电磁粒子模拟中电荷守恒的电流分配方案满足的统一公式

Abstract

Abstract:

Unified formulation of charge-conserving current assignment in the Electromagnetic Particle-in-Cell simulation in two and three dimensional Cartesian geometry is proposed. This formulation is adapt for macroparticle with constant charge distribution. From the law of charge conservation, it is revealed that the involving current densities caused by the movement of a macroparticle with certain charge distribution satisfy a linear algebraic system which is usually underdetermined. By solving the linear system, all the possible current assignments could be obtained as the general solution of the system and any charge-conserving current assignment scheme is a special solution of the system. When a macroparticle with constant shape function moving in two dimensional Cartesian geometry three possible situations are presented, and when the macroparticle moving in three dimensional Cartesian geometry, the general and simplest situation is picked. The corresponding linear algebraic systems are listed and solved, and the concrete solution expressions are given. Three prevailing charge-conserving methods as three special solutions of these expressions are listed and analysed to give the corresponding parameters.

Key words: Electromagnetic Particle-in-Cell(PIC), Charge-conserving, Current assignment

CLC Number:

O242.2

Meiyan Fu,Tiao Lu,Xiangjiang Zhu. Unified Formulation of Charge-conserving Current Assignment in Electromagnetic Particle-in-Cell Simulation[J].Acta mathematica scientia,Series A, 2020, 40(5): 1393-1408.

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Figures/Tables 12

$J$	V method $(\times \frac{q}{2\Delta t})$	E method $(\times \frac{q}{2\Delta t})$	U method $(\times \frac{q}{2\Delta t})$
$ \xi_1$	$ \Delta x_1 (1-2y^{n}-\Delta y \frac{\Delta x_1}{\Delta x}) $	$ (0.5-x^n)(0.5-y^n) $	$ (0.5-x^n)(0.5-y^n) $
$ \xi_2$	$ \Delta x_1 \Delta y {\Delta x_1}/ (\Delta x)$	$ (0.5-x^n)(0.5-y^n) $	$ (0.5-x^n)(0.5-y^n) $
$ \xi_3$	$ [\Delta y (2-\Delta x_1)\frac{\Delta x_1}{\Delta x} + (2-\Delta x \frac{\Delta y_2}{\Delta y})(0.5-y^{n}-\Delta y \frac{\Delta x_1}{\Delta x})] $	$ (2+x^n-x^{n+1})(0.5-y^n)$	$ (1.5+x^n)(0.5-y^n)$
$ \xi_4$	$ \Delta x_1 (1+2y^{n}+\Delta y \frac{\Delta x_1}{\Delta x}) $	$ (0.5-x^{n})(2+y^{n}-y^{n+1}) $	$ (0.5-x^n)(1.5+y^n) $
$ \xi_5$	$ \Delta x_2 (0.5-y^{n}-\Delta y \frac{\Delta x_1}{\Delta x}) $	$ (x^{n+1}-0.5)(0.5-y^{n}) $	$ 0 $
$ \xi_6$	$ \Delta x \frac{\Delta y_2}{\Delta y}(0.5-y^{n}-\Delta y \frac{\Delta x_1}{\Delta x}) $	$ (x^{n+1}-0.5)(0.5-y^{n}) $	$ 0 $
$ \xi_7$	$ [\Delta x_2 (1.5+y^{n}+\Delta y \frac{\Delta x_1}{\Delta x}) + (x^{n+1}-0.5-\Delta x \frac{\Delta y_2}{\Delta y})(2.5-y^{n+1})] $	$ (x^{n+1}-0.5)(2+y^{n}-y^{n+1}) $	$ (x^{n+1}-0.5)(2.5-y^{n+1}) $
$ \xi_8$	$ (y^{n+1}-0.5) (2.5-\Delta x \frac{\Delta y_2}{\Delta y}-x^{n+1}) $	$ (2+x^{n}-x^{n+1})(y^{n+1}-0.5) $	$ (2.5-x^{n+1})(y^{n+1}-0.5)$
$ \xi_9$	$ (y^{n+1}-0.5) (-0.5+\Delta x \frac{\Delta y_2}{\Delta y}+x^{n+1}) $	$ (x^{n+1}-0.5)(y^{n+1}-0.5) $	$ (x^{n+1}-0.5)(y^{n+1}-0.5) $
$ \xi_{10}$	$ (y^{n+1}-0.5) (-0.5-\Delta x \frac{\Delta y_2}{\Delta y}+x^{n+1}) $	$ (x^{n+1}-0.5)(y^{n+1}-0.5) $	$ (x^{n+1}-0.5)(y^{n+1}-0.5) $
$ \xi_{11}$	$ 0 $	$ (0.5-x^{n})(y^{n+1}-0.5) $	$ 0 $
$ \xi_{12}$	$ 0 $	$ (0.5-x^{n})(y^{n+1}-0.5) $	$ 0 $
其中: $\Delta x = x^{n+1}-x^{n}$, $\Delta y = y^{n+1}-y^{n}$, $\Delta x_1 = 0.5-x^{n}$, $\Delta y_1 = \Delta y {\Delta x_1}/{\Delta x}, $
$x_{M_1} = 0.5$, $y_{M_1} = y^{n}+\Delta y_1$, $\Delta y_2 = 0.5-y_{M_1}$, $\Delta x_2 = \Delta x {\Delta y_2}/{\Delta y}.$

References 20

1	Eastwood W . The virtual particle electromagnetic particle-mesh method. Computer Physics Communications, 1991, 64 (2): 252- 266
2	Hockney W , Eastwood W . Computer Simulation Using Particles. New York: CRC press, 1988
3	Birdsall K , Langdon A . Plasma Physics via Computer Simulation. New York: CRC press, 2004
4	Wang Y , Wang J G , Chen Z G , et al. Three-dimensional simple conformal symplectic particle-in-cell methods for simulations of high power microwave devices. Computer Physics Communications, 2016, 205, 1- 12 doi: 10.1016/j.cpc.2016.03.007
5	Goplen B , Ludeking L , Smith D , Warren G . User-configurable MAGIC for electromagnetic PIC calculations. Computer Physics Communications, 1995, 87 (1/2): 54- 86
6	Wang J G , Zhang D H , Liu C L , et al. UNIPIC code for simulations of high power microwave devices. Physics of Plasmas, 2009, 16 (3): 033108
7	Moon H , Teixeira F , Omelchenko A . Exact charge-conserving scatter-gather algorithm for particle-in-cell simulations on unstructured grids:A geometric perspective. Computer Physics Commun, 2015, 194, 43- 53 doi: 10.1016/j.cpc.2015.04.014
8	Verboncoeur P , Langdon A , Gladd N . An object-oriented electromagnetic PIC code. Computer Physics Communications, 1995, 87 (1/2): 199- 211
9	Na Dong-Yeop , Omelchenko Yuri A , Moon Haksu , et al. Axisymmetric charge-conservative electromagnetic particle simulation algorithm on unstructured grids:Application to microwave vacuum electronic devices. Journal of Computational Physics, 2017, 346, 295- 317 doi: 10.1016/j.jcp.2017.06.016
10	Gaponov-Grekhov V , Granatstein L . Applications of High-Power Microwaves. New York: Artech House Publishers, 1994
11	Li X Z , Wang J G , Sun J , et al. Experimental study on a high-power subterahertz source generated by an overmoded surface wave oscillator with fast startup. IEEE Transactions on Electron Devices, 2013, 60 (9): 2931- 2935 doi: 10.1109/TED.2013.2273489
12	Wang J G , Wang G Q , Wang D Y , et al. Experimental study on a high-power subterahertz source generated by an overmoded surface wave oscillator with fast startup. IEEE Trans Elec Devices, 2013, (9): 2931- 2935
13	Na Dong-Yeop , Teixeira Fernando L . Analysis of multipactor effects by a particle-in-cell algorithm integrated with secondary electron Emission model on irregular grids. IEEE Transactions on Plasma Science, 2019, 47 (2): 1269- 1278 doi: 10.1109/TPS.2019.2892323
14	Wang J G , Cai L B , Zhu X Q , et al. Numerical simulations of high power microwave dielectric interface breakdown involving outgassing. Physics of Plasmas, 2010, 17 (6): 063503 doi: 10.1063/1.3432715
15	Crouseilles N , Navaro P , Sonnendrücker E . Charge-conserving grid based methods for the Vlasov-Maxwell equations. Comptes Rendus Mécanique, 2014, 342 (10/11): 636- 646
16	Villasenor J , Buneman O . Rigorous charge conservation for local electromagnetic field solvers. Computer Physics Communications, 1992, 69 (2): 306- 316
17	Esirkepov T . Exact charge conservation scheme for particle-in-cell simulation with an arbitrary form-factor. Computer Physics Communications, 2001, 135 (2): 144- 153
18	Umeda T , Omura Y , Tominaga T , Matsumoto H . A new charge conservation method in electromagnetic particle-in-cell simulations. Computer Physics Communications, 2003, 156 (1): 73- 85
19	Barthelmé R , Parzani C . Numerical charge conservation in particle-in-cell codes. Numerical Methods for Hyperbolic and Kinetic Problems, 2005, 7, 7- 28
20	Yu J Q , Jin X L , Zhou W M , et al. High-order interpolation algorithms for charge conservation in particle-in-cell simulations. Communications in Computational Physics, 2013, 13 (4): 1134- 1150

$\mbox{Order}$	$D(x; x_p)$	$S_{(i)}(x_p)$
$ 0 $	$ \delta(x-x_p) \label{shapefunction_1} $,	$ \left\{\begin{array}{ll} 1, \; \mbox{if }\; \ x_p \in\left(x_i -\frac{\Delta x}{2}, x_i + \frac{\Delta x}{2} \right), \\0 \; \mbox{else}, \end{array}\right. \label{form-factor_1}$
$ 1 $	$ \left\{\begin{array}{ll} \frac{1}{\Delta x}, \mbox{if }\ x\in\left(x_p - \frac{\Delta x}{2}, \; x_p + \frac{\Delta x}{2} \right), \\0, \; \mbox{else}, \end{array}\right. \label{shapefunction_2} $	$ \left\{\begin{array}{ll} 1 - \frac{\|x_p-x_i\|}{\Delta x}, & \mbox{if }\ \|x_p - x_i\| \leq \Delta x, \\0, &\mbox{else}, \end{array}\right. \label{form-factor_2} $
$2$	$ \left\{\begin{array}{ll} \frac{1}{\Delta x}\left[1- \frac{\|x-x_p\|}{\Delta x}\right], &\mbox{if }\ \|x-x_p\| < \Delta x, \\0, &\mbox{else}, \end{array}\right. \label{shapefunction_3} $	$ \left\{\begin{array}{ll} \frac{3}{4}-\left(\frac{x_i-x_p}{\Delta x} \right)^2, &\mbox{if }\ \|\xi-x_i\| < \frac{1}{2}\Delta x, \\\frac{1}{2}\left[\frac{3}{2}+\frac{x_i-x_p}{\Delta x} \right]^2, &\mbox{if } \frac{\Delta x}{2} \leq x_p-x_i \leq \frac{3\Delta x}{2}, \\ \frac{1}{2}\left[\frac{3}{2}-\frac{x_i-x_p}{\Delta x} \right]^2, &\mbox{if } \frac{\Delta x}{2} \leq x_i-x_p \leq \frac{3\Delta x}{2}, \\ 0, &\mbox{else}. \end{array}\right. \label{form-factor_3} $

$J$	V method$(\times \frac{q}{2\Delta t})$	E method$(\times \frac{q}{2\Delta t})$	U method$(\times \frac{q}{2\Delta t})$
$\xi_1$	$ \Delta x(1-y^{n}-y^{n+1}) $	$ \Delta x(1-y^{n}-y^{n+1}) $	$ \Delta x(1-y^{n}-y^{n+1}) $
$\xi_2$	$ \Delta y(1-x^{n}-x^{n+1}) $	$ \Delta y(1-x^{n}-x^{n+1}) $	$ \Delta y(1-x^{n}-x^{n+1}) $
$\xi_3$	$ \Delta y(1+x^{n}+x^{n+1}) $	$ \Delta y(1+x^{n}+x^{n+1}) $	$ \Delta y(1+x^{n}+x^{n+1}) $
$\xi_4$	$ \Delta x(1+y^{n}+y^{n+1}) $	$ \Delta x(1+y^{n}+y^{n+1}) $	$ \Delta x(1+y^{n}+y^{n+1}) $
其中: $\Delta x = x^{n+1}-x^{n}, \ \Delta y = y^{n+1}-y^{n}.$

$J $	V method $(\times \frac{q}{2\Delta t})$	E method $(\times \frac{q}{2\Delta t})$	U method $(\times \frac{q}{2\Delta t})$
$\xi_1$	$ \Delta x_1 (1 - 2y^{n} - \Delta y\frac{\Delta x_1}{\Delta x}) $	$ \Delta x_1 (1 - 2y^{n} - \Delta y) $	$ \Delta x_1 (1 - 2y^{n} - \frac{1}{2}\Delta y) $
$\xi_2$	$ \Delta x_1 \Delta y \frac{\Delta x_1}{\Delta x} $	$ \Delta x_1 \Delta y $	$ \Delta x_1 \frac{1}{2}\Delta y $
$\xi_3$	$ \Delta y[\frac{\Delta x_1}{\Delta x}(2-\Delta x_1) + \frac{\Delta x_2}{\Delta x}(2-\Delta x_2)] $	$ \Delta y (2-\Delta x_1-\Delta x_2) $	$ \Delta y (2 - \frac{1}{2}\Delta x_1 - \frac{1}{2}\Delta x_2) $
$\xi_4$	$ \Delta x_1 (1 + 2y^{n} + \Delta y\frac{\Delta x_1}{\Delta x}) $	$ \Delta x_1 (1 + 2y^{n} + \Delta y) $	$ \Delta x_1 (1 + 2y^{n} + \frac{1}{2}\Delta y) $
$\xi_5$	$ \Delta x_2 (1 - 2y^{n} - \Delta y - \Delta y\frac{\Delta x_1}{\Delta x}) $	$ \Delta x_2 (1 - 2y^{n} - \Delta y) $	$ \Delta x_2 (1 - 2y^{n} - \frac{3}{2}\Delta y) $
$\xi_6$	$ \Delta x_2 \frac{\Delta x_2}{\Delta x} \Delta y $	$ \Delta x_2 \Delta y $	$ \Delta x_2 \frac{1}{2} \Delta y $
$\xi_7$	$ \Delta x_2 (1 + 2y^{n} + \Delta y + \Delta y\frac{\Delta x_1}{\Delta x}) $	$ \Delta x_2 (1 + 2y^{n} + \Delta y) $	$ \Delta x_2 (1 + 2y^{n} + \frac{3}{2}\Delta y) $
其中: $\Delta x = x^{n+1}-x^{n}, \ \Delta y = y^{n+1}-y^{n}, \ \Delta x_1 = 0.5-x^{n}, \ \Delta x_2 = x^{n+1}-0.5.$

Form-factor	$ -0.5 \leq x < 0.5, -0.5 \leq y < 0.5 $
$S_{(i, j)}(x, y)$	$\left(0.5-x\right)\left(0.5-y\right)$
$S_{(i, j+1)}(x, y)$	$\left(0.5-x\right)\left(0.5+y\right)$
$S_{(i+1, j)}(x, y)$	$\left(0.5+x\right)\left(0.5-y\right)$
$S_{(i+1, j+1)}(x, y)$	$\left(0.5+x\right)\left(0.5+y\right)$

Form-factor	$ -0.5 \leq x < 0.5, -0.5 \leq y < 0.5 $	$ 0.5 \leq x < 1.5, -0.5 \leq y < 0.5 $
$S_{(i, j)}(x, y)$	$\left(0.5-x\right)\left(0.5-y\right)$	$ 0 $
$S_{(i, j+1)}(x, y)$	$\left(0.5-x\right)\left(0.5+y\right)$	$ 0 $
$S_{(i+1, j)}(x, y)$	$\left(0.5+x\right)\left(0.5-y\right)$	$\left(1.5-x\right)\left(0.5-y\right)$
$S_{(i+1, j+1)}(x, y)$	$\left(0.5+x\right)\left(0.5+y\right)$	$\left(1.5-x\right)\left(0.5+y\right)$
$S_{(i+2, j) }(x, y)$	$ 0 $	$\left(x-0.5\right)\left(0.5-y\right)$
$S_{(i+1, j+1)}(x, y)$	$ 0 $	$\left(x-0.5\right)\left(0.5+y\right)$

Unified Formulation of Charge-conserving Current Assignment in Electromagnetic Particle-in-Cell Simulation

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Figures/Tables 12

References 20

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Form-factor	$ -0.5 \leq x < 0.5 $		$ 0.5 \leq x < 1.5 $
	$ -0.5 \leq y < 0.5 $	$ 0.5 \leq y < 1.5 $	$ -0.5 \leq y < 0.5 $	$ 0.5 \leq y < 1.5 $
$S_{(i, j)}(x, y)$	$\left(0.5-x\right)\left(0.5-y\right)$	$ 0 $	$ 0 $	$ 0 $
$S_{(i, j+1)}(x, y)$	$\left(0.5-x\right)\left(0.5+y\right)$	$\left(0.5-x\right)\left(1.5-y\right)$	$ 0 $	$ 0 $
$S_{(i+1, j)}(x, y)$	$\left(0.5+x\right)\left(0.5-y\right)$	$ 0 $	$\left(1.5-x\right)\left(0.5-y\right)$	$ 0 $
$S_{(i+1, j+1)}(x, y)$	$\left(0.5+x\right)\left(0.5+y\right)$	$\left(0.5+x\right)\left(1.5-y\right)$	$\left(1.5-x\right)\left(0.5+y\right)$	$\left(1.5-x\right)\left(1.5-y\right)$
$S_{(i+2, j+1)}(x, y)$	$ 0 $	$ 0 $	$ 0 $	$\left(x-0.5\right)\left(1.5-y\right)$
$S_{(i+1, j+2)}(x, y)$	$ 0 $	$\left(0.5+x\right)\left(y-0.5\right)$	$ 0 $	$\left(1.5-x\right)\left(y-0.5\right)$
$S_{(i+2, j+2)}(x, y)$	$ 0 $	$ 0 $	$ 0 $	$\left(x-0.5\right)\left(y-0.5\right)$

Form-factor(1st order)	$ -0.5 \leq x < 0.5, \ -0.5 \leq y < 0.5, \ -0.5 \leq z < 0.5 $
$S_{(i, j, k)}(x, y, z)$	$\left(0.5-x\right)\left(0.5-y\right)\left(0.5-z\right)$
$S_{(i+1, j, k)}(x, y, z)$	$\left(0.5+x\right)\left(0.5-y\right)\left(0.5-z\right)$
$S_{(i, j+1, k)}(x, y, z)$	$\left(0.5-x\right)\left(0.5+y\right)\left(0.5-z\right)$
$S_{(i+1, j+1, k)}(x, y, z)$	$\left(0.5+x\right)\left(0.5+y\right)\left(0.5-z\right)$
$S_{(i, j, k+1)}(x, y, z)$	$\left(0.5-x\right)\left(0.5-y\right)\left(0.5+z\right)$
$S_{(i+1, j, k+1)}(x, y, z)$	$\left(0.5+x\right)\left(0.5-y\right)\left(0.5+z\right)$
$S_{(i, j+1, k+1)}(x, y, z)$	$\left(0.5-x\right)\left(0.5+y\right)\left(0.5+z\right)$
$S_{(i+1, j+1, k+1)}(x, y, z)$	$\left(0.5+x\right)\left(0.5+y\right)\left(0.5+z\right)$

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