Two-dimensional Shape Functions
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+ | == Derivatives of the shape functions == | ||
+ | For isoparametric elements (those using the same shape functions to interpolate the geometry and the unknowns), we have: | ||
+ | |||
+ | |||
+ | ::<math>x=\sum_{i=1}^n N_i(\alpha,\beta) x_i \qquad y=\sum_{i=1}^n N_i(\alpha,\beta) y_i </math> | ||
+ | |||
+ | |||
+ | To obtain the derivatives of the shape functions: | ||
+ | |||
+ | |||
+ | ::<math>\frac{\partial N_i}{\partial \alpha} = \frac{\partial N_i}{\partial x} \frac{\partial x}{\partial \alpha} + \frac{\partial N_i}{\partial y} \frac{\partial y}{\partial \alpha}</math> | ||
+ | |||
+ | ::<math>\frac{\partial N_i}{\partial \beta} = \frac{\partial N_i}{\partial x} \frac{\partial x}{\partial \beta} + \frac{\partial N_i}{\partial y} \frac{\partial y}{\partial \beta}</math> | ||
+ | |||
+ | |||
+ | ::<math> | ||
+ | \begin{Bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial \alpha} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial \beta} | ||
+ | \end{Bmatrix} | ||
+ | = | ||
+ | \underbrace{ | ||
+ | \begin{bmatrix} | ||
+ | \displaystyle \frac{\partial x}{\partial \alpha} & \displaystyle \frac{\partial y}{\partial \alpha} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial x}{\partial \beta} & \displaystyle \frac{\partial y}{\partial \beta} | ||
+ | \end{bmatrix} | ||
+ | }_{\mathbf{J^{(e)}}} | ||
+ | \begin{Bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial x} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial y} | ||
+ | \end{Bmatrix} | ||
+ | = | ||
+ | \mathbf{J^{(e)}} | ||
+ | \begin{Bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial x} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial y} | ||
+ | \end{Bmatrix} | ||
+ | </math> | ||
+ | |||
+ | |||
+ | with <math>\mathbf{J^{(e)}} \,</math> the Jacobian matrix with a determinant <math>\mathbf{|J^{(e)}|} \,</math> | ||
+ | |||
+ | |||
+ | ::<math> | ||
+ | \begin{Bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial x} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial y} | ||
+ | \end{Bmatrix} | ||
+ | = | ||
+ | \begin{bmatrix} | ||
+ | \mathbf{J^{(e)}} | ||
+ | \end{bmatrix}^{-1} | ||
+ | \begin{Bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial \alpha} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial \beta} | ||
+ | \end{Bmatrix} | ||
+ | = | ||
+ | \displaystyle | ||
+ | \frac{1}{ \mathbf{|J^{(e)}|}} | ||
+ | \begin{bmatrix} | ||
+ | \displaystyle \frac{\partial y}{\partial \beta} & \displaystyle -\frac{\partial y}{\partial \alpha} \\ \quad \\ | ||
+ | \displaystyle -\frac{\partial x}{\partial \beta} & \displaystyle \frac{\partial x}{\partial \alpha} | ||
+ | \end{bmatrix} | ||
+ | \begin{Bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial \alpha} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial \beta} | ||
+ | \end{Bmatrix} | ||
+ | </math> | ||
+ | |||
+ | |||
+ | From here is easy to obtain: | ||
+ | |||
+ | |||
+ | ::<math>\partial x \partial y = \mathbf{|J^{(e)}|} \partial \alpha \partial \beta \,</math> | ||
+ | |||
+ | |||
+ | ::<math>\frac{\partial x}{\partial \alpha} = \sum_{i=1}^n \frac{\partial N_i}{\partial \alpha} x_i \qquad | ||
+ | \frac{\partial x}{\partial \beta} = \sum_{i=1}^n \frac{\partial N_i}{\partial \beta} x_i</math> | ||
+ | |||
+ | |||
+ | ::<math>\frac{\partial y}{\partial \alpha} = \sum_{i=1}^n \frac{\partial N_i}{\partial \alpha} y_i \qquad | ||
+ | \frac{\partial y}{\partial \beta} = \sum_{i=1}^n \frac{\partial N_i}{\partial \beta} y_i</math> | ||
+ | |||
+ | |||
+ | ::<math>\mathbf{J^{(e)}} = \sum_{i=1}^n | ||
+ | \begin{bmatrix} | ||
+ | \displaystyle \frac{\partial N_i}{\partial \alpha} x_i & \displaystyle \frac{\partial N_i}{\partial \alpha} y_i \\ \quad \\ | ||
+ | \displaystyle \frac{\partial N_i}{\partial \beta} x_i & \displaystyle \frac{\partial N_i}{\partial \beta} y_i | ||
+ | \end{bmatrix} | ||
+ | </math> | ||
== Shape Functions for Triangular Elements == | == Shape Functions for Triangular Elements == | ||
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− | :::[[Image: | + | :::[[Image:NaturalCoordinates_2.jpg|400 px]] |
− | Therefore: | + | Therefore: <math>N_2 = \frac{A_2}{A} = \frac{\frac{1 \times \alpha }{2}}{\frac{1 \times 1}{2}} \qquad N_3 = \frac{A_3}{A} = \frac{\frac{1 \times \beta}{2}}{\frac{1 \times 1}{2}} \qquad N_1 = \frac{A_1}{A} = \frac{A - A_2 - A_3}{A} = 1 - N_2 - N_3 </math> |
:<math>N_1=1 - \alpha - \beta \qquad N_2=\alpha \qquad N_3=\beta \,</math> | :<math>N_1=1 - \alpha - \beta \qquad N_2=\alpha \qquad N_3=\beta \,</math> | ||
+ | |||
+ | |||
+ | For isoparametric elements, we have: | ||
+ | |||
+ | |||
+ | :<math>x = \sum_{i=1}^n N_i(L_1,L_2,L_3) x_i = N_1(\alpha,\beta) x_1 + N_2(\alpha,\beta) x_2 + N_3(\alpha,\beta) x_3 = (1 - \alpha - \beta) x_1 + \alpha x_2 + \beta x_3</math> | ||
+ | |||
+ | :<math>y = \sum_{i=1}^n N_i(L_1,L_2,L_3) y_i = N_1(\alpha,\beta) y_1 + N_2(\alpha,\beta) y_2 + N_3(\alpha,\beta) y_3 = (1 - \alpha - \beta) y_1 + \alpha y_2 + \beta y_3</math> | ||
+ | |||
+ | |||
+ | To obtain the Jacobian of these shape functions: | ||
+ | |||
+ | ::<math>\mathbf{J^{(e)}} = | ||
+ | \begin{bmatrix} | ||
+ | \displaystyle \frac{\partial x}{\partial \alpha} & \displaystyle \frac{\partial y}{\partial \alpha} \\ \quad \\ | ||
+ | \displaystyle \frac{\partial x}{\partial \beta} & \displaystyle \frac{\partial y}{\partial \beta} | ||
+ | \end{bmatrix} | ||
+ | = | ||
+ | \begin{bmatrix} | ||
+ | - x_1 + x_2 & - y_1 + y_2 \\ | ||
+ | - x_1 + x_3 & - y_1 + y_3 | ||
+ | \end{bmatrix} | ||
+ | \qquad | ||
+ | \mathbf{|J^{(e)}|} = 2 A^{(e)} | ||
+ | </math> | ||
== References == | == References == | ||
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# [http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29 Barycentric Coordinates ''(Areal Coordinates)''] | # [http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29 Barycentric Coordinates ''(Areal Coordinates)''] | ||
# [http://en.wikipedia.org/wiki/Centroid Centroid] | # [http://en.wikipedia.org/wiki/Centroid Centroid] | ||
+ | # [http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant Jacobian] | ||
[[Category: Shape Functions]] | [[Category: Shape Functions]] |
Latest revision as of 18:35, 11 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 |
Derivatives of the shape functions
For isoparametric elements (those using the same shape functions to interpolate the geometry and the unknowns), we have:
To obtain the derivatives of the shape functions:
with the Jacobian matrix with a determinant
From here is easy to obtain:
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.
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 A_{1}, A_{2} and A_{3}, then:
Note that:
- A_{1} + A_{2} + A_{3} = A
- L_{1} + L_{2} + L_{3} = 1
- If P is the Centroid or Center of Mass of the triangle, then L_{1} = L_{2} = L_{3} = 1/3
For the Finite Element Method is also interesting to note that:
with x_{p} and y_{p} the coordinates of P or any other point inside the triangle (x,y). This is equivalent to the following system of equations:
that is exactly the shape functions for a triangular element of three nodes.
Natural Coordinates
It is usual to define the triangle in terms of a normalised geometry (natural coordinates) as is showed in the figure:
Therefore:
For isoparametric elements, we have:
To obtain the Jacobian of these shape functions: