A phase-field approach for detecting cavities via a Kohn-Vogelius type functional

TL;DR

This paper introduces a phase-field method for detecting cavities in a medium by minimizing a Kohn-Vogelius functional with perimeter regularization, using numerical algorithms validated through experiments.

Contribution

It proposes a novel phase-field approach with perimeter regularization for cavity detection, combining shape optimization and numerical analysis.

Findings

The algorithm effectively reconstructs cavity shapes from boundary data.

Numerical experiments demonstrate robustness and accuracy of the method.

The phase-field model successfully approximates perimeter regularization.

Abstract

We deal with the geometrical inverse problem of the shape reconstruction of cavities in a bounded linear isotropic medium by means of boundary data. The problem is addressed from the point of view of optimal control: the goal is to minimize in the class of Lipschitz domains a Kohn-Vogelius type functional with a perimeter regularization term which penalizes the perimeter of the cavity to be reconstructed. To solve numerically the optimization problem, we use a phase-field approach, approximating the perimeter functional with a Modica-Mortola relaxation and modeling the cavity as an inclusion with a very small elastic tensor. We provide a detailed analysis showing the robustness of the algorithm through some numerical experiments.