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diff --git a/doc/manual/manual-elements.tex b/doc/manual/manual-appendix-elements.tex
similarity index 79%
copy from doc/manual/manual-elements.tex
copy to doc/manual/manual-appendix-elements.tex
index 27397ea1a..b4f821915 100644
--- a/doc/manual/manual-elements.tex
+++ b/doc/manual/manual-appendix-elements.tex
@@ -1,314 +1,359 @@
-\section{Elements\index{Elements}}
+\chapter{Isoparametric Elements\index{Elements}}
+\label{app:elements}
-The base for every finite element computation is its mesh and the elements that are used within that mesh. What kind of element types can be used depends on the mesh, but also on the dimensionality of the problem (1D, 2D or 3D). In \akantu several isoparametric Langrangian element types are supported. Each of these types is discussed in more detail below, starting with the 1D-elements all the way to the 3D-elements. For each element type the coordinates of the nodes are given in the isoparametric frame of reference, together with the shape functions (and their derivatives) on these respective nodes. Also all the quadrature points within each element are listed (together with the weight that is applied on these points).
+The base for every finite element computation is its mesh and the elements that are used within that mesh. What kind of element types can be used depends on the mesh, but also on the dimensionality of the problem (1D, 2D or 3D). In \akantu several isoparametric Langrangian element types are supported. Each of these types is discussed in more detail below, starting with the 1D-elements all the way to the 3D-elements. For each element type the coordinates of the nodes are given in the isoparametric frame of reference, together with the shape functions (and their derivatives) on these respective nodes. Also all the Gaussian quadrature points within each element are listed (together with the weight that is applied on these points).
%%%%%%%%%% 1D %%%%%%%%%
-\subsection{Isoparametric Elements in 1D\index{Elements!1D}}
+\section{Isoparametric Elements in 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}.
+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 in Figure~\ref{fig:app:elements:1D}.
-\subsubsection{Segment 2\index{Elements!1D!Segment 2}}
+\begin{figure}[!htb]
+\begin{center}
+\begin{tabular}{m{0.3\textwidth}m{0.1\textwidth}m{0.3\textwidth}}
+\subfloat[\code{segment\_2}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/segment_2}
+ \label{fig:elements:segment2}
+} & &
+\subfloat[\code{segment\_3}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/segment_3}
+ \label{fig:elements:segment3}
+}
+\end{tabular}
+\end{center}
+\caption{A schematic overview of the three supported 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). This figure is the same as Figure~\ref{fig:elements:1D} and is repeated here for convience.}
+\label{fig:app:elements:1D}
+\end{figure}
+
+\subsection{Segment 2\index{Elements!1D!Segment 2}}
\begin{Element}{1D}
1 & \inelemone{-1} & $\half\left(1-\xi\right)$ & \inelemone{-\half} \\
\elemline
2 & \inelemone{1} & $\half\left(1+\xi\right)$ & \inelemone{\half} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|c}
Coord. \elemcooroned & 0 \\
\elemline
Weight & 2 \\
\end{tabular}
\end{QuadPoints}
-\subsubsection{Segment 3\index{Elements!1D!Segment 3}}
+\subsection{Segment 3\index{Elements!1D!Segment 3}}
\begin{Element}{1D}
1 & \inelemone{-1} & $\half\xi\left(\xi-1\right)$ & \inelemone{\xi-\half} \\
\elemline
2 & \inelemone{1} & $\half\xi\left(\xi+1\right)$ & \inelemone{\xi+\half} \\
\elemline
3 & \inelemone{0} & $1-\xi^{2}$ & \inelemone{-2\xi} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|cc}
Coord. \elemcooroned & $-1/\sqrt{3}$ & $1/\sqrt{3}$ \\
\elemline
Weight & 1 & 1 \\
\end{tabular}
\end{QuadPoints}
%%%%%%%%%% 2D %%%%%%%%%
-\subsection{Isoparametric Elements in 2D\index{Elements!2D}}
+\section{Isoparametric Elements in 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 are depicted in Figure~\ref{fig:app: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}.
+\begin{figure}[!htb]
+\begin{center}
+\begin{tabular}{m{0.3\textwidth}m{0.1\textwidth}m{0.3\textwidth}}
+\subfloat[\code{triangle\_3}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/triangle_3}
+ \label{fig:elements:triangle3}
+} & &
+\subfloat[\code{triangle\_6}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/triangle_6}
+ \label{fig:elements:triangle6}
+} \\
+\subfloat[\code{quadrangle\_4}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/quadrangle_4}
+ \label{fig:elements:quadrangle4}
+} & &
+\subfloat[\code{quadrangle\_8}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/quadrangle_8}
+ \label{fig:elements:quadrangle8}
+}
+\end{tabular}
+\end{center}
+\caption{A schematic overview of the four supported 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). This figure is the same as Figure~\ref{fig:elements:2D} and is repeated here for convience.}
+\label{fig:app:elements:2D}
+\end{figure}
-\subsubsection{Triangle 3\index{Elements!2D!Triangle 3}}
+\clearpage
+\subsection{Triangle 3\index{Elements!2D!Triangle 3}}
\begin{Element}{2D}
1 & \inelemtwo{0}{0} & $1-\xi-\eta$ & \inelemtwo{-1}{-1} \\
\elemline
2 & \inelemtwo{1}{0} & $\xi$ & \inelemtwo{1}{0} \\
\elemline
3 & \inelemtwo{0}{1} & $\eta$ & \inelemtwo{0}{1} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|c}
Coord. \elemcoortwod & \inquadtwo{\third}{\third} \\
\elemline
Weight & 1 \\
\end{tabular}
\end{QuadPoints}
-\subsubsection{Triangle 6\index{Elements!2D!Triangle 6}}
+\subsection{Triangle 6\index{Elements!2D!Triangle 6}}
\begin{Element}{2D}
1 & \inelemtwo{0}{0} & $-\left(1-\xi-\eta\right)\left(1-2\left(1-\xi-\eta\right)\right)$ & \inelemtwo{1-4\left(1-\xi-\eta\right)}{1-4\left(1-\xi-\eta\right)} \\
\elemline
2 & \inelemtwo{1}{0} & $-\xi\left(1-2\xi\right)$ & \inelemtwo{4\xi-1}{0} \\
\elemline
3 & \inelemtwo{0}{1} & $-\eta\left(1-2\eta\right)$ & \inelemtwo{0}{4\eta-1} \\
\elemline
4 & \inelemtwo{\half}{0} & $4\xi\left(1-\xi-\eta\right)$ & \inelemtwo{4\left(1-2\xi-\eta\right)}{-4\xi} \\
\elemline
5 & \inelemtwo{\half}{\half} & $4\xi\eta$ & \inelemtwo{4\eta}{4\xi} \\
\elemline
6 & \inelemtwo{0}{\half} & $4\eta\left(1-\xi-\eta\right)$ & \inelemtwo{-4\eta}{4\left(1-\xi-2\eta\right)} \\
\elemline
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|ccc}
Coord. \elemcoortwod & \inquadtwo{\sixth}{\sixth} & \inquadtwo{\twothird}{\sixth} & \inquadtwo{\sixth}{\twothird} \\
\elemline
Weight & \sixth & \sixth & \sixth \\
\end{tabular}
\end{QuadPoints}
-\subsubsection{Quadrangle 4\index{Elements!2D!Quadrangle 4}}
+\subsection{Quadrangle 4\index{Elements!2D!Quadrangle 4}}
\begin{Element}{2D}
1 & \inelemtwo{-1}{-1} & $\quart\left(1-\xi\right)\left(1-\eta\right)$ & \inelemtwo{-\quart\left(1-\eta\right)}{-\quart\left(1-\xi\right)} \\
\elemline
2 & \inelemtwo{1}{-1} & $\quart\left(1+\xi\right)\left(1-\eta\right)$ & \inelemtwo{\quart\left(1-\eta\right)}{-\quart\left(1-\xi\right)} \\
\elemline
3 & \inelemtwo{1}{1} & $\quart\left(1+\xi\right)\left(1+\eta\right)$ & \inelemtwo{\quart\left(1-\eta\right)}{\quart\left(1-\xi\right)} \\
\elemline
4 & \inelemtwo{-1}{1} & $\quart\left(1-\xi\right)\left(1+\eta\right)$ & \inelemtwo{-\quart\left(1-\eta\right)}{\quart\left(1-\xi\right)} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|cccc}
\elemcoortwod & \inquadtwo{-\invsqrtthree}{-\invsqrtthree} & \inquadtwo{\invsqrtthree}{-\invsqrtthree}
& \inquadtwo{\invsqrtthree}{\invsqrtthree} & \inquadtwo{-\invsqrtthree}{\invsqrtthree} \\
\elemline
Weight & 1 & 1 & 1 & 1 \\
\end{tabular}
\end{QuadPoints}
-\subsubsection{Quadrangle 8\index{Elements!2D!Quadrangle 8}}
+\clearpage
+\subsection{Quadrangle 8\index{Elements!2D!Quadrangle 8}}
\begin{Element}{2D}
1 & \inelemtwo{-1}{-1} & $\quart\left(1-\xi\right)\left(1-\eta\right)\left(-1-\xi-\eta\right)$
& \inelemtwo{\quart\left(1-\eta\right)\left(2\xi+\eta\right)}
{\quart\left(1-\xi\right)\left(\xi+2\eta\right)} \\
\elemline
2 & \inelemtwo{1}{-1} & $\quart\left(1+\xi\right)\left(1-\eta\right)\left(-1+\xi-\eta\right)$
& \inelemtwo{\quart\left(1-\eta\right)\left(2\xi-\eta\right)}
{-\quart\left(1+\xi\right)\left(\xi-2\eta\right)} \\
\elemline
3 & \inelemtwo{1}{1} & $\quart\left(1+\xi\right)\left(1+\eta\right)\left(-1+\xi+\eta\right)$
& \inelemtwo{\quart\left(1+\eta\right)\left(2\xi+\eta\right)}
{\quart\left(1+\xi\right)\left(\xi+2\eta\right)} \\
\elemline
4 & \inelemtwo{-1}{1} & $\quart\left(1-\xi\right)\left(1+\eta\right)\left(-1-\xi+\eta\right)$
& \inelemtwo{\quart\left(1+\eta\right)\left(2\xi-\eta\right)}
{-\quart\left(1-\xi\right)\left(\xi-2\eta\right)} \\
\elemline
5 & \inelemtwo{0}{-1} & $\half\left(1-\xi^{2}\right)\left(1-\eta\right)$
& \inelemtwo{-\xi\left(1-\eta\right)}
{-\half\left(1-\xi^{2}\right)} \\
\elemline
6 & \inelemtwo{1}{0} & $\half\left(1+\xi\right)\left(1-\eta^{2}\right)$
& \inelemtwo{\half\left(1-\eta^{2}\right)}
{-\eta\left(1+\xi\right)} \\
\elemline
7 & \inelemtwo{0}{1} & $\half\left(1-\xi^{2}\right)\left(1+\eta\right)$
& \inelemtwo{-\xi\left(1+\eta\right)}
{\half\left(1-\xi^{2}\right)} \\
\elemline
8 & \inelemtwo{-1}{0} & $\half\left(1-\xi\right)\left(1-\eta^{2}\right)$
& \inelemtwo{-\half\left(1-\eta^{2}\right)}
{-\eta\left(1-\xi\right)} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|ccccc}
Coord. \elemcoortwod & \inquadtwo{0}{0} & \inquadtwo{\sqrt{\tfrac{3}{5}}}{\sqrt{\tfrac{3}{5}}} & \inquadtwo{-\sqrt{\tfrac{3}{5}}}{\sqrt{\tfrac{3}{5}}}
& \inquadtwo{-\sqrt{\tfrac{3}{5}}}{-\sqrt{\tfrac{3}{5}}} & \inquadtwo{\sqrt{\tfrac{3}{5}}}{-\sqrt{\tfrac{3}{5}}} \\
\elemline
Weight & 64/81 & 25/81 & 25/81 & 25/81 & 25/81 \\
\elemline
Coord. \elemcoortwod & \inquadtwo{0}{\sqrt{\tfrac{3}{5}}} & \inquadtwo{-\sqrt{\tfrac{3}{5}}}{0}
& \inquadtwo{0}{-\sqrt{\tfrac{3}{5}}} & \inquadtwo{\sqrt{\tfrac{3}{5}}}{0} & \\
\elemline
Weight & 40/81 & 40/81 & 40/81 & 40/81 & \\
\end{tabular}
\end{QuadPoints}
%%%%%%%%%% 3D %%%%%%%%%
-\subsection{Isoparametric Elements in 3D\index{Elements!3D}}
+\section{Isoparametric Elements in 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:app:elements:3D}.
-\begin{figure}
+\begin{figure}[!htb]
\begin{center}
\begin{tabular}{m{0.3\textwidth}m{0.3\textwidth}m{0.3\textwidth}}
-\subfloat[Tetrahedron-4]{
+\subfloat[\code{tetrahedron\_4}]{
\includegraphics[width=0.3\textwidth]{figures/elements/tetrahedron_4}
\label{fig:elements:tetrahedron4}
} &
-\subfloat[Tetrahedron-10]{
+\subfloat[\code{tetrahedron\_10}]{
\includegraphics[width=0.3\textwidth]{figures/elements/tetrahedron_10}
\label{fig:elements:tetrahedron10}
} &
-\subfloat[Hexahedron-8]{
+\subfloat[\code{hexahedron\_8}]{
\includegraphics[width=0.3\textwidth]{figures/elements/hexahedron_8}
\label{fig:elements:hexahedron8}
}
\end{tabular}
-\caption{Schematic depiction of the three 3D isoparametric elements available in \akantu. In each element the node numbering used in \akantu is indicated and also the quadrature points are highlighted (gray spheres)}
+\caption{A schematic overview of the three supported 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). This figure is the same as Figure~\ref{fig:elements:3D} and is repeated here for convience.}
+\label{fig:app:elements:3D}
\end{center}
\end{figure}
-
-\subsubsection{Tetrahedron 4\index{Elements!3D!Tetrahedron 4}}
+\clearpage
+\subsection{Tetrahedron 4\index{Elements!3D!Tetrahedron 4}}
\begin{Element}{3D}
1 & \inelemthree{0}{0}{0} & $1-\xi-\eta-\zeta$ & \inelemthree{-1}{-1}{-1} \\
\elemline
2 & \inelemthree{1}{0}{0} & $\xi$ & \inelemthree{1}{0}{0} \\
\elemline
3 & \inelemthree{0}{1}{0} & $\eta$ & \inelemthree{0}{1}{0} \\
\elemline
4 & \inelemthree{0}{0}{1} & $\zeta$ & \inelemthree{0}{0}{1} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|c}
Coord. \elemcoorthreed & \inquadthree{\quart}{\quart}{\quart} \\
\elemline
Weight & \sixth \\
\end{tabular}
\end{QuadPoints}
-\subsubsection{Tetrahedron 10\index{Elements!3D!Tetrahedron 10}}
+\subsection{Tetrahedron 10\index{Elements!3D!Tetrahedron 10}}
\begin{Element}{3D}
1 & \inelemthree{0}{0}{0} & $\left(1-\xi-\eta-\zeta\right)\left(1-2\xi-2\eta-2\zeta\right)$
& \inelemthree{4\xi+4\eta+4\zeta-3}{4\xi+4\eta+4\zeta-3}{4\xi+4\eta+4\zeta-3} \\
\elemline
2 & \inelemthree{1}{0}{0} & $\xi\left(2\xi-1\right)$
& \inelemthree{4\xi-1}{0}{0} \\
\elemline
3 & \inelemthree{0}{1}{0} & $\eta\left(2\eta-1\right)$
& \inelemthree{0}{4\eta-1}{0} \\
\elemline
4 & \inelemthree{0}{0}{1} & $\zeta\left(2\zeta-1\right)$
& \inelemthree{0}{0}{4\zeta-1} \\
\elemline
5 & \inelemthree{\half}{0}{0} & $4\xi\left(1-\xi-\eta-\zeta\right)$
& \inelemthree{4-8\xi-4\eta-4\zeta}{-4\xi}{-4\xi} \\
\elemline
6 & \inelemthree{\half}{\half}{0} & $4\xi\eta$
& \inelemthree{4\eta}{4\xi}{0} \\
\elemline
7 & \inelemthree{0}{\half}{0} & $4\eta\left(1-\xi-\eta-\zeta\right)$
& \inelemthree{-4\eta}{4-4\xi-8\eta-4\zeta}{-4\eta} \\
\elemline
8 & \inelemthree{0}{0}{\half} & $4\zeta\left(1-\xi-\eta-\zeta\right)$
& \inelemthree{-4\zeta}{-4\zeta}{4-4\xi-4\eta-8\zeta} \\
\elemline
9 & \inelemthree{\half}{0}{\half} & $4\xi\zeta$
& \inelemthree{4\zeta}{0}{4\xi} \\
\elemline
10 & \inelemthree{0}{\half}{\half} & $4\eta\zeta$
& \inelemthree{0}{4\zeta}{4\eta} \\
\end{Element}
\begin{QuadPoints}
-%\newcommand\quada{\tfrac{1}{20}\left(5-\sqrt{5}\right)}
-%\newcommand\quadb{\tfrac{1}{20}\left(5+3\sqrt{5}\right)}
-\newcommand\quada{\tfrac{\left(5-\sqrt{5}\right)}{20}}
-\newcommand\quadb{\tfrac{\left(5+3\sqrt{5}\right)}{20}}
\begin{tabular}{l|cc}
Coord. \elemcoorthreed & \inquadthree{\quada}{\quada}{\quada} & \inquadthree{\quadb}{\quada}{\quada} \\
\elemline
Weight & 1/24 & 1/24 \\
\elemline
Coord. \elemcoorthreed & \inquadthree{\quada}{\quadb}{\quada} & \inquadthree{\quada}{\quada}{\quadb} \\
\elemline
Weight & 1/24 & 1/24 \\
\end{tabular}
\end{QuadPoints}
-\subsubsection{Hexahedron 8\index{Elements!3D!Hexahedron 8}}
+\clearpage
+\subsection{Hexahedron 8\index{Elements!3D!Hexahedron 8}}
\begin{Element}{3D}
1 & \inelemthree{-1}{-1}{-1} & $\eighth\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta\right)$
& \inelemthree{-\eighth\left(1-\eta\right)\left(1-\zeta\right)}
{-\eighth\left(1-\xi\right)\left(1-\zeta\right)}
{-\eighth\left(1-\xi\right)\left(1-\eta\right)} \\
\elemline
2 & \inelemthree{1}{-1}{-1} & $\eighth\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta\right)$
& \inelemthree{ \eighth\left(1-\eta\right)\left(1-\zeta\right)}
{-\eighth\left(1+\xi\right)\left(1-\zeta\right)}
{-\eighth\left(1+\xi\right)\left(1-\eta\right)} \\
\elemline
3 & \inelemthree{1}{1}{-1} & $\eighth\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta\right)$
& \inelemthree{ \eighth\left(1+\eta\right)\left(1-\zeta\right)}
{ \eighth\left(1+\xi\right)\left(1-\zeta\right)}
{-\eighth\left(1+\xi\right)\left(1+\eta\right)} \\
\elemline
4 & \inelemthree{-1}{1}{-1} & $\eighth\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta\right)$
& \inelemthree{-\eighth\left(1+\eta\right)\left(1-\zeta\right)}
{ \eighth\left(1-\xi\right)\left(1-\zeta\right)}
{-\eighth\left(1-\xi\right)\left(1+\eta\right)} \\
\elemline
5 & \inelemthree{-1}{-1}{1} & $\eighth\left(1-\xi\right)\left(1-\eta\right)\left(1+\zeta\right)$
& \inelemthree{-\eighth\left(1-\eta\right)\left(1+\zeta\right)}
{-\eighth\left(1-\xi\right)\left(1+\zeta\right)}
{ \eighth\left(1-\xi\right)\left(1-\eta\right)} \\
\elemline
6 & \inelemthree{1}{-1}{1} & $\eighth\left(1+\xi\right)\left(1-\eta\right)\left(1+\zeta\right)$
& \inelemthree{ \eighth\left(1-\eta\right)\left(1+\zeta\right)}
{-\eighth\left(1+\xi\right)\left(1+\zeta\right)}
{ \eighth\left(1+\xi\right)\left(1-\eta\right)} \\
\elemline
7 & \inelemthree{1}{1}{1} & $\eighth\left(1+\xi\right)\left(1+\eta\right)\left(1+\zeta\right)$
& \inelemthree{ \eighth\left(1+\eta\right)\left(1+\zeta\right)}
{ \eighth\left(1+\xi\right)\left(1+\zeta\right)}
{ \eighth\left(1+\xi\right)\left(1+\eta\right)} \\
\elemline
8 & \inelemthree{-1}{1}{1} & $\eighth\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)$
& \inelemthree{-\eighth\left(1+\eta\right)\left(1+\zeta\right)}
{ \eighth\left(1-\xi\right)\left(1+\zeta\right)}
{ \eighth\left(1-\xi\right)\left(1+\eta\right)} \\
\end{Element}
\begin{QuadPoints}
\begin{tabular}{l|cccc}
\elemcoortwod & \inquadthree{-\invsqrtthree}{-\invsqrtthree}{-\invsqrtthree} & \inquadthree{\invsqrtthree}{-\invsqrtthree}{-\invsqrtthree}
& \inquadthree{\invsqrtthree}{\invsqrtthree}{-\invsqrtthree} & \inquadthree{-\invsqrtthree}{\invsqrtthree}{-\invsqrtthree} \\
\elemline
Weight & 1 & 1 & 1 & 1 \\
\elemline
\elemcoortwod & \inquadthree{-\invsqrtthree}{-\invsqrtthree}{\invsqrtthree} & \inquadthree{\invsqrtthree}{-\invsqrtthree}{\invsqrtthree}
& \inquadthree{\invsqrtthree}{\invsqrtthree}{\invsqrtthree} & \inquadthree{-\invsqrtthree}{\invsqrtthree}{\invsqrtthree} \\
\elemline
Weight & 1 & 1 & 1 & 1 \\
\end{tabular}
\end{QuadPoints}
diff --git a/doc/manual/manual-elements.tex b/doc/manual/manual-elements.tex
index 27397ea1a..6e3c268ec 100644
--- a/doc/manual/manual-elements.tex
+++ b/doc/manual/manual-elements.tex
@@ -1,314 +1,125 @@
\section{Elements\index{Elements}}
-The base for every finite element computation is its mesh and the elements that are used within that mesh. What kind of element types can be used depends on the mesh, but also on the dimensionality of the problem (1D, 2D or 3D). In \akantu several isoparametric Langrangian element types are supported. Each of these types is discussed in more detail below, starting with the 1D-elements all the way to the 3D-elements. For each element type the coordinates of the nodes are given in the isoparametric frame of reference, together with the shape functions (and their derivatives) on these respective nodes. Also all the quadrature points within each element are listed (together with the weight that is applied on these points).
+The base for every finite element computation is its mesh and the elements that are used within that mesh. What kind of element types can be used depends on the mesh, but also on the dimensionality of the problem (1D, 2D or 3D). In \akantu several isoparametric Langrangian element types are supported. 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 %%%%%%%%%
\subsection{Isoparametric Elements in 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}.
+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}.
-\subsubsection{Segment 2\index{Elements!1D!Segment 2}}
-
-\begin{Element}{1D}
- 1 & \inelemone{-1} & $\half\left(1-\xi\right)$ & \inelemone{-\half} \\
-\elemline
- 2 & \inelemone{1} & $\half\left(1+\xi\right)$ & \inelemone{\half} \\
-\end{Element}
-
-\begin{QuadPoints}
-\begin{tabular}{l|c}
-Coord. \elemcooroned & 0 \\
-\elemline
-Weight & 2 \\
+\begin{figure}[!htb]
+\begin{center}
+\begin{tabular}{m{0.3\textwidth}m{0.1\textwidth}m{0.3\textwidth}}
+\subfloat[\code{segment\_2}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/segment_2}
+ \label{fig:elements:segment2}
+} & &
+\subfloat[\code{segment\_3}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/segment_3}
+ \label{fig:elements:segment3}
+}
\end{tabular}
-\end{QuadPoints}
-
-\subsubsection{Segment 3\index{Elements!1D!Segment 3}}
-
-\begin{Element}{1D}
- 1 & \inelemone{-1} & $\half\xi\left(\xi-1\right)$ & \inelemone{\xi-\half} \\
-\elemline
- 2 & \inelemone{1} & $\half\xi\left(\xi+1\right)$ & \inelemone{\xi+\half} \\
-\elemline
- 3 & \inelemone{0} & $1-\xi^{2}$ & \inelemone{-2\xi} \\
-\end{Element}
+\end{center}
+\caption{A schematic overview of the three supported 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{QuadPoints}
-\begin{tabular}{l|cc}
-Coord. \elemcooroned & $-1/\sqrt{3}$ & $1/\sqrt{3}$ \\
-\elemline
-Weight & 1 & 1 \\
+\begin{table}[!htb]
+\begin{center}
+\begin{tabular}{l||c|c|c}
+Element type & Order & \# nodes & \# quad. points \\
+\hline
+\code{segment\_2} & linear & 2 & 1 \\
+\code{segment\_3} & quadratic & 3 & 2 \\
\end{tabular}
-\end{QuadPoints}
+\end{center}
+\caption{Some basic properties of the two 1D isoparametric elements in \akantu.}
+\label{tab:elements:1D}
+\end{table}
%%%%%%%%%% 2D %%%%%%%%%
\subsection{Isoparametric Elements in 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}.
-
-\subsubsection{Triangle 3\index{Elements!2D!Triangle 3}}
+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}.
-\begin{Element}{2D}
- 1 & \inelemtwo{0}{0} & $1-\xi-\eta$ & \inelemtwo{-1}{-1} \\
-\elemline
- 2 & \inelemtwo{1}{0} & $\xi$ & \inelemtwo{1}{0} \\
-\elemline
- 3 & \inelemtwo{0}{1} & $\eta$ & \inelemtwo{0}{1} \\
-\end{Element}
-
-\begin{QuadPoints}
-\begin{tabular}{l|c}
-Coord. \elemcoortwod & \inquadtwo{\third}{\third} \\
-\elemline
-Weight & 1 \\
-\end{tabular}
-\end{QuadPoints}
-
-\subsubsection{Triangle 6\index{Elements!2D!Triangle 6}}
-
-\begin{Element}{2D}
- 1 & \inelemtwo{0}{0} & $-\left(1-\xi-\eta\right)\left(1-2\left(1-\xi-\eta\right)\right)$ & \inelemtwo{1-4\left(1-\xi-\eta\right)}{1-4\left(1-\xi-\eta\right)} \\
-\elemline
- 2 & \inelemtwo{1}{0} & $-\xi\left(1-2\xi\right)$ & \inelemtwo{4\xi-1}{0} \\
-\elemline
- 3 & \inelemtwo{0}{1} & $-\eta\left(1-2\eta\right)$ & \inelemtwo{0}{4\eta-1} \\
-\elemline
- 4 & \inelemtwo{\half}{0} & $4\xi\left(1-\xi-\eta\right)$ & \inelemtwo{4\left(1-2\xi-\eta\right)}{-4\xi} \\
-\elemline
- 5 & \inelemtwo{\half}{\half} & $4\xi\eta$ & \inelemtwo{4\eta}{4\xi} \\
-\elemline
- 6 & \inelemtwo{0}{\half} & $4\eta\left(1-\xi-\eta\right)$ & \inelemtwo{-4\eta}{4\left(1-\xi-2\eta\right)} \\
-\elemline
-\end{Element}
-
-\begin{QuadPoints}
-\begin{tabular}{l|ccc}
-Coord. \elemcoortwod & \inquadtwo{\sixth}{\sixth} & \inquadtwo{\twothird}{\sixth} & \inquadtwo{\sixth}{\twothird} \\
-\elemline
-Weight & \sixth & \sixth & \sixth \\
-\end{tabular}
-\end{QuadPoints}
-
-\subsubsection{Quadrangle 4\index{Elements!2D!Quadrangle 4}}
-
-\begin{Element}{2D}
- 1 & \inelemtwo{-1}{-1} & $\quart\left(1-\xi\right)\left(1-\eta\right)$ & \inelemtwo{-\quart\left(1-\eta\right)}{-\quart\left(1-\xi\right)} \\
-\elemline
- 2 & \inelemtwo{1}{-1} & $\quart\left(1+\xi\right)\left(1-\eta\right)$ & \inelemtwo{\quart\left(1-\eta\right)}{-\quart\left(1-\xi\right)} \\
-\elemline
- 3 & \inelemtwo{1}{1} & $\quart\left(1+\xi\right)\left(1+\eta\right)$ & \inelemtwo{\quart\left(1-\eta\right)}{\quart\left(1-\xi\right)} \\
-\elemline
- 4 & \inelemtwo{-1}{1} & $\quart\left(1-\xi\right)\left(1+\eta\right)$ & \inelemtwo{-\quart\left(1-\eta\right)}{\quart\left(1-\xi\right)} \\
-\end{Element}
-
-\begin{QuadPoints}
-\begin{tabular}{l|cccc}
-\elemcoortwod & \inquadtwo{-\invsqrtthree}{-\invsqrtthree} & \inquadtwo{\invsqrtthree}{-\invsqrtthree}
- & \inquadtwo{\invsqrtthree}{\invsqrtthree} & \inquadtwo{-\invsqrtthree}{\invsqrtthree} \\
-\elemline
-Weight & 1 & 1 & 1 & 1 \\
+\begin{figure}[!htb]
+\begin{center}
+\begin{tabular}{m{0.3\textwidth}m{0.1\textwidth}m{0.3\textwidth}}
+\subfloat[\code{triangle\_3}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/triangle_3}
+ \label{fig:elements:triangle3}
+} & &
+\subfloat[\code{triangle\_6}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/triangle_6}
+ \label{fig:elements:triangle6}
+} \\
+\subfloat[\code{quadrangle\_4}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/quadrangle_4}
+ \label{fig:elements:quadrangle4}
+} & &
+\subfloat[\code{quadrangle\_8}]{
+ \includegraphics[width=0.3\textwidth]{figures/elements/quadrangle_8}
+ \label{fig:elements:quadrangle8}
+}
\end{tabular}
-\end{QuadPoints}
-
-\subsubsection{Quadrangle 8\index{Elements!2D!Quadrangle 8}}
-
-\begin{Element}{2D}
- 1 & \inelemtwo{-1}{-1} & $\quart\left(1-\xi\right)\left(1-\eta\right)\left(-1-\xi-\eta\right)$
- & \inelemtwo{\quart\left(1-\eta\right)\left(2\xi+\eta\right)}
- {\quart\left(1-\xi\right)\left(\xi+2\eta\right)} \\
-\elemline
- 2 & \inelemtwo{1}{-1} & $\quart\left(1+\xi\right)\left(1-\eta\right)\left(-1+\xi-\eta\right)$
- & \inelemtwo{\quart\left(1-\eta\right)\left(2\xi-\eta\right)}
- {-\quart\left(1+\xi\right)\left(\xi-2\eta\right)} \\
-\elemline
- 3 & \inelemtwo{1}{1} & $\quart\left(1+\xi\right)\left(1+\eta\right)\left(-1+\xi+\eta\right)$
- & \inelemtwo{\quart\left(1+\eta\right)\left(2\xi+\eta\right)}
- {\quart\left(1+\xi\right)\left(\xi+2\eta\right)} \\
-\elemline
- 4 & \inelemtwo{-1}{1} & $\quart\left(1-\xi\right)\left(1+\eta\right)\left(-1-\xi+\eta\right)$
- & \inelemtwo{\quart\left(1+\eta\right)\left(2\xi-\eta\right)}
- {-\quart\left(1-\xi\right)\left(\xi-2\eta\right)} \\
-\elemline
- 5 & \inelemtwo{0}{-1} & $\half\left(1-\xi^{2}\right)\left(1-\eta\right)$
- & \inelemtwo{-\xi\left(1-\eta\right)}
- {-\half\left(1-\xi^{2}\right)} \\
-\elemline
- 6 & \inelemtwo{1}{0} & $\half\left(1+\xi\right)\left(1-\eta^{2}\right)$
- & \inelemtwo{\half\left(1-\eta^{2}\right)}
- {-\eta\left(1+\xi\right)} \\
-\elemline
- 7 & \inelemtwo{0}{1} & $\half\left(1-\xi^{2}\right)\left(1+\eta\right)$
- & \inelemtwo{-\xi\left(1+\eta\right)}
- {\half\left(1-\xi^{2}\right)} \\
-\elemline
- 8 & \inelemtwo{-1}{0} & $\half\left(1-\xi\right)\left(1-\eta^{2}\right)$
- & \inelemtwo{-\half\left(1-\eta^{2}\right)}
- {-\eta\left(1-\xi\right)} \\
-\end{Element}
+\end{center}
+\caption{A schematic overview of the four supported 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{QuadPoints}
-\begin{tabular}{l|ccccc}
-Coord. \elemcoortwod & \inquadtwo{0}{0} & \inquadtwo{\sqrt{\tfrac{3}{5}}}{\sqrt{\tfrac{3}{5}}} & \inquadtwo{-\sqrt{\tfrac{3}{5}}}{\sqrt{\tfrac{3}{5}}}
- & \inquadtwo{-\sqrt{\tfrac{3}{5}}}{-\sqrt{\tfrac{3}{5}}} & \inquadtwo{\sqrt{\tfrac{3}{5}}}{-\sqrt{\tfrac{3}{5}}} \\
-\elemline
-Weight & 64/81 & 25/81 & 25/81 & 25/81 & 25/81 \\
-\elemline
-Coord. \elemcoortwod & \inquadtwo{0}{\sqrt{\tfrac{3}{5}}} & \inquadtwo{-\sqrt{\tfrac{3}{5}}}{0}
- & \inquadtwo{0}{-\sqrt{\tfrac{3}{5}}} & \inquadtwo{\sqrt{\tfrac{3}{5}}}{0} & \\
-\elemline
-Weight & 40/81 & 40/81 & 40/81 & 40/81 & \\
+\begin{table}[!htb]
+\begin{center}
+\begin{tabular}{l||c|c|c}
+Element type & Order & \# nodes & \# quad. points \\
+\hline
+\code{triangle\_3} & linear & 3 & 1 \\
+\code{triangle\_6} & quadratic & 6 & 3 \\
+\hline
+\code{quadrangle\_4} & quadratic & 4 & 4 \\
+\code{quadrangle\_8} & cubic & 8 & 9 \\
\end{tabular}
-\end{QuadPoints}
+\end{center}
+\caption{Some basic properties of the four 2D isoparametric elements in \akantu.}
+\label{tab:elements:2D}
+\end{table}
%%%%%%%%%% 3D %%%%%%%%%
\subsection{Isoparametric Elements in 3D\index{Elements!3D}}
-\begin{figure}
+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\textwidth}m{0.3\textwidth}m{0.3\textwidth}}
-\subfloat[Tetrahedron-4]{
+\subfloat[\code{tetrahedron\_4}]{
\includegraphics[width=0.3\textwidth]{figures/elements/tetrahedron_4}
\label{fig:elements:tetrahedron4}
} &
-\subfloat[Tetrahedron-10]{
+\subfloat[\code{tetrahedron\_10}]{
\includegraphics[width=0.3\textwidth]{figures/elements/tetrahedron_10}
\label{fig:elements:tetrahedron10}
} &
-\subfloat[Hexahedron-8]{
+\subfloat[\code{hexahedron\_8}]{
\includegraphics[width=0.3\textwidth]{figures/elements/hexahedron_8}
\label{fig:elements:hexahedron8}
}
\end{tabular}
-\caption{Schematic depiction of the three 3D isoparametric elements available in \akantu. In each element the node numbering used in \akantu is indicated and also the quadrature points are highlighted (gray spheres)}
+\caption{A schematic overview of the three supported 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}
-
-\subsubsection{Tetrahedron 4\index{Elements!3D!Tetrahedron 4}}
-
-\begin{Element}{3D}
- 1 & \inelemthree{0}{0}{0} & $1-\xi-\eta-\zeta$ & \inelemthree{-1}{-1}{-1} \\
-\elemline
- 2 & \inelemthree{1}{0}{0} & $\xi$ & \inelemthree{1}{0}{0} \\
-\elemline
- 3 & \inelemthree{0}{1}{0} & $\eta$ & \inelemthree{0}{1}{0} \\
-\elemline
- 4 & \inelemthree{0}{0}{1} & $\zeta$ & \inelemthree{0}{0}{1} \\
-\end{Element}
-
-\begin{QuadPoints}
-\begin{tabular}{l|c}
-Coord. \elemcoorthreed & \inquadthree{\quart}{\quart}{\quart} \\
-\elemline
-Weight & \sixth \\
-\end{tabular}
-\end{QuadPoints}
-
-\subsubsection{Tetrahedron 10\index{Elements!3D!Tetrahedron 10}}
-
-\begin{Element}{3D}
- 1 & \inelemthree{0}{0}{0} & $\left(1-\xi-\eta-\zeta\right)\left(1-2\xi-2\eta-2\zeta\right)$
- & \inelemthree{4\xi+4\eta+4\zeta-3}{4\xi+4\eta+4\zeta-3}{4\xi+4\eta+4\zeta-3} \\
-\elemline
- 2 & \inelemthree{1}{0}{0} & $\xi\left(2\xi-1\right)$
- & \inelemthree{4\xi-1}{0}{0} \\
-\elemline
- 3 & \inelemthree{0}{1}{0} & $\eta\left(2\eta-1\right)$
- & \inelemthree{0}{4\eta-1}{0} \\
-\elemline
- 4 & \inelemthree{0}{0}{1} & $\zeta\left(2\zeta-1\right)$
- & \inelemthree{0}{0}{4\zeta-1} \\
-\elemline
- 5 & \inelemthree{\half}{0}{0} & $4\xi\left(1-\xi-\eta-\zeta\right)$
- & \inelemthree{4-8\xi-4\eta-4\zeta}{-4\xi}{-4\xi} \\
-\elemline
- 6 & \inelemthree{\half}{\half}{0} & $4\xi\eta$
- & \inelemthree{4\eta}{4\xi}{0} \\
-\elemline
- 7 & \inelemthree{0}{\half}{0} & $4\eta\left(1-\xi-\eta-\zeta\right)$
- & \inelemthree{-4\eta}{4-4\xi-8\eta-4\zeta}{-4\eta} \\
-\elemline
- 8 & \inelemthree{0}{0}{\half} & $4\zeta\left(1-\xi-\eta-\zeta\right)$
- & \inelemthree{-4\zeta}{-4\zeta}{4-4\xi-4\eta-8\zeta} \\
-\elemline
- 9 & \inelemthree{\half}{0}{\half} & $4\xi\zeta$
- & \inelemthree{4\zeta}{0}{4\xi} \\
-\elemline
- 10 & \inelemthree{0}{\half}{\half} & $4\eta\zeta$
- & \inelemthree{0}{4\zeta}{4\eta} \\
-\end{Element}
-
-\begin{QuadPoints}
-%\newcommand\quada{\tfrac{1}{20}\left(5-\sqrt{5}\right)}
-%\newcommand\quadb{\tfrac{1}{20}\left(5+3\sqrt{5}\right)}
-\newcommand\quada{\tfrac{\left(5-\sqrt{5}\right)}{20}}
-\newcommand\quadb{\tfrac{\left(5+3\sqrt{5}\right)}{20}}
-\begin{tabular}{l|cc}
-Coord. \elemcoorthreed & \inquadthree{\quada}{\quada}{\quada} & \inquadthree{\quadb}{\quada}{\quada} \\
-\elemline
-Weight & 1/24 & 1/24 \\
-\elemline
-Coord. \elemcoorthreed & \inquadthree{\quada}{\quadb}{\quada} & \inquadthree{\quada}{\quada}{\quadb} \\
-\elemline
-Weight & 1/24 & 1/24 \\
-\end{tabular}
-\end{QuadPoints}
-
-\subsubsection{Hexahedron 8\index{Elements!3D!Hexahedron 8}}
-
-\begin{Element}{3D}
- 1 & \inelemthree{-1}{-1}{-1} & $\eighth\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta\right)$
- & \inelemthree{-\eighth\left(1-\eta\right)\left(1-\zeta\right)}
- {-\eighth\left(1-\xi\right)\left(1-\zeta\right)}
- {-\eighth\left(1-\xi\right)\left(1-\eta\right)} \\
-\elemline
- 2 & \inelemthree{1}{-1}{-1} & $\eighth\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta\right)$
- & \inelemthree{ \eighth\left(1-\eta\right)\left(1-\zeta\right)}
- {-\eighth\left(1+\xi\right)\left(1-\zeta\right)}
- {-\eighth\left(1+\xi\right)\left(1-\eta\right)} \\
-\elemline
- 3 & \inelemthree{1}{1}{-1} & $\eighth\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta\right)$
- & \inelemthree{ \eighth\left(1+\eta\right)\left(1-\zeta\right)}
- { \eighth\left(1+\xi\right)\left(1-\zeta\right)}
- {-\eighth\left(1+\xi\right)\left(1+\eta\right)} \\
-\elemline
- 4 & \inelemthree{-1}{1}{-1} & $\eighth\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta\right)$
- & \inelemthree{-\eighth\left(1+\eta\right)\left(1-\zeta\right)}
- { \eighth\left(1-\xi\right)\left(1-\zeta\right)}
- {-\eighth\left(1-\xi\right)\left(1+\eta\right)} \\
-\elemline
- 5 & \inelemthree{-1}{-1}{1} & $\eighth\left(1-\xi\right)\left(1-\eta\right)\left(1+\zeta\right)$
- & \inelemthree{-\eighth\left(1-\eta\right)\left(1+\zeta\right)}
- {-\eighth\left(1-\xi\right)\left(1+\zeta\right)}
- { \eighth\left(1-\xi\right)\left(1-\eta\right)} \\
-\elemline
- 6 & \inelemthree{1}{-1}{1} & $\eighth\left(1+\xi\right)\left(1-\eta\right)\left(1+\zeta\right)$
- & \inelemthree{ \eighth\left(1-\eta\right)\left(1+\zeta\right)}
- {-\eighth\left(1+\xi\right)\left(1+\zeta\right)}
- { \eighth\left(1+\xi\right)\left(1-\eta\right)} \\
-\elemline
- 7 & \inelemthree{1}{1}{1} & $\eighth\left(1+\xi\right)\left(1+\eta\right)\left(1+\zeta\right)$
- & \inelemthree{ \eighth\left(1+\eta\right)\left(1+\zeta\right)}
- { \eighth\left(1+\xi\right)\left(1+\zeta\right)}
- { \eighth\left(1+\xi\right)\left(1+\eta\right)} \\
-\elemline
- 8 & \inelemthree{-1}{1}{1} & $\eighth\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)$
- & \inelemthree{-\eighth\left(1+\eta\right)\left(1+\zeta\right)}
- { \eighth\left(1-\xi\right)\left(1+\zeta\right)}
- { \eighth\left(1-\xi\right)\left(1+\eta\right)} \\
-\end{Element}
-
-\begin{QuadPoints}
-\begin{tabular}{l|cccc}
-\elemcoortwod & \inquadthree{-\invsqrtthree}{-\invsqrtthree}{-\invsqrtthree} & \inquadthree{\invsqrtthree}{-\invsqrtthree}{-\invsqrtthree}
- & \inquadthree{\invsqrtthree}{\invsqrtthree}{-\invsqrtthree} & \inquadthree{-\invsqrtthree}{\invsqrtthree}{-\invsqrtthree} \\
-\elemline
-Weight & 1 & 1 & 1 & 1 \\
-\elemline
-\elemcoortwod & \inquadthree{-\invsqrtthree}{-\invsqrtthree}{\invsqrtthree} & \inquadthree{\invsqrtthree}{-\invsqrtthree}{\invsqrtthree}
- & \inquadthree{\invsqrtthree}{\invsqrtthree}{\invsqrtthree} & \inquadthree{-\invsqrtthree}{\invsqrtthree}{\invsqrtthree} \\
-\elemline
-Weight & 1 & 1 & 1 & 1 \\
+\begin{table}[!htb]
+\begin{center}
+\begin{tabular}{l||c|c|c}
+Element type & Order & \# nodes & \# quad. points \\
+\hline
+\code{tetrahedron\_4} & linear & 4 & 1 \\
+\code{tetrahedron\_10} & quadratic & 10 & 4 \\
+\hline
+\code{hexahedron\_8} & cubic & 8 & 8 \\
\end{tabular}
-\end{QuadPoints}
-
+\end{center}
+\caption{Some basic properties of the three 3D isoparametric elements in \akantu.}
+\label{tab:elements:3D}
+\end{table}
diff --git a/doc/manual/manual.sty b/doc/manual/manual.sty
index 6af4ce730..3aa81b547 100644
--- a/doc/manual/manual.sty
+++ b/doc/manual/manual.sty
@@ -1,265 +1,275 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
%% LaTeX STYLE SHEET for AKANTU DOCUMENTATION %%
%% %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Geometry
\usepackage{a4wide}
\usepackage{geometry}
\geometry{
pdftex=true,
twoside=true,
margin=20mm,
bottom=20mm,
top=20mm,
bindingoffset=5.5mm
}
% Font encoding
\usepackage[T1]{fontenc}
\usepackage{palatino}
% Line spacing
\linespread{1.05}\selectfont
% Allow spaces to be added at the end of macro
\usepackage{xspace}
% Mathematics (including correct font encoding, after amsmath)
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{pxfonts}
\setlength\mathindent{2em}
% Some unit stuff
\usepackage[squaren]{SIunits}
% Easy tables
\usepackage{booktabs}
\usepackage{longtable}
\usepackage{rotating} % For sideways tables
\usepackage{multirow} % For larger cells
% A special environment for element tables
\newcommand\elemline{\hline}%{\noalign{\smallskip}\hline\noalign{\smallskip}}
\newcommand\elemcooroned{\ensuremath{\left(\xi\right)}}
-\newcommand\elemcoortwod{\ensuremath{\left(\xi,\eta\right)}}
-\newcommand\elemcoorthreed{\ensuremath{\left(\xi,\eta,\zeta\right)}}
+\newcommand\elemcoortwod{\ensuremath{\left(\xi\;,\;\eta\right)}}
+\newcommand\elemcoorthreed{\ensuremath{\left(\xi\;,\;\eta\;,\;\zeta\right)}}
\newcommand\elemdshapeoned{\ensuremath{\left(\partial N_i/\partial\xi\right)}}
-\newcommand\elemdshapetwod{\ensuremath{\left(\partial N_i/\partial\xi,\partial N_i/\partial\eta\right)}}
-\newcommand\elemdshapethreed{\ensuremath{\left(\partial N_i/\partial\xi,\partial N_i/\partial\eta,\partial N_i/\partial\zeta\right)}}
+\newcommand\elemdshapetwod{\ensuremath{\left(\partial N_i/\partial\xi\;,\;\partial N_i/\partial\eta\right)}}
+\newcommand\elemdshapethreed{\ensuremath{\left(\partial N_i/\partial\xi\;,\;\partial N_i/\partial\eta\;,\;\partial N_i/\partial\zeta\right)}}
\newcommand\inelemone[1]{\ensuremath{#1}}
-\newcommand\inelemtwo[2]{\ensuremath{\begin{pmatrix} {\; #1} & \!,\! & {#2 \;} \end{pmatrix}}}
-\newcommand\inelemthree[3]{\ensuremath{\begin{pmatrix} {\; #1} & \!,\! & {#2} & \!,\! & {#3 \;} \end{pmatrix}}}
+%\newcommand\inelemtwo[2]{\ensuremath{\begin{pmatrix} {\; #1} & \!,\! & {#2 \;} \end{pmatrix}}}
+%\newcommand\inelemthree[3]{\ensuremath{\begin{pmatrix} {\; #1} & \!,\! & {#2} & \!,\! & {#3 \;} \end{pmatrix}}}
+\newcommand\inelemtwo[2]{\ensuremath{\left( {\; #1} \; , \; {#2 \;} \right)}}
+\newcommand\inelemthree[3]{\ensuremath{\left( {\; #1} \; , \; {#2} \; , \; {#3 \;} \right)}}
\newcommand\inquadone[1]{\ensuremath{#1}}
\newcommand\inquadtwo[2]{\ensuremath{\left(\, #1 \, , \, #2 \,\right)}}
\newcommand\inquadthree[3]{\ensuremath{\left(\, #1 \, , \, #2 \, , \, #3 \,\right)}}
+%\newcommand\quada{\tfrac{1}{20}\left(5-\sqrt{5}\right)}
+%\newcommand\quadb{\tfrac{1}{20}\left(5+3\sqrt{5}\right)}
+\newcommand\quada{\tfrac{\left(5-\sqrt{5}\right)}{20}}
+\newcommand\quadb{\tfrac{\left(5+3\sqrt{5}\right)}{20}}
\newenvironment{Element}[1]
- {\begin{table*}[!htbp]\footnotesize\begin{center}
+ {\begin{table*}[!htbp]\footnotesize
\ifthenelse{\equal{#1}{1D}}{\renewcommand{\arraystretch}{1.50}}{}
\ifthenelse{\equal{#1}{2D}}{\renewcommand{\arraystretch}{1.60}}{}
\ifthenelse{\equal{#1}{3D}}{\renewcommand{\arraystretch}{1.70}}{}
+ \textbf{{\normalsize Element properties}}\newline\vspace*{0.5\baselineskip}
\begin{tabular}{c|c|c|c}
+ %\multicolumn{4}{l}{\textbf{{\normalsize Element properties}}} \\
Node ($i$) & Coord.
\ifthenelse{\equal{#1}{1D}}{\elemcooroned}{}
\ifthenelse{\equal{#1}{2D}}{\elemcoortwod}{}
\ifthenelse{\equal{#1}{3D}}{\elemcoorthreed}{}
& Shape function ($N_{i}$)
& {Derivative
\ifthenelse{\equal{#1}{1D}}{\elemdshapeoned}{}
\ifthenelse{\equal{#1}{2D}}{\elemdshapetwod}{}
\ifthenelse{\equal{#1}{3D}}{\elemdshapethreed}{}
}\\
\elemline
}
- {\end{tabular}\end{center}\end{table*}}
-\newenvironment{QuadPoints}
- {\begin{table*}[!htbp]\footnotesize\begin{center}
+ {\end{tabular}\end{table*}}
+\newenvironment{QuadPoints}%[1]
+ {\vspace*{-\baselineskip}
+ \begin{table*}[!htbp]\footnotesize
\renewcommand{\arraystretch}{1.50}
+ \textbf{{\normalsize Gaussian quadrature points}}\newline\vspace*{0.5\baselineskip}
}
- {\end{center}\end{table*}}
+ {\end{table*}}
% Nice coloring
\usepackage[dvipsnames,usenames,table]{xcolor}
\definecolor{RED}{rgb}{1,0,0}
\definecolor{cppbg}{HTML}{EBF2F2}
\definecolor{shellbg}{HTML}{F5EDE4}
\definecolor{commentcolor}{HTML}{101280}
% Allow for the use of listings
\usepackage{listings}
% Create an index
\usepackage{makeidx}
% Figure handling
\usepackage{graphics}
\usepackage{epsfig}
\usepackage[lofdepth,lotdepth]{subfig}
\usepackage{tikz}
\usetikzlibrary{decorations}
\renewcommand{\floatpagefraction}{.6} % default: .5
\renewcommand\topfraction{0.9} % 90% of page top can be a float (Standard 0.7)
\renewcommand\bottomfraction{0.1} % 10% of page bottom can be a float (Standard 0.3)
\renewcommand\textfraction{0.1} % only 10% of page must to be text (Standard 0.2)
% Removes parenthese around subfig number
\renewcommand*{\thesubfigure}{\alph{subfigure}}
% Create a new list style for C++
\lstdefinestyle{C++}{
language=C++, % the language of the code
basicstyle=\small\ttfamily, % Without beramono, we'd get cmtt, the teletype font.
commentstyle=\color{commentcolor}\itshape,
keywordstyle=\color{DarkOrchid}\bfseries,
% fontadjust,
% numbers=left, % where to put the line-numbers
% numberstyle=\tiny, % the size of the fonts that are used for the line-numbers
% stepnumber=2, % the step between two line-numbers. If it's 1, each line will
% be numbered
% numbersep=5pt, % how far the line-numbers are from the code
% showspaces=false, % show spaces adding particular underscores
showstringspaces=false, % underline spaces within strings
% showtabs=false, % show tabs within strings adding particular underscores
% frame=llines, % adds a frame around the code
% frame=tb,
tabsize=2, % sets default tabsize to 2 spaces
captionpos=b, % sets the caption-position to bottom
breaklines=true, % sets automatic line breaking
breakatwhitespace=false, % sets if automatic breaks should only happen at
% whitespace
% title=\lstname, % show the filename of files included with \lstinputlisting;
% also try caption instead of title
% escapeinside={\%*}{*)}, % if you want to add a comment within your code
xleftmargin=1cm,
xrightmargin=1cm,
mathescape=true,
escapechar=\%,
morekeywords={Real, UInt, Int},
columns=flexible,
keepspaces=true,
backgroundcolor=\color{cppbg}
}
% Create new list style for the shell
\lstdefinestyle{shell}{
language=bash, % the language of the code
basicstyle=\scriptsize\ttfamily, % Without beramono, we'd get cmtt, the teletype font.
showstringspaces=false, % underline spaces within strings
tabsize=2, % sets default tabsize to 2 spaces
captionpos=b, % sets the caption-position to bottom
breaklines=true, % sets automatic line breaking
breakatwhitespace=false,
xleftmargin=1cm,
xrightmargin=1cm,
escapechar=\%,
morekeywords={mkdir, make, ccmake, cmake},
columns=flexible,
keepspaces=true,
backgroundcolor=\color{shellbg}
}
% Set some derived listing environments
\lstnewenvironment{cpp}{\lstset{style=C++}}{}
\lstnewenvironment{command}{\lstset{style=shell}}{}
% Make sure outputspace is white
\makeatletter
\def\lst@outputspace{{\ifx\lst@bkgcolor\empty\color{white}\else\lst@bkgcolor\fi\lst@visiblespace}}
\makeatother
% Renow a label in the itemized lists
\renewcommand{\labelitemi}{$\mathbf{\circ}$}
% Don't care so much about overfull h-boxes
\sloppy
% Penalty adjusments
%\widowpenalty=10000 % Single lines/word on beginning of page
%\clubpenalty=10000 % Single lines/word at end of page
%\hyphenpenalty=2000 % Hyphenate words
%\tolerance=250 % To adjust the hyphenation of words, increase the tolerance to discourage hyphenation
% the higher the value, the more ugly the gaps between words
% default \tolerance=200
%\hfuzz=10000pt % threshold when an overfull hbox is reported, default \hfuzz=0.1pt
%\vfuzz=10000pt % threshold to report an overfull vbox, default \vfuzz=0.1pt
%\hbadness=10000 % threshold to report an underfull \hbox
%\vbadness=10000 % threshold to report an underfull \vbox protokolliert wird.
\emergencystretch=0pt % causes a third attempt to fix bad paragraphs and defines a maximum limit to stretch them
% Insert an empty or a blank page
\newcommand{\insertemptypage}{\newpage\hbox{}\newpage}
\newcommand{\insertblankpage}{\newpage\thispagestyle{empty}\hbox{}\newpage}
% No page number on an empty page
\let\origdoublepage\cleardoublepage
\newcommand{\clearemptydoublepage}{%
\clearpage
{\thispagestyle{empty}\origdoublepage}%
}
\let\cleardoublepage\clearemptydoublepage
% A new ruler for in chapters
\newcommand\InChapterRule{\addvspace{\baselineskip}\rule{0.3\linewidth}{0.25pt}}
% New footnote style
%\def\@fnsymbol#1{\ifcase#1\or *\or \dagger\or \ddagger\or \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger \or \ddagger\ddagger \else\@ctrerr\fi\relax}
\def\@fnsymbol#1{*\xspace\relax}
\renewcommand{\thefootnote}{\fnsymbol{footnote}} % Symbols rather than numbers
\def\footnoterule{\vspace*{0.5\baselineskip}\InChapterRule\vspace*{0.25\baselineskip}}
% Improved look of the Table of Contents
\usepackage[nottoc,notbib]{tocbibind}
\usepackage[dotinlabels]{titletoc}
\titlecontents{chapter}[1.4pc]
{\addvspace{0.6pc}\large\bfseries\filright}
{\contentslabel[\thecontentslabel.]{1.4pc}}
{\hspace{-1.4pc}}
{\hfill\contentspage}
[\addvspace{2pt}]
\titlecontents{section}[3.4pc]
{\filright}
{\contentslabel[\thecontentslabel]{2pc}}
{\hspace{-2pc}}
{\titlerule*[6pt]{.}\contentspage}
[]
\titlecontents{subsection}[5.0pc]
{\filright}
{\contentslabel[\thecontentslabel]{2.4pc}}
{}
{\titlerule*[6pt]{.}\contentspage}
[]
\setcounter{tocdepth}{2}
\newcommand\addspaceintoc{\addtocontents{toc}{\protect\addvspace{20pt}}}
% Change the appearance of the bibliography
\bibliographystyle{manual-bibliographystyle}
\renewcommand\bibname{References}
\usepackage{cite} % To sort citations: [2,10-14]
\let\oldthebibliography=\thebibliography
\let\endoldthebibliography=\endthebibliography
\renewenvironment{thebibliography}[1]
{ \begin{oldthebibliography}{#1}
% \small
\addcontentsline{toc}{chapter}{\bibname}
\setlength{\labelsep}{2mm}
\setlength{\parskip}{0\baselineskip}
\setlength{\itemsep}{0.24\baselineskip}
}
{ \end{oldthebibliography} }
% Hyperref
\usepackage{url}
\usepackage[pdftex,
bookmarks=true,
bookmarksnumbered=true,
linkbordercolor={1 1 1},
pdfborder={0 0 0},
pdfpagemode=UseOutlines
]{hyperref}
\hypersetup{
pdfauthor={Computational Solid Mechanics Laboratory - EPFL},
pdftitle={Akantu User's Guide},
pdfsubject={Open Source Finite Element Code - Akantu}
}
diff --git a/doc/manual/manual.tex b/doc/manual/manual.tex
index 321b3f8dc..5de8279b5 100644
--- a/doc/manual/manual.tex
+++ b/doc/manual/manual.tex
@@ -1,29 +1,33 @@
\documentclass[openright,a4paper,11pt,fleqn]{manual}
\usepackage{manual}
\usepackage{manual-macros}
-\newcommand{\version}{0.1}
+\newcommand{\version}{0.123}
%%% For references \todo check the coherency
%% section 3.5 -> Section 3.5
%% figure 3.5 -> Figure 3.5
%% equation 3.5 -> Equation (3.5)
% Title pages and Table of Contents (includes \begin{document})
\input{manual-titlepages}
% Introduction chapter
\input{manual-introduction}
% The planning to write the documentation
\input{manual-planning}
% The documentation chapter (split in parts)
\chapter{How to use Akantu}
\input{manual-gettingstarted}
\input{manual-elements}
\input{manual-solidmechanicsmodel}
\input{manual-structuralmechanicsmodel}
\input{manual-heattransfermodel}
+% The appendices
+\appendix
+\input{manual-appendix-elements}
+
% The backmatter material (index/bibliography, included \end{document})
\input{manual-backmatter}