Identification of cracks and cavities using the topological sensitivity boundary integral equation

Abstract

The purpose of this communication is to present a novel approach to compute the so called Topological Sensitivity (TS) of any variable or functional in elasticity using Boundary Integral Equations (BIE's), and its use as a tool for identification of defects, by itself or in conjunction with zero-order methods, like Genetic Algorithms. The TS of a cost functional provides a measure of the susceptibility of a defect being at a given location. The main contributions are summarized in the following points:Computation of the TS based on a linearized topological expansion, using Boundary Integral Equations. The TS is computed using only information of the non-damaged domain. The calculation is carried out for circular cavities or straight cracks, but the procedure is extensible to other kinds of defects. It is shown that the topological expansion provides a very accurate tool for estimating the defect sizes, even for very large flaws, relative to the domain size. Applicability of the TS for identification of defects, by itself or associated with Genetic Algorithm. This association is very advantageous since the computational time is dramatically reduced.

Inverse problems in elasticity

“…Here, its usefulness as a means of facilitating a subsequent minimization-based solution by providing a rational basis for selecting a reliable initial guess in terms of its topology, approximate size and location is considered. In [76], this idea was considered in the context of inverse elastic scattering pertaining to semi-infinite and infinite domains, where the availability of suitable fundamental solutions made it possible to establish explicit expressions for T (x o ), while developments along similar lines for 2D elastostatics are presented in [66,85]. It is instructive to compare the topological derivative concept, as applied to the direct and inverse scattering of small objects by waves, to other asymptotic approaches.…”

Asymptotic Models for the Topological Sensitivity Versus the Topological Derivative

“…Fourth, one of the main advantages of the asymptotic model for the topological sensitivity is that it provides a more accurate tool for identification of cracks and cavities in the inverse problems (see, e.g., [16,17]). At the same time, the asymptotic model based topological sensitivity ) ( 0 x S defined by (32) is computed using only information of the non-damaged domain .…”

“…Numerical results obtained by help of the topological derivative can be found in [3,12]. We refer to [16,17] for applications in inverse problems. The topological derivative was incorporated [18] into the level set method [19].…”

Crack identification by 3D time-domain elastic or acoustic topological sensitivity

“…The topological derivative field T(x, T ) is considered here as a possible crack indicator function. The heuristic of this approach, following and generalizing upon previous investigations on void identification under transient dynamical conditions [7,9] or crack identification under 2-D static conditions [10], consists in seeking the (finite) crack Γ in regions of Ω where T(x, T ) reaches its most negative values, i.e. where J(Ω ε , T ) decreases most for sufficiently small crack size ε.…”