# 2D formulation for Electrostatic Problems

From KratosWiki

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(→Stiffness Matrix K<sup>(e)</sup>) |
(→2D formulation for Triangular Elements) |
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Line 144: | Line 144: | ||

a_3 | a_3 | ||

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

− | + | </math> | |

+ | |||

+ | ::<math> | ||

\mathbf{B}= | \mathbf{B}= | ||

\begin{bmatrix} | \begin{bmatrix} | ||

Line 154: | Line 156: | ||

\displaystyle \frac{\partial N_2}{\partial y} & | \displaystyle \frac{\partial N_2}{\partial y} & | ||

\displaystyle \frac{\partial N_3}{\partial y} | \displaystyle \frac{\partial N_3}{\partial y} | ||

+ | \end{bmatrix} | ||

+ | = | ||

+ | \frac{1}{|\mathbf{J^{(e)}}|} | ||

+ | \begin{bmatrix} | ||

+ | \displaystyle \frac{\partial y}{\partial \beta} & \displaystyle -\frac{\partial y}{\partial \alpha} \\ | ||

+ | \displaystyle -\frac{\partial x}{\partial \beta} & \displaystyle \frac{\partial x}{\partial \alpha} | ||

+ | \end{bmatrix} | ||

+ | \begin{bmatrix} | ||

+ | \displaystyle \frac{\partial N_1}{\partial \alpha} & | ||

+ | \displaystyle \frac{\partial N_2}{\partial \alpha} & | ||

+ | \displaystyle \frac{\partial N_3}{\partial \alpha}\\ | ||

+ | \, \\ | ||

+ | \displaystyle \frac{\partial N_1}{\partial \beta} & | ||

+ | \displaystyle \frac{\partial N_2}{\partial \beta} & | ||

+ | \displaystyle \frac{\partial N_3}{\partial \beta} | ||

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

</math> | </math> |

## Revision as of 16:41, 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:

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