広告
・内挿関数N1の微分
4面体1次要素 の内挿関数N1の微分値を示します。
x微分
\[
\begin{aligned}
\frac{\partial N_1}{\partial x}
&=
\frac{\partial L_1}{\partial x}
=
\frac{\partial(V_1/V)}{\partial x}
=
\frac{1}{V}\frac{\partial V_1}{\partial x}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial x}\frac{1}{6}
\Big(
x_2y_3z_4-x_2z_3y_4-y_2x_3z_4+y_2z_3x_4+z_2x_3y_4-z_2y_3x_4\\[6pt]
&\qquad -x y_3z_4+x z_3y_4+x y_2z_4-x y_2z_3-x z_2y_4+x z_2y_3\\[6pt]
&\qquad +y x_3z_4-y z_3x_4-y x_2z_4+y x_2z_3+y z_2x_4-y z_2x_3\\[6pt]
&\qquad -z x_3y_4+z y_3x_4+z x_2y_4-z x_2y_3-z y_2x_4+z y_2x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
-y_3z_4+z_3y_4+y_2z_4-y_2z_3-z_2y_4+z_2y_3
\Big)
\end{aligned}
\]
y微分
\[
\begin{aligned}
\frac{\partial N_1}{\partial y}
&=
\frac{\partial L_1}{\partial y}
=
\frac{\partial(V_1/V)}{\partial y}
=
\frac{1}{V}\frac{\partial V_1}{\partial y}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial y}\frac{1}{6}
\Big(
x_2y_3z_4-x_2z_3y_4-y_2x_3z_4+y_2z_3x_4+z_2x_3y_4-z_2y_3x_4\\[6pt]
&\qquad -x y_3z_4+x z_3y_4+x y_2z_4-x y_2z_3-x z_2y_4+x z_2y_3\\[6pt]
&\qquad +y x_3z_4-y z_3x_4-y x_2z_4+y x_2z_3+y z_2x_4-y z_2x_3\\[6pt]
&\qquad -z x_3y_4+z y_3x_4+z x_2y_4-z x_2y_3-z y_2x_4+z y_2x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
x_3z_4-z_3x_4-x_2z_4+x_2z_3+z_2x_4-z_2x_3
\Big)
\end{aligned}
\]
z微分
\[
\begin{aligned}
\frac{\partial N_1}{\partial z}
&=
\frac{\partial L_1}{\partial z}
=
\frac{\partial(V_1/V)}{\partial z}
=
\frac{1}{V}\frac{\partial V_1}{\partial z}
\\[6pt]
&=
\frac{1}{V}\frac{\partial}{\partial z}\frac{1}{6}
\Big(
x_2y_3z_4-x_2z_3y_4-y_2x_3z_4+y_2z_3x_4+z_2x_3y_4-z_2y_3x_4\\[6pt]
&\qquad -x y_3z_4+x z_3y_4+x y_2z_4-x y_2z_3-x z_2y_4+x z_2y_3\\[6pt]
&\qquad +y x_3z_4-y z_3x_4-y x_2z_4+y x_2z_3+y z_2x_4-y z_2x_3\\[6pt]
&\qquad -z x_3y_4+z y_3x_4+z x_2y_4-z x_2y_3-z y_2x_4+z y_2x_3
\Big)
\\[6pt]
&=
\frac{1}{6V}\Big(
-x_3y_4+y_3x_4+x_2y_4-x_2y_3-y_2x_4+y_2x_3
\Big)
\end{aligned}
\]
| prev | | | up | | | next |
