2D formulation for Electrostatic Problems
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(Difference between revisions)
(→Stiffness Matrix K<sup>(e)</sup>) |
(→Stiffness Matrix K<sup>(e)</sup>) |
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::::<math>\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} = | ::::<math>\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} = | ||
\frac{1}{(2 A^{(e)})^2} | \frac{1}{(2 A^{(e)})^2} | ||
+ | \begin{bmatrix} | ||
+ | - y_3 + y_2 & x_3 + x_2 \\ | ||
+ | - y_1 + y_3 & - x_3 + x_1 \\ | ||
+ | - y_2 + y_1 & - x_1 + x_2 | ||
+ | \end{bmatrix} | ||
+ | \begin{bmatrix} | ||
+ | \displaystyle \varepsilon_x & 0 \\ | ||
+ | \, \\ | ||
+ | 0 & \displaystyle \varepsilon_y | ||
+ | \end{bmatrix} | ||
+ | \begin{bmatrix} | ||
+ | - y_3 + y_2 & - y_1 + y_3 & - y_2 + y_1 \\ | ||
+ | x_3 + x_2 & - x_3 + x_1 & - x_1 + x_2 | ||
+ | \end{bmatrix} | ||
+ | </math> | ||
+ | |||
+ | |||
+ | ::<math> | ||
+ | \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \partial \Omega^{(e)}= | ||
+ | \frac{2 A^{(e)}}|}{2} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} = | ||
+ | \frac{1}{2 A^{(e)}} | ||
\begin{bmatrix} | \begin{bmatrix} | ||
- y_3 + y_2 & x_3 + x_2 \\ | - y_3 + y_2 & x_3 + x_2 \\ |
Revision as of 18:42, 12 November 2009
The 2D Electrostatic Poisson's equation given by the governing PDE and its boundary conditions:
can be written as (see the General formulation for Electrostatic Problems):
with (n is the number of nodes of the element):
2D formulation for Triangular Elements
After applying the numerical integration for triangular elements by using the natural coordinates, we obtain:
Stiffness Matrix K(e)
- Failed to parse (syntax error): \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \partial \Omega^{(e)}= \frac{2 A^{(e)}}|}{2} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} = \frac{1}{2 A^{(e)}} \begin{bmatrix} - y_3 + y_2 & x_3 + x_2 \\ - y_1 + y_3 & - x_3 + x_1 \\ - y_2 + y_1 & - x_1 + x_2 \end{bmatrix} \begin{bmatrix} \displaystyle \varepsilon_x & 0 \\ \, \\ 0 & \displaystyle \varepsilon_y \end{bmatrix} \begin{bmatrix} - y_3 + y_2 & - y_1 + y_3 & - y_2 + y_1 \\ x_3 + x_2 & - x_3 + x_1 & - x_1 + x_2 \end{bmatrix}