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Basis.Table.tex
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Basis.Table.tex
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\begin{sidewaystable}%[H]
\begin{center}
{\small\begin{tabular}{|p{2in}|p{2in}|p{2in}|p{2in}|}
\hline
$i,j,k=1,2,3$ &
$i=4,5,6$ and $j,k=1,2,3$ &
$i=7,8,9$ and $j,k=1,2,3$ &
$i=10,11,12$ and $j,k=1,2,3$ \\
\hline
\begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = \delta_{i,j} \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = 0 \\
\dfrac{\partial\varphi_i}{\partial n_k} = 0
\end{split}
\end{equation*} &
\begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = \delta_{i,j+3} \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = 0 \\
\dfrac{\partial\varphi_i}{\partial n_k} = 0
\end{split}
\end{equation*} &
\begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = \delta_{i,j+6} \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = 0 \\
\dfrac{\partial\varphi_i}{\partial n_k} = 0
\end{split}
\end{equation*} &
\begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = \delta_{i,j+9} \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = 0 \\
\dfrac{\partial\varphi_i}{\partial n_k} = 0
\end{split}
\end{equation*} \\
\hline
\hline
$i=13,14,15$ and $j,k=1,2,3$ &
$i=16,17,18$ and $j,k=1,2,3$ &
$i=19,20,21$ and $k=1,2,3$ & \\ \hline
\begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = \delta_{i,j+12} \\
\dfrac{\partial\varphi_i}{\partial n_k} = 0
\end{split}
\end{equation*} &
\begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = \delta_{i,j+15} \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = 0 \\
\dfrac{\partial\varphi_i}{\partial n_k} = 0
\end{split}
\end{equation*}
& \begin{equation*}
\begin{split}
\varphi_i (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial x} (x_j,y_j) = 0 \\
\dfrac{\partial \varphi_i}{\partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x^2} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial x \partial y} (x_j,y_j) = 0 \\
\dfrac{\partial^2 \varphi_i}{\partial y^2} (x_j,y_j) = 0 \\
\dfrac{\partial\varphi_i}{\partial n_k} = \delta_{i,k-18}
\end{split}
\end{equation*} & \\
\hline
\end{tabular}}
\caption{Constraints for Argyris triangle
\cite{Argyris,Brenner,Ciarlet,Dominguez08}, where $i,j,k$ correspond to the
various degrees of freedom of the Argyris element.}
\label{tab:Constraints}
\end{center}
\end{sidewaystable}