Line segment crack recovery from incomplete boundary data

Abstract

We are concerned with non-destructive control issues, namely detection and recovery of cracks in a planar (2D) isotropic conductor from partial boundary measurements of a solution to the Laplace–Neumann problem. We first build an extension of that solution to the whole boundary, using constructive approximation techniques in classes of analytic and meromorphic functions, and then use localization algorithms based on boundary computations of the reciprocity gap.

Reconstruction of Crack Clusters in the Rectangular Domain by Ultrasonic Waves

“…Generally, the problem on cracks reconstruction was studied intensively during some decades past, because of its importance in engineering applications. This is connected with the fundamental theory of inverse problems (some interesting theoretical results with further references can be found in [5][6][7][8][9][10]). The mathematical results of this sort concern uniqueness of the solution, some others develop explicit-form analytical solutions.…”

“…Other acoustic methods are connected with measurements of dynamic wave fields over some parts of the sample's boundary, under condition that the latter is loaded by a certain oscillating force. Some theoretical works prove that the shape of the boundary can uniquely determine the geometry of the internal defects [5][6][7]. A contiguous method is founded upon measurements of the natural frequencies of the sample.…”

On the Theory of Direct and Inverse Problems in the Elastic Rectangle: Antiplane Case

“…In particular, a number of theoretical works were devoted to the inverse problems contiguous to the one studied in the present work, with applications to recognition of cracks [16][17][18][19]. Some important articles of this sort concern uniqueness of the solution; some others develop explicit-form analytical results or numerical algorithms.…”

Green’s Function Technique and Global Optimization in Reconstruction of Elliptic Objects in the Regular Triangle

“…A number of theoretical works were devoted to the inverse problems of this kind, with applications to recognition of cracks [1][2][3]. Some important papers concern uniqueness of the solution, some others develop explicitform analytical results or numerical algorithms [4,5].…”