带电粒子流的 FiPy

Question

前提

我正在尝试解决一个set of coupled PDEs，它描述了使用 FiPy 具有不同扩散系数的带电粒子的扩散。最终目标是获得物质和电场的浓度分布。

几何是一个半径为 R 的无限长圆柱体。我想使用一个非均匀网格，在畴壁附近有更多点。

带电粒子从域的中心（左边界）扩散到域的壁（右边界）。这转化为左边界处的狄利克雷边界条件 (B.C.)，其中物种浓度 = 0，以及 Neumann B.C.在右侧边界，物种通量为 0 以描述径向对称性。因为带电物质以不同的速率扩散，所以存在由空间电荷产生的电场。电场使较慢的物质加速，并使较快的物质减速，与场强成正比。

P 是带正电的物质浓度，N 是带负电的物质浓度。 E为空间电荷电场。

问题

我似乎无法从我的代码中得到一个合理的解决方案，我认为这可能与我如何将梯度/散度项转换为对流项有关：

from fipy import *
import scipy.constants as constant
from fipy.tools import numerix
import numpy as np

## Defining physical constants
pi = constant.pi
m_argon = 6.6335e-26 # kg
k_b = constant.k # J/K
e_0 = constant.epsilon_0 # F/m
q_e = constant.elementary_charge # C
m_e = constant.electron_mass # kg
planck = constant.h

def char_diff_length(L,R):
    """Characteristic diffusion length in a cylinder.
    Used for determining the ambipolar diffusion coefficient.
    ref: https://doi.org/10.6028/jres.095.035"""
    a = (pi/L)**2
    b = (2.405/R)**2
    c = (a+b)**(1/2)
    return c

def L_Debye(ne,Te):
    """Electron Debye screening length given in m.
    ne is in #/m3, Te is in K."""
    if ne < 3.3e-5:
        ne = 3.3e-5
    return (((e_0*k_b*Te)/(ne*q_e**2)))**(1/2)

## Setting system parameters
# Operation parameters
Pressure = 1.e5 # ambient pressure Pa
T_g = 400. # background gas temperature K
n_g = Pressure/k_b/T_g # gas number density #/m3
Q_std = 300. # standard volumetric flowrate in sccm
T_e_0 = 11. # plasma temperature ratio T_e/T_g here assumed to be T_e = 0.5 eV and T_g = 500 K
n_e_0 = 1.e20 # electron density in bulk plasma #/m3

# Geometric parameters
R_b = 1.e-3 # radius cylinder m
L = 1.e-1 # length of cylinder m

# Transport parameters
D_ion = 4.16e-6 #m2/s ion diffusion, obtained from https://doi.org/10.1007/s12127-020-00258-z
mu_ion = D_ion*q_e/k_b/T_g # ion electrical mobility using Einstein relation

D_e = 100.68122*D_ion #m2/s electron diffusion
mu_e = D_e*q_e/k_b/T_g # electron electrical mobility using Einstein relation

Lambda = char_diff_length(L,R_b)
debyelength_e = L_Debye(n_e_0,T_g)
gamma = (Lambda/debyelength_e)**2
delta = D_ion/D_e



def d_j(rb,n): #sets the desired spatial steps for mesh
    dj = np.zeros(n)
    for j in range(n):
        dj[j] = 2*rb*(1 - j/n)/n
    return dj

#Initializing mesh
dj = d_j(1.,100) # 100 points
mesh = CylindricalGrid1D(dr = dj)

#Declaring cell variables
N = CellVariable(mesh=mesh, value = 1., hasOld = True, name = "electron density")
P = CellVariable(mesh=mesh, value = 1., hasOld = True, name = "ion density")
H = CellVariable(mesh=mesh, value = 0., hasOld = True, name = "electric field")

#Setting boundary conditions
N.constrain(0.,mesh.facesRight) # electron density = 0 at walls
P.constrain(0.,mesh.facesRight)# ion density = 0 at walls

H.constrain(0.,mesh.facesLeft) # electric field = 0 in the center
N.faceGrad.constrain([0.],mesh.facesLeft) # flux of electron = 0 in the center
P.faceGrad.constrain([0.],mesh.facesLeft) # flux of ion = 0 in the center

if __name__ == '__main__':
    viewer = Viewer(vars=(P,N))
    viewer.plot()

eqn1 = (TransientTerm(var=P) == DiffusionTerm(coeff=delta,var=P) 
                                - ConvectionTerm(coeff=[H.cellVolumeAverage,],var=P) 
                                - ConvectionTerm(coeff=[P.cellVolumeAverage,],var=H))
eqn2 = (TransientTerm(var=N) == DiffusionTerm(var=N) 
                                + (1/delta)*(ConvectionTerm(coeff=[H.cellVolumeAverage,],var=N)
                                            +ConvectionTerm(coeff=[N.cellVolumeAverage,],var=H)))
eqn3 = (TransientTerm(var=H) == gamma*(ConvectionTerm(coeff=[delta**2,],var=P) 
                                       - ConvectionTerm(coeff=[delta,],var=N) 
                                       - H*(delta*P.cellVolumeAverage + N.cellVolumeAverage)))
P.setValue(1.)
N.setValue(1.)
H.setValue(0.)
eqn1d = eqn1 & eqn2 & eqn3



timesteps = 1e-5
steps = 100
for i in range(steps):
    P.updateOld()
    N.updateOld()
    H.updateOld()
    res = 1e10
    sweep = 0
    while res > 1e-3 and sweep < 20:
        res = eqn1d.sweep(dt=timesteps)
        sweep += 1
    if __name__ == '__main__':
        viewer.plot()

Answer 1

电场是向量，而不是标量。

H = CellVariable(rank=1, mesh=mesh, value = 0., hasOld = True, name = "electric field")

更正应该可以更清楚地将术语转换为 FiPy：

没有理由在eq1或eq2的最后一个任期内运行链式法则；它们已经是 FiPy ConvectionTerm 的规范形式。在链式规则之后，它们变成了，例如，，这两者都不是 FiPy 喜欢的形式。您可以将最后两个术语写为明确的来源，但您不应该这样做。

eqn1 = (TransientTerm(var=P) == DiffusionTerm(coeff=delta, var=P) 
                                - ConvectionTerm(coeff=H, var=P))
eqn2 = (TransientTerm(var=N) == DiffusionTerm(var=N) 
                                + (1/delta)*ConvectionTerm(coeff=H, var=N))

我不太明白eq3。它看起来有点像连续性方程的积分？我在快速扫描您引用的Phelps paper 时看不到它。无论如何，它不是 FiPy 可以接受的形式。你可以写它，但它不会很好地解决。右边的术语不是ConvectionTerms，它们只是渐变。

如果您要允许电荷分离并担心德拜长度，我认为您应该求解泊松方程。你能分享这个方程的来源吗？我们也许可以把它做成 FiPy 更喜欢的形式。

Answer 2

eq3 是修改后的泊松方程。我尝试遵循Freeman 概述的程序，其中对泊松方程执行时间导数以替换物种连续性方程。 Freeman 使用 Gear 包解决了这些方程，我只能假设它是 Fortran 上的一个包。我天真地跟着他的脚步走，因为我对数值方法不了解。

我将尝试使用标准形式的泊松方程再次求解。

编辑：我已将电场 H 更改为 1 阶张量，并修改了 eq3 并稍微更改了 gamma 的定义。其他一切都没有改变。

H = CellVariable(rank = 1, mesh=mesh, value = 0., hasOld = True, name = "electric field")

charlength = char_diff_length(L,R_b)
debyelength_e = L_Debye(n_e_0,T_g)
gamma = (debyelength_e/charlength)**2
delta = D_ion/D_e

eqn1 = (TransientTerm(var=P) == DiffusionTerm(coeff=delta,var=P) 
                                - ConvectionTerm(coeff=H,var=P))
eqn2 = (TransientTerm(var=N) == DiffusionTerm(var=N) 
                                + (1/delta)*ConvectionTerm(coeff=H,var=N))
eqn3 = (ConvectionTerm(coeff = gamma/delta, var=H) == ImplicitSourceTerm(var=P) 
                                - ImplicitSourceTerm(var=N))
P.setValue(1.)
N.setValue(1.)
H.setValue(0.)
eqn1d = eqn1 & eqn2 & eqn3

timesteps = 1e-8
steps = 100
for i in range(steps):
    P.updateOld()
    N.updateOld()
    H.updateOld()
    res = 1e10
    sweep = 0
    while res > 1e-3 and sweep < 20:
        res = eqn1d.sweep(dt=timesteps)
        sweep += 1
    if __name__ == '__main__':
        viewer.plot()

它没有给我与以前相同的错误，这是一些进步的迹象。但是，它正在吐出一个新错误：

ValueError: all the input arrays must have same number of dimensions, but the array at index 0 has 1 dimension(s) and the array at index 2 has 2 dimension(s)