Series: Finite Element Method › Shape Functions on Reference Elements

The Tri3 Element

The Tri3 element is the simplest triangular element: 3 nodes, one at each vertex of the reference triangle, with linear shape functions. The reference domain is the right triangle $\hat{T}$ with vertices $\hat{x}_0 = (0,0)$, $\hat{x}_1 = (1,0)$, $\hat{x}_2 = (0,1)$, matching the triangle reference domain.

In the library, it is implemented in elements/tri3.py by the class Tri3.

A natural coordinate system on $\hat{T}$ is provided by the barycentric coordinates $L_0$, $L_1$, $L_2$, defined by

$$L_0(\hat{x}, \hat{y}) = 1 - \hat{x} - \hat{y}, \qquad L_1(\hat{x}, \hat{y}) = \hat{x}, \qquad L_2(\hat{x}, \hat{y}) = \hat{y}.$$

Each $L_i$ equals one at vertex $i$ and zero at the other two, and they satisfy $L_0 + L_1 + L_2 = 1$ everywhere on $\hat{T}$. The Tri3 shape functions are exactly the barycentric coordinates:

$$N_0 = L_0 = 1 - \hat{x} - \hat{y}, \qquad N_1 = L_1 = \hat{x}, \qquad N_2 = L_2 = \hat{y}.$$

The partition of unity $N_0 + N_1 + N_2 = 1$ follows immediately. Since all three functions are linear, their reference-domain gradients are constant:

$$\nabla N_0 = \begin{pmatrix}-1\\-1\end{pmatrix}, \qquad \nabla N_1 = \begin{pmatrix}1\\0\end{pmatrix}, \qquad \nabla N_2 = \begin{pmatrix}0\\1\end{pmatrix}.$$

The figure below shows the three shape functions as filled contour plots over the reference triangle.

Any linear function $f$ on $\hat{T}$ is recovered exactly as $f = f(\hat{x}_0)\,N_0 + f(\hat{x}_1)\,N_1 + f(\hat{x}_2)\,N_2$. When the physical element has vertices $x_0$, $x_1$, $x_2$ in $\mathbb{R}^d$, the same combination $F_e(\hat{x}) = x_0 N_0 + x_1 N_1 + x_2 N_2$ gives the affine isoparametric mapping from $\hat{T}$ to the physical triangle.