Model Reduction for Signorini Problem and Poroelasticity Problem

Abstract

The Signorini problem involves finding a solution to an elliptic partial differential equation with a nonlinear boundary condition known as the Signorini contact condition. This problem has important applications in the study of solid mechanics and elasticity, as it can be used to model the behavior of materials subject to friction and contact. 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. In this dissertation, we 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.

Extensive research effort had been devoted to developing efficient methods for solving linear heterogeneous poroelasticity models with coefficients of high contrast. In this thesis, based on the framework of Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), we develop two methods to improve the performance of CEM-GMsFEM ap-plied to linear heterogeneous poroelasticity problem. In this dissertation, we firstly develop an on-line adaptive enrichment method to handle the missing information. The proposed method makes use of information of residual-driven error indicators to enrich the multiscale spaces for both the displacement and the pressure variables in the model. Additional online basis functions are constructed in oversampled regions and are adaptively chosen to reduce the error the most. A complete theoretical analysis of the online enrichment algorithm is provided.

We also propose a partially explicit splitting method for linear heterogeneous poroelasticity problems to correct the solution in dynamic problems without iterations. Firstly, dominant basis functions generated by the CEM-GMsFEM approach are used to capture important degrees of freedom and it is known to give contrast-independent convergence that scales with the mesh size. In typical situation, one has very few degrees of freedom in dominant basis functions. This part is treated implicitly. Secondly, we design and introduce an additional space in the complement space and these degrees are treated explicitly. We also investigate the CFL-type stability restriction for this problem and the restriction for the time step is contrast-independent.

Thorough numerical experiments are conducted to validate each method proposed in this dissertation.