# 2D formulation for Electrostatic Problems

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(→Source Vector f<sup>(e)</sup>) |
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\mathbf{N^{(e)}} = | \mathbf{N^{(e)}} = | ||

\begin{bmatrix} | \begin{bmatrix} | ||

− | N_1 | + | N_1 & N_2 & N_3 |

− | + | ||

− | + | ||

− | + | ||

− | + | ||

\end{bmatrix} | \end{bmatrix} | ||

= | = | ||

\begin{bmatrix} | \begin{bmatrix} | ||

− | L_1 | + | L_1 & L_2 & L_3 |

− | + | ||

− | + | ||

− | + | ||

− | + | ||

\end{bmatrix} | \end{bmatrix} | ||

= | = | ||

\begin{bmatrix} | \begin{bmatrix} | ||

− | 1-\alpha-\beta | + | 1-\alpha-\beta & \alpha & \beta |

− | + | ||

− | + | ||

− | + | ||

− | + | ||

\end{bmatrix} | \end{bmatrix} | ||

− | |||

\qquad | \qquad | ||

\mathbf{a^{(e)}} = | \mathbf{a^{(e)}} = |

## Revision as of 19:25, 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)}