# 2D formulation for Electrostatic Problems

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:

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

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

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

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

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

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

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

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

### 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)

$\int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)} = \int \int_{A^{(e)}} \mathbf{N^T} \rho_S d x d y = \int_0^1 \int_0^{1-\beta} |\mathbf{J^{(e)}}| \mathbf{N^T} \rho_S d \alpha d \beta =$
$= \qquad \qquad |\mathbf{J^{(e)}}| \sum_{p=1}^{n_p} \mathbf{N^T} \rho_S W_p = |\mathbf{J^{(e)}}| \mathbf{N^T} \rho_S \sum_{p=1}^{n_p} W_p = \frac{|\mathbf{J^{(e)}}|}{2} \mathbf{N^T} \rho_S$

$\oint_{\Gamma_q^{(e)}} \mathbf{N^T} \bar D_n \partial \Gamma_q^{(e)}$
$\oint_{\Gamma_V^{(e)}} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_V^{(e)}$