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: