# A simple and efficient preconditioning scheme for heaviside enriched XFEM

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## Abstract

The extended finite element method (XFEM) is an approach for solving problems with non-smooth solutions, which arise from geometric features such as cracks, holes, and material inclusions. In the XFEM, the approximate solution is locally enriched to capture the 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. Such interface configurations are often unavoidable, in particular for moving interface problems on fixed meshes. In general, the ill-conditioning reduces the performance of iterative linear solvers and impedes the convergence of solvers for nonlinear problems. This paper studies the XFEM with a Heaviside enrichment strategy for solving problems with stationary and moving material interfaces. A generalized formulation of the XFEM is combined with the level set method to implicitly define the embedded interface geometry. In order to avoid the ill-conditioning, a simple and efficient scheme based on a geometric preconditioner and constraining degrees of freedom to zero for small intersections is proposed. The 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. It is shown by numerical examples that the proposed preconditioning scheme performs well for discontinuous problems and \(C^0\)-continuous problems with both the stabilized Lagrange and Nitsche methods for enforcing the continuity constraint at the interface. Numerical examples are presented which compare the condition number and solution error with and without the proposed preconditioning scheme. The results suggest that the proposed preconditioning scheme leads to condition numbers similar to that of a body-fitted mesh using the traditional finite element method without loss of solution accuracy.

## Keywords

Level set method Extended finite element method Heaviside enrichment Ill-condition Preconditioner## Notes

### Acknowledgments

The first author acknowledges the support of the NASA Fundamental Aeronautics Program Fixed Wing Project, and the second and fourth authors acknowledges the support of the National Science Foundation under grants CMMI-0729520 and EFRI SEED-1038305. This material is based upon work of third author supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, under Award Number DE-SC0006402. The opinions and conclusions presented are those of the authors and do not necessarily reflect the views of the sponsoring organizations.

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