Passive control of singularities by topological optimization. The second order mixed shape derivatives of energy functionals for variational inequalities

Günter Leugering (1) , Jan Sokolowski (2) , Antoni Zochowski (3)

1 University of Erlangen Nuremberg, Department of Mathematics
2 IECL - Institut Élie Cartan de Lorraine
3 IBS PAN - Systems Research Institute [Warsaw]

The class of nonsmooth shape optimization problems for variational inequalities is con-sidered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of weak solutions to elliptic boundary value problems can be characterized. An example in solid mechanics is given by the Griffith's functional, which is defined in the plane elasticity to measure SIF, the so-called stress intensity factor, at the crack tips. Thus, the topological optimization can be used for passive control of singularities of weak solutions to variational inequalities. The Hadamard directional differentiability of metric projection onto the positive cone in fractional Sobolev spaces is employed to the topological sensitivity analysis of weak solutions of nonlinear elliptic boundary value problems. The first order shape derivatives of energy functionals in the direction of specific velocity fields depend on the solutions to variational inequalities in a subdomain. The domain decomposition technique is used in order to separate the unilateral boundary conditions and the energy asymptotic analysis. The topological derivatives of nonsmooth integral shape functionals for variational inequalities are derived. The singular geometrical doamin perturbations in an elastic body Ω are approximated by the regular perturbations of bilinear forms in variational inequality, without any loss of precision for the purposes of the second order shape-topological sensitivity analysis. The second-order shape-topological directional derivatives are obtained for the Laplacian and for the linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations ǫ → ωǫ ⊂ Ω centered at x ∈ Ω are replaced by regular perturbations of bilinear forms supported on the manifold ΓR = {|x − x| = R} in an elastic body, with R > ǫ > 0. The obtained expressions for topological derivatives are easy to compute and therefore useful in numerical methods of topological optimization for contact problems.

Mots clés: Signorini problem; shape sensitivity; shape functional; domain decomposition; topological sensitivity; variational inequality; Steklov-Poincaré operator; contact problems

Domaines: Mathématiques [math]

Günter Leugering, Jan Sokolowski, Antoni Zochowski. Passive control of singularities by topological optimization. The second order mixed shape derivatives of energy functionals for variational inequalities. 2015. ⟨hal-01100889⟩