Shape optimization in contact problems with Coulomb friction and a solution-dependent friction coefficient
- Authors: Beremlijski, Petr , Haslinger, Jaroslav , Outrata, Jiri , Pathó, Róbert
- Date: 2014
- Type: Text , Journal article
- Relation: SIAM Journal on Control and Optimization Vol. 52, no. 5 (2014), p. 3371-3400
- Full Text: false
- Reviewed:
- Description: The present paper deals with shape optimization in discretized two-dimensional (2D) contact problems with Coulomb friction, where the coefficient of friction is assumed to depend on the unknown solution. Discretization of the continuous state problem leads to a system of finite-dimensional implicit variational inequalities, parametrized by the so-called design variable, that determines the shape of the underlying domain. It is shown that if the coefficient of friction is Lipschitz and sufficiently small in the C0,1 -norm, then the discrete state problems are uniquely solvable for all admissible values of the design variable (the admissible set is assumed to be compact), and the state variables are Lipschitzian functions of the design variable. This facilitates the numerical solution of the discretized shape optimization problem by the so-called implicit programming approach. Our main results concern sensitivity analysis, which is based on the well-developed generalized differential calculus of B. Mordukhovich and generalizes some of the results obtained in this context so far. The derived subgradient information is then combined with the bundle trust method to compute several model examples, demonstrating the applicability and efficiency of the presented approach. © 2014 Society for Industrial and Applied Mathematics
Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction
- Authors: Haslinger, Jaroslav , Outrata, Jiri , Pathó, Róbert
- Date: 2012
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 20, no. 1 (2012), p. 31-59
- Full Text: false
- Reviewed:
- Description: The paper deals with shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems, where the coefficient of friction depends on the solution. We consider the twodimensional Signorini problem, coupled with the physically less accurate model of given friction, but assume a solution-dependent coefficient of friction. First, we investigate the shape optimization problem in the continuous, infinite-dimensional setting, followed by a suitable finite-dimensional approximation based on the finite-element method. Convergence analysis is presented as well. Next, an algebraic form of the state problem is studied, which is obtained from the discretized problem by further approximating the frictional term by a quadrature rule. It is shown that if the coefficient of friction is Lipschitz continuous with a sufficiently small modulus, then the algebraic state problem is uniquely solvable and its solution is a Lipschitz continuous function of the control variable, describing the shape of the elastic body. For the purpose of numerical solution of the shape optimization problem via the so-called implicit programming approach we perform sensitivity analysis by using the tools from the generalized differential calculus of Mordukhovich. The paper is concluded first order optimality conditions. © 2011 Springer Science+Business Media B.V.