WIAS Preprint No. 2908, (2021)

Two-scale topology optimization with heterogeneous mesostructures based on a local volume constraint


  • Ebeling-Rump, Moritz
  • Hömberg, Dietmar
  • Lasarzik, Robert

2020 Mathematics Subject Classification

  • 49Q10 74P05 49Q20 65M60 74P10


  • Additive manufacturing, topology optimization, linear elasticity, phase field method, optimality conditions, numerical simulations




A new approach to produce optimal porous mesostructures and at the same time optimizing the macro structure subject to a compliance cost functional is presented. It is based on a phase-field formulation of topology optimization and uses a local volume constraint (LVC). The main novelty is that the radius of the LVC may depend both on space and a local stress measure. This allows for creating optimal topologies with heterogeneous mesostructures enforcing any desired spatial grading and accommodating stress concentrations by stress dependent pore size. The resulting optimal control problem is analysed mathematically, numerical results show its versatility in creating optimal macroscopic designs with tailored mesostructures.

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