A Simple and Efficient Preconditioning Scheme for Heaviside Enriched XFEM

TL;DR

This paper introduces a simple, efficient preconditioning scheme for XFEM that improves numerical stability when dealing with small interface intersections, enhancing its applicability to nonlinear problems.

Contribution

A novel geometric preconditioning scheme for XFEM that is easy to implement, computationally inexpensive, and effective in handling ill-conditioning due to small interface volumes.

Findings

Reduces system ill-conditioning in XFEM with small interface intersections.

Preconditioner can be constructed prior to system assembly, saving computational effort.

Applicable to nonlinear problems with fixed and moving interfaces.

Abstract

The eXtended Finite Element Method (XFEM) is an approach for solving problems with non-smooth solutions. In the XFEM, the approximate solution is locally enriched to capture discontinuities without requiring a mesh which conforms to the geometric features. One drawback of the XFEM is that an ill-conditioned system of equations results when the ratio of volumes on either side of the interface in an element is small. In this paper, to avoid this ill-conditioning, a simple and efficient scheme based on a geometric preconditioner and constraining degrees of freedom to zero for small intersections is proposed. This geometric preconditioner is computed from the nodal basis functions, and therefore may be constructed prior to building the system of equations. This feature and the low-cost of constructing the preconditioning matrix makes it well suited for nonlinear problems with fixed and moving interfaces.