Series: Finite Element Method › Shape Functions on Reference Elements

The Quad9 Element

The Quad9 element is a 9-node biquadratic quadrilateral. It lives on the same reference square as Quad4 but adds a node at the midpoint of each edge and one at the center, giving four corner nodes, four edge-midpoint nodes, and one interior node:

$$\hat{x}_0 = (0,0), \quad \hat{x}_1 = (1,0), \quad \hat{x}_2 = (1,1), \quad \hat{x}_3 = (0,1),$$

$$\hat{x}_4 = \bigl(\tfrac{1}{2},0\bigr), \quad \hat{x}_5 = \bigl(1,\tfrac{1}{2}\bigr), \quad \hat{x}_6 = \bigl(\tfrac{1}{2},1\bigr), \quad \hat{x}_7 = \bigl(0,\tfrac{1}{2}\bigr), \quad \hat{x}_8 = \bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr).$$

In the library, it is implemented in elements/quad9.py by the class Quad9.

The shape functions are built as tensor products of the 1D quadratic Lagrange basis polynomials on $[0,1]$ introduced in the Line3 post:

$$\ell_0(t) = (1-t)(1-2t), \qquad \ell_1(t) = t(2t-1), \qquad \ell_2(t) = 4t(1-t).$$

Each Quad9 shape function is $N_i(\hat{x},\hat{y}) = \ell_a(\hat{x})\,\ell_b(\hat{y})$ for the index pair $(a,b)$ that places node $i$ at $(\hat{x}_a, \hat{y}_b)$:

Node	$(\hat{x},\hat{y})$	$N_i$
0	$(0,0)$	$\ell_0(\hat{x})\,\ell_0(\hat{y})$
1	$(1,0)$	$\ell_1(\hat{x})\,\ell_0(\hat{y})$
2	$(1,1)$	$\ell_1(\hat{x})\,\ell_1(\hat{y})$
3	$(0,1)$	$\ell_0(\hat{x})\,\ell_1(\hat{y})$
4	$(\tfrac{1}{2},0)$	$\ell_2(\hat{x})\,\ell_0(\hat{y})$
5	$(1,\tfrac{1}{2})$	$\ell_1(\hat{x})\,\ell_2(\hat{y})$
6	$(\tfrac{1}{2},1)$	$\ell_2(\hat{x})\,\ell_1(\hat{y})$
7	$(0,\tfrac{1}{2})$	$\ell_0(\hat{x})\,\ell_2(\hat{y})$
8	$(\tfrac{1}{2},\tfrac{1}{2})$	$\ell_2(\hat{x})\,\ell_2(\hat{y})$

The gradients follow directly from the product rule. For any node with $N_i = \ell_a(\hat{x})\,\ell_b(\hat{y})$:

$$\frac{\partial N_i}{\partial \hat{x}} = \ell_a'(\hat{x})\,\ell_b(\hat{y}), \qquad \frac{\partial N_i}{\partial \hat{y}} = \ell_a(\hat{x})\,\ell_b'(\hat{y}),$$

where $\ell_0'(t) = 4t - 3$, $\ell_1'(t) = 4t - 1$, $\ell_2'(t) = 4 - 8t$ are the derivatives from Line3.

The nine functions form a partition of unity and span the full biquadratic space $\mathbb{Q}_2 = \mathrm{span}\{1, \hat{x}, \hat{y}, \hat{x}^2, \hat{x}\hat{y}, \hat{y}^2, \hat{x}^2\hat{y}, \hat{x}\hat{y}^2, \hat{x}^2\hat{y}^2\}$, so Quad9 can represent any biquadratic function exactly and achieves second-order accuracy. The face element is Line3, since each edge carries three nodes (two corners plus the edge midpoint) that together form a quadratic 1D element.