広告
・内挿関数N4の微分
4面体1次要素 の内挿関数N4の微分値を示します。
x微分
\[
\begin{aligned}
\frac{\partial N_4}{\partial x}
&=
\frac{\partial L_4}{\partial x}
=
\frac{\partial(V_4/V)}{\partial x}
=
\frac{1}{V}\frac{\partial V_4}{\partial x}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial x}\frac{1}{6}
\Big(
x_2y_3z-x_2z_3y-y_2x_3z+y_2z_3x+z_2x_3y-z_2y_3x\\[6pt]
&\qquad -x_1y_3z+x_1z_3y+x_1y_2z-x_1y_2z_3-x_1z_2y+x_1z_2y_3\\[6pt]
&\qquad +y_1x_3z-y_1z_3x-y_1x_2z+y_1x_2z_3+y_1z_2x-y_1z_2x_3\\[6pt]
&\qquad -z_1x_3y+z_1y_3x+z_1x_2y-z_1x_2y_3-z_1y_2x+z_1y_2x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
y_2z_3-z_2y_3-y_1z_3+y_1z_2+z_1y_3-z_1y_2
\Big)
\end{aligned}
\]
y微分
\[
\begin{aligned}
\frac{\partial N_4}{\partial y}
&=
\frac{\partial L_4}{\partial y}
=
\frac{\partial(V_4/V)}{\partial y}
=
\frac{1}{V}\frac{\partial V_4}{\partial y}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial y}\frac{1}{6}
\Big(
x_2y_3z-x_2z_3y-y_2x_3z+y_2z_3x+z_2x_3y-z_2y_3x\\[6pt]
&\qquad -x_1y_3z+x_1z_3y+x_1y_2z-x_1y_2z_3-x_1z_2y+x_1z_2y_3\\[6pt]
&\qquad +y_1x_3z-y_1z_3x-y_1x_2z+y_1x_2z_3+y_1z_2x-y_1z_2x_3\\[6pt]
&\qquad -z_1x_3y+z_1y_3x+z_1x_2y-z_1x_2y_3-z_1y_2x+z_1y_2x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
-x_2z_3+z_2x_3+x_1z_3-x_1z_2-z_1x_3+z_1x_2
\Big)
\end{aligned}
\]
z微分
\[
\begin{aligned}
\frac{\partial N_4}{\partial z}
&=
\frac{\partial L_4}{\partial z}
=
\frac{\partial(V_4/V)}{\partial z}
=
\frac{1}{V}\frac{\partial V_4}{\partial z}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial z}\frac{1}{6}
\Big(
x_2y_3z-x_2z_3y-y_2x_3z+y_2z_3x+z_2x_3y-z_2y_3x\\[6pt]
&\qquad -x_1y_3z+x_1z_3y+x_1y_2z-x_1y_2z_3-x_1z_2y+x_1z_2y_3\\[6pt]
&\qquad +y_1x_3z-y_1z_3x-y_1x_2z+y_1x_2z_3+y_1z_2x-y_1z_2x_3\\[6pt]
&\qquad -z_1x_3y+z_1y_3x+z_1x_2y-z_1x_2y_3-z_1y_2x+z_1y_2x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
x_2y_3-y_2x_3-x_1y_3+x_1y_2+y_1x_3-y_1x_2
\Big)
\end{aligned}
\]
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