# Two-dimensional Shape Functions

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 Revision as of 11:06, 4 November 2009 (view source)JMora (Talk | contribs) (→References)← Older edit Revision as of 11:32, 4 November 2009 (view source)JMora (Talk | contribs) (→Areal Coordinates)Newer edit → Line 122: Line 122: In order to generalise the procedure to obtain the shape functions, the areal coordinates is a very useful transformation. In order to generalise the procedure to obtain the shape functions, the areal coordinates is a very useful transformation. + + + In a triangle, areal or barycentric coordinates are defined as each of the partial subareas obtained by dividing the triangle in three sections. + + + That is, if we use a inner point '''''P''''' as the common vertex of the triangle of area '''A''' divided in three subareas '''A1''', '''A2''' and '''A3''', then: + + + [[Image:ArealCoordinates.jpg]] + + + + + are also known as area coordinates, because the coordinates of P with respect to triangle ABC are proportional to the (signed) areas of PBC, PCA and PAB. Areal and trilinear coordinates are used for similar purposes in geometry. == References == == References ==

## Revision as of 11:32, 4 November 2009

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

That is, a complete polynomial of nth 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:

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

The element area is computed as the half of the determinant of the coordinates matrix:

$2 ·A^{(e)} = \begin{vmatrix} 1 & x_1^{(e)} & y_1^{(e)} \\ 1 & x_2^{(e)} & y_2^{(e)} \\ 1 & x_3^{(e)} & y_3^{(e)} \end{vmatrix}$

Finally, the different parameters can be expressed in terms of the nodal local coordinates as:

$a_i^{(e)}=x_j^{(e)}y_k^{(e)}-x_k^{(e)}y_j^{(e)}$
$b_i^{(e)}=y_j^{(e)}-y_k^{(e)}$
$c_i^{(e)}=x_k^{(e)}-x_j^{(e)}$

with  $i=1,2,3; \quad j=2,3,1; \quad k=3,1,2 \,$

### Areal Coordinates

In order to generalise the procedure to obtain the shape functions, the areal coordinates is a very useful transformation.

In a triangle, areal or barycentric coordinates are defined as each of the partial subareas obtained by dividing the triangle in three sections.

That is, if we use a inner point P as the common vertex of the triangle of area A divided in three subareas A1</, A2</ and A3</, then:

are also known as area coordinates, because the coordinates of P with respect to triangle ABC are proportional to the (signed) areas of PBC, PCA and PAB. Areal and trilinear coordinates are used for similar purposes in geometry.