# 2D formulation for Electrostatic Problems

From KratosWiki

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m (→2D formulation for Triangular Elements) |
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</math> | </math> | ||

+ | |||

+ | ::<math> | ||

+ | \mathbf{B}= | ||

+ | = | ||

+ | \frac{1}{2 A^{(e)}}} | ||

+ | \begin{bmatrix} | ||

+ | - y_1 + y_3 & - y_2 + y_1 \\ | ||

+ | - x_3 + x_1 & - x_1 + x_2 | ||

+ | \end{bmatrix} | ||

+ | \begin{bmatrix} | ||

+ | -1 & 1 & 0 \\ | ||

+ | -1 & 0 & 1 | ||

+ | \end{bmatrix} | ||

+ | = | ||

+ | \frac{1}{2 A^{(e)}}} | ||

+ | \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 18:11, 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:

**Failed to parse (syntax error): \mathbf{B}= = \frac{1}{2 A^{(e)}}} \begin{bmatrix} - y_1 + y_3 & - y_2 + y_1 \\ - x_3 + x_1 & - x_1 + x_2 \end{bmatrix} \begin{bmatrix} -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} = \frac{1}{2 A^{(e)}}} \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}**

### Stiffness Matrix K^{(e)}