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An iterative constraint energy minimizing generalized multiscale finite element method for contact problem

Zishang Li, Changqing Ye, Eric T. Chung

TL;DR

The paper addresses Signorini-type contact problems with high-contrast coefficients by transforming the variational inequality into a non-smooth unconstrained problem via a penalty term and solving it iteratively with an Iterative Constraint Energy Minimizing Generalized Multiscale Finite Element Method (ICEM-GMsFEM). The method builds multiscale spaces through local spectral problems and constraint energy minimization on oversampled domains, enriched with boundary correctors, and uses a semismooth Newton framework to handle nonlinearity on the contact boundary. Theoretical results establish superlinear convergence of the semismooth Newton iteration and derive error estimates for the multiscale solution, while numerical experiments on heterogeneous media validate fast convergence and accuracy under various parameter settings. The approach offers an efficient, scalable tool for multiscale contact problems in engineering applications, with potential extensions to dynamic and frictional contact regimes.

Abstract

This work presents an Iterative Constraint Energy Minimizing Generalized Multiscale Finite Element Method (ICEM-GMsFEM) for solving the contact problem with high contrast coefficients. The model problem can be characterized by a variational inequality, where we add a penalty term to convert this problem into a non-smooth and non-linear unconstrained minimizing problem. The characterization of the minimizer satisfies the variational form of a mixed Dirilect-Neumann-Robin boundary value problem. So we apply CEM-GMsFEM iteratively and introduce special boundary correctors along with multiscale spaces to achieve an optimal convergence rate. Numerical results are conducted for different highly heterogeneous permeability fields, validating the fast convergence of the CEM-GMsFEM iteration in handling the contact boundary and illustrating the stability of the proposed method with different sets of parameters. We also prove the fast convergence of the proposed iterative CEM-GMsFEM method and provide an error estimate of the multiscale solution under a mild assumption.

An iterative constraint energy minimizing generalized multiscale finite element method for contact problem

TL;DR

The paper addresses Signorini-type contact problems with high-contrast coefficients by transforming the variational inequality into a non-smooth unconstrained problem via a penalty term and solving it iteratively with an Iterative Constraint Energy Minimizing Generalized Multiscale Finite Element Method (ICEM-GMsFEM). The method builds multiscale spaces through local spectral problems and constraint energy minimization on oversampled domains, enriched with boundary correctors, and uses a semismooth Newton framework to handle nonlinearity on the contact boundary. Theoretical results establish superlinear convergence of the semismooth Newton iteration and derive error estimates for the multiscale solution, while numerical experiments on heterogeneous media validate fast convergence and accuracy under various parameter settings. The approach offers an efficient, scalable tool for multiscale contact problems in engineering applications, with potential extensions to dynamic and frictional contact regimes.

Abstract

This work presents an Iterative Constraint Energy Minimizing Generalized Multiscale Finite Element Method (ICEM-GMsFEM) for solving the contact problem with high contrast coefficients. The model problem can be characterized by a variational inequality, where we add a penalty term to convert this problem into a non-smooth and non-linear unconstrained minimizing problem. The characterization of the minimizer satisfies the variational form of a mixed Dirilect-Neumann-Robin boundary value problem. So we apply CEM-GMsFEM iteratively and introduce special boundary correctors along with multiscale spaces to achieve an optimal convergence rate. Numerical results are conducted for different highly heterogeneous permeability fields, validating the fast convergence of the CEM-GMsFEM iteration in handling the contact boundary and illustrating the stability of the proposed method with different sets of parameters. We also prove the fast convergence of the proposed iterative CEM-GMsFEM method and provide an error estimate of the multiscale solution under a mild assumption.
Paper Structure (17 sections, 4 theorems, 70 equations, 13 figures, 2 algorithms)

This paper contains 17 sections, 4 theorems, 70 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Penalty method
  5. Semismooth Newton Method
  6. Numerical Method
  7. Conversion of the contact problem
  8. One step CEM-GMsFEM solver
  9. Iterative CEM-GMsFEM Algorithm
  10. Numerical experiments
  11. Model problem 1
  12. Model problem 2
  13. For testing accuracy
  14. For different initial guesses and parameters
  15. For interdependent parameters
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $R: V\to \R$. Then, for $u\in V$, the following statements are equivalent: (a) $R$ is semismooth at $u$. (b) $R$ is locally Lipschitz continuous at $u$, $R'(u;\cdot)$ exists, and for any $G\in \partial R(u + d)$, (c) $R$ is locally Lipschitz continuous at $u$, $R'(u;\cdot)$ exists, and for any $G\in \partial R(u + d)$,

Figures (13)

  • Figure 1: A deformable body in contact with a rigid obstacle.
  • Figure 2: Illustration of meshes, fine element, coarse element, and oversampling domain.
  • Figure 3: The permeability fields (a) medium A; (b) medium B; (c) medium C.
  • Figure 4: The source function (a) $f_1$; (b) $f_2$; (c) $f_3$.
  • Figure 5: The solutions after iterations with medium A, using the source function $f_1$ and the initial value $u_0^\mathup{cem}=u_0^\mathup{fe}=u_0^0$. The first row shows the contour images, and the second row displays the 3D images. The iteration number: (a)(e)$k$=1; (b)(f)$k$=2; (c)(g)$k$=3; (d)(h)$k$=8.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Definition 2.1
  • Theorem 5.1
  • proof
  • Lemma 5.2: See Ye2023
  • Theorem 5.4
  • proof