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[[Image:Example_of_2D_meshExample of 2D mesh.png |thumb|2D [[Finite element method|FEM]] [[mesh]], the triangles are the elements, the [[vertex (graph theory)|vertices]] are the [[nodesnode (graph theory)|node]]{{dn|date=June 2012}}s. The [[finite element method]] ([[Finite element method|FEM]]) has been the tool of choice since itscivil inceptionengineer [[Ray W. Clough]] in 1940 derived the 1940'sstiffness matrix of a 3-node triangular finite element (and coined the name). The precursors of FEM were elements built-up from bars ([[Alexander Hrennikoff|Hrennikoff]], [[RichardJohn CourantArgyris|CourantArgyris]], Turner) forand thea simulationconceptual ofvariation structuralapproach mechanicssuggested by R. [[Richard Courant|Courant]]. Today, the [[Finite element method|FEM]] is used to model a much wider range of physical phenomena. ]]
The '''extended finite element method''' ('''XFEM)'''), alsois knowna asnumerical technique based on the '''generalized finite element method''' ('''GFEM)''') orand the '''[[Partition of unity|partition of unity method]]''' ('''PUM)'''). is a numerical technique thatIt extends the classical [[finite element method]] (FEM) approach by enriching the solution space for solutions to [[differential equationsequation]]s with discontinuous functions.
== History ==
The extended finite element method (XFEM) was developed in 1999 by [[Ted Belytschko]] and collaborators<ref>{{cite journal | first1=Nicolas | last1=Moës | first2=John | last2=Dolbow | first3=Ted | last3=Belytschko | title=A finite element method for crack growth without remeshing | journal=International Journal for Numerical Methods in Engineering | year=1999 | issue=1 | pages= 131–150 | volume=46}}</ref>, to help alleviate shortcomings of the finite element method and has been used to model the propagation of various discontinuities: strong ([[Fracture|cracks]]) and weak (material interfaces). The idea behind XFEM is to retain most advantages of meshfree methods while alleviating their negative sides. ▼
The extended finite element method (XFEM) was developed in 1999 by [[Ted Belytschko]] and collaborators,<ref>
{{cite journal
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 are 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. Subsequent research has illustrated the more general use of the method for problems involving [[Mathematical singularity|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. ▼ | first1= Nicolas
| last1= Moës
| first2= John
| last2= Dolbow
| first3= Ted
| last3= Belytschko
| title= A finite element method for crack growth without remeshing
| journal= International Journal for Numerical Methods in Engineering
| year= 1999
| issue= 1
| pages= 131–150
| volume=46
| doi= 10.1002/(sici)1097-0207(19990910)46:1<131::aid-nme726>3.3.co;2-a
| url= https://hal.archives-ouvertes.fr/hal-01004829/file/Moes99.pdf
}}
</ref>
▲The extended finite element method (XFEM) was developed in 1999 by [[Ted Belytschko]] and collaborators<ref>{{cite journal | first1=Nicolas | last1=Moës | first2=John | last2=Dolbow | first3=Ted | last3=Belytschko | title=A finite element method for crack growth without remeshing | journal=International Journal for Numerical Methods in Engineering | year=1999 | issue=1 | pages= 131–150 | volume=46}}</ref>, to help alleviate shortcomings of the finite element method and has been used to model the propagation of various discontinuities: strong ([[Fracture|cracks]]) and weak (material interfaces). The idea behind XFEM is to retain most advantages of meshfree methods while alleviating their negative sides.
▲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 are 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. Subsequent research has illustrated the more general use of the method for problems involving [[Mathematical singularity|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==
There exists several research codes implementing this technique to various degrees.
* getfem[[GetFEM++]]
* xfem++
* openxfem++
* [https://blogs.princeton.edu/prevost/dynaflow/dynaflow-description/ Dynaflow]
* [https://git.gem.ec-nantes.fr/ eXlibris]
* [https://github.com/ngsxfem/ngsxfem ngsxfem]
XFEM has also been implemented in code ASTER,like Morfeo, and'''Altair [[AbaqusRadioss]]<ref>{{cite''', press release |url=http://www.simulia.com/news/pr_090519_DSS.html |title=Abaqus 6.9 Offers New Capabilities for Fracture and FailureASTER, High-Performance ComputingMorfeo, and Noise and Vibration |publisher=Dassault Systèmes |date=2009-05-19}}</ref> <ref>{{cite press release |url=http://www.simulia.com/news/pr_100524_DSS.html |title=Dassault Systèmes Announces New Multiphysics Technology in [[Abaqus Release 6.10 from SIMULIA |publisher=Dassault Systèmes |date=2010-05-24}}</ref>]]. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available ([[ANSYS]], [[SAMCEF]], [[OOFELIE]], etc.).
==References==
[[Category:Continuum mechanics]]
[[Category:Finite element method]]
[[Category:Mechanics]]
[[fr:Méthode des éléments finis étendus]]
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