A geometric projection method for designing three-dimensional open lattices with inverse homogenization

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Topology optimization is a methodology for assigning material or void to each point in a design domain in a way that extremizes some objective function, such as the compliance of a structure under given loads, subject to various imposed constraints, such as an upper bound on the mass of the structure. Geometry projection is a means to parameterize the topology optimization problem, by describing the design in a way that is independent of the mesh used for analysis of the design's performance; it results in many fewer design parameters, necessarily resolves the ill-posed nature of the topology optimization problem, and provides sharp descriptions of the material interfaces. We extend previous geometric projection work to 3 dimensions and design unit cells for lattice materials using inverse homogenization. We perform a sensitivity analysis of the geometric projection and show it has smooth derivatives, making it suitable for use with gradient-based optimization algorithms. The technique is demonstrated by designing unit cells comprised of a single constituent material plus void space to obtain light, stiff materials with cubic and isotropic material symmetry. We also design a single-constituent isotropic material with negative Poisson's ratio and a light, stiff material comprised of 2 constituent solids plus void space.

Original languageEnglish (US)
Pages (from-to)1564-1588
Number of pages25
JournalInternational Journal for Numerical Methods in Engineering
Issue number11
StatePublished - Dec 14 2017


  • elasticity
  • finite element methods
  • topology design

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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