# 2D formulation for Electrostatic Problems

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(→Stiffness Matrix K<sup>(e)</sup>) |
(→Source Vector f<sup>(e)</sup>) |
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− | ::Linear case ('''n<sub>p</sub>'''=1 integration point): | + | ::'''Linear case''' ('''n<sub>p</sub>'''=1 integration point): |

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

− | ::<math> | + | ::::<math> |

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

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

## Revision as of 19:33, 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}