# 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.

## Key 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.