\chapter{Elements\index{Elements}}
\label{chap:elements}
The base for every Finite-Elements computation is its mesh and the elements that
are used within that mesh. The element types that can be used depend on the
mesh, but also on the dimensionality of the problem (1D, 2D or 3D). In \akantu,
several isoparametric Lagrangian element types are supported (and one
serendipity element). Each of these types is discussed in some detail below,
starting with the 1D-elements all the way to the 3D-elements. More detailed
information (shape function, location of Gaussian quadrature points, and so on)
can be found in Appendix~\ref{app:elements}.

%%%%%%%%%% 1D %%%%%%%%%
\section{Isoparametric Elements\index{Elements!Isoparametric}}

\subsection*{1D\index{Elements!1D}}

In \akantu, there are two types of isoparametric elements defined in 1D. These
element types are called \code{\_segment\_2} and \code{\_segment\_3}, and are
depicted schematically in Figure~\ref{fig:elements:1D}. Some of the basic
properties of these elements are listed in Table~\ref{tab:elements:1D}.

\begin{figure}[!htb]
\begin{center}
\begin{tabular}{m{0.31\textwidth}m{0.1\textwidth}m{0.31\textwidth}}
\subfloat[\code{\_segment\_2}]{
  \includegraphics[width=0.29\textwidth,clip,trim=1cm 0 1.5cm 0]{figures/elements/segment_2}
  \label{fig:elements:segment2}
} & &
\subfloat[\code{\_segment\_3}]{
  \includegraphics[width=0.29\textwidth,clip,trim=1cm 0 1.5cm 0]{figures/elements/segment_3}
  \label{fig:elements:segment3}
}
\end{tabular}
\end{center}
\caption{Schematic overview of the two 1D element types in \akantu. In each
  element, the node numbering as used in \akantu is indicated and also the
  quadrature points are highlighted (gray circles).}
\label{fig:elements:1D}
\end{figure}

\begin{table}[!htb]
\begin{center}
\begin{tabular}{l|ccc}
\toprule
Element type & Order & \# nodes & \# quad. points \\
\midrule
\texttt{\_segment\_2} & linear & 2 & 1 \\
\texttt{\_segment\_3} & quadratic & 3 & 2 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Some basic properties of the two 1D isoparametric elements in \akantu.}
\label{tab:elements:1D}
\end{table}

%%%%%%%%%% 2D %%%%%%%%%
\subsection*{2D\index{Elements!2D}}

In \akantu, there are four types of isoparametric elements defined in 2D. These
element types are called \code{\_triangle\_3}, \code{\_triangle\_6},
\code{\_quadrangle\_4} and \code{\_quadrangle\_8}, and all of them are depicted
in Figure~\ref{fig:elements:2D}. As with the 1D elements, some of the most basic
properties of these elements are listed in Table~\ref{tab:elements:2D}. It is
important to note that the first element is linear, the next two quadratic and
the last one cubic. Furthermore, the last element type (\code{\_quadrangle\_8})
is not a Lagrangian but a serendipity element.

\begin{figure}[!htb]
\begin{center}
\begin{tabular}{m{0.31\textwidth}m{0.1\textwidth}m{0.31\textwidth}}
\subfloat[\code{\_triangle\_3}]{
  \includegraphics[width=0.29\textwidth]{figures/elements/triangle_3}
  \label{fig:elements:triangle3}
} & &
\subfloat[\code{\_triangle\_6}]{
  \includegraphics[width=0.29\textwidth]{figures/elements/triangle_6}
  \label{fig:elements:triangle6}
} \\
\subfloat[\code{\_quadrangle\_4}]{
  \includegraphics[width=0.29\textwidth]{figures/elements/quadrangle_4}
  \label{fig:elements:quadrangle4}
} & &
\subfloat[\code{\_quadrangle\_8}]{
  \includegraphics[width=0.29\textwidth]{figures/elements/quadrangle_8}
  \label{fig:elements:quadrangle8}
}
\end{tabular}
\end{center}
\caption{Schematic overview of the four 2D element types in \akantu. In each
  element, the node numbering as used in \akantu is indicated and also the
  quadrature points are highlighted (gray circles).}
\label{fig:elements:2D}
\end{figure}

\begin{table}[!htb]
\begin{center}
\begin{tabular}{l|ccc}
\toprule
Element type & Order & \# nodes & \# quad. points \\
\midrule
\texttt{\_triangle\_3} & linear & 3 & 1 \\
\texttt{\_triangle\_6} & quadratic & 6 & 3 \\
\hline
\texttt{\_quadrangle\_4} & quadratic & 4 & 4 \\
\texttt{\_quadrangle\_8} & cubic & 8 & 9 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Some basic properties of the four 2D isoparametric elements in \akantu.}
\label{tab:elements:2D}
\end{table}

%%%%%%%%%% 3D %%%%%%%%%
\subsection*{3D\index{Elements!3D}}

In \akantu, there are three types of isoparametric elements defined in 3D. These
element types are called \code{\_tetrahedron\_4}, \code{\_tetrahedron\_10} and
\code{\_hexahedron\_8}, and all of them are depicted schematically in
Figure~\ref{fig:elements:3D}. As with the 1D and 2D elements some of the most
basic properties of these elements are listed in Table~\ref{tab:elements:3D}.

\begin{figure}[!htb]
\begin{center}
\begin{tabular}{m{0.3\linewidth}m{0.3\linewidth}m{0.3\linewidth}}
\subfloat[\code{\_tetrahedron\_4}]{
  \includegraphics[width=0.9\linewidth]{figures/elements/tetrahedron_4}
  \label{fig:elements:tetrahedron4}
} &
\subfloat[\code{\_tetrahedron\_10}]{
  \includegraphics[width=0.9\linewidth]{figures/elements/tetrahedron_10}
  \label{fig:elements:tetrahedron10}
} &
\subfloat[\code{\_hexahedron\_8}]{
  \includegraphics[width=0.9\linewidth]{figures/elements/hexahedron_8}
  \label{fig:elements:hexahedron8}
}
\end{tabular}
\caption{Schematic overview of the three 3D element types in \akantu. In each
  element, the node numbering as used in \akantu is indicated and also the
  quadrature points are highlighted (gray spheres).}
\label{fig:elements:3D}
\end{center}
\end{figure}

\begin{table}[!htb]
\begin{center}
\begin{tabular}{l|ccc}
\toprule
Element type & Order & \# nodes & \# quad. points  \\
\midrule
\texttt{\_tetrahedron\_4} & linear & 4 & 1  \\
\texttt{\_tetrahedron\_10} & quadratic & 10 & 4  \\
\hline
\texttt{\_hexahedron\_8} & cubic & 8 & 8  \\
\bottomrule
\end{tabular}
\end{center}
\caption{Some basic properties of the three 3D isoparametric elements in \akantu.}
\label{tab:elements:3D}
\end{table}


%%%%%%%%%% COHESIVE ELEMENTS %%%%%%%%%
\IfFileExists{manual-cohesive_elements.tex}{\input{manual-cohesive_elements}}{}

%%%%%%%%%% STRUCTURAL ELEMENTS %%%%%%%%%
\IfFileExists{manual-structuralmechanicsmodel-elements.tex}{\input{manual-structuralmechanicsmodel-elements}}{}

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