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arXiv · 2209.15491

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

Peter Gangl·Michael H. Gfrerer
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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.

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Peter Gangl, Michael H. Gfrerer. 2022-09-30. A unified approach to shape and topological sensitivity analysis of discretized optimal design problems. https://doi.org/10.1007/s00245-023-10016-2

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