A semi-explicit algorithm for the reconstruction of 3D planar cracks

Abstract

This paper deals with a semi-explicit algorithm to reconstruct two-dimensional (2D) segment cracks, or three-dimensional (3D) planar cracks, in the framework of overspecified boundary data. The algorithm is based on the reciprocity gap concept, introduced by Andrieux and Ben Abda, which provides explicitly the line (or the plane) support of the cracks. A numerical reconstruction of the cracks, which are actually the support of the solution jump across this plane, is then performed by computing the Fourier expansion of the solution jump itself. After the numerical analysis of the method, some numerical results are presented and commented on.

Physics-driven quantized consensus for distributed diffusion source estimation using sensor networks

“…This is the basis of the reciprocity gap (RG) method [37] used in non-destructive testing of solids [37,38]. Essentially, if u(x, t) is known and since we choose and therefore also know k (x), the left hand side (lhs) of (12) can be computed.…”

Identifiability and stability results of one emerging crack in heterogeneous media by one boundary measurements

“…Starting from the variational problem (12) and performing the change of variables y = F h (x), u h is the solution of…”

Line segment cracks identification via the reciprocity gap principle and Fourier transform

“…When complete data is available on the boundary S. Andrieux and A. Ben Abda introduced in [11] and [12] the reciprocity gap concept which turned out to be a relevent tool for recovering 3D-planar cracks in the case of Laplace equation and elastostatic system [13]. The proof of the uniqueness result is constructive and semi-explicit algorithms were built on it [14] , [2]and [1].…”