A DOFs condensation based algorithm for solving saddle point systems in contact computation

TL;DR

This paper introduces an efficient DOFs condensation algorithm for saddle point systems in contact mechanics, reducing system size and enabling faster solutions with PCG, demonstrated through numerical experiments.

Contribution

It develops a DOFs condensation technique leveraging the mortar matrix structure to transform large saddle point systems into smaller, symmetric positive definite systems for efficient solving.

Findings

Reduced system size after DOFs condensation

System becomes symmetric positive definite

Numerical results confirm efficiency and effectiveness

Abstract

In contact mechanics computation, the constraint conditions on the contact surfaces are typically enforced by the Lagrange multiplier method, resulting in a saddle point system. The mortar finite element method is usually employed to discretize the variational form on the meshed contact surfaces, leading to a large-scale discretized saddle point system. Due to the indefiniteness of the discretized system, it is a challenge to solve the saddle point algebraic system. For two-dimensional tied contact problem, an efficient DOFs condensation technique is developed. The essential of the proposed method is to carry out the DOFs elimination by using the tridiagonal characteristic of the mortar matrix. The scale of the linear system obtained after DOFs elimination is smaller, and the matrix is symmetric positive definite. By using the preconditioned conjugate gradient (PCG) method, the linear system can be solved efficiently. Numerical results show the effectiveness of the method.