2D formulation for Electrostatic Problems

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Line 85: Line 85:
  
  
 
 
::<math>\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}
 
</math>
 
 
 
 
This is:
 
 
 
::<math>
 
    {
 
    \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
 
    }
 
  </math>
 
 
 
with <math>\alpha = \frac{1}{r}</math> the infinit condition factor and <math>\bar \mathbf{q}</math> the field produced whe <math>V \,</math> is fixed by <math>\bar V \,</math>.
 
 
 
The weak form of this expression can be obtained using the integration by parts. In addition, if &nbsp; &nbsp; <math> \bar W = - W \,</math>:
 
 
 
::<math>
 
    {
 
    \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
 
    }
 
  </math>
 
 
 
Remembering that:
 
 
 
::<math> \hat V (x,y,z) = \sum_{i=0}^n N_i (x,y,z) a_i = \mathbf{N} \mathbf{a}^{(e)}</math>
 
 
 
::<math>\nabla \hat V = \nabla \mathbf{N} \mathbf{a}^{(e)} = \mathbf{B} \mathbf{a}^{(e)}</math>
 
 
 
is the gradient potential with:
 
 
 
::<math>\mathbf{B}= \left [ \mathbf{B_1, B_2 ... B_n} \right ] </math>
 
 
 
and:
 
 
 
::<math>\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}
 
</math>
 
 
 
The electric field and electric displacement field can be written as follows:
 
 
 
::<math>\mathbf{q} = -  \mathbf{B} \mathbf{a}^{(e)}  \qquad  \mathbf{q'} = - \mathbf{\varepsilon} \mathbf{B} \mathbf{a}^{(e)}</math>
 
 
 
 
We will now use the Galerkin Method <math>W_i(x) \equiv N_i(x) \,</math>. So, finally, the integral expression ready to create the matricial system of equations is:
 
 
 
::<math>
 
    {
 
    \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
 
    }
 
  </math>
 
 
 
 
::<math>\mathbf{K} \mathbf{a} \,= \mathbf{f}</math>
 
 
 
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.
 
 
 
::<math>\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)}
 
</math>
 
 
::<math>\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)}
 
  </math>
 
  
  

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}
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