Homec4science
Diffusion akantu (features/30-implementing-a-checkpoint-restart-mechanism) Browse

rAKA/doc/manual/manual-appendix-elements.tex
5b6d038f664ffeatures/30-implementin…

Last Change
Raw File

manual-appendix-elements.tex
View Options

\chapter{Shape Functions\index{Elements}}
\label{app:elements}
Schmatic overview of all the element types defined in \akantu is described in Chapter~\ref{chap:elements}. In this appendix, more detailed information (shape function, location of Gaussian quadrature points, and so on) of each of these types is listed. 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 assigned (together with the weight that is applied on these points). The graphical representations of all the element types can be found in Chapter~\ref{chap:elements}.
%%%%%%%%%% 1D %%%%%%%%%
%\section{Isoparametric Elements in 1D\index{Elements!1D}}
\section{1D-Shape Functions\index{Elements!1D}}
\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}{lc}
Coord. \elemcooroned & 0 \\
\elemline
Weight & 2 \\
\end{QuadPoints}
\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}{lcc}
Coord. \elemcooroned & $-1/\sqrt{3}$ & $1/\sqrt{3}$ \\
\elemline
Weight & 1 & 1 \\
\end{QuadPoints}
%%%%%%%%%% 2D %%%%%%%%%
%\section{Isoparametric Elements in 2D\index{Elements!2D}}
\section{2D-Shape Functions\index{Elements!2D}}
\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}{lc}
Coord. \elemcoortwod & \inquadtwo{\third}{\third} \\
\elemline
Weight & 1 \\
\end{QuadPoints}
\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}{lccc}
Coord. \elemcoortwod & \inquadtwo{\sixth}{\sixth} & \inquadtwo{\twothird}{\sixth} & \inquadtwo{\sixth}{\twothird} \\
\elemline
Weight & \sixth & \sixth & \sixth \\
\end{QuadPoints}
\clearpage
\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}{lcccc}
\elemcoortwod & \inquadtwo{-\invsqrtthree}{-\invsqrtthree} & \inquadtwo{\invsqrtthree}{-\invsqrtthree}
& \inquadtwo{\invsqrtthree}{\invsqrtthree} & \inquadtwo{-\invsqrtthree}{\invsqrtthree} \\
\elemline
Weight & 1 & 1 & 1 & 1 \\
\end{QuadPoints}
\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}{lccccc}
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{QuadPoints}
\clearpage
%%%%%%%%%% 3D %%%%%%%%%
\section{3D-Shape Functions\index{Elements!3D}}
%\section{Isoparametric Elements in 3D\index{Elements!3D}}
\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}{lc}
Coord. \elemcoorthreed & \inquadthree{\quart}{\quart}{\quart} \\
\elemline
Weight & \sixth \\
\end{QuadPoints}
\subsection{Tetrahedron 10\index{Elements!3D!Tetrahedron 10}}
\begin{Element}{3D}
1 & \inelemthree{0}{0}{0} & $\left(1-\xi-\eta-\zeta\right)\left(1-2\xi-2\eta-2\zeta\right)$
& \inelemthreecolumn{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}{lcc}
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{QuadPoints}
\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}{lcccc}
\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{QuadPoints}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "manual"
%%% End:
c4science · Help