Phase-field topology optimization with periodic microstructure

TL;DR

This paper develops a topology optimization framework for elastic materials with periodic microstructures, combining phase-field modeling and homogenization to optimize both macro and micro-scale properties.

Contribution

It introduces a variational bilevel phase-field formulation for microstructured topology optimization, with analysis of homogenized limits and optimality conditions.

Findings

Characterization of the homogenized problem via Gamma-convergence

Derivation of first-order optimality conditions at multiple levels

Analysis of the sharp-interface limit for microstructure design

Abstract

Progresses in additive manufacturing technologies allow the realization of finely graded microstructured materials with tunable mechanical properties. This paves the way to a wealth of innovative applications, calling for the combined design of the macroscopic mechanical piece and its underlying microstructure. In this context, we investigate a topology optimization problem for an elastic medium featuring a periodic microstructure. The optimization problem is variationally formulated as a bilevel minimization of phase-field type. By resorting to Gamma-convergence techniques, we characterize the homogenized problem and investigate the corresponding sharp-interface limit. First-order optimality conditions are derived, both at the homogenized phase-field and at the sharp-interface level.