Reconstruction of cracks in an inhomogeneous anisotropic medium using point sources

Abstract

We consider the inverse problem, in two and three dimensions, of identifying elastic cracks embedded in an inhomogeneous anisotropic elastic medium using point sources. The observable data is given by the near-field measurements of the outgoing Green's function for the related stationary system. We give a reconstruction algorithm for this inverse problem.

An expansion theorem for two‐dimensional elastic waves and its application

“…The following key lemma verifies that the Dirichlet-to-Neumann map can be constructed by G(x, y) on |x| = |y| = R. Lemma 2 (Nakamura et al [13,Lemma 5.3]) − e is injective and ( − e ) = I .…”

“…The problem is the same as Inverse Problem 2 on p. 209 of Reference [8] and Section 5.2 on p. 608 of Reference [13]. We will not repeat the proof again and refer the readers to the above articles.…”

“…However, the approaches developed in References EXPANSION THEOREM FOR 2D ELASTIC WAVES 1859 [7,8] essentially rely on an Atkinson-Wilcox-type expansion for three-dimensional Helmholtz equation. Having an Atkinson-Wilcox-type expansion for two-dimensional elastic wave at hand, the same arguments in References [7,8,12,13] can be applied to the two-dimensional inverse elastic wave scattering.…”

“…Step 2: As the same arguments in Reference [13] where the inhomogeneous anisotropic elasticity system is considered, we show that the Dirichlet-to-Neumann map on *B R can be constructed by the measurements G(x, y) on |x| = |y| = R. For the reader's convenience, the similar notations in Reference [13] will be used. Define the Dirichlet-to-Neumann map : g(x) = −…”

Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems

“…The domain D stands for the region of the inclusion or cavity embedded in Ω. In the degenerate case where the domain D represents the crack, the inverse problem of identifying D by near-field measurements was consider in [16] and [17]. To simply our presentation, we will not discuss this matter here.…”