General formulation for Electrostatic Problems

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

 A(V) = \vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0  ~~ in ~ \Omega

 B(V) = 
  \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}

We will apply the residual formulation based on the Weighted Residual Method (WRM).

    \int_{\Omega} W(x,y,z) r_{\Omega} \partial \Omega 
    + \oint_{\Gamma} \overline{W}(x,y,z) r_{\Gamma} \partial \Gamma=0


W(x,y,z) \, and \overline{W}(x,y,z) the weighting functions.
\frac{}{} r_{\Omega} = A(\hat V) \ne 0 \quad in \quad \Omega
\frac{}{} r_{\Gamma} = B(\hat V) \ne 0 \quad in \quad \Gamma

Where \hat V \, is the numerical approach of the unknown V \,:

 V (x,y,z) \cong \hat V (x,y,z) = \sum_{i=0}^n N_i (x,y,z) a_i

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 ]


     \frac{\partial N_i}{\partial x} \\ 
     \, \\
     \frac{\partial N_i}{\partial y} \\ 
     \, \\
     \frac{\partial N_i}{\partial z} 

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.

    \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)}
    \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)}
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