有限要素法・流体力学による数値計算

\[ \phi= \frac{\partial P}{\partial\tau} +V_x\frac{\partial P}{\partial x} +V_y\frac{\partial P}{\partial y} +\frac{1}{Ma^2}\left( \frac{\partial V_x}{\partial X} +\frac{\partial V_y}{\partial Y} \right)=0 \]

\[ \begin{aligned} &\int_V[N]^T\phi\,dV =\int_V \begin{bmatrix}N_1\\N_1\\N_3\end{bmatrix}\phi\,dV\\[6pt] &={}\int_V \begin{bmatrix}0\\0\\0\end{bmatrix}dV\\[6pt] &={}\begin{bmatrix}0\\0\\0\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} V_x &=N_1V_{x1}+N_2V_{x2}+N_3V_{x3}\\[6pt] &=\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix} \begin{Bmatrix}V_{x1}&V_{x2}&V_{x3}\end{Bmatrix}\\[6pt] &=[N]^T\{V_x\},\\[6pt] V_y&=[N]^T\{V_y\}\\[6pt] P&=[N]^T\{P\} \end{aligned} \]

\[ \begin{aligned} &\int_S[N]^T\phi\,dS\\[6pt] &={}\int_S[N]^T\left\{ \frac{\partial P}{\partial\tau} +V_x\frac{\partial P}{\partial x} +V_y\frac{\partial P}{\partial y} +\frac{1}{Ma^2}\left( \frac{\partial V_x}{\partial X} +\frac{\partial V_y}{\partial Y} \right) \right\}dS \end{aligned} \]

\[ \begin{aligned} &={}\int_S[N]^T\frac{\partial P}{\partial\tau}\,dS +\int_S[N]^TV_x\frac{\partial P}{\partial X}\,dS +\int_S[N]^TV_y\frac{\partial P}{\partial Y}\,dS\\[6pt] &+\frac{1}{Ma^2}\int_S[N]^T\frac{\partial V_x}{\partial X}\,dS +\frac{1}{Ma^2}\int_S[N]^T\frac{\partial V_y}{\partial Y}\,dS \end{aligned} \]

\[ \begin{aligned} &={}\int_S[N]^T[N]dS\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S[N]^T\frac{\partial[N]\{P\}^{\Delta\tau+\tau}}{\partial X}dS +V_y\int_S[N]^T\frac{\partial[N]\{P\}^{\Delta\tau+\tau}}{\partial Y}dS\\[6pt] &+\frac{1}{Ma^2}\int_S[N]^T\frac{\partial[N]\{V_x\}^{\Delta\tau+\tau}}{\partial X}dS +\frac{1}{Ma^2}\int_S[N]^T\frac{\partial[N]\{V_y\}^{\Delta\tau+\tau}}{\partial Y}dS \end{aligned} \]

\[ \begin{aligned} &={}\int_S[N]^T[N]dS\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S[N]^T\frac{\partial[N]}{\partial X}dS\,\{P\}^{\Delta\tau+\tau} +V_y\int_S[N]^T\frac{\partial[N]}{\partial Y}dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S[N]^T\frac{\partial[N]}{\partial X}dS\,\{V_x\}^{\Delta\tau+\tau} +\frac{1}{Ma^2}\int_S[N]^T\frac{\partial[N]}{\partial Y}dS\,\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix} \begin{bmatrix}N_1&N_2&N_3\end{bmatrix}dS\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial X}&\dfrac{\partial N_2}{\partial X}&\dfrac{\partial N_3}{\partial X} \end{bmatrix}dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial Y}&\dfrac{\partial N_2}{\partial Y}&\dfrac{\partial N_3}{\partial Y} \end{bmatrix}dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial X}&\dfrac{\partial N_2}{\partial X}&\dfrac{\partial N_3}{\partial X} \end{bmatrix}dS\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S \begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial Y}&\dfrac{\partial N_2}{\partial Y}&\dfrac{\partial N_3}{\partial Y} \end{bmatrix}dS\,\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_S \begin{bmatrix} N_1N_1&N_1N_2&N_1N_3\\ N_2N_1&N_2N_2&N_2N_3\\ N_3N_1&N_3N_2&N_3N_3 \end{bmatrix}dS\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial X}&N_1\dfrac{\partial N_2}{\partial X}&N_1\dfrac{\partial N_3}{\partial X}\\ N_2\dfrac{\partial N_1}{\partial X}&N_2\dfrac{\partial N_2}{\partial X}&N_2\dfrac{\partial N_3}{\partial X}\\ N_3\dfrac{\partial N_1}{\partial X}&N_3\dfrac{\partial N_2}{\partial X}&N_3\dfrac{\partial N_3}{\partial X} \end{bmatrix}dS\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_S \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial Y}&N_1\dfrac{\partial N_2}{\partial Y}&N_1\dfrac{\partial N_3}{\partial Y}\\ N_2\dfrac{\partial N_1}{\partial Y}&N_2\dfrac{\partial N_2}{\partial Y}&N_2\dfrac{\partial N_3}{\partial Y}\\ N_3\dfrac{\partial N_1}{\partial Y}&N_3\dfrac{\partial N_2}{\partial Y}&N_3\dfrac{\partial N_3}{\partial Y} \end{bmatrix}dS\,\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &+\frac{1}{Ma^2}\int_S \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial X}&N_1\dfrac{\partial N_2}{\partial X}&N_1\dfrac{\partial N_3}{\partial X}\\ N_2\dfrac{\partial N_1}{\partial X}&N_2\dfrac{\partial N_2}{\partial X}&N_2\dfrac{\partial N_3}{\partial X}\\ N_3\dfrac{\partial N_1}{\partial X}&N_3\dfrac{\partial N_2}{\partial X}&N_3\dfrac{\partial N_3}{\partial X} \end{bmatrix}dS\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial Y}&N_1\dfrac{\partial N_2}{\partial Y}&N_1\dfrac{\partial N_3}{\partial Y}\\ N_2\dfrac{\partial N_1}{\partial Y}&N_2\dfrac{\partial N_2}{\partial Y}&N_2\dfrac{\partial N_3}{\partial Y}\\ N_3\dfrac{\partial N_1}{\partial Y}&N_3\dfrac{\partial N_2}{\partial Y}&N_3\dfrac{\partial N_3}{\partial Y} \end{bmatrix}dS\,\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_S \begin{bmatrix} L_1L_1&L_1L_2&L_1L_3\\ L_2L_1&L_2L_2&L_2L_3\\ L_3L_1&L_3L_2&L_3L_3 \end{bmatrix}dS\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial X}&L_1\dfrac{\partial L_2}{\partial X}&L_1\dfrac{\partial L_3}{\partial X}\\ L_2\dfrac{\partial L_1}{\partial X}&L_2\dfrac{\partial L_2}{\partial X}&L_2\dfrac{\partial L_3}{\partial X}\\ L_3\dfrac{\partial L_1}{\partial X}&L_3\dfrac{\partial L_2}{\partial X}&L_3\dfrac{\partial L_3}{\partial X} \end{bmatrix}dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_S \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial Y}&L_1\dfrac{\partial L_2}{\partial Y}&L_1\dfrac{\partial L_3}{\partial Y}\\ L_2\dfrac{\partial L_1}{\partial Y}&L_2\dfrac{\partial L_2}{\partial Y}&L_2\dfrac{\partial L_3}{\partial Y}\\ L_3\dfrac{\partial L_1}{\partial Y}&L_3\dfrac{\partial L_2}{\partial Y}&L_3\dfrac{\partial L_3}{\partial Y} \end{bmatrix}dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial X}&L_1\dfrac{\partial L_2}{\partial X}&L_1\dfrac{\partial L_3}{\partial X}\\ L_2\dfrac{\partial L_1}{\partial X}&L_2\dfrac{\partial L_2}{\partial X}&L_2\dfrac{\partial L_3}{\partial X}\\ L_3\dfrac{\partial L_1}{\partial X}&L_3\dfrac{\partial L_2}{\partial X}&L_3\dfrac{\partial L_3}{\partial X} \end{bmatrix}dS\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial Y}&L_1\dfrac{\partial L_2}{\partial Y}&L_1\dfrac{\partial L_3}{\partial Y}\\ L_2\dfrac{\partial L_1}{\partial Y}&L_2\dfrac{\partial L_2}{\partial Y}&L_2\dfrac{\partial L_3}{\partial Y}\\ L_3\dfrac{\partial L_1}{\partial Y}&L_3\dfrac{\partial L_2}{\partial Y}&L_3\dfrac{\partial L_3}{\partial Y} \end{bmatrix}dS\,\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_S \begin{bmatrix} L_1L_1&L_1L_2&L_1L_3\\L_2L_1&L_2L_2&L_2L_3\\L_3L_1&L_3L_2&L_3L_3 \end{bmatrix}dS\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_S\frac{1}{2A} \begin{bmatrix}L_1c_{1x}&L_1c_{2x}&L_1c_{3x}\\L_2c_{1x}&L_2c_{2x}&L_2c_{3x}\\L_3c_{1x}&L_3c_{2x}&L_3c_{3x}\end{bmatrix} dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_S\frac{1}{2A} \begin{bmatrix}L_1c_{1y}&L_1c_{2y}&L_1c_{3y}\\L_2c_{1y}&L_2c_{2y}&L_2c_{3y}\\L_3c_{1y}&L_3c_{2y}&L_3c_{3y}\end{bmatrix} dS\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_S\frac{1}{2A} \begin{bmatrix}L_1c_{1x}&L_1c_{2x}&L_1c_{3x}\\L_2c_{1x}&L_2c_{2x}&L_2c_{3x}\\L_3c_{1x}&L_3c_{2x}&L_3c_{3x}\end{bmatrix} dS\,\{V_x\}^{\Delta\tau+\tau}3\\[6pt] &+\frac{1}{Ma^2}\int_S\frac{1}{2A} \begin{bmatrix}L_1c_{1y}&L_1c_{2y}&L_1c_{3y}\\L_2c_{1y}&L_2c_{2y}&L_2c_{3y}\\L_3c_{1y}&L_3c_{2y}&L_3c_{3y}\end{bmatrix} dS\,\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \int_S L_1^pL_2^qL_3^r\,dS =\frac{p!q!r!}{(p+q+r+2)!}\,2A \] \[ \int_S L_iL_j\,dS= \begin{cases} \dfrac{1!1!}{(1+1+2)!}\,2A=\dfrac{2}{4!}A=\dfrac{1}{12}A &(i\ne j)\\[8pt] \dfrac{2!}{(1+1+2)!}\,2A=\dfrac{4}{4!}V=\dfrac{1}{6}A &(i=j) \end{cases} \] \[ \int_S L_i\,dS =\frac{1!}{(1+2)!}\,2A =\frac{2}{3!}A =\frac{1}{3}A \]

\[ \begin{aligned} &={}\frac{A}{12}\begin{bmatrix}2&1&1\\1&2&1\\1&1&2\end{bmatrix} \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\frac{1}{2A}\frac{A}{3} \begin{bmatrix}c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\end{bmatrix} \{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\frac{1}{2A}\frac{A}{3} \begin{bmatrix}c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\end{bmatrix} \{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\frac{1}{2A}\frac{A}{3} \begin{bmatrix}c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\end{bmatrix} \{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\frac{1}{2A}\frac{A}{3} \begin{bmatrix}c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\end{bmatrix} \{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\frac{A}{12}\begin{bmatrix}2&1&1\\1&2&1\\1&1&2\end{bmatrix} \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\frac{1}{6} \begin{bmatrix}c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\frac{1}{6} \begin{bmatrix}c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\frac{1}{6} \begin{bmatrix}c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\\c_{1x}&c_{2x}&c_{3x}\end{bmatrix}\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\frac{1}{6} \begin{bmatrix}c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\\c_{1y}&c_{2y}&c_{3y}\end{bmatrix}\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}[C] \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x[C_x]\{P\}^{\Delta\tau+\tau} +V_y[C_y]\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}[C_x]\{V_x\}^{\Delta\tau+\tau} +\frac{1}{Ma^2}[C_y]\{V_y\}^{\Delta\tau+\tau} \end{aligned} \]

\[ =\begin{bmatrix}0\\0\\0\end{bmatrix} \]

\[ \begin{aligned} &[C]\frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau} +V_x[C_x]\{P\}^{\Delta\tau+\tau} +V_y[C_y]\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}[C_x]\{V_x\}^{\Delta\tau+\tau} +\frac{1}{Ma^2}[C_y]\{V_y\}^{\Delta\tau+\tau} =[0] \end{aligned} \]

\[ \begin{aligned} &\frac{[C]}{\Delta\tau}\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_x[C_x]\{P\}^{\Delta\tau+\tau} +V_y[C_y]\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}[C_x]\{V_x\}^{\Delta\tau+\tau} +\frac{1}{Ma^2}[C_y]\{V_y\}^{\Delta\tau+\tau} =\frac{[C]}{\Delta\tau}\{P\}^{\tau} \end{aligned} \]

\[ \begin{aligned} &\left( \frac{[C]}{\Delta\tau} +V_x[C_x] +V_y[C_y] \right)\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\left( [C_x]\{V_x\}^{\Delta\tau+\tau} +[C_y]\{V_y\}^{\Delta\tau+\tau} \right) =\frac{[C]}{\Delta\tau}\{P\}^{\tau} \end{aligned} \]