Adaptive optimal control of Signorini’s problem

Abstract

ABSTRACT. In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini's problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton's method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini's contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini's contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini's problem in two and three dimensions.

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

“…There are various studies related to problems describing contact of elastic bodies with rigid or elastic obstacles, see for example [1][2][3][4][5][6][7][8][9][10]. For an overview of contact problems we refer to [11][12][13][14][15].…”

Optimal control of friction coefficient in Signorini contact problems

“…In order to handle these difficulties, we shall approximate the given problem by a family of penalized control problems governed by a family of iterative variational equality, obtained by fixed point strategy, and then we shall approximate each penalized problem by a family of regularized problems governed by a variational equation, this common approach is performed, for instance, in References 30‐33.…”