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


and the boundary conditions


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.



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


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


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


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


Using the Lagrange polynomials we get


and finally


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*}

  • $\begingroup$ Thanks! What about the boundary conditions!? $\endgroup$
    – Mat Demon
    Nov 24 '19 at 12:18
  • $\begingroup$ The boundary conditions do not come into play for the shape functions calculation. $\endgroup$
    – user33403
    Nov 24 '19 at 12:56

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