2D formulation for Electrostatic Problems

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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 . \displaystyle  \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}} 
  \approx \displaystyle - \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_S \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_S \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} 
     \displaystyle \frac{\partial N_i}{\partial x} \\ 
     \, \\
     \displaystyle \frac{\partial N_i}{\partial y} 
   \end{bmatrix}
\qquad
\mathbf{\varepsilon}=
   \begin{bmatrix} 
     \varepsilon_x & 0 \\ 
     \, \\
     0 & \varepsilon_y 
   \end{bmatrix}


\alpha = \frac{1}{|r-r_0|^{exp}} \qquad with \quad exp=0.5, 1, 2...




2D formulation for Triangular Elements

After applying the numerical integration for triangular elements by using the natural coordinates, we obtain:



   \mathbf{N^{(e)}} = 
   \begin{bmatrix} 
     N_1 & N_2 & N_3 
   \end{bmatrix}
=
   \begin{bmatrix} 
     L_1 & L_2 & L_3 
   \end{bmatrix}
=
   \begin{bmatrix} 
     (1-\alpha-\beta) & \alpha & \beta 
   \end{bmatrix}
\qquad
   \mathbf{a^{(e)}} = 
   \begin{bmatrix} 
     a_1 \\ 
     \, \\
     a_2 \\ 
     \, \\
     a_3 
   \end{bmatrix}



  \frac{\partial N_1}{\partial \alpha}=-1 \qquad 
  \frac{\partial N_2}{\partial \alpha}=1  \qquad 
  \frac{\partial N_3}{\partial \alpha}=0  \qquad
  \frac{\partial N_1}{\partial \beta}=-1  \qquad 
  \frac{\partial N_2}{\partial \beta}=0   \qquad 
  \frac{\partial N_3}{\partial \beta}=1


NaturalCoordinates 2.jpg



  x = N_1 x_1 + N_2 x_2 + N_3 x_3 = ( 1 - \alpha - \beta) x_1 + \alpha x_2 + \beta x_3 \,



  y = N_1 y_1 + N_2 y_2 + N_3 y_3 = ( 1 - \alpha - \beta) y_1 + \alpha y_2 + \beta y_3 \,


\mathbf{J^{(e)}} = 
 \begin{bmatrix}
  \displaystyle \frac{\partial x}{\partial \alpha} & \displaystyle \frac{\partial y}{\partial \alpha} \\ \quad \\
  \displaystyle \frac{\partial x}{\partial \beta} & \displaystyle \frac{\partial y}{\partial \beta}
 \end{bmatrix}
=
 \begin{bmatrix}
  - x_1 + x_2 & - y_1 + y_2 \\
  - x_1 + x_3 & - y_1 + y_3
 \end{bmatrix}
\qquad
\mathbf{|J^{(e)}|} = 2 A^{(e)}


\mathbf{B(\alpha,\beta)}=\mathbf{J^{(e)}} \mathbf{B(x,y)} \qquad \mathbf{B(x,y)}= \mathbf{[J^{(e)}]^{-1}} \mathbf{B(\alpha,\beta)}



   \mathbf{B}= 
   \begin{bmatrix} 
     \displaystyle \frac{\partial N_1}{\partial x} & 
     \displaystyle \frac{\partial N_2}{\partial x} & 
     \displaystyle \frac{\partial N_3}{\partial x}\\ 
     \, \\
     \displaystyle \frac{\partial N_1}{\partial y} & 
     \displaystyle \frac{\partial N_2}{\partial y} & 
     \displaystyle \frac{\partial N_3}{\partial y}
   \end{bmatrix}
=
   \frac{1}{|\mathbf{J^{(e)}}|}
   \begin{bmatrix}
     \displaystyle  \frac{\partial y}{\partial \beta} & \displaystyle -\frac{\partial y}{\partial \alpha} \\ 
     \displaystyle -\frac{\partial x}{\partial \beta} & \displaystyle  \frac{\partial x}{\partial \alpha}
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \frac{\partial N_1}{\partial \alpha} & 
     \displaystyle \frac{\partial N_2}{\partial \alpha} & 
     \displaystyle \frac{\partial N_3}{\partial \alpha}\\ 
     \, \\
     \displaystyle \frac{\partial N_1}{\partial \beta} & 
     \displaystyle \frac{\partial N_2}{\partial \beta} & 
     \displaystyle \frac{\partial N_3}{\partial \beta}
   \end{bmatrix}



   \mathbf{B}
   =
   \frac{1}{2 A^{(e)}}
   \begin{bmatrix}
     - y_1 + y_3 & - y_2 + y_1 \\ 
     - x_3 + x_1 & - x_1 + x_2   
   \end{bmatrix}
   \begin{bmatrix} 
     -1 & 1 & 0 \\
     -1 & 0 & 1
   \end{bmatrix}
   =
   \frac{1}{2 A^{(e)}}
   \begin{bmatrix}
     - y_3 + y_2 & - y_1 + y_3 & - y_2 + y_1 \\ 
     x_3 - x_2   & - x_3 + x_1 & - x_1 + x_2   
   \end{bmatrix}


Stiffness Matrix K(e)


  \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B}  \partial \Omega^{(e)}=
  \int \int_{A^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} d x d y = 
  \int_0^1 \int_0^{1-\beta} |\mathbf{J^{(e)}}| \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} d \alpha d \beta =

  = \qquad \qquad |\mathbf{J^{(e)}}| \sum_{p=1}^{n_p} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} W_p =
  |\mathbf{J^{(e)}}| \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \sum_{p=1}^{n_p} W_p =
  \frac{|\mathbf{J^{(e)}}|}{2} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B}


\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
   \begin{bmatrix} 
     \displaystyle \frac{\partial N_1}{\partial x} & 
     \displaystyle \frac{\partial N_1}{\partial y} \\
     \, \\
     \displaystyle \frac{\partial N_2}{\partial x} & 
     \displaystyle \frac{\partial N_2}{\partial y} \\ 
     \, \\
     \displaystyle \frac{\partial N_3}{\partial x} & 
     \displaystyle \frac{\partial N_3}{\partial y}
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \varepsilon_x & 0 \\
     \, \\
     0 & \displaystyle \varepsilon_y 
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \frac{\partial N_1}{\partial x} & 
     \displaystyle \frac{\partial N_2}{\partial x} & 
     \displaystyle \frac{\partial N_3}{\partial x}\\ 
     \, \\
     \displaystyle \frac{\partial N_1}{\partial y} & 
     \displaystyle \frac{\partial N_2}{\partial y} & 
     \displaystyle \frac{\partial N_3}{\partial y}
   \end{bmatrix}


\mathbf{B(x,y)^T} \mathbf{\varepsilon} \mathbf{B(x,y)} =
\mathbf{B(\alpha,\beta)^T} \mathbf{[[J^{(e)}]^{-1}]^T} \mathbf{\varepsilon} \mathbf{[J^{(e)}]^{-1}} \mathbf{B(\alpha,\beta)}


\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
   \frac{1}{|\mathbf{J^{(e)}}|^2}
   \begin{bmatrix} 
     \displaystyle \frac{\partial N_1}{\partial \alpha} & 
     \displaystyle \frac{\partial N_1}{\partial \beta} \\
     \, \\
     \displaystyle \frac{\partial N_2}{\partial \alpha} & 
     \displaystyle \frac{\partial N_2}{\partial \beta} \\ 
     \, \\
     \displaystyle \frac{\partial N_3}{\partial \alpha} & 
     \displaystyle \frac{\partial N_3}{\partial \beta}
   \end{bmatrix}
   \begin{bmatrix}
     \displaystyle  \frac{\partial y}{\partial \beta} & \displaystyle -\frac{\partial x}{\partial \beta} \\ 
     \displaystyle -\frac{\partial y}{\partial \alpha} & \displaystyle  \frac{\partial x}{\partial \alpha}
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \varepsilon_x & 0 \\
     \, \\
     0 & \displaystyle \varepsilon_y 
   \end{bmatrix}
   \begin{bmatrix}
     \displaystyle  \frac{\partial y}{\partial \beta} & \displaystyle -\frac{\partial y}{\partial \alpha} \\ 
     \displaystyle -\frac{\partial x}{\partial \beta} & \displaystyle  \frac{\partial x}{\partial \alpha}
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \frac{\partial N_1}{\partial \alpha} & 
     \displaystyle \frac{\partial N_2}{\partial \alpha} & 
     \displaystyle \frac{\partial N_3}{\partial \alpha}\\ 
     \, \\
     \displaystyle \frac{\partial N_1}{\partial \beta} & 
     \displaystyle \frac{\partial N_2}{\partial \beta} & 
     \displaystyle \frac{\partial N_3}{\partial \beta}
   \end{bmatrix}


\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
   \frac{1}{(2 A^{(e)})^2}
   \begin{bmatrix} 
     -1 & -1 \\
      1 &  0 \\
      0 &  1 
   \end{bmatrix}
   \begin{bmatrix}
     - y_1 + y_3 & - x_3 + x_1 \\ 
     - y_2 + y_1 & - x_1 + x_2
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \varepsilon_x & 0 \\
     \, \\
     0 & \displaystyle \varepsilon_y 
   \end{bmatrix}
   \begin{bmatrix}
     - y_1 + y_3 & - y_2 + y_1 \\ 
     - x_3 + x_1 & - x_1 + x_2
   \end{bmatrix}
   \begin{bmatrix} 
     -1 & 1 & 0 \\
     -1 & 0 & 1
   \end{bmatrix}


\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
   \frac{1}{(2 A^{(e)})^2}
   \begin{bmatrix} 
     - y_3 + y_2 &   x_3 + x_2 \\
     - y_1 + y_3 & - x_3 + x_1 \\
     - y_2 + y_1 & - x_1 + x_2 
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \varepsilon_x & 0 \\
     \, \\
     0 & \displaystyle \varepsilon_y 
   \end{bmatrix}
   \begin{bmatrix} 
     - y_3 + y_2 & - y_1 + y_3 & - y_2 + y_1 \\
       x_3 + x_2 & - x_3 + x_1 & - x_1 + x_2
   \end{bmatrix}



  \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B}  \partial \Omega^{(e)}=
  A^{(e)} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
   \frac{1}{4 A^{(e)}}
   \begin{bmatrix} 
     - y_3 + y_2 &   x_3 + x_2 \\
     - y_1 + y_3 & - x_3 + x_1 \\
     - y_2 + y_1 & - x_1 + x_2 
   \end{bmatrix}
   \begin{bmatrix} 
     \displaystyle \varepsilon_x & 0 \\
     \, \\
     0 & \displaystyle \varepsilon_y 
   \end{bmatrix}
   \begin{bmatrix} 
     - y_3 + y_2 & - y_1 + y_3 & - y_2 + y_1 \\
       x_3 + x_2 & - x_3 + x_1 & - x_1 + x_2
   \end{bmatrix}


\oint_{\Gamma_{\infty}^{(e)}} \mathbf{N^T} \alpha \mathbf{N} \partial \Gamma_{\infty}^{(e)}

Source Vector f(e)


  \int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)} =
  \int \int_{A^{(e)}} \mathbf{N^T} \rho_S d x d y =
  \int_0^1 \int_0^{1-\beta} |\mathbf{J^{(e)}}| \mathbf{N^T} \rho_S d \alpha d \beta =
  |\mathbf{J^{(e)}}| \sum_{p=1}^{n_p} \mathbf{N^T} \rho_S W_p =


Linear case (np=1 integration point):


N=\left [ \frac{1}{3} \quad \frac{1}{3} \quad \frac{1}{3}\right ] \qquad W_i=\frac{1}{2}\,

  \int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)}
  = 2 A^{(e)} \left [ \frac{1}{6} \quad  \frac{1}{6} \quad \frac{1}{6}\right ]^T \rho_S
  = A^{(e)} \frac{\rho_S}{3} \left [ 1 \quad  1 \quad 1 \right ]^T


Quadratic case (np=3 integration points):


p=1 \qquad N=\left [ \frac{1}{2} \quad \frac{1}{2} \quad 0\right ] \qquad W_1=\frac{1}{6}\,
p=2 \qquad N=\left [ 0 \quad \frac{1}{2} \quad \frac{1}{2}\right ] \qquad W_2=\frac{1}{6}\,
p=3 \qquad N=\left [ \frac{1}{2} \quad 0 \quad \frac{1}{2}\right ] \qquad W_3=\frac{1}{6}\,



  \int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)}
  = 2 A^{(e)} \left ( \left [ \frac{1}{2} \quad  \frac{1}{2} \quad 0 \right ]^T 
    + \left [ 0 \quad \frac{1}{2} \quad  \frac{1}{2} \right ]^T 
    + \left [ \frac{1}{2} \quad  0 \quad \frac{1}{2} \right ]^T \right )
   \frac{1}{6} \rho_S
  = A^{(e)} \frac{\rho_S}{3} \left [ 1 \quad  1 \quad 1 \right ]^T




\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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