A Bem- and Adjoint Variable-based Approach to Crack Shape Sensitivity Analysis

This communication addresses a computation strategy, based on the adjoint variable approach and BIE/BEM formulations of the direct problem, for evaluating crack or void shape sensitivities of objective functionals. Boundary-only expressions for such sensitivities are sought, in the context of linear elastodynamics. In the case of a void, boundary-only expressions for sensitivities of integral functionals defined on (part of) the external boundary are easy to obtain by the standard adjoint variable approach. When the void degenerates to a crack, the previous result ceases to be applicable, however, because non-integrable terms arise due to crack-tip singularities. We show, for two classes of crack perturbations, that boundaryonly sensitivity expressions using an adjoint state can still be obtained: (1) simple transformations (translation, rotation or expansion of the crack) of arbitrarily shaped domains, and (2) general two-dimensional geometries and crack perturbations. In the latter case, the shape sensitivity is expressed using the primal and adjoint stress intensity factors. Numerical tests of the latter kind of sensitivity expression are presented for a 2-D body with an internal crack, in plane-strain elastodynamics. The influence of crack shape perturbations on an objective functional is examined. The sensitivity results obtained using the present strategy compare well with finite difference evaluations.