# 2D formulation for Electrostatic Problems

(Difference between revisions)
 Revision as of 15:24, 30 October 2009 (view source)JMora (Talk | contribs)← Older edit Revision as of 15:24, 30 October 2009 (view source)JMora (Talk | contribs) Newer edit → Line 8: Line 8: \begin{cases} \begin{cases} \left . V - \bar V = 0 \right |_{\Gamma_{V}}  & in ~ \Gamma_{\varphi} \\ \left . V - \bar V = 0 \right |_{\Gamma_{V}}  & in ~ \Gamma_{\varphi} \\ + \, \\ \left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}}  & in ~ \Gamma_{q} \\ \left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}}  & in ~ \Gamma_{q} \\ − \left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r^{exponent}} & in ~ \Gamma_{\infty} + \, \\ + \left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r^{exp}} & in ~ \Gamma_{\infty} \end{cases} \end{cases} [/itex] [/itex] Line 85: Line 87: − ::$\alpha = \frac{1}{|r-r_0|^{exponent}} \qquad with \quad exponent=0.5, 1, 2...$ + ::$\alpha = \frac{1}{|r-r_0|^{exp}} \qquad with \quad exp=0.5, 1, 2...$

## Revision as of 15:24, 30 October 2009

The 2D Electrostatic Poisson's equation given by the governing PDE and its boundary conditions: $A(V) = \left[ \frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial}{\partial y} \right) \right]V(x,y) + \rho_S = 0 ~~ in ~ \Omega$ $B(V) = \begin{cases} \left . V - \bar V = 0 \right |_{\Gamma_{V}} & in ~ \Gamma_{\varphi} \\ \, \\ \left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}} & in ~ \Gamma_{q} \\ \, \\ \left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r^{exp}} & in ~ \Gamma_{\infty} \end{cases}$

can be written as (see the General formulation for Electrostatic Problems): ${ \int_{\Omega} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \mathbf{a} \partial \Omega + \oint_{\Gamma_{\infty}} \mathbf{N^T} \alpha \mathbf{N} \mathbf{a} \partial \Gamma_{\infty} = \int_{\Omega} \mathbf{N^T} \rho_v \partial \Omega - \oint_{\Gamma_q} \mathbf{N^T} \bar D_n \partial \Gamma_q - \oint_{\Gamma_V} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V }$ $\mathbf{K} \mathbf{a} \,= \mathbf{f}$ $\mathbf{K}^{(e)}= \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \partial \Omega^{(e)} + \oint_{\Gamma_{\infty}^{(e)}} \mathbf{N^T} \alpha \mathbf{N} \partial \Gamma_{\infty}^{(e)}$ $\mathbf{f}^{(e)}= \int_{\Omega^{(e)}} \mathbf{N^T} \rho_v \partial \Omega^{(e)} - \oint_{\Gamma_q^{(e)}} \mathbf{N^T} \bar D_n \partial \Gamma_q^{(e)} - \oint_{\Gamma_V^{(e)}} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V^{(e)}$

with (n is the number of nodes of the element): $V (x,y) \cong \hat V (x,y) = \sum_{i=0}^n N_i (x,y) a_i = \mathbf{N}^{(e)} · \mathbf{a}^{(e)}$ $\mathbf{N^{(e)}} = \begin{bmatrix} N_1 \\ \, \\ N_2 \\ \, \\ \vdots \\ \, \\ N_n \end{bmatrix} \qquad \mathbf{a^{(e)}} = \begin{bmatrix} a_1 \\ \, \\ a_2 \\ \, \\ \vdots \\ \, \\ a_n \end{bmatrix} \qquad \mathbf{B}= \left [ \mathbf{B_1 B_2 ... B_n} \right ] \qquad \mathbf{B_i}= \begin{bmatrix} \frac{\partial N_i}{\partial x} \\ \, \\ \frac{\partial N_i}{\partial y} \end{bmatrix}$ $\alpha = \frac{1}{|r-r_0|^{exp}} \qquad with \quad exp=0.5, 1, 2...$