Coupling pdes across various domains using fipy

问题

I am completely new to fipy. I am trying to solve the following set of pdes across two different domains using fipy. The variables are p,n and ψ and q,Dn,Dp,un,up,e and N are all constants. N and e change values from domain 1 to domain 2. Domain 2 is a rectangular domain stacked above a similar rectangular domain 1. The variables p and n are to be solved for in domain 1 and ψ is to be solved for in both the domains- domain1 and domain 2.

qDn∇2n − qun∇.(n∇ψ) = q(n-10**11)/10**(-6), in Domain 1
qDp∇2p + qup∇.(p∇ψ) = -q(p-10**21)/10**(-6), in Domain 1
∇2ψ = −(p − n- N)/e in Domain  1
e∇2ψ = 0 in Domain 2

The code that I have written for solving the pdes has been attached below.

! pip install pyparse
from fipy import *
L= 10**(-6)
h= 20**(-6)
tox= 0.1*10**(-6)
q=1.6*10**(-19)
un=0.14
up=0.045
Vth=0.026
Dp= up*Vth
Dn=un*Vth
p0= 10**(21)
n0= 10**(11)
e0=8.854*10**(-12)

mesh1= Grid2D(dx= L/100,nx=100,dy=h/200,ny=200)
mesh2= Grid2D(dx= L/100,nx=100,dy=tox/10,ny=10)
mesh3= mesh1+(mesh2+[[0],[h]]) # final mesh

x,y= mesh3.cellCenters
N= 10**21*(y<=h) # N changes from Domain 1 to Domain 2
e= 11.9*e0*(y<=h)+ 3.9*e0*(y>h) # e changes from Domain 1 to Domain 2


p1=CellVariable(name='hole',mesh=mesh3,hasOld=True,value=p0)
n1=CellVariable(name='electron',mesh=mesh3,hasOld=True,value=n0)
psi=CellVariable(name='potential',mesh=mesh3,hasOld=True,value=1)

p=p1*(y<=h) # for domain separation
n=n1*(y<=h) # for domain separation

mask1=((y==0))
mask2=(y==h)
mask3=(y==(h+tox))

# 1e50 is the large value 
# boundary conditions are p(x,0)= p0, n(x,0)= n0, psi(x,0)= 0, p(x,h)= p0*exp(-psi/Vth), n(x,h)= n0*exp(psi/Vth), psi(h+tox)= 5 
eq1=(DiffusionTerm(coeff=q*Dn,var=n)-ConvectionTerm(coeff=q*un*psi.faceGrad,var=n)==ImplicitSourceTerm(coeff=q*(n-10**5)/10**(-6),var=n)-ImplicitSourceTerm(mask1*1e50)-ImplicitSourceTerm(mask2*1e50)+mask1*1e50*n0+mask2*1e50*n0*numerix.exp(psi/Vth))
eq2=(DiffusionTerm(coeff=q*Dp,var=p)+ConvectionTerm(coeff=q*up*psi.faceGrad,var=p)==ImplicitSourceTerm(coeff=-q*(p-10**15)/10**(-6),var=p)-ImplicitSourceTerm(mask1*1e50)-ImplicitSourceTerm(mask2*1e50)+mask1*1e50*p0+mask2*1e50*p0*numerix.exp(psi/Vth))
eq3=(DiffusionTerm(coeff=e,var=psi)==ImplicitSourceTerm(coeff=-q*(p-n-N),var=psi)-ImplicitSourceTerm(mask1*1e50)-ImplicitSourceTerm(mask2*1e50)-ImplicitSouceTerm(mask3*1e50)+mask1*1e50*0+mask2*1e50*psi+mask3*1e50*5)
eq= eq1 & eq2 & eq3


for t in range (50):
 p.updateOld()
 n.updateOld()
 psi.updateOld()
 eq.solve(dt=10) # Since the equation does not have any transient term, a large value of dt=5 has been chosen

Now, I am getting the following error: ExplicitVariableError: Terms with explicit Variables cannot mix with Terms with implicit Variables.

I am also not sure whether the coupling between the equations actually work out considering the way I have written the code.

Please note that I am actually solving a MOSCAP using the above pdes. Any help regarding this would be highly appreciated.

回答1:

FiPy does not support this usage. One set of equations govern one set of variables defined on one domain. n, p, and psi must all exist on the same mesh in order to be able to write eq3.

Define all equations on mesh3.
Use an internal constraint on n and p.
You will not be able to use mesh1.facesTop and mesh1.facesBottom. Define the internal boundary parametrically.

Note: e should be moved to the diffusion coefficient in eq3. This is both physically correct (divergence of the electric displacement field goes like the charge) and necessary to account for the step in permittivity at the boundary between your subdomains.

来源：https://stackoverflow.com/questions/61913292/coupling-pdes-across-various-domains-using-fipy

标签

python

pde

fipy