広告
3角形1次要素
無次元化した熱収支式は次式となります。
\[
\phi=\frac{\partial\Theta}{\partial\tau}
+V_x\frac{\partial\Theta}{\partial X}
+V_y\frac{\partial\Theta}{\partial Y}
+\frac{\partial q_x^{*}}{\partial X}
+\frac{\partial q_y^{*}}{\partial Y}=0
\]
内挿関数Niを重み関数に使用すると離散化式は次式となります。
\[
\int_S
\begin{bmatrix}
N_1\\
N_2\\
N_3
\end{bmatrix}
\phi\,dS
=
\int_S
\begin{bmatrix}
0\\
0\\
0
\end{bmatrix}
dS
=
\begin{bmatrix}
0\\
0\\
0
\end{bmatrix}
\]
速度、温度は、内挿関数Niを用いてそれぞれ次式で表されます。
\[
\begin{aligned}
V_x&=N_1V_{x,1}+N_2V_{x,2}+N_3V_{x,3}
=\begin{bmatrix}N_1&N_2&N_3\end{bmatrix}
\begin{Bmatrix}V_{x,1}\\V_{x,2}\\V_{x,3}\end{Bmatrix}
=[N]\{V_x\}=0\\[6pt]
V_y&=N_1V_{y,1}+N_2V_{y,2}+N_3V_{y,3}=0\\[6pt]
\Theta&=N_1\Theta_1+N_2\Theta_2+N_3\Theta_3
\end{aligned}
\]
従って、離散化式は、次式となります。
\[
\begin{aligned}
&\int_S
\begin{bmatrix}
N_1\\N_2\\N_3
\end{bmatrix}
\left(
\frac{\partial\Theta}{\partial\tau}
+V_x\frac{\partial\Theta}{\partial X}
+V_y\frac{\partial\Theta}{\partial Y}
+\frac{\partial q_x^{*}}{\partial X}
+\frac{\partial q_y^{*}}{\partial Y}
\right)dS\\[6pt]
&=
\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial\Theta}{\partial\tau}dS
+V_x\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial\Theta}{\partial X}dS
+V_y\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial\Theta}{\partial Y}dS\\[6pt]
&\quad+
\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial q_x^{*}}{\partial X}dS
+
\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial q_y^{*}}{\partial Y}dS
\end{aligned}
\]
展開すると
\[
\begin{aligned}
&\int_S
\begin{bmatrix}
N_1\\N_2\\N_3
\end{bmatrix}
\left(
\frac{\partial\Theta}{\partial\tau}
+V_x\frac{\partial\Theta}{\partial X}
+V_y\frac{\partial\Theta}{\partial Y}
+\frac{\partial q_x^{*}}{\partial X}
+\frac{\partial q_y^{*}}{\partial Y}
\right)dS\\[6pt]
&=
\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial\Theta}{\partial\tau}dS
+V_x\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial\Theta}{\partial X}dS
+V_y\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial\Theta}{\partial Y}dS\\[6pt]
&\quad+
\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial q_x^{*}}{\partial X}dS
+
\int_S
\begin{bmatrix}N_1\\N_2\\N_3\end{bmatrix}
\frac{\partial q_y^{*}}{\partial Y}dS
\end{aligned}
\]
\[
\begin{aligned}
&={}
\int_S[N]^T\frac{\partial\Theta}{\partial\tau}dS
+V_x\int_S[N]^T\frac{\partial\Theta}{\partial X}dS
+V_y\int_S[N]^T\frac{\partial\Theta}{\partial Y}dS\\[6pt]
&+\int_S[N]^T\frac{\partial q_x^{*}}{\partial X}dS
+\int_S[N]^T\frac{\partial q_y^{*}}{\partial Y}dS
\end{aligned}
\]
グリーン・ガウスの定理より
\[
\begin{aligned}
&={}
\int_S[N]^T
\frac{[N]\left(\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}\right)}
{\Delta\tau}dS\\[6pt]
&+V_x\int_S[N]^T
\frac{\partial[N]\{\Theta\}}{\partial X}dS
+V_y\int_S[N]^T
\frac{\partial[N]\{\Theta\}}{\partial Y}dS\\[6pt]
&+\int_L[N]^Tq_x^{*}n_x\,dL
-\int_S\left[\frac{\partial N}{\partial X}\right]^Tq_x^{*}dS\\[6pt]
&+\int_L[N]^Tq_y^{*}n_y\,dL
-\int_S\left[\frac{\partial N}{\partial Y}\right]^Tq_y^{*}dS
\end{aligned}
\]
界面に作用する伝熱の項をまとめると
\[
\begin{aligned}
&={}
\int_S[N]^T[N]dS
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}\\[6pt]
&+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{\Theta\}
+V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{\Theta\}\\[6pt]
&-\int_S\left[\frac{\partial N}{\partial X}\right]^Tq_x^{*}dS
-\int_S\left[\frac{\partial N}{\partial Y}\right]^Tq_y^{*}dS\\[6pt]
&+\int_L[N]^T(q_x^{*}n_x+q_y^{*}n_y)dL
\end{aligned}
\]
熱流束を展開して
\[
\begin{aligned}
&={}
\int_S[N]^T[N]dS
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}\\[6pt]
&+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{\Theta\}
+V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{\Theta\}\\[6pt]
&-\int_S\left[\frac{\partial N}{\partial X}\right]^T
\left(-\frac{1}{Pe}\frac{\partial\Theta}{\partial X}\right)dS\\[6pt]
&-\int_S\left[\frac{\partial N}{\partial Y}\right]^T
\left(-\frac{1}{Pe}\frac{\partial\Theta}{\partial Y}\right)dS\\[6pt]
&+Nu(\Theta-\Theta_h)\int_L[N]^T dL
\end{aligned}
\]
項ごとに分解して
\[
\begin{aligned}
&={}
\int_S[N]^T[N]dS
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}\\[6pt]
&+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{\Theta\}
+V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\int_S\left[\frac{\partial N}{\partial X}\right]^T
\frac{\partial\Theta}{\partial X}dS
+\frac{1}{Pe}\int_S\left[\frac{\partial N}{\partial Y}\right]^T
\frac{\partial\Theta}{\partial Y}dS\\[6pt]
&+Nu(\Theta-\Theta_h)\int_L[N]^T dL
\end{aligned}
\]
温度を内挿関数で表示して
\[
\begin{aligned}
&={}
\int_S[N]^T[N]dS
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}\\[6pt]
&+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{\Theta\}
+V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\int_S\left[\frac{\partial N}{\partial X}\right]^T
\frac{\partial[N]\{\Theta\}}{\partial X}dS
+\frac{1}{Pe}\int_S\left[\frac{\partial N}{\partial Y}\right]^T
\frac{\partial[N]\{\Theta\}}{\partial Y}dS\\[6pt]
&+Nu(\Theta-\Theta_h)\int_L[N]^T dL
\end{aligned}
\]
温度は定数なので積分の外に出す
\[
\begin{aligned}
&={}
\int_S[N]^T[N]dS
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}\\[6pt]
&+V_x\int_S[N]^T\left[\frac{\partial N}{\partial X}\right]dS\{\Theta\}
+V_y\int_S[N]^T\left[\frac{\partial N}{\partial Y}\right]dS\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\int_S\left[\frac{\partial N}{\partial X}\right]^T
\left[\frac{\partial N}{\partial X}\right]dS\{\Theta\}
+\frac{1}{Pe}\int_S\left[\frac{\partial N}{\partial Y}\right]^T
\left[\frac{\partial N}{\partial Y}\right]dS\{\Theta\}\\[6pt]
&+Nu(\Theta-\Theta_h)\int_L[N]^T dL
\end{aligned}
\]
行列式で表す
\[
=
\int_S
\begin{bmatrix}
N_1N_1&N_1N_2&N_1N_3\\
N_2N_1&N_2N_2&N_2N_3\\
N_3N_1&N_3N_2&N_3N_3
\end{bmatrix}
dS\,
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}
\]
対流項
\[
\begin{aligned}
&+V_x\int_S
\begin{bmatrix}
N_1\frac{\partial N_1}{\partial X}&N_1\frac{\partial N_2}{\partial X}&N_1\frac{\partial N_3}{\partial X}\\
N_2\frac{\partial N_1}{\partial X}&N_2\frac{\partial N_2}{\partial X}&N_2\frac{\partial N_3}{\partial X}\\
N_3\frac{\partial N_1}{\partial X}&N_3\frac{\partial N_2}{\partial X}&N_3\frac{\partial N_3}{\partial X}
\end{bmatrix}
dS\{\Theta\}\\[6pt]
&+V_y\int_S
\begin{bmatrix}
N_1\frac{\partial N_1}{\partial Y}&N_1\frac{\partial N_2}{\partial Y}&N_1\frac{\partial N_3}{\partial Y}\\
N_2\frac{\partial N_1}{\partial Y}&N_2\frac{\partial N_2}{\partial Y}&N_2\frac{\partial N_3}{\partial Y}\\
N_3\frac{\partial N_1}{\partial Y}&N_3\frac{\partial N_2}{\partial Y}&N_3\frac{\partial N_3}{\partial Y}
\end{bmatrix}
dS\{\Theta\}
\end{aligned}
\]
伝導伝熱の項
\[
\begin{aligned}
&+\frac{1}{Pe}\int_S
\begin{bmatrix}
\frac{\partial N_1}{\partial X}\frac{\partial N_1}{\partial X}&
\frac{\partial N_1}{\partial X}\frac{\partial N_2}{\partial X}&
\frac{\partial N_1}{\partial X}\frac{\partial N_3}{\partial X}\\
\frac{\partial N_2}{\partial X}\frac{\partial N_1}{\partial X}&
\frac{\partial N_2}{\partial X}\frac{\partial N_2}{\partial X}&
\frac{\partial N_2}{\partial X}\frac{\partial N_3}{\partial X}\\
\frac{\partial N_3}{\partial X}\frac{\partial N_1}{\partial X}&
\frac{\partial N_3}{\partial X}\frac{\partial N_2}{\partial X}&
\frac{\partial N_3}{\partial X}\frac{\partial N_3}{\partial X}
\end{bmatrix}
dS\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\int_S
\begin{bmatrix}
\frac{\partial N_1}{\partial Y}\frac{\partial N_1}{\partial Y}&
\frac{\partial N_1}{\partial Y}\frac{\partial N_2}{\partial Y}&
\frac{\partial N_1}{\partial Y}\frac{\partial N_3}{\partial Y}\\
\frac{\partial N_2}{\partial Y}\frac{\partial N_1}{\partial Y}&
\frac{\partial N_2}{\partial Y}\frac{\partial N_2}{\partial Y}&
\frac{\partial N_2}{\partial Y}\frac{\partial N_3}{\partial Y}\\
\frac{\partial N_3}{\partial Y}\frac{\partial N_1}{\partial Y}&
\frac{\partial N_3}{\partial Y}\frac{\partial N_2}{\partial Y}&
\frac{\partial N_3}{\partial Y}\frac{\partial N_3}{\partial Y}
\end{bmatrix}
dS\{\Theta\}
\end{aligned}
\]
対流伝熱の項
\[
+Nu(\Theta-\Theta_h)\int_L
\begin{bmatrix}
N_1\\N_2
\end{bmatrix}dL
\]
内挿関数を形状関数に直す
\[
=
\int_S
\begin{bmatrix}
L_1L_1&L_1L_2&L_1L_3\\
L_2L_1&L_2L_2&L_2L_3\\
L_3L_1&L_3L_2&L_3L_3
\end{bmatrix}
dS\,
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}
\]
対流項
\[
\begin{aligned}
&+V_x\int_S\frac{1}{2A}
\begin{bmatrix}
L_1c_{1x}&L_1c_{2x}&L_1c_{3x}\\
L_2c_{1x}&L_2c_{2x}&L_2c_{3x}\\
L_3c_{1x}&L_3c_{2x}&L_3c_{3x}
\end{bmatrix}
dS\{\Theta\}\\[6pt]
&+V_y\int_S\frac{1}{2A}
\begin{bmatrix}
L_1c_{1y}&L_1c_{2y}&L_1c_{3y}\\
L_2c_{1y}&L_2c_{2y}&L_2c_{3y}\\
L_3c_{1y}&L_3c_{2y}&L_3c_{3y}
\end{bmatrix}
dS\{\Theta\}
\end{aligned}
\]
伝導伝熱の項
\[
\begin{aligned}
&+\frac{1}{Pe}\int_S\frac{1}{4A^2}
\begin{bmatrix}
c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}\\
c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}\\
c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}
\end{bmatrix}
dS\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\int_S\frac{1}{4A^2}
\begin{bmatrix}
c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}\\
c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}\\
c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}
\end{bmatrix}
dS\{\Theta\}
\end{aligned}
\]
対流伝熱の項
\[
+Nu(\Theta-\Theta_h)\int_L
\begin{bmatrix}
L_1\\L_2
\end{bmatrix}dL
\]
ここで、面積積分、体積積分の公式より
\[
\begin{aligned}
\int_S L_1^pL_2^qL_3^r\,dS
&=\frac{p!q!r!}{(p+q+r+2)!}\,2A\\[6pt]
\int_S L_iL_j\,dS
&=
\begin{cases}
\displaystyle\frac{1!1!}{(1+1+2)!}\,2A
=\frac{2}{4!}A
=\frac{1}{12}A &(i\ne j)\\
\displaystyle\frac{2!}{(1+1+2)!}\,2A
=\frac{4}{4!}V
=\frac{1}{6}A &(i=j)
\end{cases}\\[6pt]
\int_S L_i\,dS
&=\frac{1!}{(1+2)!}\,2A
=\frac{2}{3!}A
=\frac{1}{3}A\\[6pt]
\int_L L_1^pL_2^q\,dL
&=\frac{p!q!}{(p+q)!}L\\[6pt]
\int_L L_i\,dL
&=\frac{1}{1!}L=L
\end{aligned}
\]
形状関数を積分すると
\[
=
\frac{1}{12}A
\begin{bmatrix}
2&1&1\\
1&2&1\\
1&1&2
\end{bmatrix}
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}
\]
対流項
\[
\begin{aligned}
&+V_x\frac{1}{2A}\frac{A}{3}
\begin{bmatrix}
c_{1x}&c_{2x}&c_{3x}\\
c_{1x}&c_{2x}&c_{3x}\\
c_{1x}&c_{2x}&c_{3x}
\end{bmatrix}\{\Theta\}\\[6pt]
&+V_y\frac{1}{2A}\frac{A}{3}
\begin{bmatrix}
c_{1y}&c_{2y}&c_{3y}\\
c_{1y}&c_{2y}&c_{3y}\\
c_{1y}&c_{2y}&c_{3y}
\end{bmatrix}\{\Theta\}
\end{aligned}
\]
伝導伝熱の項
\[
\begin{aligned}
&+\frac{1}{Pe}\frac{1}{4A^2}A
\begin{bmatrix}
c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}\\
c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}\\
c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}
\end{bmatrix}\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\frac{1}{4A^2}A
\begin{bmatrix}
c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}\\
c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}\\
c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}
\end{bmatrix}\{\Theta\}
\end{aligned}
\]
対流伝熱の項
\[
+Nu(\Theta-\Theta_h)L
\begin{bmatrix}
1\\1
\end{bmatrix}
\]
係数をまとめて
\[
=
\frac{1}{12}A
\begin{bmatrix}
2&1&1\\
1&2&1\\
1&1&2
\end{bmatrix}
\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}
{\Delta\tau}
\]
対流項
\[
\begin{aligned}
&+V_x\frac{1}{6}
\begin{bmatrix}
c_{1x}&c_{2x}&c_{3x}\\
c_{1x}&c_{2x}&c_{3x}\\
c_{1x}&c_{2x}&c_{3x}
\end{bmatrix}\{\Theta\}\\[6pt]
&+V_y\frac{1}{6}
\begin{bmatrix}
c_{1y}&c_{2y}&c_{3y}\\
c_{1y}&c_{2y}&c_{3y}\\
c_{1y}&c_{2y}&c_{3y}
\end{bmatrix}\{\Theta\}
\end{aligned}
\]
伝導伝熱の項
\[
\begin{aligned}
&+\frac{1}{Pe}\frac{1}{4A}
\begin{bmatrix}
c_{1x}c_{1x}&c_{1x}c_{2x}&c_{1x}c_{3x}\\
c_{2x}c_{1x}&c_{2x}c_{2x}&c_{2x}c_{3x}\\
c_{3x}c_{1x}&c_{3x}c_{2x}&c_{3x}c_{3x}
\end{bmatrix}\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\frac{1}{4A}
\begin{bmatrix}
c_{1y}c_{1y}&c_{1y}c_{2y}&c_{1y}c_{3y}\\
c_{2y}c_{1y}&c_{2y}c_{2y}&c_{2y}c_{3y}\\
c_{3y}c_{1y}&c_{3y}c_{2y}&c_{3y}c_{3y}
\end{bmatrix}\{\Theta\}
\end{aligned}
\]
対流伝熱の項
\[
+Nu(\Theta-\Theta_h)n_iL
\begin{bmatrix}
1\\1
\end{bmatrix}
\]
行列式をまとめると
\[
\begin{aligned}
&={}[C]\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}{\Delta\tau}
+V_x[C_x]\{\Theta\}
+V_y[C_y]\{\Theta\}\\[6pt]
&+\frac{1}{Pe}[S_{xx}]\{\Theta\}
+\frac{1}{Pe}[S_{yy}]\{\Theta\}\\[6pt]
&+Nu(\Theta-\Theta_h)L
\begin{bmatrix}1\\1\end{bmatrix}
\end{aligned}
\]
行列ごとにまとめると
\[
\begin{aligned}
&={}[C]\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}{\Delta\tau}
+\left(V_x[C_x]+V_y[C_y]\right)\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\left([S_{xx}]+[S_{yy}]\right)\{\Theta\}\\[6pt]
&+Nu(\Theta-\Theta_h)L
\begin{bmatrix}1\\1\end{bmatrix}\\[6pt]
&={}
\begin{bmatrix}0\\0\\0\end{bmatrix}
\end{aligned}
\]
最終的に次式が導出されます。
\[
\begin{aligned}
&[C]\frac{\{\Theta\}^{\tau+\Delta\tau}-\{\Theta\}^{\tau}}{\Delta\tau}
+\left(V_x[C_x]+V_y[C_y]\right)\{\Theta\}\\[6pt]
&+\frac{1}{Pe}\left([S_{xx}]+[S_{yy}]\right)\{\Theta\}\\[6pt]
&+Nu(\Theta-\Theta_h)L
\begin{bmatrix}1\\1\end{bmatrix}
=
\begin{bmatrix}0\\0\\0\end{bmatrix}
\end{aligned}
\]
既知の項を右辺に移項します。下式は陰解法の場合の離散化式になります。
\[
\begin{aligned}
&\frac{[C]}{\Delta\tau}\{\Theta\}^{\tau+\Delta\tau}
+\left(V_x[C_x]+V_y[C_y]\right)\{\Theta\}^{\tau+\Delta\tau}\\[6pt]
&+\frac{1}{Pe}\left([S_{xx}]+[S_{yy}]\right)\{\Theta\}^{\tau+\Delta\tau}\\[6pt]
&={}
\frac{[C]}{\Delta\tau}\{\Theta\}^{\tau}
-Nu(\Theta-\Theta_h)L
\begin{bmatrix}1\\1\end{bmatrix}
\end{aligned}
\]
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