# 2D formulation for Electrostatic Problems

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|\mathbf{J^{(e)}}| \sum_{p=1}^{n_p} \mathbf{N^T} \rho_S W_p = | |\mathbf{J^{(e)}}| \sum_{p=1}^{n_p} \mathbf{N^T} \rho_S W_p = | ||

</math> | </math> | ||

+ | |||

+ | :Linear case ('''n<sub>p</sub>'''=1 integration point): | ||

+ | |||

+ | ::<math>N=\left [ \frac{1}{3} \frac{1}{3} \frac{1}{3}\right ] \qquad W_i=\frac{1}{2}\,</math> | ||

::<math> | ::<math> | ||

− | + | \int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)} = | |

− | \frac{ | + | = 2 A^{(e)} \left [ \frac{1}{6} \frac{1}{6} \frac{1}{6}\right ]^T \rho_S = |

+ | = A^{(e)} \left [ \frac{\rho_S}{3} \frac{\rho_S}{3} \frac{\rho_S}{3}\right ]^T = | ||

</math> | </math> | ||

## Revision as of 19:31, 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)}

### Source Vector f^{(e)}

- Linear case (
**n**=1 integration point):_{p}