2D formulation for Magnetostatic Problems

From KratosWiki

Revision as of 10:01, 2 February 2010 by JMora (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

The 2D Magnetostatic Poisson's equation given by the governing PDE and its boundary conditions:

$A(A_z) = \left[ \frac{\partial}{\partial x}\cdot \left( \frac{1}{\mu_y} \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left( \frac{1}{\mu_x} \frac{\partial }{\partial y}\right) \right] A_z(x,y)+ J_z(x,y)=0 ~~ in ~ \Omega$

$B(A_z) = \begin{cases} \left . A_z - \bar A_z = 0 \right |_{\Gamma_{A_z}} & in ~ \Gamma_{\varphi} \\ \, \\ \left . \hat n \vec{H} - \bar H_n = 0 \right |_{\Gamma_{q}} & in ~ \Gamma_{q} \\ \, \\ \left . \displaystyle \frac{\partial A_z}{\partial r} \right |_{\Gamma_{\infty}} \approx \displaystyle - \frac{A_z}{r^{exp}} & in ~ \Gamma_{\infty} \end{cases}$

can be written as (see the General formulation for Magnetostatic Problems):

${ \int_{\Omega} \mathbf{B^T} \mathbf{D} \mathbf{B} \mathbf{a} \partial \Omega + \oint_{\Gamma_{\infty}} \mathbf{N^T} \alpha \mathbf{N} \mathbf{a} \partial \Gamma_{\infty} = \int_{\Omega} \mathbf{N^T} J_S \partial \Omega - \oint_{\Gamma_q} \mathbf{N^T} \bar q_n \partial \Gamma_q - \oint_{\Gamma_V} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_{\varphi} }$

$\mathbf{K} \mathbf{a} \,= \mathbf{f}$

$\mathbf{K}^{(e)}= \int_{\Omega^{(e)}} \mathbf{B^T} \mathbf{D} \mathbf{B} \partial \Omega^{(e)} + \oint_{\Gamma_{\infty}^{(e)}} \mathbf{N^T} \alpha \mathbf{N} \partial \Gamma_{\infty}^{(e)}$

Failed to parse (unknown function\J): \mathbf{f}^{(e)}= \int_{\Omega^{(e)}} \mathbf{N^T} \J_S \partial \Omega^{(e)} - \oint_{\Gamma_q^{(e)}} \mathbf{N^T} \bar q_n \partial \Gamma_q^{(e)} - \oint_{\Gamma_{{A_z}^{(e)}}} \mathbf{n^T} \mathbf{N^T} \mathbf{q_n} \partial \Gamma_{\varphi^{(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)

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

Linear case (n_p=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 (n_p=3 integration points):

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

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

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

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

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

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

2D formulation for Magnetostatic Problems

2D formulation for Triangular Elements

Stiffness Matrix K^(e)

Source Vector f^(e)

Views

Personal tools

Navigation

Search

Toolbox

Categories

2D formulation for Magnetostatic Problems

2D formulation for Triangular Elements

Stiffness Matrix K(e)

Source Vector f(e)

Views

Personal tools

Navigation

Search

Toolbox

Categories

Stiffness Matrix K^(e)

Source Vector f^(e)