Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiscale modeling for contact problem with high-contrast heterogeneous coefficients with primary-dual formulation

Zishang Li, Changqing Ye, Eric T. Chung

TL;DR

This paper addresses Signorini-type contact problems in media with highly heterogeneous and contrasting coefficients by marrying constrained energy minimizing GMsFEM (CEM-GMsFEM) with a primal-dual active-set strategy. It builds a contrast-robust reduced-order space using local spectral problems to form an auxiliary space, extends basis functions through energy minimization on oversampled domains, and updates basis functions only along the contact boundary during active-set evolution. Theoretical results establish finite-step convergence and error bounds that account for the contrast and coarse discretization, while numerical experiments demonstrate robust performance and accurate capture of fine-scale features near contact boundaries. The approach avoids penalty terms, preserves constraint satisfaction, and shows promise for scalable simulations in high-contrast multiscale contact problems.

Abstract

In this paper, we propose a novel iterative multiscale framework for solving high-contrast contact problems of Signorini type. The method integrates the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with a primal-dual active set strategy derived from semismooth Newton methods. First, local spectral problems are employed to construct an auxiliary multiscale space, from which energy minimizing multiscale basis functions are derived on oversampled domains, yielding a contrast-robust reduced-order approximation of the underlying partial differential equation. The multiscale bases are updated iteratively, but only at contact boundary, during the active set evolution process. Rigorous analysis is provided to establish error estimates and finite step convergence of the iterative scheme. Numerical experiments on heterogeneous media with high-contrast coefficients demonstrate that the proposed approach is both robust and efficient in capturing fine-scale features near contact boundaries.

Multiscale modeling for contact problem with high-contrast heterogeneous coefficients with primary-dual formulation

TL;DR

This paper addresses Signorini-type contact problems in media with highly heterogeneous and contrasting coefficients by marrying constrained energy minimizing GMsFEM (CEM-GMsFEM) with a primal-dual active-set strategy. It builds a contrast-robust reduced-order space using local spectral problems to form an auxiliary space, extends basis functions through energy minimization on oversampled domains, and updates basis functions only along the contact boundary during active-set evolution. Theoretical results establish finite-step convergence and error bounds that account for the contrast and coarse discretization, while numerical experiments demonstrate robust performance and accurate capture of fine-scale features near contact boundaries. The approach avoids penalty terms, preserves constraint satisfaction, and shows promise for scalable simulations in high-contrast multiscale contact problems.

Abstract

In this paper, we propose a novel iterative multiscale framework for solving high-contrast contact problems of Signorini type. The method integrates the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with a primal-dual active set strategy derived from semismooth Newton methods. First, local spectral problems are employed to construct an auxiliary multiscale space, from which energy minimizing multiscale basis functions are derived on oversampled domains, yielding a contrast-robust reduced-order approximation of the underlying partial differential equation. The multiscale bases are updated iteratively, but only at contact boundary, during the active set evolution process. Rigorous analysis is provided to establish error estimates and finite step convergence of the iterative scheme. Numerical experiments on heterogeneous media with high-contrast coefficients demonstrate that the proposed approach is both robust and efficient in capturing fine-scale features near contact boundaries.
Paper Structure (10 sections, 10 theorems, 68 equations, 9 figures, 6 tables, 1 algorithm)

This paper contains 10 sections, 10 theorems, 68 equations, 9 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Semismooth Newton method
  5. Numerical method
  6. Analysis
  7. Numerical experiments
  8. Model problem 1
  9. Model problem 2
  10. Conclusion

Key Result

Theorem 2.1

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

Figures (9)

  • Figure 1: Illustration of meshes, fine element, coarse element, and oversampling domain.
  • Figure 2: The permeability fields (a) medium A; (b) medium B.
  • Figure 3: The source function (a) $f_1$; (b) $f_2$.
  • Figure 4: The multiscale solutions $u^\mathup{cem}$ after iterations with medium A, using the source function $f_1$: (a)the 3D images; (b)the contour images;
  • Figure 5: The multiscale solutions $u_k^\mathup{cem}$ and $\lambda_k$ along $\Gamma_{\rm{C}}$ with medium A and source function $f_1$
  • ...and 4 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.3
  • Corollary 4.4
  • ...and 4 more