四面体与六面体单元形函数推导及建模应用

1、推导四节点四面体单元和八节点六面体单元的形函数。

四节点四面体单元的形函数：
$h_i(x,y,z)$，$i = 1 - 4$

$$
\begin{aligned}
h_1(x,y,z) &= s_{11} + s_{21}x + s_{31}y + s_{41}z \
h_2(x,y,z) &= s_{12} + s_{22}x + s_{32}y + s_{42}z \
h_3(x,y,z) &= s_{13} + s_{23}x + s_{33}y + s_{43}z \
h_4(x,y,z) &= s_{14} + s_{24}x + s_{34}y + s_{44}z
\end{aligned}
$$

其中
$$
s_{11} = \frac{1}{6V} \det\begin{bmatrix} x_2 & y_2 & z_2 \ x_3 & y_3 & z_3 \ x_4 & y_4 & z_4 \end{bmatrix}
$$
$$
V = \frac{1}{6} \det\begin{bmatrix}
1 & x_1 & y_1 & z_1 \
1 & x_2 & y_2 & z_2 \
1 & x_3 & y_3 & z_3 \
1 & x_4 & y_4 & z_4
\end{bmatrix}
$$

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八节点六面体单元的形函数：
$$
\begin{aligned}
h_1(x,y,z) &= \frac{1}{8abc}(1 - x)(1 - y)(1 - z) \
h_2(x,y,z) &= \frac{1}{8abc}(1 + x)(1 - y)(1 + z) \
h_3(x,y,z) &= \frac{