Discussion of "Second order topological sensitivity analysis" by J. Rocha de Faria et al

TL;DR

This paper discusses the second order topological sensitivity analysis of potential energy in a 2-D Laplace problem, highlighting errors in the original article's second-order expansion calculations.

Contribution

It critically examines and identifies inaccuracies in the second-order terms of the topological energy perturbation expansion in the original work.

Findings

The first-order topological derivative is well-established.

The second-order expansion contains errors as shown in this analysis.

Corrected formulas are proposed for accurate second-order sensitivity.

Abstract

The article by J. Rocha de Faria et al. under discussion is concerned with the evaluation of the perturbation undergone by the potential energy of a domain $\Omega$ (in a 2-D, scalar Laplace equation setting) when a disk $B_{\epsilon}$ of small radius $\epsilon$ centered at a given location $\hat{\boldsymbol{x}\in\Omega$ is removed from $\Omega$, assuming either Neumann or Dirichlet conditions on the boundary of the small `hole' thus created. In each case, the potential energy $\psi(\Omega_{\epsilon})$ of the punctured domain $\Omega_{\epsilon}=\Omega\setminus\B_{\epsilon}$ is expanded about $\epsilon=0$ so that the first two terms of the perturbation are given. The first (leading) term is the well-documented topological derivative of $\psi$. The article under discussion places, logically, its main focus on the next term of the expansion. However, it contains incorrrect results, as shown in this discussion. In what follows, equations referenced with Arabic numbers refer to those of the article under discussion.