2D formulation for Electrostatic Problems
From KratosWiki
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)
- Failed to parse (syntax error): \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \partial \Omega^{(e)}= \frac{2 A^{(e)}}|}{2} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} = \frac{1}{2 A^{(e)}} \begin{bmatrix} - y_3 + y_2 & x_3 + x_2 \\ - y_1 + y_3 & - x_3 + x_1 \\ - y_2 + y_1 & - x_1 + x_2 \end{bmatrix} \begin{bmatrix} \displaystyle \varepsilon_x & 0 \\ \, \\ 0 & \displaystyle \varepsilon_y \end{bmatrix} \begin{bmatrix} - y_3 + y_2 & - y_1 + y_3 & - y_2 + y_1 \\ x_3 + x_2 & - x_3 + x_1 & - x_1 + x_2 \end{bmatrix}