有限要素法・流体力学による数値計算

\[ \begin{aligned} [\vec n^{\,*}\cdot\vec\sigma^{\,*}] &=\left[\sum_{i=1}^{3}\vec\delta_i n_i\cdot \sum_{j=1}^{3}\sum_{k=1}^{3}\vec\delta_j\vec\delta_k\sigma_{jk}^{*}\right]\\[6pt] &=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3} [\vec\delta_i\cdot\vec\delta_j\vec\delta_k]n_i\sigma_{jk}^{*}\\[6pt] &=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3} \delta_{ij}\vec\delta_k n_i\sigma_{jk}^{*}\\[6pt] &=\sum_{i=1}^{3}\sum_{k=1}^{3}\vec\delta_k n_i\sigma_{ik}^{*} \qquad(\because i=j)\\[6pt] &=\sum_{k=1}^{3}\vec\delta_k\sum_{i=1}^{3}n_i\sigma_{ik}^{*}\\[6pt] &=\sum_{k=1}^{3}\vec\delta_k (n_x\sigma_{xk}^{*}+n_y\sigma_{yk}^{*}+n_z\sigma_{zk}^{*})\\[6pt] &=\vec\delta_x(n_x\sigma_{xx}^{*}+n_y\sigma_{yx}^{*}+n_z\sigma_{zx}^{*})\\[6pt] &\quad+\vec\delta_y(n_x\sigma_{xy}^{*}+n_y\sigma_{yy}^{*}+n_z\sigma_{zy}^{*})\\[6pt] &\quad+\vec\delta_z(n_x\sigma_{xz}^{*}+n_y\sigma_{yz}^{*}+n_z\sigma_{zz}^{*})\\[6pt] &=\vec\delta_x\frac{-2K^{*}}{We}n_x +\vec\delta_y\frac{-2K^{*}}{We}n_y +\vec\delta_z\frac{-2K^{*}}{We}n_z \end{aligned} \]

\[ [\vec n\cdot\vec\sigma]=-2\gamma K\vec n \]

\[ p_{bd}=2\gamma K \]

\[ \begin{aligned} [\vec n^{\,*}\cdot\vec\sigma^{\,*}] &=-\frac{2\gamma K^{*}}{\rho v_0^2x_0}\vec n\\[6pt] &=-\frac{2K^{*}}{We}\vec n\\[6pt] &=\vec\delta_x\frac{-2K^{*}}{We}n_x +\vec\delta_y\frac{-2K^{*}}{We}n_y +\vec\delta_z\frac{-2K^{*}}{We}n_z \end{aligned} \]

\[ \begin{aligned} P_{bd}^{*} &=\frac{P_{bd}}{\rho v_0^2}\\[6pt] &=\frac{2\gamma K}{\rho v_0^2}\\[6pt] &=\frac{2\gamma K^{*}}{\rho v_0^2x_0}\\[6pt] &=\frac{2K^{*}}{We}\\[6pt] &=\frac{2K^{*}}{Oh^2} \end{aligned} \]