Shape and Topology Optimization involving the eigenvalues of an elastic structure: A multi-phase-field approach

TL;DR

This paper develops a multi-phase-field method for optimizing elastic structures based on eigenvalues, enabling topology changes and multiple materials, with proven mathematical properties and optimality conditions.

Contribution

It introduces a novel multi-phase-field framework for eigenvalue-based topology optimization, including existence, differentiability, and optimality conditions.

Findings

Proved continuity and differentiability of eigenvalues in the model.

Established existence of global minimizers for the optimization problem.

Derived first-order necessary optimality conditions.

Abstract

A cost functional involving the eigenvalues of an elastic structure, that is described by a multi-phase-field equation, is optimized. This allows us to handle topology changes and multiple materials. We prove continuity and differentiability of the eigenvalues and we establish the existence of a global minimizer to our optimization problem. We further derive first-order necessary optimality conditions for local minimizers. Moreover, an optimization problem combining eigenvalue and compliance optimization is also discussed.