Asymptotic analysis of variational inequalities with applications to optimum design in elasticity

Contact problems with given friction are considered for plane elasticity in the framework of shape-topological optimization. The asymptotic analysis of the second kind variational inequalities in plane elasticity is performed for the purposes of shapetopological optimization. To this end, the saddle point formulation for the associated Lagrangian is introduced for the variational inequality. The non-smooth term in the energy functional is replaced by pointwise constraints for the multipliers. The one term expansion of the strain energy with respect to the small parameter which governs the size of the singular perturbation of geometrical domain is obtained. The topological derivatives of energy functional are derived in closed form adapted to the numerical methods of shape-topological optimization. In general, the topological derivative (TD) of the elastic energy is defined through a limit passage when the small parameter governing the size of the topological perturbation goes to zero. TD can be used as a steepestdescent direction in an optimization process like in any method based on the gradient of the cost functional. In this paper, we deal with the topological asymptotic analysis in the context of contact problems with given friction. Since the problem is nonlinear, the domain decomposition technique combined with the Steklov-Poincaré pseudodifferential boundary operator is used for asymptotic analysis purposes with respect to the small parameter associated with the size of the topological perturbation. As a fundamental result, the expansion of the strain energy coincides with the expansion of the SteklovPoincaré operator on the boundary of the truncated domain, leading to the expression for TD. Finally, the obtained TD is applied in the context of topology optimization of mechanical structures under contact condition with given friction.