4.1.2 Linear Shape Functions on Triangular Elements

Figure 4.1: Triangular element.

The triangular element and the corresponding notations used in this work are shown in Fig. <4.1>. The well known linear approximation of an unknown function $\phi$ within a triangle is given by

$$\phi(x,y) = a + bx + cy.$$ (4.25)

The linear element form functions will be obtained from this approximation. Assume the values of $\phi$ on the nodes of the triangle are known. The nodes are numbered by $1$ , $2$ and $3$ as in Fig. <4.1>, and the corresponding field values are $\phi_1$ , $\phi_2$ and $\phi_3$ . The coefficients $a$ , $b$ and $c$ are determined by the system

$$\begin{split}\phi_1& = a + bx_1 + cy_1 \phi_2& = a + bx_2 + cy_2 \phi_3& = a + bx_3 + cy_3, \end{split}$$ (4.26)

where $x_i$ and $y_i$ are the coordinates of the i-th node in the element. Solving (4.26) for $a$ , $b$ , and $c$ leads to

$$\begin{split}a & = \frac{1}{J}\left\vert \begin{array}{ccc} \... ...\phi_1 + (x_1 - x_3)\phi_2 + (x_2 - x_1)\phi_3}{J}. \end{split}$$ (4.27)

$J$ is the so called Jacobi determinant

$$J = \left\vert \begin{array}{ccc} 1 & x_1 & y_1 1 & x_2 & y_2 ... ...ot\vec{e}_z = \left(\vec{r}_{31}\times\vec{r}_{12}\right)\cdot\vec{e}_z = 2F_e,$$ (4.28)

where $F_e$ is the area of the triangular element and $\vec{r}_{ij} = \vec{r}_j - \vec{r}_i$ with $\vec{r}_i = x_i\vec{e}_x + y_i\vec{e}_y$ . With back substituting of $a$ , $b$ , and $c$ from (4.27) into (4.25) $\phi$ is written in the form

$$\begin{split}\phi(x,y)& = \frac{x_2y_3 - x_3y_2 + (y_2 - y_3)... ...phi_3 = \sum^3_{i=1}\lambda^e_i(x,y) \phi_i, \end{split}$$ (4.29)

which gives the element shape functions $\lambda^e_i$ .