- Extended finite element method
The

**extended finite element method (XFEM)**, also known as**generalized finite element method (GFEM)**or**partition of unity method (PUM)**is a numerical technique that extends the classicalfinite element method (FEM) approach by extending the solution space for solutions to differential equations with discontinuous functions.**History**The extended finite element method (XFEM) was developed in in 1999 by

Ted Belytschko and collaborators, to help alleviate the above shortcomings of the finite element method and has been used to model the propagation of various discontinuities: strong (crack s) and weak (material interfaces). The idea behind XFEM is to retain most advantages of meshfree methods while alleviating their negative sides.**Rationale**The extended finite element method was developed to ease difficulties in solving problems with localized features that are not efficiently resolved by mesh refinement. One of the initial applications was the modelling of

fracture s in a material. In this original implementation, discontinuous basis functions are added to standard polynomial basis functions for nodes that belonged to elements that where intersected by a crack to provide a basis that included crack opening displacements. A key advantage of XFEM is that in such problems the finite element mesh does not need to be updated to track the crack path; a downside is that cracks can only follow mesh edges. Subsequent research has illustrated the more general use of the method for problems involving singularities, material interfaces, regular meshing of microstructural features such as voids, and other problems where a localized feature can be described by an appropriate set of basis functions.**Principle**Enriched finite element methods extend, or enrich, theapproximation space so that it is able to naturally reproduce thechallenging feature associated with the problem of interest: thediscontinuity, singularity,

boundary layer , etc. It was shown thatfor some problems, such an embedding of the problem's feature into the approximationspace can significantly improve convergence rates and accuracy.Moreover, treating problems with discontinuities with eXtendedFinite Element Methods suppresses the need to mesh and remesh thediscontinuity surfaces, thus alleviating the computational costs and projection errorsassociated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges.**Existing XFEM codes**There exists several research codes implementing this technique to various degrees.

* getfem++

* xfem++

* openxfem++XFEM was also implemented in code ASTER and in Morfeo and is being taken up by industry, with a few plugins and actual core implementations available (

ANSYS ,ABAQUS ,SAMCEF ,OOFELIE , etc.).

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Discrete element method**— A discrete element method (DEM), also called a distinct element method is any of family of numerical methods for computing the motion of a large number of particles of micrometre scale size and above. Though DEM is very closely related to… … Wikipedia**Method of lines**— The method of lines (MOL, NMOL, NUMOL) (Schiesser, 1991; Hamdi, et al., 2007; Schiesser, 2009 ) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. MOL allows standard, general purpose… … Wikipedia**Extended Euclidean algorithm**— The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the greatest common divisor (GCD) of integers a and b : it also finds the integers x and y in Bézout s identity: ax + by = gcd(a, b). ,(Typically either x or… … Wikipedia**Crank–Nicolson method**— In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations.[1] It is a second order method in time, implicit in time, and is numerically … Wikipedia**Spectral method**— Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have… … Wikipedia**Multigrid method**— Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in (but not… … Wikipedia**Discontinuous Galerkin method**— Discontinuous Galerkin methods (DG methods) in mathematics form a class of numerical methods for solving partial differential equations. They combine features of the finite element and the finite volume framework and have been successfully… … Wikipedia**Collocation method**— In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite dimensional space of candidate solutions… … Wikipedia**Schwarz alternating method**— In mathematics, the Schwarz alternating method, named after Hermann Schwarz, is an iterative method to find the solution of a partial differential equations on a domain which is the union of two overlapping subdomains, by solving the equation on… … Wikipedia**Neumann–Dirichlet method**— In mathematics, the Neumann–Dirichlet method is a domain decomposition preconditioner which involves solving Neumann boundary value problem on one subdomain and Dirichlet boundary value problem on another, adjacent across the interface between… … Wikipedia