\section{Global enrichment XFEM}\label{Global enrichment XFEM}

\begin{frame}[fragile]
\frametitle{Global enrichment XFEM}

An XFEM variant is introduced which:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
\item Enables the application of geometrical enrichment to 3D.
\item Extends dof gathering to 3D through global enrichment.
\item Employs weight function blending.
\item Employs enrichment function shifting.
\end{itemize}

\end{frame}

%\subsection{Related works}\label{Related works}
%
%\begin{frame}
%\frametitle{Related works}
%Similar concepts to the ones introduced herein can be found:
%
%\vspace{\baselineskip}
%
%\begin{itemize}
%\setlength\itemsep{2em}
%\item In the work of Laborde et al. 
%\begin{itemize}
%\item[$\rightarrow$] dof gathering
%\end{itemize}
%\cite{Laborde2005}
%
%\item In the work of Langlois et al.
%\begin{itemize}
%\item[$\rightarrow$] discretization along the crack front
%\end{itemize}
%\cite{Langlois2014}
%
%\item In the s-finite element method
%\begin{itemize}
%\item[$\rightarrow$] superimposed mesh
%\end{itemize}
%\cite{Fish1992}
%
%\end{itemize}
%
%\end{frame}
%
%\begin{frame}
%
%\frametitle{Related works}
%\begin{itemize}
%\setlength\itemsep{2em}
%
%\item In the corrected XFEM
%\begin{itemize}
%\item[$\rightarrow$] weight function blending
%\end{itemize}
%\cite{Fries2008}
%
%\item In the work of Ventura et al.
%\begin{itemize}
%\item[$\rightarrow$] weight function blending
%\item[$\rightarrow$] enrichment function shifting
%\end{itemize}
%\cite{Ventura2009}
%
%\item Our previous work
%\begin{itemize}
%\item[$\rightarrow$] global enrichment approach
%\end{itemize}
%\cite{Agathos2015}
%
%\end{itemize}
%
%\end{frame}

%\subsection{Crack representation}\label{Crack representation}
%
%\begin{frame}[fragile]
%\frametitle{Crack representation}
%
%Level set functions:
%
%\vspace{\baselineskip}
%
%\begin{itemize}
%\setlength\itemsep{2em}
  %\item $\phi\left(\mathbf{x}\right)$ is the signed distance from the crack surface.
	%\item $\psi\left(\mathbf{x}\right)$ is a signed distance function such that:
	%
	%\vspace{\baselineskip}
	%
	%\begin{itemize}
	%\setlength\itemsep{1em}
	%\item[$\rightarrow$] $\nabla\phi\cdot\nabla\psi=0$
	%\item[$\rightarrow$] $\phi\left(\mathbf{x}\right)=0$ and $\psi\left(\mathbf{x}\right)=0$ defines the crack front
	%\end{itemize}
%\end{itemize}
%
%\vspace{\baselineskip}
%
%Polar coordinates:
%
%\begin{equation*}\label{eq:polar_coords}
	%r=\sqrt{\phi^2+\psi^2}, \qquad \theta=\arctan\left(\frac{\phi}{\psi}\right)
%\end{equation*}
%
%\end{frame}
%
%
%\begin{frame}[fragile]
%\frametitle{Crack representation}
%
%\begin{figure}
		%\includegraphics[width=0.90\textwidth]{../Figures/lsets.eps}
%\end{figure}
%
%\end{frame}

\subsection{Definition of the Front Elements}\label{Definition of the Front Elements}

\begin{frame}
\frametitle{Front elements}
A superimposed mesh is used to provide a p.u. basis.

\vspace{\baselineskip}

Desired properties:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{1em}
\item Satisfaction of the partition of unity property.

\item Spatial variation only along the direction of the crack front.

\end{itemize}

\end{frame}


%\begin{frame}
%\frametitle{Front elements}
%
%Special elements are employed which are both:
%
%\vspace{\baselineskip}
%
%\begin{itemize}
%\setlength\itemsep{1em}
%\item 1D $\rightarrow$ shape functions vary only along one dimension
%\item 3D $\rightarrow$ they are defined in a three-dimensional domain
%
%\end{itemize}
%
%\end{frame}

\begin{frame}
\frametitle{Front elements}

\begin{columns}
\begin{column}{0.50\textwidth}
\begin{figure}
	\centering
		\includegraphics[width=1.00\textwidth]{../Figures/front_discr.eps}
	\label{fig:front_discr}
\end{figure}

\end{column}

\begin{column}{0.50\textwidth}

\begin{itemize}
\setlength\itemsep{1em}
\item A set of nodes along the crack front is defined.
\item Each element is defined by two nodes.
\item A good starting point for front element thickness is $h$.
\end{itemize}

\end{column}

\end{columns}

\end{frame}

\begin{frame}
\frametitle{Front elements}

Volume corresponding to two consecutive front elements.

\begin{figure}
	\centering
		\includegraphics[width=0.6\textwidth]{../Figures/front_elem_3d.eps}
	\label{fig:front_el_3d}
\end{figure}

Different element colors correspond to different front elements.

\end{frame}

%\begin{frame}
%\frametitle{Front element parameter}
%
%A function similar to the level sets is defined which varies along the crack front.
%
%\begin{figure}
	%\centering
		%\includegraphics[width=0.9\textwidth]{../Figures/front_param.eps}
	%\label{fig:front param}
%\end{figure}
%
%\end{frame}

\begin{frame}
\frametitle{Front element shape functions}

Linear 1D shape functions are used:

\begin{equation*}
	\mathbf{N}^g\left(\xi\right)=\left[\frac{1-\xi}{2} \ \ \frac{1+\xi}{2}\right]
\end{equation*}

\vspace{\baselineskip}

where $\xi$ is the local coordinate of the superimposed element.

\vspace{\baselineskip}

Those functions:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
\item form a partition of unity.

\item are used to weight tip enrichment functions.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Front element shape functions}

Definition of the front element parameter used for shape function evaluation.

\begin{figure}
	\centering
		\includegraphics[width=0.8\textwidth]{../Figures/front_param_3.eps}
	
	\label{fig:front param_3}
\end{figure}

\end{frame}


\subsection{Tip enrichment}\label{Tip enrichment}

\begin{frame}
\frametitle{Tip enrichment functions}

Tip enrichment functions used:

\begin{equation*}\label{eq:tip_enr_functions}
	F_j\left(\mathbf{x}\right) \equiv F_j\left(r,\theta\right)= \left[\sqrt{r} \sin \frac{\theta}{2}, \sqrt{r} \cos \frac{\theta}{2}, \sqrt{r} \sin \frac{\theta}{2} \sin \theta, \sqrt{r} \cos \frac{\theta}{2} \sin \theta\right]
\end{equation*}

%\end{frame}

%\begin{frame}
%\frametitle{Tip enrichment}

\vspace{\baselineskip}

Tip enriched part of the displacements:

\begin{equation*}\label{eq:tip_enr}
	\mathbf{u_t}\left(\mathbf{x}\right)= \sum_{K\in \mathcal{N}^s} N_K^g\left(\mathbf{x}\right) \sum_{j} F_{j}\left(\mathbf{x}\right) \mathbf{c}_{Kj}
\end{equation*}

where 

\vspace{\baselineskip}

\begin{itemize}
	\setlength\itemsep{2em}
	\item $N_K^g$ are the global shape functions
	\item $\mathcal{N}^s$ is the set of superimposed nodes
\end{itemize}

\end{frame}

%\subsection{Jump enrichment}\label{Jump enrichment}
%
%\begin{frame}
%\frametitle{Jump enrichment function}
%
%Jump enrichment functions:
%
%\begin{equation*}\label{eq:jump_enr_functions}
	%H(\phi)=\left\{
	%\begin{array}{lr}
     %&1  \quad  \text{for} \ \phi > 0 \\
    %-&1  \quad  \text{for} \ \phi < 0
  %\end{array}\right.
%\end{equation*}
%
%\end{frame}

\subsection{Weight function blending}\label{weight function blending}

\begin{frame}
\frametitle{Topological enrichment}

The weight function assumes the form:

\begin{equation*}\label{eq:weight_topo}
	\varphi\left(\mathbf{x}\right) = \sum_{T\in \mathcal{N}^{t1}} N_T\left(\mathbf{x}\right)
\end{equation*}

where

\vspace{\baselineskip}

\begin{itemize}
	\setlength\itemsep{2em}
	\item $N_T$ are the FE shape functions.
	\item $\mathcal{N}^{t1}$ is a set including all nodes belonging to elements that contain the crack front.
\end{itemize}

\vspace{\baselineskip}

This definition is identical to the one of Fries~\cite{Fries2008}.

\end{frame}

\begin{frame}
\frametitle{Geometrical enrichment}

The weight function is defined as in~\cite{Ventura2009}. Nodal values:

\begin{equation*}\label{eq:weight_nodal}
	\varphi_I=\left\{
	\begin{array}{l r}
     1, & g_I < 0 \\
     \left(1-g_I\right)^n,  & 0 \leq g_I \leq 1 \\
		 0, & g_I > 1
  \end{array}\right.
\end{equation*}

where

\vspace{\baselineskip}

\begin{itemize}
	\setlength\itemsep{1.5em}
	\item $r_e$  is the enrichment radius.
	\item $r_i$ is an additional distance such that $r_i<r_e$.
	\item $r_I$ are the nodal values of parameter $r$.
	\item $g_I=\frac{r_I-r_i}{r_e-r_i}$
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Geometrical enrichment}

Weight function values are obtained through FE interpolation:

\vspace{\baselineskip}

\begin{equation*}\label{eq:weight_geo}
	\varphi\left(\mathbf{x}\right) = \sum_{T\in \mathcal{N}^t} N_T\left(\mathbf{x}\right) \varphi_T
\end{equation*}

\vspace{\baselineskip}

where $\mathcal{N}^t$ is the set of tip enriched nodes.

\end{frame}

\begin{frame}
\frametitle{Weight functions}

Weight functions for a) topological and b) geometrical enrichment.

\begin{figure}
	\centering
		\includegraphics[width=0.7\textwidth]{../Figures/weight_func_.eps}
	\label{fig:weights}
\end{figure}

\end{frame}

\begin{frame}
\frametitle{Weight functions}

Weight function for jump enrichment:

\begin{equation*}\label{eq:weight_jump}
	\bar{\varphi}\left(\mathbf{x}\right) = 1-\varphi\left(\mathbf{x}\right)
\end{equation*}

\end{frame}


\begin{frame}
\frametitle{Weight functions}

Enrichment strategies used for tip and jump enrichment.

\begin{figure}
	\centering
		\includegraphics[width=0.9\textwidth]{../Figures/enrichment_2.eps}
	\label{fig:enrichment}
\end{figure}

\end{frame}

%\subsection{Enrichment function shifting}\label{enrichment function shifting}


\subsection{Displacement approximation}\label{Displacement approximation}

\begin{frame}[shrink]
\frametitle{Displacement approximation}

\begin{align*}
		\mathbf{u}\left(\mathbf{x}\right) &=\sum_{I\in \mathcal{N}} N_I\left(\mathbf{x}\right) \mathbf{u}_I +\bar{\varphi}\left(\mathbf{x}\right) \sum_{J\in \mathcal{N}^j} N_J\left(\mathbf{x}\right) (H\left(\mathbf{x}\right)-    H_J) \mathbf{b}_J + \nonumber \\
		&+ \varphi\left(\mathbf{x}\right) \left(\sum_{K\in \mathcal{N}^s} N_K^g\left(\mathbf{x}\right) \sum_{j} F_{j}\left(\mathbf{x}\right) -\right. \nonumber \\
		& \left. - \sum_{T\in \mathcal{N}^t} N_T\left(\mathbf{x}\right)\sum_{K\in \mathcal{N}^s} N_K^g\left(\mathbf{x}_T\right) \sum_{j} F_{j}\left(\mathbf{x}_T\right)\right) \mathbf{c}_{Kj}		
	\end{align*}

where:

\begin{description}
\setlength\itemsep{1em}
\item[$\mathcal{N}$] is the set of all nodes in the FE mesh.
\item[$\mathcal{N}^j$] is the set of jump enriched nodes.
\item[$\mathcal{N}^t$] is the set of tip enriched nodes.
\item[$\mathcal{N}^s$] is the set of nodes in the superimposed mesh.
\end{description}

\end{frame}