# 2D formulation for Electrostatic Problems

Line 71: | Line 71: | ||

\, \\ | \, \\ | ||

a_n | a_n | ||

+ | \end{bmatrix} | ||

+ | \qquad | ||

+ | \mathbf{B}= \left [ \mathbf{B_1 B_2 ... B_n} \right ] | ||

+ | \qquad | ||

+ | \mathbf{B_i}= | ||

+ | \begin{bmatrix} | ||

+ | \frac{\partial N_i}{\partial x} \\ | ||

+ | \, \\ | ||

+ | \frac{\partial N_i}{\partial y} | ||

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

</math> | </math> |

## Revision as of 15:07, 30 October 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):

This is:

with the infinit condition factor and the field produced whe is fixed by .

The weak form of this expression can be obtained using the integration by parts. In addition, if :

Remembering that:

is the gradient potential with:

and:

The electric field and electric displacement field can be written as follows:

We will now use the Galerkin Method . So, finally, the integral expression ready to create the matricial system of equations is:

Note that **K** is a coefficients matrix that depends on the geometrical and physical properties of the problem, **a** is the vector with the * n* unknowns to be obtained and

**f**is a vector that depends on the source values and boundary conditions.