On the importance of smoothness, interface resolution and numerical sensitivities in shape and topological sensitivity analysis

TL;DR

This paper examines how discretization choices in PDE constraints affect shape and topological derivatives, highlighting the importance of interface resolution and basis function smoothness for convergence.

Contribution

It compares standard and enriched discretization methods, demonstrating that only the enriched method ensures convergence of topological derivatives.

Findings

Standard method's shape derivative regularity depends on basis smoothness.

Interface location must be included in the ansatz for point-wise shape derivative convergence.

Only the enriched method guarantees convergence of the topological derivative.

Abstract

In this paper we investigate the influence of the discretization of PDE constraints on shape and topological derivatives. To this end, we study a tracking-type functional and a two-material Poisson problem in one spatial dimension. We consider the discretization by a standard method and an enriched method. In the standard method we use splines of degree $p$ such that we can control the smoothness of the basis functions easily, but do not take any interface location into consideration. This includes for p=1 the usual hat basis functions. In the enriched method we additionally capture the interface locations in the ansatz space by enrichment functions. For both discretization methods shape and topological sensitivity analysis is performed. It turns out that the regularity of the shape derivative depends on the regularity of the basis functions. Furthermore, for point-wise convergence of the shape derivative the interface has to be considered in the ansatz space. For the topological derivative we show that only the enriched method converges.