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5px Times}span.s2 {font: 9.5px Helvetica}Chapter 11.1 IntroductionC Losed form or analytical solutions to partial differential equations (PDEs) are notobtainable for most problems. Hence, scientists and engineers have developednumerical methods, such as the finite difference method (FDM) 50, the finite elementmethod (FEM) 5, 70, meshless methods 24, 34, spectral methods 13, boundary elementmethods 57, discrete element methods 43, Lattice Boltzmann methods 63etc., A popular and widely used approach to the solution of PDEs is the FEM. FEMbased computational mechanics plays a prominent role in all fields of science and engineering.FEM does not operate on the differential equations; instead, the continuousboundary and initial value problems are reformulated into equivalent variational forms.

The FEMrequires the domain to be subdivided into non-overlapping regions, called theelements. In FEM, individual elements are connected together by a topological map,called a mesh and local polynomial representation is used for the fields within the element.The solution obtained is a function of the quality of mesh and the fundamentalrequirement is that the mesh has to conform to the geometry. The main advantage ofthe FEM is that it can handle complex boundaries without much difficulty. Despite itspopularity, FEM suffers from certain drawbacks. Note that some solution methodologiesaim at addressing more than one limitation of the FEM, for example, meshfreemethods 24, 34 and the recently proposed Smoothed Finite Element Method (SFEM)12, 35, 46.

It has been noted that the FEM with piecewise polynomials are inefficientto deal with singularities or high gradients in the domain. One strategy is to enrich theFEM approximation basis with additional functions 61. Some of the proposed techniquescan be combined with enrichment techniques to solve problems involving highgradients or singularities. Such numerical technique based on the generalized finite elementmethod (GFEM) and the partition of unity method (PUM) is called the extendedfinite element method (XFEM). It extends the classical finite element method(FEM)approach by enriching the solution space for solutions to differential equations withpronounced non-smooth characteristics in small parts of the computational domain,for example near discontinuities and singularities.

In these cases, standard numericalmethods such as the FEM or FVM often exhibit poor accuracy. The XFEM offers significantadvantages by enabling optimal convergence rates for these applications. Theterm ‘enrichment’ implies augmenting or supplementing a basis of piecewise polyno