A convex analysis approach to multi-material topology optimization

TL;DR

This paper introduces a convex analysis-based method for multi-material topology optimization, enabling the design of structures with discrete material choices through convex penalties and validated on PDE control problems.

Contribution

It proposes a novel convex analysis framework for multi-material topology optimization that handles discrete material constraints within PDE control problems.

Findings

Effective convex penalty formulation for discrete material control

Successful numerical implementation using semi-smooth Newton method

Validated approach on potential and diffusion coefficient control problems

Abstract

This work is concerned with optimal control of partial differential equations where the control enters the state equation as a coefficient and should take on values only from a given discrete set of values corresponding to available materials. A "multi-bang" framework based on convex analysis is proposed where the desired piecewise constant structure is incorporated using a convex penalty term. Together with a suitable tracking term, this allows formulating the problem of optimizing the topology of the distribution of material parameters as minimizing a convex functional subject to a (nonlinear) equality constraint. The applicability of this approach is validated for two model problems where the control enters as a potential and a diffusion coefficient, respectively. This is illustrated in both cases by numerical results based on a semi-smooth Newton method.