From KratosWiki
(Difference between revisions)


Line 257: 
Line 257: 
   
 :<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> 
− 
 
− 
 
−  === 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(L_1,L_2,L_3) x_i \qquad y = \sum_{i=1}^n N_i(L_1,L_2,L_3) 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>
 
   
   
Revision as of 19:29, 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 (threenode 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:


References
 Pascal's triangle
 Barycentric Coordinates (Areal Coordinates)
 Centroid
 Jacobian