Identification of Cavities and Inclusions in Linear Elasticity with a Phase-Field Approach

Abstract

In this paper we deal with the inverse problem of determining cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement’s measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modelled as inclusions with a very small elasticity tensor.

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

“…hence the continuity of the functional J δ D . Using the same arguments in [8], it is straightforward to prove the following existence result.…”

“…In the inverse problem context, applications of a phase-field approach have been proposed in [33,70,71] for a linear elliptic equation, in [14,15] for a semilinear elliptic equation, and very recently in [8] for the Lamé system and in [58] for a quasilinear Maxwell system.…”

“…The convexification of the cavity is due to the presence of the Modica-Mortola relaxation that approximates the perimeter of the cavity. In figure 9, we finally provide a numerical experiment for a comparison between the results given by J sum δ,ε , as defined in (6.10), and the misfit functional studied in [8] (see the section titled 'numerical examples'), which is, in the notation adopted in this paper, equal to where u δ N,i , for i = 1, . .…”

“…Recently, a phase-field approach has been proposed in [33] and then applied also in [15] for the identification of inclusions in the framework of a linear and a semilinear elliptic equation, respectively. The same approach has been also extended to the detection of cavities in the case of a semilinear elliptic equation in [14] and of linear elasticity in [8]. All these papers propose an algorithm rephrasing the inverse problem as an optimization procedure, where the goal is to minimize a suitable misfit functional, defined on the boundary of Ω, with the addition of a regularization term which involves a relaxation of the perimeter of the domain to be reconstructed.…”

On the reconstruction of cavities in a nonlinear model arising from cardiac electrophysiology

“…From a numerical standpoint, we can compare our strategy with other existing approaches in the literature related with the linear counterpart of the problem, namely the cavity detection in the linear conductivity problem. In such a context, phase-field techniques have been studied for the reconstruction of cavities (and cracks) in the conductivity case in [52] and in the elasticity case in [6] and [1]. Among the several alternative strategies, we can perform a main distinction between algorithms which have been originally developed for inclusion detection and later extended to the cavity case, and algorithms specifically suited for the reconstruction of cavities.…”