\label{xfemoverview}

Consider $\Omega \subset \mathbb{R}^2$, the reference configuration of a cracked linearly isotropic elastic body (see \fref{fig:bodywcrack}). The boundary of $\Omega$, denoted by $\Gamma$, is partitioned into three parts $\Gamma_u, \Gamma_n$ and $\Gamma_c$, where Dirichlet condition is prescribed on $\Gamma_u$ and Neumann condition is prescribed on $\Gamma_n$ and $\Gamma_c$.  The governing equilibrium equations for a 2D elasticity problem with internal boundary  $\Gamma_c$ defined in the domain $\Omega$ and bounded by $\Gamma$ is
\begin{equation}
\nabla_s^{\rm T} \bvsig + \bb = \mathbf{0} ~\hspace{0.5cm} \in ~\hspace{0.5cm} \Omega
\label{eqn:EE}
\end{equation}
where $\nabla_s(\cdot)$ is the symmetric part of the gradient operator, $\mathbf{0}$ is a null vector, $\bvsig$ is the stress tensor and $\bb$ is the body force. The boundary conditions for this problem are:
\begin{eqnarray}
\bvsig \cdot \cn = \overline{\mathbf{t}} \hspace{0.5cm} \textup{on} ~\hspace{0.5cm} \Gamma_t \nonumber \\
\uu = \overline{\uu} \hspace{0.5cm} \textup{on} ~\hspace{0.5cm} \Gamma_u \nonumber \\
\bvsig \cdot \cn = \overline{\mathbf{t}} \hspace{0.5cm} \textup{on} ~\hspace{0.5cm}  \Gamma_c
\end{eqnarray}
where $\overline{\uu} = (\overline{u}_x,\overline{u}_y)^{\rm T}$ is the prescribed displacement vector on the essential boundary $\Gamma_u$; $\overline{\mathbf{t}} = (\overline{t}_x,\overline{t}_y)^{\rm T}$ is the prescribed traction vector on the natural boundary $\Gamma_t$ and $\cn$ is the outward normal vector. In this study, it is assumed that the displacements remain small and the strain-displacement relation is given by $\bveps = \nabla_s \uu$. Let us assume a linear elastic behaviour, the constitutive relation is given by $\bvsig = \dd \colon \bveps$, where $\dd$ is a fourth order elasticity tensor. The weak formulation of the static problem is then given by:
\begin{equation}
\textup{find} ~\hspace{0.3cm} \uu^h \in \mathcal{U}^h \hspace{0.3cm} \textup{such~~that} \hspace{0.3cm} \forall \vv^h \in \mathcal{V}^h \hspace{0.3cm} a(\uu^h,\vv^h) = \ell(\vv^h)
\label{eqn:variweakform1}
\end{equation}
where $\mathcal{U}^h \subset \mathcal{U}$ and $\mathcal{V}^h \subset \mathcal{V}$.
\begin{figure}[htpb]
\centering

\scalebox{0.7}{\input{./Figures/probdes.pstex_t}}
\caption{Two-dimensional elastic body with a crack.}
\label{fig:bodywcrack}
\end{figure}

\subsection{eXtended Finite Element Method (XFEM)}
In the XFEM, the classical FEM polynomial approximation space is augmented by a set of functions called the `enrichment functions'. This is achieved through the framework of the partition of unity. XFEM is classified as one of the partition of unity methods (PUMs). A partition of unity in a domain $\Omega$ is a set of functions $N_I$ such that
\begin{equation}
\sum\limits_{I \in \mathcal{N}^{\rm {fem}}} N_I(\xx) = 1, \hspace{0.5cm} \xx \in \Omega
\end{equation}
where $\mathcal{N}^{\rm{fem}}$ is the set of nodes in the FE mesh. Using this property, any function $\varphi$ can be reproduced by a product of the PU shape functions with $\varphi$. Let $\uu^h \subset \mathcal{U}$, the XFEM approximation can be decomposed into the standard part $\uu^h_{\rm{fem}}$ and an enriched part $\uu^h_{\rm{enr}}$ as:
\begin{eqnarray}
\uu^h(\xx) &=& \uu^h_{\rm{fem}}(\xx) + \textup{Enrichment functions}
\label{eqn:xfemgeneralform}
\end{eqnarray}
where $\mathcal{N}^{\rm{fem}}$ is the set of all the nodes in the FE mesh. The modification of the displacement approximation does not introduce a new form of the discretized finite element equilibrium equations, but leads to an enlarged problem to solve. A generic form for the displacement approximation in case of the linear elastic fracture mechanics (LEFM) takes the form~\cite{belytschkoblack1999}:
\begin{equation}
\uu^h(\xx)=\sum\limits_{I \in \mathcal{N}^{\rm{fem}}} N_I(\xx)\qq_I + \sum\limits_{J \in \mathcal{N}^{\rm c}} N_J(\xx)\vartheta(\xx)\aaa_J + \sum\limits_{K \in
\mathcal{N}^{\rm f}} N_K (\xx)\sum_{\alpha=1}^n B_{\alpha}(r,\theta)\bb^{\alpha}_K ,\label{eqn:uS2}
\end{equation}
where $\mathcal{N}^{\rm c}$ is the set of nodes whose shape function support is cut by the crack interior (`squared' nodes in \fref{fig:xfemelementcategory}) and $\mathcal{N}^{\rm f}$ is the set of nodes whose shape function support is cut by the crack tip (`circled' nodes in \fref{fig:xfemelementcategory}). $\vartheta$ and $B_{\alpha}$ are the enrichment functions chosen to capture the displacement jump across the crack surface and the singularity at the crack tip and $n$ is the total number of asymptotic functions. $\aaa_J$ and $\bb^{\alpha}_K$ are the nodal degrees of freedom corresponding to functions $\vartheta$ and $B_{\alpha}$, respectively. $n$ is the total number of near-tip asymptotic functions and $(r,\theta)$ are the local crack tip coordinates. A priori knowledge of the shape functions is required for the successful application of the method. Since its inception, this has attracted researchers to investigate the enrichment functions that can accurately describe the asymptotic fields ahead of the crack tip~\cite{asadpouremohammadi2007,hattorirojas-diaz2012,asharimohammadi2011,bouhalashao2013}.
\begin{figure}[htpb]
\centering
\includegraphics[angle=0,width=0.4\columnwidth]{./Figures/xfem_crack}
\caption{A typical FE mesh with an internal discontinuity. Nodes are enriched with a Heaviside functions whose nodal support is cut by the crack interior (`\emph{squared}' nodes) and with set of functions that span the asymptotic fields whose nodal support is cut by the crack tip (`\emph{circled}' nodes).} 
\label{fig:xfemelementcategory}
\end{figure}
{"config": "arxiv", "file": "1406.3869/XFEMintro.tex"}