Two-dimensional Shape Functions

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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:

$f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le n$

with: $\qquad p=\frac{(n+1)(n+2)}{2}$ the number of terms.

More specifically:

polynomial order n	number of terms p	$f(x,y) \,$
Constant: $0 \,$	$1 \,$	$\alpha \,$
Linear: $1 \,$	$3 \,$	$\alpha_1+\alpha_2 x + \alpha_3 y \,$
Quadratic: $2 \,$	$6 \,$	$\alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,$

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
	$1 \,$	$1 \,$
Linear	$x \qquad y \,$	$3 \,$
Quadratic	$x^2 \qquad 2 x y \qquad y^2\,$	$6 \,$
Cubic	$x^3 \qquad 3 x^2 y \qquad 3 x y^2 \qquad y^3\,$	$10 \,$
Quartic	$x^4 \qquad 4 x^3 y \qquad 6 x^2 y^2 \qquad 4 x y^3 \qquad y^4\,$	$15 \,$

Shape Functions for Triangular Elements

The Three Node Linear Triangle

The solution $\varphi^{(e)} (x,y)$ for each triangular element can be approached by their corresponding $\hat \varphi^{(e)} (x,y)$ to be expressed using the shape functions:

$\varphi^{(e)}(x,y) \cong \hat \varphi^{(e)} (x,y) = \sum_{i=1}^n N_i (x,y) \varphi^{(e)}_i$

If the shape functions are lineal polynomials (three-node triangular element, n=3), and remembering:

$N_i^{(e)}(x_j,y_j) = \begin{cases} 1, & i = j \\ 0, & i \ne j \end{cases}$

this expression can be written as:

$N_i^{(e)} (x,y) = \frac{1}{2 A^{(e)}} \left [ a_i^{(e)} + b_i^{(e)} x + c_i^{(e)} y \right ] \qquad$ with $A^{(e)} \,$ the element area and $i=1, 2, 3 \,$

And the system of equations is:

$\begin{bmatrix} 1 & x_1^{(e)} & y_1^{(e)} \\ 1 & x_2^{(e)} & y_2^{(e)} \\ 1 & x_3^{(e)} & y_3^{(e)} \end{bmatrix} \begin{bmatrix} a_1^{(e)} & a_2^{(e)} & a_3^{(e)} \\ b_1^{(e)} & b_2^{(e)} & b_3^{(e)} \\ c_1^{(e)} & c_2^{(e)} & c_3^{(e)} \end{bmatrix} = 2 ·A^{(e)}· \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

The element area is computed as the half of the determinant of the coordinates matrix:

$2 ·A^{(e)} = \begin{vmatrix} 1 & x_1^{(e)} & y_1^{(e)} \\ 1 & x_2^{(e)} & y_2^{(e)} \\ 1 & x_3^{(e)} & y_3^{(e)} \end{vmatrix}$

Finally, the different parameters can be expressed in terms of the nodal local coordinates as:

$a_i^{(e)}=x_j^{(e)}y_k^{(e)}-x_k^{(e)}y_j^{(e)}$

$b_i^{(e)}=y_j^{(e)}-y_k^{(e)}$

$c_i^{(e)}=x_k^{(e)}-x_j^{(e)}$

with $i=1,2,3; \quad j=2,3,1; \quad k=3,1,2 \,$

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 A₁, A₂ and A₃, then:

$L_1=\frac{A_1}{A}=\frac{\mathbf{CP}}{\mathbf{C}\mathbf{C'}} \qquad L_2=\frac{A_2}{A}=\frac{\mathbf{B}\mathbf{P}}{\mathbf{B}\mathbf{B'}} \qquad L_3=\frac{A_3}{A}=\frac{\mathbf{A}\mathbf{P}}{\mathbf{A}\mathbf{A'}}$

Note that:

A₁ + A₂ + A₃ = A
L₁ + L₂ + L₃ = 1
If P is the Centroid or Center of Mass of the triangle, then L₁ = L₂ = L₃ = 1/3

For the Finite Element Method is also interesting to note that:

$x_p=L_1 x_1 + L_2 x_2 + L_3 x_3 \,$

$y_p=L_1 y_1 + L_2 y_2 + L_3 y_3 \,$

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:

$\begin{bmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} L_1 \\ L_2 \\ L_3 \end{bmatrix} = \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$

$L_1= \frac{ \begin{vmatrix} x & x_2 & x_3 \\ y & y_2 & y_3 \\ 1 & 1 & 1 \end{vmatrix} } { \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{vmatrix} } = \frac{x (y_2-y_3)-y (x_2-x_3)+(x_2 y_3 - x_3 y_2)}{2A} = \frac{(x_2 y_3 - x_3 y_2) + x (y_2-y_3) + y (x_3 - x_2)}{2A}$

$L_2= \frac{ \begin{vmatrix} x_1 & x & x_3 \\ y_1 & y & y_3 \\ 1 & 1 & 1 \end{vmatrix} } { \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{vmatrix} } = \frac{x_1 (y-y_3)-y_1 (x-x_3)+(x y_3 - x_3 y)}{2A} = \frac{(x_3 y_1 - x_1 y_3) + x (y_3 - y_1) + y (x_1 - x_3)}{2A}$

$L_3= \frac{ \begin{vmatrix} x_1 & x_2 & x \\ y_1 & y_2 & y \\ 1 & 1 & 1 \end{vmatrix} } { \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{vmatrix} } = \frac{x_1 (y_2-y)-y_1 (x_2-x)+(x_2 y - x y_2)}{2A} = \frac{(x_1 y_2 - x_2 y_1) + x (y_1 - y_2) + y (x_2 - x_1)}{2A}$

$L_i (x,y) = \frac{1}{2 A} \left [ a_i + b_i x + c_i y \right ] = N_i(x,y)$

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: $N_2 = \frac{A_2}{A} = \frac{\frac{1 \times \alpha }{2}}{\frac{1 \times 1}{2}} \qquad N_3 = \frac{A_3}{A} = \frac{\frac{1 \times \beta}{2}}{\frac{1 \times 1}{2}} \qquad N_1 = \frac{A_1}{A} = \frac{A - A_2 - A_3}{A} = 1 - N_2 - N_3$

$N_1=1 - \alpha - \beta \qquad N_2=\alpha \qquad N_3=\beta \,$

$\mathbf{J^{(e)}} = \begin{bmatrix} - x_1 + x_2 & - y_1 + y_2 \\ - x_1 + x_3 & - y_1 + y_3 \end{bmatrix} \qquad \mathbf{|J^{(e)}|} = 2 A^{(e)}$

References

@@ Line 257: / Line 257: @@
 :<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>

Two-dimensional Shape Functions

Revision as of 19:29, 4 November 2009

Contents

Shape Functions for Triangular Elements

The Three Node Linear Triangle

Areal Coordinates

Natural Coordinates

References

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