Two-dimensional Shape Functions
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(Difference between revisions)
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− | + | == Shape Functions for Triangular Elements == | |
+ | === The Three Node Linear Triangle === | ||
− | :<math> | + | :The solution <math> \varphi^{(e)} (x,y)</math> for each triangular element can be approached by their corresponding <math>\hat \varphi^{(e)} (x,y)</math> to be expressed using the shape functions: |
+ | ::<math> \varphi^{(e)}(x,y) \cong \hat \varphi^{(e)} (x,y) = \sum_{i=1}^n N_i (x,y) \varphi^{(e)}_i </math> | ||
− | + | :If the shape functions are lineal polynomials (three-node triangular element, n=3), and remembering: | |
+ | ::<math> | ||
+ | N_i^{(e)}(x_j,y_j) = | ||
+ | \begin{cases} | ||
+ | 1, & i = j \\ | ||
+ | 0, & i \ne j | ||
+ | \end{cases} | ||
+ | </math> | ||
− | + | :this expression can be written as: | |
+ | ::<math>N_i^{(e)} (x,y) = \frac{1}{2 A^{(e)}} \left [ a_i^{(e)} + b_i^{(e)} x + c_i^{(e)} y \right ] \qquad</math> with <math>A^{(e)} \,</math> the element area and <math>i=1, 2, 3 \,</math> | ||
+ | |||
+ | :And the system of equations is: | ||
+ | |||
+ | |||
+ | ::<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | |||
+ | |||
+ | :The element area is computed as the half of the determinant of the coordinates matrix: | ||
+ | |||
+ | |||
+ | ::<math>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} | ||
+ | </math> | ||
+ | |||
+ | |||
+ | :[[Image:Three node triangle.jpg]] | ||
+ | |||
+ | |||
+ | :Finally, the different parameters can be expressed in terms of the nodal local coordinates as: | ||
+ | |||
+ | |||
+ | ::<math>a_i^{(e)}=x_j^{(e)}y_k^{(e)}-x_k^{(e)}y_j^{(e)}</math> | ||
+ | |||
+ | ::<math>b_i^{(e)}=y_j^{(e)}-y_k^{(e)}</math> | ||
+ | |||
+ | ::<math>c_i^{(e)}=x_k^{(e)}-x_j^{(e)}</math> | ||
+ | |||
+ | |||
+ | :with <math>i=1,2,3; \quad j=2,3,1; \quad k=3,1,2 \,</math> | ||
Revision as of 10:39, 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 |
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