Sensitivity analysis for shape perturbation of cavity or internal crack using BIE and adjoint variable approach

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 (translations, 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, arbitary 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.