# 2D formulation for Electrostatic Problems

(Difference between revisions)
 Revision as of 15:08, 30 October 2009 (view source)JMora (Talk | contribs)← Older edit Revision as of 15:09, 30 October 2009 (view source)JMora (Talk | contribs) Newer edit → Line 85: Line 85: − − − ::$\mathbf{B_i}= − \begin{bmatrix} − \frac{\partial N_i}{\partial x} \\ − \, \\ − \frac{\partial N_i}{\partial y} \\ − \, \\ − \frac{\partial N_i}{\partial z} − \end{bmatrix} −$ − − − − This is: − − − ::$− { − \int_{\Omega} W \left [ \nabla^T \mathbf{\varepsilon} \nabla \hat V + \rho_v \right ] \partial\Omega − + \oint_{\Gamma} \overline{W} \left [\mathbf{n}^T \mathbf{\varepsilon} \nabla \hat V + \bar \mathbf{q} + \alpha V \right ] \partial \Gamma=0 − } −$ − − − with $\alpha = \frac{1}{r}$ the infinit condition factor and $\bar \mathbf{q}$ the field produced whe $V \,$ is fixed by $\bar V \,$. − − − The weak form of this expression can be obtained using the integration by parts. In addition, if     $\bar W = - W \,$: − − − ::$− { − \int_{\Omega} \nabla^T W^T \mathbf{\varepsilon} \nabla \hat V \partial \Omega + − \oint_{\Gamma_{\infty}} W^T \alpha V \partial \Gamma_{\infty} = − \int_{\Omega} W^T \rho_v \partial \Omega - − \oint_{\Gamma_q} W^T \bar D_n \partial \Gamma_q - − \oint_{\Gamma_V} W^T \mathbf{q_n} \partial \Gamma_V − } −$ − − − Remembering that: − − − ::$\hat V (x,y,z) = \sum_{i=0}^n N_i (x,y,z) a_i = \mathbf{N} \mathbf{a}^{(e)}$ − − − ::$\nabla \hat V = \nabla \mathbf{N} \mathbf{a}^{(e)} = \mathbf{B} \mathbf{a}^{(e)}$ − − − is the gradient potential with: − − − ::$\mathbf{B}= \left [ \mathbf{B_1, B_2 ... B_n} \right ]$ − − − and: − − − ::$\mathbf{B_i}= − \begin{bmatrix} − \frac{\partial N_i}{\partial x} \\ − \, \\ − \frac{\partial N_i}{\partial y} \\ − \, \\ − \frac{\partial N_i}{\partial z} − \end{bmatrix} −$ − − − The electric field and electric displacement field can be written as follows: − − − ::$\mathbf{q} = - \mathbf{B} \mathbf{a}^{(e)} \qquad \mathbf{q'} = - \mathbf{\varepsilon} \mathbf{B} \mathbf{a}^{(e)}$ − − − − We will now use the Galerkin Method $W_i(x) \equiv N_i(x) \,$. So, finally, the integral expression ready to create the matricial system of equations is: − − − ::$− { − \int_{\Omega} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \mathbf{a} \partial \Omega + − \oint_{\Gamma_{\infty}} \mathbf{N^T} \alpha \mathbf{N} \mathbf{a} \partial \Gamma_{\infty} = − \int_{\Omega} \mathbf{N^T} \rho_v \partial \Omega - − \oint_{\Gamma_q} \mathbf{N^T} \bar D_n \partial \Gamma_q - − \oint_{\Gamma_V} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V − } −$ − − − − ::$\mathbf{K} \mathbf{a} \,= \mathbf{f}$ − − − Note that '''K''' is a coefficients matrix that depends on the geometrical and physical properties of the problem, '''a''' is the vector with the '''''n''''' unknowns to be obtained and '''f''' is a vector that depends on the source values and boundary conditions. − − − ::$\mathbf{K}^{(e)}= − \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \partial \Omega^{(e)} + − \oint_{\Gamma_{\infty}^{(e)}} \mathbf{N^T} \alpha \mathbf{N} \partial \Gamma_{\infty}^{(e)} −$ − − ::$\mathbf{f}^{(e)}= − \int_{\Omega^{(e)}} \mathbf{N^T} \rho_v \partial \Omega^{(e)} - − \oint_{\Gamma_q^{(e)}} \mathbf{N^T} \bar D_n \partial \Gamma_q^{(e)} - − \oint_{\Gamma_V^{(e)}} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V^{(e)} −$

## Revision as of 15:09, 30 October 2009

The 2D Electrostatic Poisson's equation given by the governing PDE and its boundary conditions:

$A(V) = \left[ \frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial}{\partial y} \right) \right]V(x,y) + \rho_S = 0 ~~ in ~ \Omega$

$B(V) = \begin{cases} \left . V - \bar V = 0 \right |_{\Gamma_{V}} & in ~ \Gamma_{\varphi} \\ \left . \hat n \vec{D} - \bar D_n = 0 \right |_{\Gamma_{q}} & in ~ \Gamma_{q} \\ \left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r} & in ~ \Gamma_{\infty} \end{cases}$

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):

$V (x,y) \cong \hat V (x,y) = \sum_{i=0}^n N_i (x,y) a_i = \mathbf{N}^{(e)} · \mathbf{a}^{(e)}$

$\mathbf{N^{(e)}} = \begin{bmatrix} N_1 \\ \, \\ N_2 \\ \, \\ \vdots \\ \, \\ N_n \end{bmatrix} \qquad \mathbf{a^{(e)}} = \begin{bmatrix} a_1 \\ \, \\ a_2 \\ \, \\ \vdots \\ \, \\ a_n \end{bmatrix} \qquad \mathbf{B}= \left [ \mathbf{B_1 B_2 ... B_n} \right ] \qquad \mathbf{B_i}= \begin{bmatrix} \frac{\partial N_i}{\partial x} \\ \, \\ \frac{\partial N_i}{\partial y} \end{bmatrix}$