Four-noded rectangular element shape functions

Question

I works on a project where I need to compute a modal analysis of an acoustic cavity. The cavity is rigid which translates the problem to the following equation

$$\frac{\partial^2p}{\partial{}x^2}+\frac{\partial^2p}{\partial{}y^2}+k^2p=0$$

and the boundary conditions

$$\partial_xp(y=0)=0,\quad\partial_xp(y=L_y)=0,\quad\partial_yp(x=0)=0,\quad\partial_yp(x=L_x).$$

I'm trying to solve the problem with the finite element method as especially with four-noded rectangular elements. I don't how to get the shape function for this rectangular 2D mesh since we only learnt the triangular mesh in class.

Regards.

Answer 1

I think the easiest way to calculate the shape functions is to go through the Lagrange polynomials. The pressure $p$ in element $e$ reads

$$p^e(x,y)\approx\sum_{i=1}^{i=4}N_i(x,y)p_i.$$

Let's start with the shape function of the first node $N_1(x,y)$. We want to generate a function with the properties

$$N_1(x_1,y_1)=1,\qquad{}N_1(x_{i\neq{}1},y_i{\neq{}1})=0.$$

If we proceed by separation of variable we can write the shape function as

$$N_1(x,y)=\ell_x(x)\ell_y(y)$$

and then we must find two functions $\ell_x(x)$ and $\ell_y(y)$ verifying

$$\ell_x(x_1)=1,\qquad{}\ell_y(y_1)=1.$$

Using the Lagrange polynomials we get

$$\ell_x(x)=\frac{x-x_2}{x_1-x_2},\qquad\ell_y(y)=\frac{y-y_4}{y_1-y_4}$$

and finally

$$N_1(x,y)=\frac{x-x_2}{x_1-x_2}\frac{y-y_4}{y_1-y_4}=\frac{1}{\Delta{}x\Delta{}y}(x-x_2)(y-y_4).$$

Using the same approach for the other shape functions leads to \begin{align*} &N_2=-\frac{1}{\Delta{}x\Delta{}y}(x-x_1)(y-y_3),\\ &N_3=\frac{1}{\Delta{}x\Delta{}y}(x-x_4)(y-y_2),\\ &N_4=-\frac{1}{\Delta{}x\Delta{}y}(x-x_3)(y-y_1). \end{align*}