有限要素法・流体力学による数値計算

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

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

\[ \begin{aligned} V_1&=\frac{1}{6}\det \begin{pmatrix} 1&x&y&z\\ 1&x_2&y_2&z_2\\ 1&x_3&y_3&z_3\\ 1&x_4&y_4&z_4 \end{pmatrix}\\[6pt] &=\frac{1}{6}\Big( x_2y_3z_4-x_2z_3y_4-y_2x_3z_4+y_2z_3x_4+z_2x_3y_4-z_2y_3x_4\\[6pt] &\qquad-xy_3z_4+xz_3y_4+xy_2z_4-xy_2z_3-xz_2y_4+xz_2y_3\\[6pt] &\qquad+yx_3z_4-yz_3x_4-yx_2z_4+yx_2z_3+yz_2x_4-yz_2x_3\\[6pt] &\qquad-zx_3y_4+zy_3x_4+zx_2y_4-zx_2y_3-zy_2x_4+zy_2x_3 \Big) \end{aligned} \]

\[ \begin{aligned} V_2&=\frac{1}{6}\det \begin{pmatrix} 1&x_1&y_1&z_1\\ 1&x&y&z\\ 1&x_3&y_3&z_3\\ 1&x_4&y_4&z_4 \end{pmatrix}\\[6pt] &=\frac{1}{6}\Big( xy_3z_4-xz_3y_4-yx_3z_4+yz_3x_4+zx_3y_4-zy_3x_4\\[6pt] &\qquad-x_1y_3z_4+x_1z_3y_4+x_1yz_4-x_1yz_3-x_1zy_4+x_1zy_3\\[6pt] &\qquad+y_1x_3z_4-y_1z_3x_4-y_1xz_4+y_1xz_3+y_1zx_4-y_1zx_3\\[6pt] &\qquad-z_1x_3y_4+z_1y_3x_4+z_1xy_4-z_1xy_3-z_1yx_4+z_1yx_3 \Big) \end{aligned} \]

\[ \begin{aligned} V_3&=\frac{1}{6}\det \begin{pmatrix} 1&x_1&y_1&z_1\\ 1&x_2&y_2&z_2\\ 1&x&y&z\\ 1&x_4&y_4&z_4 \end{pmatrix}\\[6pt] &=\frac{1}{6}\Big( x_2yz_4-x_2zy_4-y_2xz_4+y_2zx_4+z_2xy_4-z_2yx_4\\[6pt] &\qquad-x_1yz_4+x_1zy_4+x_1y_2z_4-x_1y_2z-x_1z_2y_4+x_1z_2y\\[6pt] &\qquad+y_1xz_4-y_1zx_4-y_1x_2z_4+y_1x_2z+y_1z_2x_4-y_1z_2x\\[6pt] &\qquad-z_1xy_4+z_1yx_4+z_1x_2y_4-z_1x_2y-z_1y_2x_4+z_1y_2x \Big) \end{aligned} \]

\[ \begin{aligned} V_4&=\frac{1}{6}\det \begin{pmatrix} 1&x_1&y_1&z_1\\ 1&x_2&y_2&z_2\\ 1&x_3&y_3&z_3\\ 1&x&y&z \end{pmatrix}\\[6pt] &=\frac{1}{6}\Big( x_2y_3z-x_2z_3y-y_2x_3z+y_2z_3x+z_2x_3y-z_2y_3x\\[6pt] &\qquad-x_1y_3z+x_1z_3y+x_1y_2z-x_1y_2z_3-x_1z_2y+x_1z_2y_3\\[6pt] &\qquad+y_1x_3z-y_1z_3x-y_1x_2z+y_1x_2z_3+y_1z_2x-y_1z_2x_3\\[6pt] &\qquad-z_1x_3y+z_1y_3x+z_1x_2y-z_1x_2y_3-z_1y_2x+z_1y_2x_3 \Big) \end{aligned} \]