Regularized bie formulations for first- and second-order shape sensitivity of elastic fields

The subject of this paper is the formulation of boundary integral equations for first- and second-order shape sensitivities of boundary elastic fields in three-dimensional bodies. Here the direct differentiation approach is considered. It relies on the repeated application of the material derivative concept to the governing regularized (i.e. weakly singular) displacement boundary integral equation (RDBIE) for an elastostatic state on a given domain. As a result, governing BIEs, which are also weakly singular, are obtained for the elastic sensitivities up to the second order. They are formulated so as to allow a straightforward implementation; in particular no strongly singular integral is involved. It is shown that the actual computation of shape sensitivities using usual BEM discretization uses the already built and factored discrete integral operators and needs only to set up additional right-hand sides and additional backsubstitutions. Some relevant discretization aspects are discussed.