Two-dimensional Shape Functions
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:
with: 
the number of terms.
More specifically:
| polynomial order n | number of terms p | 
|
|---|---|---|
Constant:  |
 |
|
| Linear: |
 |
|
Quadratic:  |
 |
|
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 |
|---|---|---|
| |
| |
| Linear |  |
|
| Quadratic |  |
|
| Cubic |  |
|
| Quartic |  |
|
Contents |
Shape Functions for Triangular Elements
The Three Node Linear Triangle
- The solution
for each triangular element can be approached by their corresponding 
to be expressed using the shape functions:
- If the shape functions are lineal polynomials (three-node triangular element, n=3), and remembering:
- 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](/images/math/3/a/b/3abefaeda8e986df0f7a7f6dea181d4a.png)
with 
the element area and 
- And the system of equations is:
- The element area is computed as the half of the determinant of the coordinates matrix:
- Finally, the different parameters can be expressed in terms of the nodal local coordinates as:
- with
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 of the triangle of area A as the common vertex of the three subareas A1, A2 and A3, then:
- Failed to parse (unknown function\matbf): L_1=\frac{A_1}{A}=\frac{\matbf{CP}}{\matbf{CC'}} \qquad L_2=\frac{A_2}{A}=\frac{\matbf{BP}}{\matbf{BB'}} \qquad L_3=\frac{A_3}{A}=\frac{\matbf{AP}}{\matbf{AA'}}
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.
