A unified approach to shape and topological sensitivity analysis of discretized optimal design problems

TL;DR

This paper presents a unified sensitivity analysis method for shape and topological changes in discretized PDE-constrained design problems, enabling seamless optimization without distinguishing update types.

Contribution

It introduces a combined sensitivity concept for shape and topology, relating level set perturbations to design modifications in a discretized setting.

Findings

Validated sensitivities against continuous derivatives.

Applied in a level-set-based optimization algorithm.

Connected discrete sensitivities with classical shape and topological derivatives.

Abstract

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction-diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made.