Topological sensitivities via a Lagrangian approach for semi-linear problems

TL;DR

This paper introduces a Lagrangian-based method for efficiently computing topological derivatives in semilinear elliptic problems, with potential extensions to more complex nonlinear and evolutionary problems.

Contribution

It presents a novel, general framework using averaged adjoint states and weak convergence for calculating topological sensitivities in semilinear PDEs.

Findings

Method converges using weakly converging subsequences

Framework simplifies topological sensitivity computation

Potential extension to evolutionary and nonlinear problems

Abstract

In this paper we present a methodology that allows the efficient computation of the topological derivative for semilinear elliptic problems within the averaged adjoint Lagrangian framework. The generality of our approach should also allow the extension to evolutionary problems and other nonlinear problems. Our strategy relies on a rescaled differential quotient of the averaged adjoint state variable which we show converges weakly to a function satisfying an equation defined in the whole space. A unique feature and advantage of this framework is that we only need to work with weakly converging subsequences of the differential quotient. This allows the computation of the topological sensitivity within a simple functional analytic framework under mild assumptions.