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.
|polynomial order n||number of terms p|
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|
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:
- 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:
In order to generalise the procedure to obtain the shape functions, the areal coordinates is a very useful transformation.