A unified approach to shape and topological sensitivity analysis of discretized optimal design problems
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.
Keywords
Cite
@article{arxiv.2209.15491,
title = {A unified approach to shape and topological sensitivity analysis of discretized optimal design problems},
author = {Peter Gangl and Michael H. Gfrerer},
journal= {arXiv preprint arXiv:2209.15491},
year = {2026}
}