Shape Sensitivity Analysis of Unilateral Problems

Abstract

After a short introduction we turn to the sensitivity analysis of an abstract variational inequality and we apply this result to the sensitivity analysis of an obstacle problem: $\Omega _t $ is a domain built by a speed vector field V, $K(\Omega _t )$ a closed convex subset of the Sobolev space $H^1 (\Omega _t )$, $z_t $ is the element of $K(\Omega _t )$ solution of an elliptic variational inequality. We characterize the shape derivative $z' = \dot z - \nabla z$. $V(0)$ (where $\dot z$ is the material derivative, $\dot z = (({d / {dt}})z_t \circ T_t )_{t = 0} $) as the solution of another variational inequality well posed on a new convex subset $S_V (\Omega )$ of $H^1 (\Omega )$ depending on the speed vector field V only by boundary expression (see Theorem 2). Finally we characterize the material derivative $\dot {\bf u}$ of the solution of the Signorini variational inequality associated with planar linear elasticity. For this purpose the material derivative has been generalized to vectors situation by $\dot {\bf u} = {d / {dt}}(DT_t^{ - 1} .\dot {\bf u}_t \circ T_t )_{t = 0} $. This material derivative is the solution of a variational inequality posed on a convex subset $S(\Omega )$ of $H^1 (\Omega _t )^2 $.