有限要素法・流体力学による数値計算

\[ V=\frac{1}{6} \begin{vmatrix} 1&x_1&y_1&z_1\\ 1&x_2&y_2&z_2\\ 1&x_3&y_3&z_3\\ 1&x_4&y_4&z_4 \end{vmatrix} \]

\[ \begin{aligned} \det \begin{pmatrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{43}&a_{42}&a_{43}&a_{44} \end{pmatrix} &=\sum_{\sigma\in\sigma_4}\varepsilon(\sigma) a_{1\sigma(1)}a_{2\sigma(2)}a_{3\sigma(3)}a_{4\sigma(4)}\\[6pt] &=\sum_{\sigma\in\sigma_4}\varepsilon \begin{pmatrix}1&2&3&4\\\sigma(1)&\sigma(2)&\sigma(3)&\sigma(4)\end{pmatrix} a_{1\sigma(1)}a_{2\sigma(2)}a_{3\sigma(3)}a_{4\sigma(4)}\\[6pt] &=\varepsilon\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}a_{11}a_{22}a_{33}a_{44}+\varepsilon\begin{pmatrix}1&2&3&4\\1&2&4&3\end{pmatrix}a_{11}a_{22}a_{34}a_{43}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\1&3&2&4\end{pmatrix}a_{11}a_{23}a_{32}a_{44}+\varepsilon\begin{pmatrix}1&2&3&4\\1&3&4&2\end{pmatrix}a_{11}a_{23}a_{34}a_{42}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\1&4&2&3\end{pmatrix}a_{11}a_{24}a_{32}a_{43}+\varepsilon\begin{pmatrix}1&2&3&4\\1&4&3&2\end{pmatrix}a_{11}a_{24}a_{33}a_{42}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\2&1&3&4\end{pmatrix}a_{12}a_{21}a_{33}a_{44}+\varepsilon\begin{pmatrix}1&2&3&4\\2&1&4&3\end{pmatrix}a_{12}a_{21}a_{34}a_{43}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\2&3&1&4\end{pmatrix}a_{12}a_{23}a_{31}a_{44}+\varepsilon\begin{pmatrix}1&2&3&4\\2&3&4&1\end{pmatrix}a_{12}a_{23}a_{34}a_{41}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\2&4&1&3\end{pmatrix}a_{12}a_{24}a_{31}a_{43}+\varepsilon\begin{pmatrix}1&2&3&4\\2&4&3&1\end{pmatrix}a_{12}a_{24}a_{33}a_{41}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\3&1&2&4\end{pmatrix}a_{13}a_{21}a_{32}a_{44}+\varepsilon\begin{pmatrix}1&2&3&4\\3&1&4&2\end{pmatrix}a_{13}a_{21}a_{34}a_{42}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\3&2&1&4\end{pmatrix}a_{13}a_{22}a_{31}a_{44}+\varepsilon\begin{pmatrix}1&2&3&4\\3&2&4&1\end{pmatrix}a_{13}a_{22}a_{34}a_{41}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\3&4&1&2\end{pmatrix}a_{13}a_{24}a_{31}a_{42}+\varepsilon\begin{pmatrix}1&2&3&4\\3&4&2&1\end{pmatrix}a_{13}a_{24}a_{32}a_{41}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\4&1&2&3\end{pmatrix}a_{14}a_{21}a_{32}a_{43}+\varepsilon\begin{pmatrix}1&2&3&4\\4&1&3&2\end{pmatrix}a_{14}a_{21}a_{33}a_{42}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\4&2&1&3\end{pmatrix}a_{14}a_{22}a_{31}a_{43}+\varepsilon\begin{pmatrix}1&2&3&4\\4&2&3&1\end{pmatrix}a_{14}a_{22}a_{33}a_{41}\\[6pt] &\quad+\varepsilon\begin{pmatrix}1&2&3&4\\4&3&1&2\end{pmatrix}a_{14}a_{23}a_{31}a_{42}+\varepsilon\begin{pmatrix}1&2&3&4\\4&3&2&1\end{pmatrix}a_{14}a_{23}a_{32}a_{41}\\[6pt] \end{aligned} \]

\[ \begin{aligned} =\varepsilon\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}a_{11}a_{22}a_{33}a_{44}+\varepsilon\left((34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{11}a_{22}a_{34}a_{43}\\[6pt] &\quad+\varepsilon\left((23)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{11}a_{23}a_{32}a_{44}+\varepsilon\left((23)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{11}a_{23}a_{34}a_{42}\\[6pt] &\quad+\varepsilon\left((24)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{11}a_{24}a_{32}a_{43}+\varepsilon\left((24)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{11}a_{24}a_{33}a_{42}\\[6pt] &\quad+\varepsilon\left((12)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{12}a_{21}a_{33}a_{44}+\varepsilon\left((12)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{12}a_{21}a_{34}a_{43}\\[6pt] &\quad+\varepsilon\left((12)(23)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{12}a_{23}a_{31}a_{44}+\varepsilon\left((12)(23)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{12}a_{23}a_{34}a_{41}\\[6pt] &\quad+\varepsilon\left((12)(24)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{12}a_{24}a_{31}a_{43}+\varepsilon\left((12)(24)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{12}a_{24}a_{33}a_{41}\\[6pt] &\quad+\varepsilon\left((13)(23)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{13}a_{21}a_{32}a_{44}+\varepsilon\left((13)(23)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{13}a_{21}a_{34}a_{42}\\[6pt] &\quad+\varepsilon\left((13)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{13}a_{22}a_{31}a_{44}+\varepsilon\left((13)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{13}a_{22}a_{34}a_{41}\\[6pt] &\quad+\varepsilon\left((13)(24)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{13}a_{24}a_{31}a_{42}+\varepsilon\left((13)(24)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{13}a_{24}a_{32}a_{41}\\[6pt] &\quad+\varepsilon\left((14)(24)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{14}a_{21}a_{32}a_{43}+\varepsilon\left((14)(24)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{14}a_{21}a_{33}a_{42}\\[6pt] &\quad+\varepsilon\left((14)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{14}a_{22}a_{31}a_{43}+\varepsilon\left((14)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{14}a_{22}a_{33}a_{41}\\[6pt] &\quad+\varepsilon\left((14)(23)(34)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{14}a_{23}a_{31}a_{42}+\varepsilon\left((14)(23)\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}\right)a_{14}a_{23}a_{32}a_{41}\\[6pt] \end{aligned} \]

\[ \begin{aligned} =(-1)^{0}a_{11}a_{22}a_{33}a_{44}+(-1)^{1}a_{11}a_{22}a_{34}a_{43}+(-1)^{1}a_{11}a_{23}a_{32}a_{44}\\[6pt] &\quad+(-1)^{2}a_{11}a_{23}a_{34}a_{42}+(-1)^{2}a_{11}a_{24}a_{32}a_{43}+(-1)^{1}a_{11}a_{24}a_{33}a_{42}\\[6pt] &\quad+(-1)^{1}a_{12}a_{21}a_{33}a_{44}+(-1)^{2}a_{12}a_{21}a_{34}a_{43}+(-1)^{2}a_{12}a_{23}a_{31}a_{44}\\[6pt] &\quad+(-1)^{3}a_{12}a_{23}a_{34}a_{41}+(-1)^{3}a_{12}a_{24}a_{31}a_{43}+(-1)^{2}a_{12}a_{24}a_{33}a_{41}\\[6pt] &\quad+(-1)^{2}a_{13}a_{21}a_{32}a_{44}+(-1)^{3}a_{13}a_{21}a_{34}a_{42}+(-1)^{1}a_{13}a_{22}a_{31}a_{44}\\[6pt] &\quad+(-1)^{2}a_{13}a_{22}a_{34}a_{41}+(-1)^{2}a_{13}a_{24}a_{31}a_{42}+(-1)^{3}a_{13}a_{24}a_{32}a_{41}\\[6pt] &\quad+(-1)^{3}a_{14}a_{21}a_{32}a_{43}+(-1)^{2}a_{14}a_{21}a_{33}a_{42}+(-1)^{2}a_{14}a_{22}a_{31}a_{43}\\[6pt] &\quad+(-1)^{1}a_{14}a_{22}a_{33}a_{41}+(-1)^{3}a_{14}a_{23}a_{31}a_{42}+(-1)^{2}a_{14}a_{23}a_{32}a_{41}\\[6pt] \\[6pt] =a_{11}a_{22}a_{33}a_{44}-a_{11}a_{22}a_{34}a_{43}-a_{11}a_{23}a_{32}a_{44}\\[6pt] &\quad+a_{11}a_{23}a_{34}a_{42}+a_{11}a_{24}a_{32}a_{43}-a_{11}a_{24}a_{33}a_{42}\\[6pt] &\quad-a_{12}a_{21}a_{33}a_{44}+a_{12}a_{21}a_{34}a_{43}+a_{12}a_{23}a_{31}a_{44}\\[6pt] &\quad-a_{12}a_{23}a_{34}a_{41}-a_{12}a_{24}a_{31}a_{43}+a_{12}a_{24}a_{33}a_{41}\\[6pt] &\quad+a_{13}a_{21}a_{32}a_{44}-a_{13}a_{21}a_{34}a_{42}-a_{13}a_{22}a_{31}a_{44}\\[6pt] &\quad+a_{13}a_{22}a_{34}a_{41}+a_{13}a_{24}a_{31}a_{42}-a_{13}a_{24}a_{32}a_{41}\\[6pt] &\quad-a_{14}a_{21}a_{32}a_{43}+a_{14}a_{21}a_{33}a_{42}+a_{14}a_{22}a_{31}a_{43}\\[6pt] &\quad-a_{14}a_{22}a_{33}a_{41}-a_{14}a_{23}a_{31}a_{42}+a_{14}a_{23}a_{32}a_{41}\\[6pt] \end{aligned} \]

\[ \begin{aligned} V&=\frac{1}{6}\det \begin{pmatrix} 1&x_1&y_1&z_1\\ 1&x_2&y_2&z_2\\ 1&x_3&y_3&z_3\\ 1&x_4&y_4&z_4 \end{pmatrix}\\[6pt] &=\frac{1}{6}\Big( x_2y_3z_4-x_2z_3y_4-y_2x_3z_4+y_2z_3x_4+z_2x_3y_4-z_2y_3x_4\\[6pt] &\qquad-x_1y_3z_4+x_1z_3y_4+x_1y_2z_4-x_1y_2z_3-x_1z_2y_4+x_1z_2y_3\\[6pt] &\qquad+y_1x_3z_4-y_1z_3x_4-y_1x_2z_4+y_1x_2z_3+y_1z_2x_4-y_1z_2x_3\\[6pt] &\qquad-z_1x_3y_4+z_1y_3x_4+z_1x_2y_4-z_1x_2y_3-z_1y_2x_4+z_1y_2x_3 \Big) \end{aligned} \]

\[ \begin{aligned} &a_{11}=1,\ a_{12}=x_1,\ a_{13}=y_1,\ a_{14}=z_1\\[6pt] &a_{21}=1,\ a_{22}=x_2,\ a_{23}=y_2,\ a_{24}=z_2\\[6pt] &a_{31}=1,\ a_{32}=x_3,\ a_{33}=y_3,\ a_{34}=z_3\\[6pt] &a_{41}=1,\ a_{42}=x_4,\ a_{43}=y_4,\ a_{44}=z_4 \end{aligned} \]