInternational Journal for Numerical Methods in Engineering
Issue number9
StatePublished - Jun 1 2018


  • additive manufacturing
  • multimaterial optimization
  • topology optimization

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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