Shape Optimization in Contact Problems with Coulomb Friction

Abstract

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 C 0,1-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.

Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction

“…A similar situation arises in discretized contact problems with Coulomb friction, cf. [1] and [2]. The GE considered here differs, however, considerably from the corresponding GE in the case of 2D contact problems with Coulomb friction.…”

Shape optimisation with the level set method for contact problems in linearised elasticity

“…Subgradients were computed for the problems written as variational inequalities in [33] and [51]. It yields a better accuracy on the normal force and gives good results for optimisation with Coulomb friction [7]. Note…”

“…For this last model of friction, the uniqueness of the contact solution is not ensured for the continuous model and examples of non-uniqueness can be built. Consequently, in [7] and [8], the authors analyse the derivation of the discretised problem, which admits a unique solution for small friction coefficients, by using subgradient calculus. Eventually, a thorough review of other results in shape optimisation for contact problems can be found in [31].…”

Contact Optimization Problems for Stationary and Sliding Conditions

“…We have solved Problem 10 defined by (23) for different values of parameter q. The calculated torque is presented in Fig.…”

“…Optimization problems with frictional contact were investigated in [23][24][25][26][27][28][29][30]. Special methods (level set method, evolutionary approach) were also used for topology optimization [31,32].…”