An energy gap functional: Cavity identification in linear elasticity

Abstract

The aim of this work is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.

Cavities Identification from Partially Overdetermined Boundary Data in Linear Elasticity

“…The computation of the topological gradient exposed below is inspired by the paper [10] and let us point out that the main contribution in this paper relies not only on the detection of cavities from partially overdetermined boundary data but also on the use of the error functional (13), also known as an energetic least-squares functional [3,4,8]. Indeed, the aim here is to derive an asymptotic expansion for the cost functional j (13) following the same procedure outlined in the previous section, that is, to study the variation of the design functional  with respect to the creation of small hole.…”

“…In the context of geometrical inverse problem, it is a question about overdetermined boundary data, namely data provided by measurements distributed on the exterior boundary of the domain of interest [3,4]. To the author's best knowledge, all geometric inverse problems in linear elasticity, investigated in the literature, have in common to be defined by complete overdetermined boundary data [3][4][5][6] with the exception of a recent work [7], where data appear to be partial.…”

“…Given the normal component of the normal stress imposed g and the displacement field f measured on the boundary ϒ Where n ϒ and n are the outward unit normals to the boundary of \ Ω  . The stress tensor σ and the strain tensor ε are given by In order to solve our geometrical inverse problem (1), we propose a Dirichlet-Neumann approach by the means of a self regularization technique, namely the Kohn-Vogelius formulation [3,4,7,8]. Indeed, it leads to define two forward problems.…”

“…A logical sequel of the work [7] is to combine the shape gradient method with the level set one which is one of promising techniques that can be designed to such shape reconstruction problem [3,4,8]. However, one can remark that although the level set method is able to automatically handle topology changes and reconstruct the precise geometry of the cavities [3,4,8], it requires an iterative procedure where at each iteration step, one needs to solve a sequence of forward solutions.…”

“…However, one can remark that although the level set method is able to automatically handle topology changes and reconstruct the precise geometry of the cavities [3,4,8], it requires an iterative procedure where at each iteration step, one needs to solve a sequence of forward solutions. As a consequence, the level set method seems to be a slow and expensive process.…”