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In this work, we have exposed a recent method for modeling crack growth without re-meshing. The main advantage of this method is its capability in modeling discontinuities independently, so the mesh is prepared without any considering the existence of discontinuities. The paper covers the formulation and implementation of XFEM, and discusses various aspects of the approach (enrichments functions, level set representation, numerical integration…). Numerical experiments show the effectiveness and robustness of the XFEM implementation.

The method finite element is widespread in applications of industrial design, and much of various software packages based on techniques of FEM were developed. It proved appropriate for the study of the fracture mechanics. However, modelling the propagation of a crack by a finite element mesh proves to be difficult because of the topology alteration of the mesh. Besides, the singularity of the crack end has to be represented exactly by the approximation [

Recently a new class has been proposed that simulates the singular nature of discrete models within a geometrically continuous mesh of finite elements. The extended finite element method XFEM has emerged from this class of problems, and is based on the concept of partition of unity for enriching the classical finite element approximation to include the effects of singular or discontinuous fields around a crack [

The method of X-FEM originators were Belytschko and Black [

A significant advance of the extended finite element method was given by its coupling with level set methods (LSM): The LSM is employed to represent both the crack position and that of the crack ends. The X-FEM is employed to calculate the fields of stress and displacement that is important to determine the crack growth ratio [

The results of the X-FEM method have been so encouraging that some authors have immediately seized the opportunity to apply this method for solving many kinds of problems where disconti-nuities and moving boundaries are to be modeled.

In the X-FEM method, a standard displacement based finite element approximation is enriched by additional functions using the framework of partition of unity (

where

• N_{i} is the shape function associated to node i

• I is the set of all nodes of the domain

• J is the set of nodes whose shape function support is cut by a crack

• K is the set of nodes whose shape function support contains the crack front

• u_{i} are the classical degrees of freedom (i.e. displacement) for node i

• b_{j} account for the jump in the displacement field across the crack at node j. If the crack is aligned with the mesh, b_{j} represent the opening of the crack

• H(x) is the Heaviside function

• c_{kl} are the additional degrees of freedom associated with the crack tip enrichment functions F_{l}

• F_{l} is an enrichment which corresponds to the four asymptotic functions in the development expansion of the crack tip displacement field in a linear elastic solid (

The nodes whose the corresponding shape function support contains the crack tip are enriched by singular functions that can model the singular behavior of the displacement field at the crack tip.

The crack tip enrichment functions in isotropic elasticity Fi(r, q) are obtained from the asymptotic displacement fields:

Note that the third singular function F_{3} is the only enrichment function which is discontinuous across the crack. Thus, the discontinuity of the displacement field at in the singular enrichment zone is only modeled by F_{3} on the element containing the crack tip.

The nodes whose the corresponding shape function support is totally cut by the crack, are enriched by an Heaviside function (

The function of Heaviside jump is a discontinuous function through the surface of slit and constant on both slit sides: +1 on a side and –1 on the other.

For the slit cut elements that are enriched with the jump function H(x), Moes [

The description of discontinuities in the context of the extended finite element method is often realized by the level-set method. The Method of level set is a numerical design of Osher [

A crack is described by two level sets (

• a normal level set, y(x), which the signed distance to the crack surface

• a tangent level set f(x), which is the signed distance to the plane including the crack front and perpendicular to the crack surface.

In a given element, y_{min} and y_{max}, respectively, be the minimum and maximum nodal values of y on the nodes of that element. Similarly, let f_{min} and f_{max}, respectively be the minimum and maximum nodal values of f on the nodes of an element:

• If f < 0 and y_{min} y_{max} £ 0, then the crack cuts through the element and the nodes of the element are to be enriched with H(x).

• If in that element f_{min} f_{max} £ 0 and y_{min} y_{max} £ 0, then the tip lies within that element, and its nodes are to be enriched Fi(r, q).

One can apply the method of finite extended element within one finite element code with relatively slight alterations: variable degrees numbers of freedom per node; interaction of mesh geometry (a manner to detect elements intersecting with discontinuity geometry); matrices of enriched rigidity; numerical integration. Sukumar and Prévost [

cracks in isotropic and bi-material media. Nisto et al.’s suggestion [

The XFEM method can be implemented within a finite element code with relatively small modifica-tions: variable number of degrees of freedom per node; mesh geometry interaction; enriched stiff-ness matrices; numerical integration [

1) Input data: defining various object entities (crack, holes, inclusions, interfaces…), enrichment types and crack growth law.

2) Nodal degrees of freedom: a part from the classical degrees of freedom, additional unknown enriched degrees of freedom is introduced via the displacement approximation.

3) Mesh-geometry interactions: This sub-category detects the selection of the enriched nodes, then touches upon the computation of enrichment functions, and detects the partitioning of the finite elements that are intersected by the crack.

4) Assembly procedure: The stiffness matrix and force vector assembly are done on an element level, which is similar to classical finite element implementation. The distinction herein is that the dimensions of the element stiffness matrix can differ from element (unenriched) to element enric-hed).

5) Post-processing: This sub-category addresses the main objectives of a fracture analysis by determining the interaction integral, and controlling the crack growth criteria.

The task of incorporating the X-FEM capabilities within a general-purpose finite element program can be broken down into the following schema (Figures 7 and 8,

The Figures 9-12 show tow examples of crack growth modeling without re-meshing obtained by X-FEM code.

The extended finite element method (X-FEM) uses the partition of unity to remove the need to mesh physical surfaces or to remesh them as they evolve. It allows to model cracks, material inclusions and holes on non conforming meshes. The methodology of X-FEM that differs from that of the traditional method of finite element is of very particular concern since it does not force discontinuities to go with the borders. It solves the technological problems in the various complex fields accurately; the thing that can hardly be achieved impossible when using the traditional method of finite element alone. In this work, we present the basic concepts and the implantation of the X-FEM. The work discusses general algorithms for implementing an efficient X-FEM. A numerical experiment is provided to demonstrate the effectiveness

and robustness of the X-FEM implementation.