2D formulation for Electrostatic Problems

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   \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^{exp}} & 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>
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\mathbf{B_i}=
 
\mathbf{B_i}=
 
   \begin{bmatrix}  
 
   \begin{bmatrix}  
−
     \frac{\partial N_i}{\partial x} \\  
+
     \displaystyle \frac{\partial N_i}{\partial x} \\  
 
     \, \\
 
     \, \\
−
     \frac{\partial N_i}{\partial y}  
+
     \displaystyle \frac{\partial N_i}{\partial y}  
 +
  \end{bmatrix}
 +
\qquad
 +
\mathbf{\varepsilon}=
 +
  \begin{bmatrix}
 +
    \varepsilon_x & 0 \\
 +
    \, \\
 +
    0 & \varepsilon_y
 
   \end{bmatrix}
 
   \end{bmatrix}
 
</math>
 
</math>
Line 91: Line 99:
  
  
 +
 +
 +
 +
 +
== 2D formulation for Triangular Elements ==
 +
 +
 +
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>
 +
  \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>
 +
 +
 +
::<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>
 +
 +
 +
::[[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>
 +
 +
 +
::<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}=
 +
  \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}
 +
</math>
 +
 +
 +
::<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>
 +
 +
 +
 +
=== 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>
 +
 +
::<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>\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}
 +
</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{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>
 +
 +
 +
::::<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{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>
 +
  \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}
 +
</math>
 +
 +
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::<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>
  
  

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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