Extended finite element method
Encyclopedia
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 classical finite element method
(FEM) approach by enriching the solution space for solutions to differential equations with discontinuous functions.
and collaborators, to help alleviate shortcomings of the finite element method and has been used to model the propagation of various discontinuities: strong (cracks
) and weak (material interfaces). The idea behind XFEM is to retain most advantages of meshfree methods while alleviating their negative sides.
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 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.
approximation space so that it is able to naturally reproduce the
challenging feature associated with the problem of interest: the
discontinuity, singularity
, boundary layer
, etc. It was shown that
for some problems, such an embedding of the problem's feature into the approximation
space can significantly improve convergence rates and accuracy.
Moreover, treating problems with discontinuities with eXtended
Finite Element Methods suppresses the need to mesh and remesh the
discontinuity surfaces, thus alleviating the computational costs and projection errors
associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges.
XFEM has also been implemented in code ASTER, Morfeo, and Abaqus
. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF
, OOFELIE, etc.).
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
(FEM) approach by enriching the solution space for solutions to differential equations with discontinuous functions.
History
The extended finite element method (XFEM) was developed in 1999 by Ted BelytschkoTed Belytschko
Ted Bohdan Belytschko is an American mechanical engineer. He is Walter P. Murphy Professor and McCormick Professor of Computational Mechanics at Northwestern University. He works in field of computational solid mechanics and is known for development of methods like element-free Galerkin methods...
and collaborators, to help alleviate shortcomings of the finite element method and has been used to model the propagation of various discontinuities: strong (cracks
Fracture
A fracture is the separation of an object or material into two, or more, pieces under the action of stress.The word fracture is often applied to bones of living creatures , or to crystals or crystalline materials, such as gemstones or metal...
) 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 fractureFracture
A fracture is the separation of an object or material into two, or more, pieces under the action of stress.The word fracture is often applied to bones of living creatures , or to crystals or crystalline materials, such as gemstones or metal...
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 singularities
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
, 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 the
challenging feature associated with the problem of interest: the
discontinuity, singularity
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
, boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...
, etc. It was shown that
for some problems, such an embedding of the problem's feature into the approximation
space can significantly improve convergence rates and accuracy.
Moreover, treating problems with discontinuities with eXtended
Finite Element Methods suppresses the need to mesh and remesh the
discontinuity surfaces, thus alleviating the computational costs and projection errors
associated 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 has also been implemented in code ASTER, Morfeo, and Abaqus
Abaqus
Abaqus FEA is a suite of software applications for finite element analysis and computer-aided engineering, originally released in 1978...
. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF
SAMCEF
SAMCEF is a finite element analysis software package dedicated to mechanical virtual prototyping. SAMCEF development started in 1965 at the University of Liège and is still developed and sold by SAMTECH, a Belgian company the HQ of which is located in Liège, Belgium.-Software features:SAMCEF...
, OOFELIE, etc.).