Two-dimensional Shape Functions
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# [http://en.wikipedia.org/wiki/Pascal%27s_triangle Pascal's triangle] | # [http://en.wikipedia.org/wiki/Pascal%27s_triangle Pascal's triangle] | ||
+ | # [http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29 Barycentric Coordinates ''(Areal Coordinates)''] | ||
[[Category: Shape Functions]] | [[Category: Shape Functions]] |
Revision as of 11:06, 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 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:
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:
- 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.