Two-dimensional Shape Functions
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That is, if we use a inner point '''''P'''''
That is, if we use a inner point '''''P''''' of the triangle of area '''A''' three subareas '''A<sub>1</sub>''', '''A<sub>2</sub>''' and '''A<sub>3</sub>''', then:
Revision as of 11:34, 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:
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.
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:
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.