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Sounding of finite solid bodies by way of topological derivative
Author(s) -
Bonnet Marc,
Guzina Bojan B.
Publication year - 2004
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1153
Subject(s) - finite element method , sensitivity (control systems) , boundary value problem , directional derivative , mathematics , computation , topology (electrical circuits) , boundary (topology) , computer science , mathematical analysis , algorithm , physics , engineering , combinatorics , thermodynamics , electronic engineering
This paper is concerned with an application of the concept of topological derivative to elastic‐wave imaging of finite solid bodies containing cavities. Building on the approach originally proposed in the (elastostatic) theory of shape optimization, the topological derivative, which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity‐free (reference) body, is used as a void indicator through an assembly of sampling points where it attains negative values. The computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary‐value problems for the reference body, which are here formulated as boundary integral equations. For a comprehensive treatment of the subject, formulas for topological sensitivity are obtained using three alternative methodologies, namely (i) direct differentiation approach, (ii) adjoint field method, and (iii) limiting form of the shape sensitivity analysis. The competing techniques are further shown to lead to distinct computational procedures. Methodologies (i) and (ii) are implemented within a BEM‐based platform and validated against an analytical solution. A set of numerical results is included to illustrate the utility of topological derivative for 3D elastic‐wave sounding of solid bodies; an approach that may perform best when used as a pre‐conditioning tool for more accurate, gradient‐based imaging algorithms. Despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution, the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques. Copyright © 2004 John Wiley & Sons, Ltd.