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

Xiaoyu Duan, Hengbin An, Zeyao Mo

TL;DR

This work tackles the computational bottleneck of saddle point systems arising from Lagrange multiplier enforcement in 2D tied contact problems discretized by the mortar method. It introduces a DOFs condensation strategy that eliminates Lagrange multipliers and slave displacements by exploiting the tridiagonal block structure of the mortar matrix, yielding a symmetric positive definite reduced system solved efficiently by an AMG-preconditioned CG method. Numerical results on three models show that the transformed system markedly outperforms the original saddle point system in robustness and speed, with stable iteration counts as problem size grows and favorable parallel scalability. The approach provides a practical, scalable path for large-scale contact computations, with potential extension to three-dimensional problems.

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.

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

TL;DR

This work tackles the computational bottleneck of saddle point systems arising from Lagrange multiplier enforcement in 2D tied contact problems discretized by the mortar method. It introduces a DOFs condensation strategy that eliminates Lagrange multipliers and slave displacements by exploiting the tridiagonal block structure of the mortar matrix, yielding a symmetric positive definite reduced system solved efficiently by an AMG-preconditioned CG method. Numerical results on three models show that the transformed system markedly outperforms the original saddle point system in robustness and speed, with stable iteration counts as problem size grows and favorable parallel scalability. The approach provides a practical, scalable path for large-scale contact computations, with potential extension to three-dimensional problems.

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.
Paper Structure (17 sections, 2 theorems, 52 equations, 5 figures, 6 tables, 2 algorithms)

This paper contains 17 sections, 2 theorems, 52 equations, 5 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical model for tied contact problem
  3. Discretization and the linear system
  4. Discretization based on mortar method
  5. Representation of the linear system for multi-body contact
  6. Condensation method for saddle point linear systems
  7. Elimination by condensation
  8. Solving matrix equations based on block Thomas algorithm
  9. Matrix representation
  10. Condensation algorithm
  11. Numerical Results
  12. Test models
  13. Algorithms setup
  14. Solution comparison for transformed and untransformed linear systems
  15. Algorithm scalability
  16. ...and 2 more sections

Key Result

Proposition 4.1

If the original matrix $\boldsymbol{\mathcal{A}}$ is nonsingular, then the condensed matrix $\hat{\boldsymbol{\mathcal{A}}}$ is also nonsingular.

Figures (5)

  • Figure 1: 2D two-body tied contact model.
  • Figure 2: Node number of slave contact surface.
  • Figure 3: Matrix pattern of slave mortar matrix $\mathbf{D}$.
  • Figure 4: Tied contact example models.
  • Figure 5: Deformation of example models.

Theorems & Definitions (4)

  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof