Sharp interface limit for a phase field model in structural optimization

TL;DR

This paper develops a phase field approach for shape and topology optimization in structural design, establishing the connection to sharp interface models through $mma$-convergence, and demonstrates its effectiveness with numerical examples.

Contribution

It introduces a phase field formulation for structural optimization and rigorously links it to sharp interface models via $mma$-convergence, including convergence of optimality conditions.

Findings

The phase field model converges to a sharp interface limit.

Optimality conditions are consistent between diffuse and sharp interface models.

Numerical results show applicability to complex structural problems.

Abstract

We formulate a general shape and topology optimization problem in structural optimization by using a phase field approach. This problem is considered in view of well-posedness and we derive optimality conditions. We relate the diffuse interface problem to a perimeter penalized sharp interface shape optimization problem in the sense of $\Gamma$-convergence of the reduced objective functional. Additionally, convergence of the equations of the first variation can be shown. The limit equations can also be derived directly from the problem in the sharp interface setting. Numerical computations demonstrate that the approach can be applied for complex structural optimization problems.