Variation Boundary Integral Equation for Flaw Shape Identification

In this communication a Variation Boundary Integral Equation (BIE) for the solution of identification inverse problems is presented. This equation relates the variation of the fields along the boundary with the variation of the geometry of a flaw, whose position and shape are unknown beforehand. The Variation BIE is obtained linearizing the difference between the standard BIE for the actual configuration (actual flaw) and the standard BIE for the assumed configuration. The resulting Variation BIE has not been completely derived before, to the authors knowledge. The solution of the ensuing Variation BIE is tackled by a procedure that avoids altogether the solution of a nonlinear minimization problem. The variations of the design variables (geometry) have been written in terms of a virtual strain field applied to the flaw. This approach can be applied to flaws of any shape and location. Several numerical applications are solved with the proposed formulation. INTRODUCTION The study of inverse problems in engineering has become and active area of research in the last two decades (Tanaka and Dulikravich, 1998; Delaunay and Jarny, 1996; Zabaras, Woodbury and Raynaud, 1993). There are many important topics in engineering which involve the solution of this kind of problems: non-destructive evaluation of materials and structures, characterAddress all correspondence to this author. ization of material properties, medical imaging or even diagnosis, etc. In this paper we consider the identification inverse problem in the case of an acoustic field over a two-dimensional domain. The aim in the identification inverse problem is to compute an inaccessible part of the boundary of the domain, usually an internal flaw, using experimental data as additional information. Since the objective is to find part of the boundary, an approach based on Boundary Integral Equations appears as a very sensible alternative. The derivation of the gradient for such problems has been tackle by several researches by either the adjoint variable approach or direct differentiation. The first method has been employed by several authors in different applications, i.e. Aithal and Saigal (1995) for two-dimensional thermal problems; Bonnet (1995a) for 3D inverse scattering problems by hard and penetrable obstacles; Meric (1995) for shape optimization in potential fields. The direct differentiation approach has been extensively used as well, for regular, strongly, and even hypersingular BIE (Mellings and Alliabadi, 1993; Bonnet, 1995b; Matsumoto et al., 1993; Nishimura and Kobayashi, 1991; Kirsch, 1993). Zeng and Saigal (1992) developed a formulation for potential fields based on variations. Tanaka and Matsuda (1989) developed a similar approach years earlier using Taylor expansions of the kernels and densities in the BIE. In these two papers, the authors propose an approach different to the minimization of a cost functional, but failed to demonstrate its reliability, due to mathematical inconsistencies (Tanaka and Matsuda 1989) or simply 1 Copyright  1999 by ASME because no numerical application is carried out (Zeng and Saigal, 1992). Gallego and Suarez (1999a) developed the variation Boundary Integral Equation (δBIE) and presented some numerical results using the non-minimization approach. Nevertheless, the δBIE should be equivalent to the sensitivity integral equations obtained by the direct differentiation approach, and therefore, the δBIE can be used to compute the gradient of a given cost function, although no effort has been undertaken in this direction yet. In this communication we demonstrate that the δBIE is a reliable method to solve identification inverse problems for acoustic fields. First the δBIE is reviewed and a numerical procedure for its solution by collocation boundary element discussed. The virtual strain field is introduced for the geometrical parameter representation, which allows for greater flexibility for the shape of the assumed flaw. An iterative procedure for the solution of the identification inverse problem is proposed, and finally, some numerical applications are presented. INVERSE PROBLEM FOR LINEAR ACOUSTIC MEDIA: BASIC EQUATIONS In this section the basic equations for the solution of harmonic wave propagation problems in a linear acoustic media are reviewed. In the last paragraph the identification inverse problem is briefly described. Differential equation statement Consider the well known problem of an acoustic field, ∇u(x)+ k2u(x) = 0; x 2Ω (1) subject to essential and/or natural boundary conditions, u(x) = ū; x 2 Γu q(x) = q̄; x 2 Γq where, u(x) is the acoustic field in Ω; q(x) = ∂u=∂n is the flux at a point x on the boundary Γ whose outward normal is n(x); k = c is the wave number, ω the frequency of the wave and c its velocity; ū and q̄ represent known values of the field and flux on Γu and Γq respectively, where Γu[Γq = Γ and Γu\Γq = / 0. To solve the problem stated in equation (1) one needs to know (Kubo, 1988): the domain Ω and its boundary Γ, the differential operator (∇ + k2 in this case), boundary conditions (ū and q̄, and their supports Γu and Γq), and material properties (c). If one or several of these items are not completely known the problem stated in equation (1) will not be well-posed and will not have solution or if any, will not be unique. An inverse problem can therefore be stated whose objective is to find the missing information, using some additional data. Depending on the missing information different kinds of inverse problems can be stated. In this paper we deal with the so called identification inverse problem whose objective is to find part of the domain or its boundary. Integral equation statement In the identification problem part of the boundary, say Γ̃h, is the main unknown of the problem. Therefore the statement of the problem in terms of Boundary Integral Equations (BIE) appears as the most promising approach. The acoustic problem, stated in differential form in equation (1), can be written in terms of BIE (Dominguez, 1993) by the equation,