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

Lukáš Pospíšil, Martin Čermák, David Horák and Jakub Kružík
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Lukáš Pospíšil: Department of Mathematics, Faculty of Civil Engineering, VSB-TU Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava, Czech Republic
Martin Čermák: Department of Mathematics, Faculty of Civil Engineering, VSB-TU Ostrava, Ludvíka Podéště 1875/17, 708 00 Ostrava, Czech Republic
David Horák: Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VSB-TU Ostrava, 17 Listopadu 2172/15, 708 00 Ostrava, Czech Republic
Jakub Kružík: Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VSB-TU Ostrava, 17 Listopadu 2172/15, 708 00 Ostrava, Czech Republic

Sustainability, 2020, vol. 12, issue 20, 1-11

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.

Keywords: contact problems; linear elasticity; tresca friction; SPG-QP; quadratic programming (search for similar items in EconPapers)
JEL-codes: O13 Q Q0 Q2 Q3 Q5 Q56 (search for similar items in EconPapers)
Date: 2020
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