2D formulation for Electrostatic Problems
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− | ::<math> V (x,y | + | ::<math> V (x,y) \cong \hat V (x,y) = \sum_{i=0}^n N_i (x,y) a_i = \mathbf{N}^{(e)} · \mathbf{a}^{(e)}</math> |
Revision as of 15:08, 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.