2D formulation for Electrostatic Problems

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   \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} & in ~ \Gamma_{\infty}
+
   \left . \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} \approx - \frac{V}{r^{exponent}} & in ~ \Gamma_{\infty}
 
\end{cases}
 
\end{cases}
 
</math>
 
</math>

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^{exponent}} & 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|^{exponent}} \qquad with \quad exponent=0.5, 1, 2...
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