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

Lukáš Pospíšil ,Martin Čermák,David Horák,Jakub Kružík

doi:10.3390/su12208674

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.

Highlights

One of the key components of the structural design is the contact problem
In the case of the linear elasticity contact problem with the Tresca friction, the discretized minimization problem of the energy functional with the additional non-linear term representing the response of displacement caused by friction and constraints representing the non-penetration, can be reformulated using the dualization into the Quadratic programming (QP) problem with bound and separable quadratic inequality constraints [1,12,14]
The contact problems of linear elasticity are characterized by the bounded spectrum independent of the used mesh [22]

Summary

Introduction

One of the key components of the structural design is the contact problem. This problem is crucial, for example, during the design process of steel components, namely welded and screwed contacts. In the case of the linear elasticity contact problem with the Tresca (given) friction, the discretized minimization problem of the energy functional with the additional non-linear term representing the response of displacement caused by friction and constraints representing the non-penetration, can be reformulated using the dualization into the QP problem with bound and separable quadratic inequality constraints [1,12,14]. The contact problems of linear elasticity are characterized by the bounded spectrum independent of the used mesh [22] The utilization of this property has been proven and practically applied/demonstrated only in the case of active-set algorithms, namely the MPRGP algorithm (Modified Proportioning with Reduced Gradient Projections, [4,8,12,23,24]). We examine the projected gradient descent method [26,27,28], especially SPG-QP (Spectral Projected Gradients [29] for QP [5])

Problem Definition

Numerical Solution

Results

Conclusions