広告
・内挿関数N2の微分
3角形1次要素 の内挿関数N2の微分値を示します。
x微分
\[
\begin{aligned}
\frac{\partial N_2}{\partial x}
&=\frac{\partial L_2}{\partial x}
=\frac{\partial(S_2/S)}{\partial x}
=\frac{1}{S}\frac{\partial S_2}{\partial x}\\[6pt]
&=\frac{1}{S}\frac{\partial}{\partial x}\frac{1}{2}
(xy_3-yx_3-x_1y_3+x_1y+y_1x_3-y_1x)\\[6pt]
&=\frac{1}{2S}(y_3-y_1)
\end{aligned}
\]
y微分
\[
\begin{aligned}
\frac{\partial N_2}{\partial y}
&=\frac{\partial L_2}{\partial y}
=\frac{\partial(S_2/S)}{\partial y}
=\frac{1}{S}\frac{\partial S_2}{\partial y}\\[6pt]
&=\frac{1}{S}\frac{\partial}{\partial x}\frac{1}{2}
(xy_3-yx_3-x_1y_3+x_1y+y_1x_3-y_1x)\\[6pt]
&=\frac{1}{2S}(-x_3+x_1)
\end{aligned}
\]
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