4.2.1 Linear Shape Functions on Tetrahedral Elements

Figure 4.7: Tetrahedral element.

The unknown field in the tetrahedral element (Fig. <4.7>) is approximated by the linear function

$$\phi(x,y,z) = a + bx + cy + dz.$$ (4.58)

Analogously to the two-dimensional case the four coefficients $a$ , $b$ , $c$ and $d$ are obtained assuming that the field values $\phi_1$ , $\phi_2$ , $\phi_3$ and $\phi_4$ on the four vertexes of the tetrahedron are known.

$$\begin{split}\phi_1& = a + bx_1 + cy_1 + dz_1 \phi_2& = a ... ... + cy_3 + dz_3 \phi_4& = a + bx_4 + cy_4 + dz_4. \end{split}$$ (4.59)

The coordinates $x_i$ , $y_i$ and $z_i$ correspond to the vertex $i$ . From (4.59) the coefficients $a$ , $b$ , $c$ and $d$ become

$$\begin{split}a = \frac{1}{J} \left\vert\begin{array}{cccc} \p... ...uad a_1\phi_1 + a_2\phi_2 + a_3\phi_3 + a_4\phi_4 \end{split}$$ (4.60)

$$\begin{split}b = \frac{1}{J} \left\vert\begin{array}{cccc} 1 ... ...uad b_1\phi_1 + b_2\phi_2 + b_3\phi_3 + b_4\phi_4 \end{split}$$ (4.61)

$$\begin{split}c = \frac{1}{J}\left\vert\begin{array}{cccc} 1 &... ...uad c_1\phi_1 + c_2\phi_2 + c_3\phi_3 + c_4\phi_4 \end{split}$$ (4.62)

$$\begin{split}d = \frac{1}{J}\left\vert\begin{array}{cccc} 1 &... ...ad d_1\phi_1 + d_2\phi_2 + d_3\phi_3 + d_4\phi_4. \end{split}$$ (4.63)

The Jacoby determinant for the three-dimensional case is given by

$$J = \left\vert \begin{array}{cccc} 1 & x_1 & y_1 & z_1 1 & x_2... ... z_4 \end{array} \right\vert = (\vec{r}_1\times\vec{r}_2)\cdot\vec{r}_3 = 6V_e,$$ (4.64)

where $V_e$ is the volume of the tetrahedron. Equations (4.60) to (4.63) define the auxiliary coefficients $a_i$ , $b_i$ , $c_i$ and $d_i$ ($i\in[1;4]$ ), which are used to write (4.58) as

$$\begin{split}\phi(x,y,z) & = (a_1 + b_1x + c_1y + d_1z)\phi_1... ..._4y + d_4z)\phi_4 = \sum^4_{i=1}\lambda^e_i \phi_i \end{split}$$ (4.65)

to introduce the element shape functions $\lambda^e_i$ .