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The eXtended Finite Element Method (XFEM)
The aim of this writting is to give a brief introduction to the eXtended Finite Element Method (XFEM) and investigation of its practical applications.
Firstly introduced in 1999 by the work of Black and Belytschko, XFEM is a local partition of unity (PUM) enriched finite element method. By local, it means that only a region near the discontinuties such as cracks, holes, material interfaces are enriched. The most important concept in this method is "enrichment" which means that the displacement approximation is enriched (incorporated) by additional problem-specific functions. For example, for crack modelling, the Heaviside function is used to enrich nodes whose support cut by the crack face whereas the near tip asymptotic functions are used to model the crack tip singularity (nodes whose support containes the tip are enriched).
From its appearance, XFEM has been used to model several applied mechanics problems. Quasi static crack propagation in 2D and 3D were introduced by the work of Dolbow, Sukumar, Moes; branched and intersecting cracks by Daux et al(2000); dynamic fracture was studied by Belytschko's group and Alain Combescure's group. The later group also have introduced many interesting results in confined plastic fracture mechanics.
Computer implementation aspects of new proposed methods always make the method itself apparent. In this context, one can cite the work of Sukumar and Prevost on Fortran implementation and the work of Bordas and Nguyen on object-oriented programming which was published recently on IJNME.
All of the previous work are about homogeneous materials. For heterogeneous materials in the XFEM framework, the paper of Sukumar on polycristall structure, Dolbow on functionally graded material and one of Moes are the only work (to the author's knowledge) in this field.
XFEM allows the discontinuties not align with the finite element mesh, then crack propagation simulation without remeshing. However, the numerical integration of elements cut by the discontinuity require special treatment. The most commonly adopted method is to divide element into subdomains on two sides of the line of discontinuty. This method is flexible but not appropriate for history-dependent material where the projection of variables from old Gauss points to new ones are inevitable. In order to replace this method, Giulio Ventura proposed one method without element decomposition into subdomains. However, until now, this method is limited to elements completely cut by discontinuity.
Due to enrichment, in the XFEM, additional degrees of freedom are introduced which can make the implementation of XFEM into available commercial FE code difficult. Recently Belytschko and colloborators proposed a new method without additional d.o.fs by using the moving least square. However, it is obvious that MLS shape functions are very heavy computational, which makes this method not so much attractive (am I wrong? :-)).
Beside the original XFEM version, many instances have been introduced such as the cohesive segment method of Wells where only the Heaviside is adopted to enrich the displacement field.
Another interesting topic involved in the development of the XFEM is the method used to represent the geometry of discontinuties (noting that they are not meshed in XFEM). It is the Level Set Method (LSM) which have been proved to be the most reliable method. Problem is that, often, LSM is used over the uniform Cartesian grid while the finite element mesh is usualy unstructured mesh. Therefore, one could use a structured grid for the level sets and unstructured mesh for mechanical fields. It is in fact the work of Duddu and Bordas in the biofilm application. However, to me, many details have not been introduced and 3D application is still of interest.
I believe that there are still many XFEM's application which are not cited here. It is only due to limited knowledge of the author. I hope that we can cite all application of XFEM to access its strength as well as its limits.