\section{Vector Level Sets}\label{vector level sets}

\begin{frame}
\frametitle{Vector Level Sets}

A method for the representation of 3D cracks is introduced which:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
	\item Produces level set functions using geometric operations.
	\item Does not require integration of evolution equations.
\end{itemize}

\vspace{\baselineskip}

Similar methods:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
	\item 2D Vector level sets~\cite{Ventura2003}.
	\item Hybrid implicit-explicit crack representation~\cite{Fries2012}.
\end{itemize}

\end{frame}

\subsection{Crack representation}\label{crack representation}

\begin{frame}[fragile]
\frametitle{Crack front}

\begin{columns}
\begin{column}{0.60\textwidth}

Crack front at time $t$:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
	\item Ordered series of line segments $\textbf{t}_i$
	\item Set of points $\textbf{x}_i$
\end{itemize}

\end{column}

\begin{column}{0.40\textwidth}

\begin{figure}
	\centering
		\includegraphics[width=0.80\textwidth]{../Figures/crack_front_t.eps}
	\label{fig:crack_front_t}
\end{figure}

\end{column}
\end{columns}

\end{frame}


\begin{frame}[fragile]
\frametitle{Crack front advance}

\begin{columns}
\begin{column}{0.60\textwidth}

Crack front at time $t+1$:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
	\item Crack advance vectors $\textbf{s}_i^t$ at points $\textbf{x}_i$
	\item New set of points $\textbf{x}_i^{t+1}=\textbf{x}_i^{t}+\textbf{s}_i^t$
\end{itemize}

\end{column}

\begin{column}{0.40\textwidth}

\begin{figure}
	\centering
		\includegraphics[width=1.00\textwidth]{../Figures/crack_front_t1.eps}
	\label{fig:crack_front_t1}
\end{figure}

\end{column}
\end{columns}

\end{frame}


\begin{frame}[fragile]
\frametitle{Crack surface advance}

\begin{columns}
\begin{column}{0.60\textwidth}

Crack surface advance:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
	\item Sequence of four sided bilinear segments.
	\item Vertexes: $\textbf{x}_i^{t}$, $\textbf{x}_{i+1}^{t}$, $\textbf{x}_{i+1}^{t+1}$, $\textbf{x}_{i}^{t+1}$
\end{itemize}

\end{column}

\begin{column}{0.40\textwidth}

\begin{figure}
	\centering
		\includegraphics[width=1.00\textwidth]{../Figures/crack_adv_segment.eps}
	\label{fig:crack_front_t1}
\end{figure}

\end{column}
\end{columns}

\end{frame}


\begin{frame}
\frametitle{Kink wedges}

Discontinuities (\textit{kink wedges}) are present:

\begin{itemize}
	\item Along the crack front (a).
	\item Along the advance vectors (b).
\end{itemize}

\begin{figure}
	\centering
		\includegraphics[width=0.90\textwidth]{../Figures/kink_wedges.eps}
	\label{fig:crack_front_t1}
\end{figure}

\end{frame}


\subsection{Level set functions}\label{level set functions}


\begin{frame}[fragile]
\frametitle{Level set functions}

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

Definition of the level set functions at a point $\mathbf{P}$:

\vspace{\baselineskip}

\begin{description}[$\mathbf{f}$]
\setlength\itemsep{2em}
\item[$\mathbf{f}$] distance from the crack surface.
\item[$\mathbf{g}$] distance from the crack front.
\end{description}

\end{column}

\begin{column}{0.50\textwidth}

\begin{figure}
	\centering
		\includegraphics[width=1.00\textwidth]{../Figures/ls_def.eps}
	\label{fig:crack_front_t1}
\end{figure}

\end{column}
\end{columns}

\end{frame}

\subsection{Point projection}\label{point projection}

\begin{frame}
\frametitle{Point projection}

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

Element parametric equations $\phi\left(u,v\right)$, $u,v\in\left[-1,1\right]$:

\begin{equation*}\label{eq:quad_parametric_eq}
\left\{
	\begin{array}{lr}
    \phi_x&=g_1 \left( u,v \right) x_{i}^{t} + g_2 \left( u,v \right) x_{i+1}^{t} + g_3 \left( u,v \right) x_{i+1}^{t+1} + g_4 \left( u,v \right) x_{i}^{t+1} \\
		\phi_y&=g_1 \left( u,v \right) y_{i}^{t} + g_2 \left( u,v \right) y_{i+1}^{t} + g_3 \left( u,v \right) y_{i+1}^{t+1} + g_4 \left( u,v \right) y_{i}^{t+1} \\
		\phi_z&=g_1 \left( u,v \right) z_{i}^{t} + g_2 \left( u,v \right) z_{i+1}^{t} + g_3 \left( u,v \right) z_{i+1}^{t+1} + g_4 \left( u,v \right) z_{i}^{t+1} \\
  \end{array}\right.
\end{equation*}

where $g_i \left( u,v \right)$, $u,v\in\left[-1,1\right]$ are linear shape functions.

\end{frame}


\begin{frame}
\frametitle{Point projection}

Equation of the tangent plane $\Pi_0$ at $(u_0,v_0)$:

\vspace{\baselineskip}

\begin{equation*}\label{eq:quad_tangent_plane}
	\text{det}\left[
	\begin{array}{c c c}
	x-\phi_x\left(u_0,v_0\right)    &   y-\phi_y\left(u_0,v_0\right)  &  z-\phi_z\left(u_0,v_0\right)  \\
	\phi_{x,u}\left(u_0,v_0\right)  &  \phi_{y,u}\left(u_0,v_0\right) & \phi_{z,u}\left(u_0,v_0\right) \\
	\phi_{x,v}\left(u_0,v_0\right)  &  \phi_{y,v}\left(u_0,v_0\right) & \phi_{z,v}\left(u_0,v_0\right)
	\end{array}\right]=0
\end{equation*}

\vspace{\baselineskip}

Normal vector to the parametric surface at $(u_0,v_0)$:

\vspace{\baselineskip}

\begin{equation*}\label{eq:quad_tangent_plane}
	\mathbf{n}\left(u_0,v_0\right)=\left(A,B,C\right)
\end{equation*}

\vspace{\baselineskip}

where $A,B,C$ are the minors of the previous matrix at $(u_0,v_0)$.

\end{frame}


\begin{frame}
\frametitle{Point projection}

Point $\textbf{P}$ can be expressed as:

\begin{equation*}\label{eq:point_projection}
	\textbf{P}=\textbf{P}'\left(u,v\right)+\lambda \textbf{n}\left(u,v\right)
\end{equation*}

where:

\begin{description}[$\mathbf{P}'$]
%\setlength\itemsep{2em}
\item[$\mathbf{P}'$] the projection of the point to the surface.
\item[$\lambda$] unknown parameter.
\end{description}

\vspace{\baselineskip}

The above is solved for $u, v$ and $\lambda$ to obtain the projection.

%\vspace{\baselineskip}

%If $u,v\notin\left[-1,1\right]$ 

\end{frame}

\subsection{Evaluation of the level set functions}\label{evaluation of the level set functions}

\begin{frame}
\frametitle{Evaluation of the level set functions}

At each step $t$:

\begin{itemize}
\setlength\itemsep{2em}
	\item For each point all crack advance segments are tested.
	\item If for a certain element $u,v\in\left[-1,1\right]$ then the point is projected on that element.
	\item If $u\notin\left[-1,1\right]$ for all elements then the projection lies on the advance vector.
	\item If $v\notin\left[-1,1\right]$ for all elements then the projection lies either:
	
	\begin{itemize}
		\item[$\rightarrow$] at a previous crack advance segment
		\item[$\rightarrow$] at the crack front at time $t-1$ or $t$
	\end{itemize}
\end{itemize}

\end{frame}


\begin{frame}
\frametitle{Evaluation of the level set functions}

Level set function $\mathbf{f}$:

\begin{equation*}\label{eq:function_f}
	\mathbf{f}=\mathbf{P}-\mathbf{P}'
\end{equation*}

where $\mathbf{P}'$ is either:

\begin{itemize}
	\item Projection to an element of the crack surface
	\item Closest point projection to a kink wedge
\end{itemize}

\vspace{\baselineskip}
\vspace{\baselineskip}

Level set function $\mathbf{g}$:

\begin{equation*}\label{eq:function_g}
	\mathbf{g}=\mathbf{P}-\mathbf{P}'
\end{equation*}

where $\mathbf{P}'$ is a closest point projection to the crack front

\end{frame}