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Mixed Finite Element Method for Multi-layer Elastic Contact Systems

Zhizhuo Zhang, Mikaël Barboteu, Xiaobing Nie, Serge Dumont, Mahmoud Abdel-Aty, Jinde Cao

TL;DR

This work addresses the numerical solution of multi-layer elastic contact problems under interlayer Tresca friction by formulating a mixed finite element method (MFEM) that yields a saddle-point problem with interlayer Lagrange multipliers. It establishes an equivalent mixed formulation, proves convergence of the MFEM solution to the continuous solution, and derives a priori error estimates with rate \(\hat{k}=3/4\) under regularity. An algebraic dual form is developed to enable efficient implementation, illustrated via a GetFEM-based workflow. Numerical experiments on 3D and 2D layered systems compare MFEM with a Layer Decomposition approach, verify convergence, and demonstrate the predicted error behavior and practical performance of the method.

Abstract

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is given. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter $H$ is verified.

Mixed Finite Element Method for Multi-layer Elastic Contact Systems

TL;DR

This work addresses the numerical solution of multi-layer elastic contact problems under interlayer Tresca friction by formulating a mixed finite element method (MFEM) that yields a saddle-point problem with interlayer Lagrange multipliers. It establishes an equivalent mixed formulation, proves convergence of the MFEM solution to the continuous solution, and derives a priori error estimates with rate under regularity. An algebraic dual form is developed to enable efficient implementation, illustrated via a GetFEM-based workflow. Numerical experiments on 3D and 2D layered systems compare MFEM with a Layer Decomposition approach, verify convergence, and demonstrate the predicted error behavior and practical performance of the method.

Abstract

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is given. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter is verified.
Paper Structure (15 sections, 6 theorems, 115 equations, 19 figures, 1 table)

This paper contains 15 sections, 6 theorems, 115 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. The physical model and variational inequality
  4. Equivalent formulation of
  5. Finite element approximation and convergence analysis
  6. Finite element spaces and discrete problem
  7. Convergence analysis
  8. Error estimates
  9. Implementation of Algorithm
  10. Numerical examples and comparisons
  11. Experimental model
  12. Mesh and Finite Element Space
  13. Comparison between LDM and MFEM
  14. Numerical validation of the error estimation
  15. Conclusion

Key Result

Lemma 1

In problem 4:prb:var.ieq, the stress ${\sigma}^{i}_{N}$ and $\boldsymbol{\sigma}^{i}_{T}$ on contact zone $\Gamma^{i}_{c}$ satisfy:

Figures (19)

  • Figure 1: Three-dimensional n-layer contact system
  • Figure 2: Two-dimensional 3-layer contact system
  • Figure 3: The physical model of three-layer elastic contact system.
  • Figure 4: Displacement nephogram along the X-axis of the $\Omega^1$ obtained by MFEM.
  • Figure 5: Displacement nephogram along the Y-axis of the $\Omega^1$ obtained by MFEM.
  • ...and 14 more figures

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Lemma 2
  • ...and 7 more