\section{Numerical Examples} \label{numerical examples}

\subsection{Convergence study} \label{convergence study}

\begin{frame}
\frametitle{Convergence study}

\begin{figure}
	\centering
		\includegraphics[width=0.9\textwidth]{../Figures/penny_crack_l_2.eps}
	\label{fig:penny_crack}
\end{figure}

\begin{itemize}

\item Uniform normal and shear loads of magnitude 1 are applied at $\Gamma_c^t$.

\item Problem dimensions: $L_x = L_y = 2 L_z = 0.4 $ units and $a=0.1$ unit.

\item Material parameters: $E=100$ units and $\nu=0.3$.

\item Mesh consists of $n_x \times n_y \times n_z$ hexahedral elements, $n_x=n_y=2 n_z=n$ and $n\in\left\{21, 41, 61, 81, 101\right\}$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{$L_2$ and energy norms}

\begin{figure}
	\centering
		\includegraphics[width=0.5\textwidth]{../Figures/conv_topo_p.eps}
	\label{fig:3d_topo}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[width=0.85\textwidth]{../Figures/conv_geo_p.eps}
	\label{fig:3d_geo}
\end{figure}

\end{frame}

\begin{frame}
\frametitle{$L_2$ and energy norms}

\begin{block}{Convergence rates}

\begin{table}
	\centering
	\setlength{\tabcolsep}{2pt}
	
	\begin{tabular}{|c|c|c|c|}
		\hline
		                        &   $r_e=0.00$   &   $r_e=0.02$    &    $r_e=0.04$ \\ 
		\hline
		XFEM  E                 &      0.492     &      0.911      &       1.015   \\
		%\hline
		XFEM  $L_2$             &      1.009     &      1.824      &       1.976   \\
		%\hline
		%GE-XFEM  E              &      0.558     &      1.057      &       0.988   \\
		%\hline
		%GE-XFEM  $L_2$          &      1.535     &      1.753      &       1.448   \\
		%\hline
		CGE-XFEM  E             & \textbf{0.635} & \textbf{0.957}  & \textbf{1.014}\\
		%\hline
		CGE-XFEM  $L_2$         & \textbf{1.265} & \textbf{1.890}  & \textbf{1.930}\\
		\hline
	\end{tabular}
	\label{tab:conv_rates_3d}
\end{table}

\end{block}

\end{frame}


\begin{frame}
\frametitle{Stress intensity factors}

\begin{figure}
	\centering
		\includegraphics[width=1\textwidth]{../Figures/SIFS_04_c.eps}
	\label{fig:3d_SIF_04}
\end{figure}
\end{frame}


\begin{frame}
\frametitle{Conditioning}

Condition numbers for three different enrichment radii.

\vspace{\baselineskip}

\begin{figure}
	\centering
		\includegraphics[width=0.9\textwidth]{../Figures/cond_num.eps}
	\label{fig:3d_it}
\end{figure}

\end{frame}


\subsection{Crack propagation} \label{crack propagation}

\begin{frame}
\frametitle{Edge crack in a beam}

Edge crack in a beam under three point bending.

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

Geometrical parameters:

$L=260$ mm, $L_1=240$ mm, $H=60$ mm, $d=10$ mm, $\alpha=20$ mm

Material parameters:

$E=2.1 \times 10^5$ N/mm$^2$, $\nu=0.0$

\end{frame}

\begin{frame}
\frametitle{Edge crack in a beam}

An unstructured tetrahedral mesh is used:

\vspace{\baselineskip}

\begin{figure}
	\centering
		\includegraphics[width=0.9\textwidth]{../Figures/mesh2.jpg}
	\label{fig:beam_notch_straight}
\end{figure}

\end{frame}

\begin{frame}
\frametitle{Edge crack in a beam}

Energy release rates along the crack front:

\vspace{\baselineskip}

\begin{figure}
	\centering
		\includegraphics[width=0.7\textwidth]{../Figures/beam_3pb_plane_strain_g.eps}
	\label{fig:beam_notch_straight}
\end{figure}

\end{frame}

\begin{frame}
\frametitle{Inclined edge crack in a beam}

Inclined edge crack in a beam under three point bending.

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

Geometrical parameters:

$L=260$ mm, $L_1=240$ mm, $H=60$ mm, $d=10$ mm, $\alpha=20$ mm, $\beta=45^{\circ}$

Material parameters:

$E=2.1 \times 10^5$ N/mm$^2$, $\nu=0.3$

\end{frame}

\begin{frame}
\frametitle{Inclined edge crack in a beam}

\begin{figure}
	\centering
		\includegraphics<1>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_0.jpg}
		\includegraphics<2>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_1.jpg}
		\includegraphics<3>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_2.jpg}
		\includegraphics<4>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_3.jpg}
		\includegraphics<5>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_4.jpg}
		\includegraphics<6>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_5.jpg}
		\includegraphics<7>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_6.jpg}
		\includegraphics<8>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_7.jpg}
		\includegraphics<9>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_8.jpg}
		\includegraphics<10>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_9.jpg}
		\includegraphics<11>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_10.jpg}
		\includegraphics<12>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_11.jpg}
		\includegraphics<13>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_12.jpg}
		\includegraphics<14>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_13.jpg}
		\includegraphics<15>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_14.jpg}
		\includegraphics<16>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_15.jpg}
		\includegraphics<17>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_16.jpg}
		\includegraphics<18>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_17.jpg}
		\includegraphics<19>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_18.jpg}
		\includegraphics<20>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_19.jpg}
		\includegraphics<21>[width=1.00\textwidth]{../Propagation_Example/Beam_3pb_incl_20.jpg}
%	\label{fig:beam_prop}
\end{figure}

\end{frame}


\begin{frame}
\frametitle{Inclined edge crack in a beam}

\begin{figure}
	\centering
	\subfigure[]{\includegraphics[height=6cm]{../Figures/mesh_deformed_zoom.jpg}} \qquad \quad
	\subfigure[]{\includegraphics[height=6cm]{../Figures/crack_final.jpg}} \qquad \quad
	\subfigure[]{\includegraphics[height=6cm]{../Figures/crack_final_2.jpg}}
	\label{fig:beam_3pb_def_magn}
\end{figure}

\end{frame}

\subsection{Detection of a penny crack in a unit cube}

\begin{frame}
\frametitle{Penny crack in a cube}

Geometry and sensors:

\begin{columns}
	\begin{column}{0.50\textwidth}

	\begin{figure}
		\centering
			\includegraphics[width=0.90\textwidth]{../Figures/example1_geometry.eps}
	\end{figure}
	
	\end{column}
	
	\begin{column}{0.50\textwidth}
	
	\begin{figure}
		\centering
			\includegraphics[width=0.90\textwidth]{../Figures/example1_sensors.eps}
	\end{figure}
	
	\end{column}
	
\end{columns}

\end{frame}


\begin{frame}
\frametitle{Penny crack in a cube}

Optimization problem convergence:

\begin{figure}
	\centering
		\includegraphics[width=0.80\textwidth]{../Figures/ff.eps}
\end{figure}

\end{frame}


\begin{frame}
\frametitle{Penny crack in a cube}

Best solution after different numbers of iterations

\begin{figure}
	\centering
		\includegraphics[width=0.70\textwidth]{../Figures/crack_locations.eps}
\end{figure}

\end{frame}