広告
・内挿関数N2の微分
4面体1次要素 の内挿関数N2の微分値を示します。
x微分
\[
\begin{aligned}
\frac{\partial N_2}{\partial x}
&=
\frac{\partial L_2}{\partial x}
=
\frac{\partial(V_2/V)}{\partial x}
=
\frac{1}{V}\frac{\partial V_2}{\partial x}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial x}\frac{1}{6}
\Big(
x y_3z_4-x z_3y_4-y x_3z_4+y z_3x_4+z x_3y_4-z y_3x_4\\[6pt]
&\qquad -x_1y_3z_4+x_1z_3y_4+x_1y z_4-x_1y z_3-x_1z y_4+x_1z y_3\\[6pt]
&\qquad +y_1x_3z_4-y_1z_3x_4-y_1x z_4+y_1x z_3+y_1z x_4-y_1z x_3\\[6pt]
&\qquad -z_1x_3y_4+z_1y_3x_4+z_1x y_4-z_1x y_3-z_1y x_4+z_1y x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
y_3z_4-z_3y_4-y_1z_4+y_1z_3+z_1y_4-z_1y_3
\Big)
\end{aligned}
\]
y微分
\[
\begin{aligned}
\frac{\partial N_2}{\partial y}
&=
\frac{\partial L_2}{\partial y}
=
\frac{\partial(V_2/V)}{\partial y}
=
\frac{1}{V}\frac{\partial V_2}{\partial y}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial y}\frac{1}{6}
\Big(
x y_3z_4-x z_3y_4-y x_3z_4+y z_3x_4+z x_3y_4-z y_3x_4\\[6pt]
&\qquad -x_1y_3z_4+x_1z_3y_4+x_1y z_4-x_1y z_3-x_1z y_4+x_1z y_3\\[6pt]
&\qquad +y_1x_3z_4-y_1z_3x_4-y_1x z_4+y_1x z_3+y_1z x_4-y_1z x_3\\[6pt]
&\qquad -z_1x_3y_4+z_1y_3x_4+z_1x y_4-z_1x y_3-z_1y x_4+z_1y x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
-x_3z_4+z_3x_4+x_1z_4-x_1z_3-z_1x_4+z_1x_3
\Big)
\end{aligned}
\]
z微分
\[
\begin{aligned}
\frac{\partial N_2}{\partial z}
&=
\frac{\partial L_2}{\partial z}
=
\frac{\partial(V_2/V)}{\partial z}
=
\frac{1}{V}\frac{\partial V_2}{\partial z}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial z}\frac{1}{6}
\Big(
x y_3z_4-x z_3y_4-y x_3z_4+y z_3x_4+z x_3y_4-z y_3x_4\\[6pt]
&\qquad -x_1y_3z_4+x_1z_3y_4+x_1y z_4-x_1y z_3-x_1z y_4+x_1z y_3\\[6pt]
&\qquad +y_1x_3z_4-y_1z_3x_4-y_1x z_4+y_1x z_3+y_1z x_4-y_1z x_3\\[6pt]
&\qquad -z_1x_3y_4+z_1y_3x_4+z_1x y_4-z_1x y_3-z_1y x_4+z_1y x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
x_3y_4-y_3x_4-x_1y_4+x_1y_3+y_1x_4-y_1x_3
\Big)
\end{aligned}
\]
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