Two-dimensional Shape Functions

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Shape functions are selected to fit as exact as possible the Finite Element Solution. If this solution is a combination of polynomial functions of n^th order, these functions should include a complete polynomial of equal order.

That is, a complete polynomial of n^th order can be written as:

$f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le n$

with: $\qquad p=\frac{(n+1)(n+2)}{2}$ the number of terms.

More specifically:

polynomial order n	number of terms p	$f(x,y) \,$
Constant: $0 \,$	$1 \,$	$\alpha \,$
Linear: $1 \,$	$3 \,$	$\alpha_1+\alpha_2 x + \alpha_3 y \,$
Quadratic: $2 \,$	$6 \,$	$\alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,$

A quick way to easily obtain the terms of a complete polynomial is by using the Pascal's triangle:

order n	new polynomial terms	number of terms p
	$1 \,$	$1 \,$
Linear	$x \qquad y \,$	$3 \,$
Quadratic	$x^2 \qquad 2 x y \qquad y^2\,$	$6 \,$
Cubic	$x^3 \qquad 3 x^2 y \qquad 3 x y^2 \qquad y^3\,$	$10 \,$
Quartic	$x^4 \qquad 4 x^3 y \qquad 6 x^2 y^2 \qquad 4 x y^3 \qquad y^4\,$	$15 \,$

Shape Functions for Triangular Elements

The Three Node Linear Triangle

The solution $\varphi^{(e)} (x,y)$ for each triangular element can be approached by their corresponding $\hat \varphi^{(e)} (x,y)$ to be expressed using the shape functions:

$\varphi^{(e)}(x,y) \cong \hat \varphi^{(e)} (x,y) = \sum_{i=1}^n N_i (x,y) \varphi^{(e)}_i$

If the shape functions are lineal polynomials (three-node triangular element, n=3), and remembering:

$N_i^{(e)}(x_j,y_j) = \begin{cases} 1, & i = j \\ 0, & i \ne j \end{cases}$

this expression can be written as:

$N_i^{(e)} (x,y) = \frac{1}{2 A^{(e)}} \left [ a_i^{(e)} + b_i^{(e)} x + c_i^{(e)} y \right ] \qquad$ with $A^{(e)} \,$ the element area and $i=1, 2, 3 \,$

And the system of equations is:

$\begin{bmatrix} 1 & x_1^{(e)} & y_1^{(e)} \\ 1 & x_2^{(e)} & y_2^{(e)} \\ 1 & x_3^{(e)} & y_3^{(e)} \end{bmatrix} \begin{bmatrix} a_1^{(e)} & a_2^{(e)} & a_3^{(e)} \\ b_1^{(e)} & b_2^{(e)} & b_3^{(e)} \\ c_1^{(e)} & c_2^{(e)} & c_3^{(e)} \end{bmatrix} = 2 ·A^{(e)}· \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$