Non-Monotone Projected Gradient Method in Linear Elasticity Contact Problems with Given Friction

Abstract

We are focusing on the algorithms for solving the large-scale convex optimization problem in linear elasticity contact problems discretized by Finite Element method (FEM). The unknowns of the problem are the displacements of the FEM nodes, the corresponding objective function is defined as a convex quadratic function with symmetric positive definite stiffness matrix and additional non-linear term representing the friction in contact. The feasible set constraints the displacement subject to non-penetration conditions. The dual formulation of this optimization problem is well-known as a Quadratic Programming (QP) problem and can be considered as a most basic non-linear optimization problem. Understanding these problems and the development of efficient algorithms for solving them play the crucial role in the large-scale problems in practical applications. We shortly review the theory and examine the behavior and the efficiency of Spectral Projected Gradient method modified for QP problems (SPG-QP) on the solution of a toy example in MATLAB environment.