# 2D formulation for Electrostatic Problems

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== 2D formulation for Triangular Elements == | == 2D formulation for Triangular Elements == | ||

+ | |||

+ | After applying the [[Numerical_Integration#Numerical_Integration_for_Isoparametric_Triangular_Domains numerical integration for triangular elements]] by using the [[Two-dimensional_Shape_Functions#Natural_Coordinates natural coordinates]], we obtain: | ||

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

+ | \begin{bmatrix} | ||

+ | \displaystyle \frac{\partial x}{\partial \alpha} & \displaystyle \frac{\partial y}{\partial \alpha} \\ \quad \\ | ||

+ | \displaystyle \frac{\partial x}{\partial \beta} & \displaystyle \frac{\partial y}{\partial \beta} | ||

+ | \end{bmatrix} | ||

+ | = | ||

+ | \begin{bmatrix} | ||

+ | - x_1 + x_2 & - y_1 + y_2 \\ | ||

+ | - x_1 + x_3 & - y_1 + y_3 | ||

+ | \end{bmatrix} | ||

+ | \qquad | ||

+ | \mathbf{|J^{(e)}|} = 2 A^{(e)} | ||

+ | </math> | ||

## Revision as of 19:30, 11 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#Numerical_Integration_for_Isoparametric_Triangular_Domains numerical integration for triangular elements by using the Two-dimensional_Shape_Functions#Natural_Coordinates natural coordinates, we obtain: