Combined symmetric BEM and semi-smooth Newton method for the contact problems in linear elasticity of Yukawa type

Abstract

The semi-smooth Newton method and the boundary element method are developed and analyzed for the solution of 2-D Signorini contact problems in linear elasticity of Yukawa type. First we consider the contact problem with Tresca friction. This leads to a constrained non-differentiable minimization problem where the solvability is in general problematic. But, by utilizing the Fenchel duality theory, the dual formulation in terms of contact stresses turns out to be a quadratic optimization problem with a smooth functional. The regularization of the dual problem motivated by the augmented Lagrangian is suitable for the application of the generalized Newton method. Applying the boundary integral equation method, the problem is reduced to the boundary curve. The corresponding boundary integral equations are approximated by using a Galerkin method with the help of B-splines on the boundary curve (BEM). This yields an algebraic system of linear equations with dense stiffness matrix but which is symmetric. The symmetry property of the stiffness matrix enables the application of efficient iterative solution strategies for the linear systems at each Newton step. Second, the methods are carried over to the Coulomb friction problem by means of a fixed point approach. In particular, the analysis of the algorithm is presented and some numerical examples, which show a remarkable efficiency and reliability of the semi-smooth Newton method, are given.