# A shape optimization algorithm for cellular composites

## TL;DR

This paper introduces a mesh deformation method with gradient penalization for PDE-constrained shape optimization, enabling scalable and stable optimization of cellular composite shapes under tension.

## Contribution

It presents a novel mesh deformation technique with gradient penalization that prevents mesh degeneration and enhances scalability in PDE-constrained shape optimization.

## Key findings

- Gradient penalization effectively prevents mesh degeneration.
- The method scales well with parallel multigrid solvers.
- Application to cellular composites under tension demonstrates practical utility.

## Abstract

We propose and investigate a mesh deformation technique for PDE constrained shape optimization. Introducing a gradient penalization to the inner product for linearized shape spaces, mesh degeneration can be prevented within the optimization iteration allowing for the scalability of employed solvers. We illustrate the approach by a shape optimization for cellular composites with respect to linear elastic energy under tension. The influence of the gradient penalization is evaluated and the parallel scalability of the approach demonstrated employing a geometric multigrid solver on hierarchically distributed meshes.