# 2D formulation for Electrostatic Problems

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\frac{\partial N_3}{\partial \beta}=1 | \frac{\partial N_3}{\partial \beta}=1 | ||

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

+ | |||

+ | |||

+ | ::<math> | ||

+ | x = N_1 x_1 + N_2 x_2 + N_3 x_3 = ( 1 - \alpha - \beta) x_1 + \alpha x_2 + \beta x_3 | ||

+ | </math> | ||

+ | |||

+ | |||

+ | ::<math> | ||

+ | y = N_1 y_1 + N_2 y_2 + N_3 y_3 = ( 1 - \alpha - \beta) y_1 + \alpha y_2 + \beta y_3 | ||

+ | </math> | ||

+ | |||

::<math>\mathbf{J^{(e)}} = | ::<math>\mathbf{J^{(e)}} = |

## Revision as of 18:06, 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:

*x*=*N*_{1}*x*_{1}+*N*_{2}*x*_{2}+*N*_{3}*x*_{3}= (1 − α − β)*x*_{1}+ α*x*_{2}+ β*x*_{3}

*y*=*N*_{1}*y*_{1}+*N*_{2}*y*_{2}+*N*_{3}*y*_{3}= (1 − α − β)*y*_{1}+ α*y*_{2}+ β*y*_{3}

### Stiffness Matrix K^{(e)}