Iterative Contact-resolving Hybrid Methods for Multiscale Contact Mechanics

TL;DR

The paper addresses nonlinear elastic contact with high-contrast coefficients by introducing an iterative, two-subdomain framework that confines nonlinear contact constraints to a small region while the large subdomain remains linear. It integrates four discretization strategies, including standard/mixed FEM and CEM-GMsFEM variants, and employs Robin-type transmission to couple subproblems, enabling scalable solvers. The authors establish convergence results for the discrete schemes and validate the methods through extensive numerical experiments, including nearly incompressible materials. The combination of penalty-based variational formulations, mixed methods for direct stress computation, and multiscale model reduction offers robust, accurate, and computationally efficient solutions for complex multiscale contact mechanics problems.

Abstract

Modeling contact mechanics with high contrast coefficients presents significant mathematical and computational challenges, especially in achieving strongly symmetric stress approximations. Due to the inherent nonlinearity of contact problems, conventional methods that treat the entire domain as a monolithic system often lead to high global complexity. To address this, we develop an iterative contact-resolving hybrid method by localizing nonlinear contact constraints within a smaller subdomain, while the larger subdomain is governed by a linear system. Our system employs variational inequality theory, minimization principles, and penalty methods. More importantly, we propose four discretization types within the two-subdomain framework, ranging from applying standard/mixed FEM across the entire domain to combining standard/mixed multiscale methods in the larger subdomain with standard/mixed FEM in the smaller one. % The standard finite element method and standard constraint energy minimizing generalized multiscale finite element method are simple and easy to demonstrate. By employing a multiscale reduction technique, the method avoids excessive degrees of freedom inherent in conventional methods in the larger domain, while the mixed formulation enables direct stress computation, ensures local momentum conservation, and resists locking in nearly incompressible materials. Convergence analysis and the corresponding algorithms are provided for all cases. Extensive numerical experiments are presented to validate the effectiveness of the approaches.

Figures (10)

• Figure 2.1: Simplified 2D model
• Figure 5.1: Young’s Modulus of the test models in $\Omega_1$
• Figure 5.2: (a)-(c): components of stress reference solution in Test model 1 (i.e. $(\underline{\bm{\sigma}}_h)_{11},(\underline{\bm{\sigma}}_h)_{12},(\underline{\bm{\sigma}}_h)_{22}$); (d)-(f): components of final stress iteration solution for Algorithm \ref{['an iterative dd algorithm for continuous setting in mixed form']} in Test model 1 (i.e. $(\underline{\bm{\sigma}}^n)_{11},(\underline{\bm{\sigma}}^n)_{12},(\underline{\bm{\sigma}}^n)_{22}$) with $h=1/64$, $\beta=1$.
• Figure 5.3: (a)-(b): components of displacement reference solution in Test model 1 (i.e. $(\mathbf{u}_h)_1,(\mathbf{u}_h)_2$); (c)-(d): components of final displacement iteration solution for Algorithm \ref{['an iterative dd algorithm for continuous setting in mixed form']} in Test model 1 (i.e. $(\mathbf{u}^n)_1,(\mathbf{u}^n)_2$) with $h=1/64$, $\beta=1$
• Figure 5.4: Contact values of final stress and displacement iteration solutions for Algorithm \ref{['an iterative dd algorithm for continuous setting in mixed form']} in Test model 1

Theorems & Definitions (7)

• Theorem 3.1
• Theorem 3.2
• proof : proof of Theorem \ref{['equivalent discrete form']}
• Lemma 4.1
• Lemma 4.2
• proof
• proof : proof of Theorem \ref{['convergence in discrete setting']}