2D formulation for Electrostatic Problems

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−
The Electrostatic Poisson's equation given by the governing PDE and its boundary conditions:
+
The 2D Electrostatic Poisson's equation given by the governing PDE and its boundary conditions:
  
  
−
:<math>\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]\varphi(x,y) = f(x,y)</math>
+
::<math> 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 </math>
−
 
+
−
 
+
−
 
+
−
::<math> A(V) = \vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0  ~~ in ~ \Omega </math>
+
  
  
Line 12: Line 8:
 
\begin{cases}  
 
\begin{cases}  
 
   \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 . \displaystyle  \frac{\partial V}{\partial r} \right |_{\Gamma_{\infty}}  
 +
  \approx \displaystyle - \frac{V}{r^{exp}} & in ~ \Gamma_{\infty}
 
\end{cases}
 
\end{cases}
 
</math>
 
</math>
  
  
−
We will apply the [[Poisson's_Equation#Weighted_Residual_Method_.28WRM.29 | residual formulation]] based on the Weighted Residual Method (WRM).
+
can be written as (see the [[General_formulation_for_Electrostatic_Problems | General formulation for Electrostatic Problems]]):
  
  
 
::<math>  
 
::<math>  
 
     {
 
     {
−
     \int_{\Omega} W(x,y,z) r_{\Omega} \partial \Omega  
+
     \int_{\Omega} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \mathbf{a}  \partial \Omega +
−
     + \oint_{\Gamma} \overline{W}(x,y,z) r_{\Gamma} \partial \Gamma=0
+
     \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
 
     }
 
     }
 
   </math>
 
   </math>
  
  
 +
::<math>\mathbf{K} \mathbf{a} \,= \mathbf{f}</math>
  
−
with:
 
  
−
::<math>W(x,y,z) \,</math> and <math>\overline{W}(x,y,z)</math> the weighting functions.
+
::<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>\frac{}{} r_{\Omega} = A(\hat V) \ne 0 \quad in \quad \Omega</math>
+
::<math>\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)}
 +
  </math>
  
−
::<math>\frac{}{} r_{\Gamma} = B(\hat V) \ne 0 \quad in \quad \Gamma</math>
 
  
  
  
−
Where <math>\hat V \,</math> is the numerical approach of the unknown <math>V \,</math>:
+
with ('''''n''''' is the number of nodes of the element):
  
  
−
::<math> V (x,y,z) \cong \hat V (x,y,z) = \sum_{i=0}^n N_i (x,y,z) a_i </math>
+
::<math> V (x,y) \cong \hat V (x,y) = \sum_{i=0}^n N_i (x,y) a_i = \mathbf{N}^{(e)} · \mathbf{a}^{(e)}</math>
  
  
−
This is:
+
::<math>\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}
 +
</math>
  
  
−
::<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>
 
  
 +
::<math>\alpha = \frac{1}{|r-r_0|^{exp}} \qquad with \quad exp=0.5, 1, 2...</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:
+
== 2D formulation for Triangular Elements ==
  
  
−
::<math> \hat V (x,y,z) = \sum_{i=0}^n N_i (x,y,z) a_i = \mathbf{N} \mathbf{a}^{(e)}</math>
+
After applying the [[Numerical_Integration#Numerical_Integration_for_Isoparametric_Triangular_Domains | numerical integration for triangular elements]] by using the [[Two-dimensional_Shape_Functions#Natural_Coordinates | natural coordinates]], we obtain:
  
  
−
::<math>\nabla \hat V = \nabla \mathbf{N} \mathbf{a}^{(e)} = \mathbf{B} \mathbf{a}^{(e)}</math>
+
::<math>
 +
  \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}
 +
</math>
  
  
−
is the gradient potential with:
+
::<math>
 +
  \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
 +
</math>
  
  
−
::<math>\mathbf{B}= \left [ \mathbf{B_1, B_2 ... B_n} \right ] </math>
+
::[[Image:NaturalCoordinates 2.jpg|300px]]
 +
 
 +
 
 +
::<math>
 +
  x = N_1 x_1 + N_2 x_2 + N_3 x_3 = ( 1 - \alpha - \beta) x_1 + \alpha x_2 + \beta x_3 \,
 +
</math>
 +
 
 +
 
 +
::<math>
 +
  y = N_1 y_1 + N_2 y_2 + N_3 y_3 = ( 1 - \alpha - \beta) y_1 + \alpha y_2 + \beta y_3 \,
 +
</math>
 +
 
 +
 
 +
::<math>\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)}
 +
</math>
  
  
−
and:
+
::<math>\mathbf{B(\alpha,\beta)}=\mathbf{J^{(e)}} \mathbf{B(x,y)} \qquad \mathbf{B(x,y)}= \mathbf{[J^{(e)}]^{-1}} \mathbf{B(\alpha,\beta)}</math>
  
  
−
::<math>\mathbf{B_i}=
+
::<math>
 +
  \mathbf{B}=  
 
   \begin{bmatrix}  
 
   \begin{bmatrix}  
−
     \frac{\partial N_i}{\partial x} \\  
+
     \displaystyle \frac{\partial N_1}{\partial x} &
 +
    \displaystyle \frac{\partial N_2}{\partial x} &
 +
    \displaystyle \frac{\partial N_3}{\partial x}\\  
 
     \, \\
 
     \, \\
−
     \frac{\partial N_i}{\partial y} \\  
+
     \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}\\  
 
     \, \\
 
     \, \\
−
     \frac{\partial N_i}{\partial z}  
+
     \displaystyle \frac{\partial N_1}{\partial \beta} &
 +
    \displaystyle \frac{\partial N_2}{\partial \beta} &
 +
    \displaystyle \frac{\partial N_3}{\partial \beta}
 
   \end{bmatrix}
 
   \end{bmatrix}
 
</math>
 
</math>
  
  
−
The electric field and electric displacement field can be written as follows:
+
::<math>
 +
  \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}
 +
</math>
  
  
−
::<math>\mathbf{q} = -  \mathbf{B} \mathbf{a}^{(e)}  \qquad  \mathbf{q'} = - \mathbf{\varepsilon} \mathbf{B} \mathbf{a}^{(e)}</math>
 
  
 +
=== Stiffness Matrix K<sup>(e)</sup> ===
  
 +
::<math>
 +
  \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 =
 +
</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>
 +
  = \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}
 +
</math>
  
  
−
::<math>  
+
::::<math>\mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
−
    {
+
  \begin{bmatrix}  
−
    \int_{\Omega} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} \mathbf{a} \partial \Omega +
+
    \displaystyle \frac{\partial N_1}{\partial x} &
−
    \oint_{\Gamma_{\infty}} \mathbf{N^T} \alpha \mathbf{N} \mathbf{a} \partial \Gamma_{\infty} =
+
    \displaystyle \frac{\partial N_1}{\partial y} \\
−
    \int_{\Omega} \mathbf{N^T} \rho_v \partial \Omega -
+
    \, \\
−
    \oint_{\Gamma_q} \mathbf{N^T} \bar D_n \partial \Gamma_q -
+
    \displaystyle \frac{\partial N_2}{\partial x} &
−
    \oint_{\Gamma_V} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V
+
    \displaystyle \frac{\partial N_2}{\partial y} \\
−
    }
+
    \, \\
−
  </math>
+
    \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}
 +
</math>
  
  
 +
::::<math>\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)}</math>
  
−
::<math>\mathbf{K} \mathbf{a} \,= \mathbf{f}</math>
 
  
 +
::::<math>\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}
 +
</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{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}
 +
</math>
  
−
::<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>\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}
 
</math>
 
</math>
  
−
::<math>\mathbf{f}^{(e)}=
+
 
−
    \int_{\Omega^{(e)}} \mathbf{N^T} \rho_v \partial \Omega^{(e)} -
+
::::<math>
−
     \oint_{\Gamma_q^{(e)}} \mathbf{N^T} \bar D_n \partial \Gamma_q^{(e)} -
+
  \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B}  \partial \Omega^{(e)}=
−
    \oint_{\Gamma_V^{(e)}} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V^{(e)}
+
  A^{(e)} \mathbf{B^T} \mathbf{\varepsilon} \mathbf{B} =
−
  </math>
+
  \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}
 +
</math>
 +
 
 +
 
 +
::<math>\oint_{\Gamma_{\infty}^{(e)}} \mathbf{N^T} \alpha \mathbf{N} \partial \Gamma_{\infty}^{(e)}</math>
 +
 
 +
=== Source Vector f<sup>(e)</sup> ===
 +
 
 +
::<math>
 +
  \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 =
 +
</math>
 +
 
 +
 
 +
:::'''Linear case''' ('''n<sub>p</sub>'''=1 integration point):
 +
 
 +
 
 +
::::<math>N=\left [ \frac{1}{3} \quad \frac{1}{3} \quad \frac{1}{3}\right ] \qquad W_i=\frac{1}{2}\,</math>
 +
 
 +
::::<math>
 +
  \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
 +
</math>
 +
 
 +
 
 +
 
 +
:::'''Quadratic case''' ('''n<sub>p</sub>'''=3 integration points):
 +
 
 +
 
 +
::::<math>p=1 \qquad N=\left [ \frac{1}{2} \quad \frac{1}{2} \quad 0\right ] \qquad W_1=\frac{1}{6}\,</math>
 +
::::<math>p=2 \qquad N=\left [ 0 \quad \frac{1}{2} \quad \frac{1}{2}\right ] \qquad W_2=\frac{1}{6}\,</math>
 +
::::<math>p=3 \qquad N=\left [ \frac{1}{2} \quad 0 \quad \frac{1}{2}\right ] \qquad W_3=\frac{1}{6}\,</math>
 +
 
 +
 
 +
::::<math>
 +
  \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
 +
</math>
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
::<math>\oint_{\Gamma_q^{(e)}} \mathbf{N^T} \bar D_n \partial \Gamma_q^{(e)}</math>
 +
 
 +
::<math>\oint_{\Gamma_V^{(e)}} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V^{(e)}</math>
  
  
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[[Category: Electrostatic Application]]
 
[[Category: Electrostatic Application]]
−
[[Category: Theory]]
+
[[Category: Electrostatic Theory]]

Latest revision as of 10:47, 27 November 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 . \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)}
Retrieved from "https://kratos-wiki.cimne.upc.edu/index.php/2D_formulation_for_Electrostatic_Problems"
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