Shape-topological differentiability of energy functionals for unilateral problems in domains with cracks and applications - Leugering, Günter; Sokolowski, Jan; Zochowski, Antoni (Springer Verlag, 2015)
A review of results on first order shape-topological differentiability of energy functionals for a class of variational inequalities of elliptic type is presented. The velocity method in shape sensitivity analysis for solutions of elliptic unilateral problems is established in the monograph (Sokołowski and Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). The shape and material derivatives of solutions to frictionless contact problems in solid mechanics are obtained. In this way the shape gradients of the associated integral functionals are derived within the framework of nonsmooth analysis. In the case of the energy type functionals classical differentiability results can be obtained, because the shape differentiability of solutions is not required to obtain the shape gradient of the shape functional (Sokołowski and Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). Therefore, for cracks the strong continuity of solutions with respect to boundary variations is sufficient in order to obtain first order shape differentiability of the associated energy functional. This simple observation which is used in Sokołowski and Zolésio (Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) for the shape differentiability of multiple eigenvalues is further applied in Khludnev and Sokołowski (Eur. J. Appl. Math. 10:379–394, 1999; Eur. J. Mech. A Solids 19:105–120, 2000) to derive the first order shape gradient of the energy functional with respect to perturbations of the crack tip. A domain decomposition technique in shape-topology sensitivity analysis for problems with unilateral constraints on the crack faces (lips) is presented for the shape functionals.We introduce the Griffith shape functional as the distributed shape derivative of the elastic energy evaluated in a domain with a crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the crack. As a result, the directional shape and topological derivatives of the nonsmooth Griffith shape functional are obtained with respect to boundary variations of an inclusion.
Keywords: Conical differential of metric projection; Dirichlet Sobolev space; Griffith criterium for crack propagation; Hadamard shape differentiability; Nonsmooth analysis; Shape gradient; Shape Hessian; Signorini variational inequality