Series: Finite Element Method › Shape Functions on Reference Elements

The Tri6 Element

The Tri6 element is a 6-node quadratic triangle. It lives on the same reference triangle as Tri3 but adds a node at the midpoint of each edge, giving three vertex nodes and three edge-midpoint nodes:

$$\hat{x}_0 = (0,0), \quad \hat{x}_1 = (1,0), \quad \hat{x}_2 = (0,1),$$

$$\hat{x}_3 = \bigl(\tfrac{1}{2},0\bigr), \quad \hat{x}_4 = \bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr), \quad \hat{x}_5 = \bigl(0,\tfrac{1}{2}\bigr).$$

In the library, it is implemented in elements/tri6.py by the class Tri6.

The shape functions are built from the barycentric coordinates $L_0 = 1 - \hat{x} - \hat{y}$, $L_1 = \hat{x}$, $L_2 = \hat{y}$ introduced in the Tri3 post. The pattern mirrors the quadratic Lagrange basis polynomials on $[0, 1]$: vertex functions use $L_i(2L_i - 1)$ and edge-midpoint functions use $4L_i L_j$:

$$N_0 = L_0(2L_0 - 1), \quad N_1 = L_1(2L_1 - 1), \quad N_2 = L_2(2L_2 - 1),$$

$$N_3 = 4L_0 L_1, \quad N_4 = 4L_1 L_2, \quad N_5 = 4L_0 L_2.$$

Node $i$ of each edge is shared between the two adjacent vertex functions; the edge-midpoint function reaches its maximum of one at the midpoint and vanishes at both endpoints. The figure below shows all six nodes on the reference triangle.

The gradients follow from the chain rule applied to the barycentric coordinates. For the vertex functions,

$$\nabla N_0 = -(4L_0 - 1)\begin{pmatrix}1\\1\end{pmatrix}, \qquad \nabla N_1 = (4L_1 - 1)\begin{pmatrix}1\\0\end{pmatrix}, \qquad \nabla N_2 = (4L_2 - 1)\begin{pmatrix}0\\1\end{pmatrix},$$

and for the edge-midpoint functions,

$$\nabla N_3 = 4\begin{pmatrix}1 - 2\hat{x} - \hat{y}\\-\hat{x}\end{pmatrix}, \qquad \nabla N_4 = 4\begin{pmatrix}\hat{y}\\\hat{x}\end{pmatrix}, \qquad \nabla N_5 = 4\begin{pmatrix}-\hat{y}\\1 - \hat{x} - 2\hat{y}\end{pmatrix}.$$

The six shape functions form a partition of unity and span the full space of quadratic polynomials on the triangle, so Tri6 can represent any quadratic function exactly.