Partition of Unity Methods

4 Advanced Topics

Stéphane P. A. Bordas¹, Alexander Menk², and Sundararajan Natarajan³

¹ University of Luxembourg, Luxembourg, UK² Robert Bosch GmbH, Germany³ Indian Institute of Technology Madras, India

University of Luxembourg Luxembourg UK Robert Bosch GmbH, Germany Indian Institute of Technology Madras India

The previous chapters aimed to convey an understanding of the general concept of the extended finite element method (XFEM) and its advantages and drawbacks. In this chapter, different aspects and advancements of the XFEM will be analyzed in greater detail. We start by discussing some of the more advanced integration techniques for the enriched elements. Although it is possible to integrate enrichment functions sufficiently accurately with traditional Gaussian quadrature performed on subcells, the techniques discussed in this chapter are more efficient in certain situations. We continue by discussing the choice of the enriched domain and a correction to unwanted artifacts introduced by the enrichments. This is done by taking the same one-dimensional (1D) example discussed in Chapter 2. Both the topics are closely related to the convergence of the XFEM itself. For large system of equations, iterative methods are an ideal choice to obtain a numerical solution. The condition number of the equations plays an important role if the convergence and the stability of these methods are considered. We, therefore, finish the chapter by discussing preconditioning techniques for ...