تجزیه و تحلیل حساسیت برای اغتشاش شکل حفره و یا شکاف های داخلی با استفاده از BIE و الحاقی روش متغیر

This paper deals with the application of the adjoint variable approach to sensitivity analysis of objective functions used for defect detection from knowledge of supplementary boundary data, in connection with the use of BIE/BEM formulations for the relevant forward problem. The main objective is to establish expressions for crack shape sensitivity, based on the adjoint variable approach, that are suitable for BEM implementation. In order to do so, it is useful to consider first the case of a cavity defect, for which such boundary-only sensitivity expressions are obtained for general initial geometry and shape perturbations. The analysis made in the cavity defect case is then seen to break down in the limiting case of a crack. However, a closer analysis reveals that sensitivity formulas suitable for BEM implementation can still be established. First, particular sensitivity formulas are obtained for special shape transformations (translation, rotation or expansion of the crack) for either two- or three-dimensional geometries which, except for the case of crack expansion together with dynamical governing equations, are made only of surface integrals (three-dimensional geometries) or line integrals (two-dimensional geometries). Next, arbitrary shape transformations are accommodated by using an additive decomposition of the transformation velocity over a tubular neighbourhood of the crack front, which leads to sensitivity formulas. This leads to sensitivity formulas involving integrals on the crack, the tubular neighbourhood and its boundary. Finally, the limiting case of the latter results when the tubular neighbourhood shrinks around the crack front is shown to yield a sensitivity formula involving the stress intensity factors of both the forward and the adjoint solutions. Classical path-independent integrals are recovered as special cases. The main exposition is done in connection with the scalar transient wave equation. The results are then extended to the linear time-domain elastodynamics framework. Linear static governing equations are contained as obvious special cases. Numerical results for crack shape sensitivity computation are presented for two-dimensional time-domain elastodynamics.

The consideration of sensitivity analysis of integral functionals with respect to shape parameters arises in many situations where a geometrical domain plays a primary role; shape optimization and inverse problems are the most obvious, as well as possibly the most important, of such instances. It is well known that, apart from resorting to approximative techniques such as finite differences, shape sensitivity evaluation can be dealt with using either the direct differentiation approach or the adjoint variable approach (see, e.g. Burczyński, 1993b), the present paper being focused on the latter. Besides, consideration of shape changes in otherwise (i.e. for fixed shape) linear problems makes it very attractive to use boundary integral equation (BIE) formulations, which constitute the minimal modelling as far as the geometrical support of unknown field variables is concerned. In the BIE context, the direct differentiation approach rests primarily upon the material differentiation of the governing integral equations. This step has been studied by many researchers, from BIE formulation in either singular form (Barone and Yang, 1989; Mellings and Aliabadi, 1995) or regularized form (Bonnet, 1995b; Matsumoto et al., 1993; Nishimura et al., 1992; Nishimura, 1995). Following this approach, the process of sensitivity computation needs the solution of as many new boundary-value problems as the numbers of shape parameters present. The fact that they all involve the same, original, governing operator reduces the computational effort to the building of new right-hand sides and the solution of linear systems by backsubstitution. The usual material differentiation formula for surface integrals is shown in Bonnet (1997) to be still valid when applied to strongly singular or hypersingular formulations. Thus, the direct differentiation approach is in particular applicable in the presence of cracks. The adjoint variable approach is even more attractive, since it requires the solution of only one new boundary-value problem (the so-called adjoint problem) per integral functional present (often only one), whatever the number of shape parameters. In connection with BIE formulations alone, the adjoint variable approach has been successfully applied to many shape sensitivity problems (see, e.g. Aithal and Saigal, 1995; Bonnet, 1995a; Burczyński, 1993a; Burczyński and Fedelinski, 1992; Burczyński et al., 1995; Choi and Kwak, 1988; Meric, 1995). This relies heavily upon the possibility of formulating the final, analytical expression of the shape sensitivity of a given integral functional as a boundary integral that involves the values taken by the primary and adjoint states on the boundary. However, obtaining this boundary-only expression raises mathematical difficulties when the geometrical domain under consideration contains cracks or other geometrical singularities; non-integrable terms associated with, e.g. crack tip singularity of field variables appear in some expressions. This paper deals with the formulation of the adjoint variable method applied to sensitivity analysis, in connection with the use of BIE formulations for the transient wave equation. Typical problems where this approach is useful are inverse problems of cavity or crack detection from transient wave measurements on a part of the external boundary, where the integral functionals considered express the gap between measured and computed data on the external boundary, e.g. in the form of a least-squares distance. However, the sensitivity results are derived for more general boundary integral functionals. The formulation of the adjoint problem and the corresponding boundary-only formula for the shape sensitivity of the functional are established for the case of an unknown cavity. The latter is then shown to become inconsistent in the limit when the cavity becomes a crack, due to the non-integrability of a certain domain integral, causing an integration-by-parts process to break down. However, resting on the analysis made for the case of a cavity, functional shape sensitivity expressions consistent with the use of BIE formulations and applicable to crack identification problems are derived in three different forms. Firstly, simple shape transformations (translations, rotations, expansion) are considered. Secondly, a sensitivity formula involving integrals on the crack, on an arbitrary tubular neighbourhood of the crack front and on its boundary is derived. Thirdly, the limiting case of the latter result when the tubular neighbourhood shrinks around the crack front is shown to yield a sensitivity formula involving the stress intensity factors of both the forward and the adjoint solutions. All sensitivity results presented here are obtained from the formulation of the continuous problem, i.e. are not directly obtained from the BIE formulations but are tailored for use in conjunction with the BEM. It is also possible to define adjoint problems and sensitivity results directly from the BIE formulations (Bonnet, 2001).

In the present work a shape sensitivity analysis for identification of internal cavities or cracks has been presented. The main motivation of this paper was to explore the adjoint variable approach, in the presence of cracks and in connection with BIE formulations of the forward problem. First, a general formulation for the sensitivity with respect to the shape of a cavity of objective functionals expressed as boundary integrals has been derived using the material derivative-adjoint variable approach. The sensitivity of the functional has been expressed as a boundary integral. In the case of a crack, the previous boundary-only expression is not applicable. However, revisiting the discussion of the cavity problem, it has been shown that for two classes of crack perturbations the adjoint variable approach to sensitivity analysis is still applicable in the presence of cracks. Firstly, when the domain transformations considered consist of translation, rotation or expansion of the crack, the functional sensitivity is expressed as an integral over an arbitrary surface surrounding the crack, supplemented for the case of crack expansion in dynamics by a domain integral over the crack front neighbourhood enclosed by this surface. This applies for arbitrary geometries, either three- and two-dimensional. Earlier works on path-independent integral approach to sensitivity analysis are thus revisited and generalized. Secondly, sensitivity formulas applicable to arbitrary shape perturbations were established by means of an additive decomposition of the transformation velocity over a tubular neighbourhood of the crack front. Thirdly, the limiting case of the latter results when the tubular neighbourhood shrinks around the crack front has been shown to yield a boundary-only sensitivity formula involving the stress intensity factors of both the forward and the adjoint solutions. All these results were obtained in connection with both scalar wave and elastodynamic problems formulated in the time domain. The analysis conducted in this paper is applicable without difficulty to objective functions of the form: equation(56) where the last integral might for instance be used to formulate some a priori information about the defect (for instance by penalizing high curvatures to avoid recovering oscillatory shapes). Since this last integral depends on Γ in an explicit manner, one simply needs to invoke the differentiation formula (12). As a result, the contribution should be added to each of the sensitivity formulas (20), , , , and , , , and , (33), (41), (43), (44), , and , (51). It is important to stress that Eq. (51) provides the sensitivity of an integral functional to a perturbation of a fixed crack configuration, not a crack propagation, hence the use of expansions (50), valid for a crack which does not physically propagate. Eq. (51) is also applicable, with straightforward modifications, to elastostatics and elastodynamics in the frequency domain. For instance, in elastostatics, is the potential energy at equilibrium for the particular choice , in Eq. (5). For this special case, the adjoint solution turns out to be , i.e. KIv=KIu/2, etc. In Eq. (51), the factor of θν(s) turns out to be, as expected, minus the energy release rate G(s), i.e. minus the J1-integral, whereas the factor of θn(s) is the three-dimensional generalization of the J2-integral ( Budiansky and Rice, 1973; Bui, 1978). Finally, with the choice and , where are the boundary traces of a pre-selected auxiliary elastodynamic state with final homogeneous conditions, one finds that and that the factor of θν(s) in (51) is the three-dimensional generalization of the so-called H-integral ( Bui and Maigre, 1988).