# 2D formulation for Electrostatic Problems

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+ | ::::<math>\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} = | ||

+ | \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> | ||

## Revision as of 18:39, 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)}