A Multiscale Method for the Heterogeneous Signorini Problem

TL;DR

This paper introduces a hybrid multiscale finite element method for efficiently solving the Signorini problem with heterogeneous properties, combining spectral decomposition and specialized basis functions to handle unilateral contact conditions.

Contribution

It develops a novel multiscale approach based on GMsFEM that incorporates boundary-specific basis functions for the Signorini problem, with proven spectral convergence.

Findings

Method achieves spectral convergence.

Numerical results validate theoretical analysis.

Handles unilateral contact conditions effectively.

Abstract

In this paper, we develop a multiscale method for solving the Signorini problem with a heterogeneous field. The Signorini problem is encountered in many applications, such as hydrostatics, thermics, and solid mechanics. It is well-known that numerically solving this problem requires a fine computational mesh, which can lead to a large number of degrees of freedom. The aim of this work is to develop a new hybrid multiscale method based on the framework of the generalized multiscale finite element method (GMsFEM). The construction of multiscale basis functions requires local spectral decomposition. Additional multiscale basis functions related to the contact boundary are required so that our method can handle the unilateral condition of the Signorini type naturally. A complete analysis of the proposed method is provided and a result of the spectral convergence is shown. Numerical results are provided to validate our theoretical findings.