Passive Control of Singularities by Topological Optimization: The Second-Order Mixed Shape Derivatives of Energy Functionals for Variational Inequalities

A class of nonsmooth shape optimization problems for variational inequalities is considered. 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 plane elasticity to measure SIF, the so-called stress intensity factor, at the crack tips. Thus, topological optimization can be used for passive control of singularities of weak solutions to variational inequalities. The Hadamard directional differentiability of metric the 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. A 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. Singular geometrical domain perturbations in an elastic body Ω $$\Omega$$ are approximated by 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 linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations 𝜖 → ω 𝜖 ⊂ Ω $$\epsilon \rightarrow \omega _{\epsilon }\subset \Omega$$ centred at x ̂ ∈ Ω $$\hat{x} \in \Omega$$ are replaced by regular perturbations of bilinear forms supported on the manifold Γ R = { | x − x ̂ | = R } $$\Gamma _{R} =\{ \vert x -\hat{ x}\vert = R\}$$ in an elastic body, with R > ε > 0. The obtained expressions for topological derivatives are easy to compute and therefore useful in numerical methods of topological optimization for contact problems.