def buildPhaseEquation(phase, theta):
mPhiVar = phase - 0.5 + temperature * phase * (1 - phase)
thetaMag = theta.getOld().getGrad().getMag()
implicitSource = mPhiVar * (phase - (mPhiVar < 0))
implicitSource += (2 * s + epsilon**2 * thetaMag) * thetaMag
return TransientTerm(phaseTransientCoeff) == \
ExplicitDiffusionTerm(alpha**2) \
- ImplicitSourceTerm(implicitSource) \
+ (mPhiVar > 0) * mPhiVar * phase
def define_ode(self, current_time):
x, y = self.mesh.faceCenters
# Internal source specificatio - currently no functional
internal_source_value = self.parameter.internal_source_value
internal_source_region = self.parameter.internal_source_region
internal_source_mask = (
(x > internal_source_region.xmin) &
(x < internal_source_region.xmax) &
(y > internal_source_region.ymin) &
(y < internal_source_region.ymax)
)
# Get convection data
convection = self.define_convection_variable(current_time)
eq = TransientTerm() == - ConvectionTerm(coeff=convection) \
+ DiffusionTerm(coeff=self.parameter.Diffusivity)\
- ImplicitSourceTerm(coeff=self.parameter.Decay)\
# + ImplicitSourceTerm(coeff=internal_source_value*internal_source_mask) # Internal source not working
return eq
phase = CellVariable(name='PhaseField', mesh=mesh, value=1.)
from fipy.variables.modularVariable import ModularVariable
theta = ModularVariable(name='Theta', mesh=mesh, value=1.)
theta.setValue(0., where=mesh.getCellCenters()[..., 0] > L / 2.)
from fipy.terms.implicitSourceTerm import ImplicitSourceTerm
mPhiVar = phase - 0.5 + temperature * phase * (1 - phase)
thetaMag = theta.getOld().getGrad().getMag()
implicitSource = mPhiVar * (phase - (mPhiVar < 0))
implicitSource += (2 * s + epsilon**2 * thetaMag) * thetaMag
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.explicitDiffusionTerm import ExplicitDiffusionTerm
phaseEq = TransientTerm(phaseTransientCoeff) == \
ExplicitDiffusionTerm(alpha**2) \
- ImplicitSourceTerm(implicitSource) \
+ (mPhiVar > 0) * mPhiVar * phase
if __name__ == '__main__':
import fipy.viewers
phaseViewer = fipy.viewers.make(vars=phase)
phaseViewer.plot()
for step in range(steps):
phaseEq.solve(phase, dt=timeStepDuration)
phaseViewer.plot()
raw_input('finished')
shift = 1.
KMVar = CellVariable(mesh=mesh, value=params['KM'] * shift, hasOld=1)
KCVar = CellVariable(mesh=mesh, value=params['KC'] * shift, hasOld=1)
TMVar = CellVariable(mesh=mesh, value=params['TM'] * shift, hasOld=1)
TCVar = CellVariable(mesh=mesh, value=params['TC'] * shift, hasOld=1)
P3Var = CellVariable(mesh=mesh, value=params['P3'] * shift, hasOld=1)
P2Var = CellVariable(mesh=mesh, value=params['P2'] * shift, hasOld=1)
RVar = CellVariable(mesh=mesh, value=params['R'], hasOld=1)
PN = P3Var + P2Var
KMscCoeff = params['chiK'] * (RVar + 1) * (1 - KCVar -
KMVar.getCellVolumeAverage())
KMspCoeff = params['lambdaK'] / (1 + PN / params['kappaK'])
KMEq = TransientTerm() - KMscCoeff + ImplicitSourceTerm(KMspCoeff)
TMscCoeff = params['chiT'] * (1 - TCVar - TMVar.getCellVolumeAverage())
TMspCoeff = params['lambdaT'] * (KMVar + params['zetaT'])
TMEq = TransientTerm() - TMscCoeff + ImplicitSourceTerm(TMspCoeff)
TCscCoeff = params['lambdaT'] * (TMVar * KMVar).getCellVolumeAverage()
TCspCoeff = params['lambdaTstar']
TCEq = TransientTerm() - TCscCoeff + ImplicitSourceTerm(TCspCoeff)
PIP2PITP = PN / (PN / params['kappam'] + PN.getCellVolumeAverage() /
params['kappac'] + 1) + params['zetaPITP']
P3spCoeff = params['lambda3'] * (TMVar + params['zeta3T'])
P3scCoeff = params['chi3'] * KMVar * (PIP2PITP /
(1 + KMVar / params['kappa3']) +
def buildMetalIonDiffusionEquation(ionVar=None,
distanceVar=None,
depositionRate=1,
transientCoeff=1,
diffusionCoeff=1,
metalIonMolarVolume=1):
r"""
The `MetalIonDiffusionEquation` solves the diffusion of the metal
species with a source term at the electrolyte interface. The governing
equation is given by,
.. math::
\frac{\partial c}{\partial t} = \nabla \cdot D \nabla c
where,
.. math::
D = \begin{cases}
D_c & \text{when $\phi > 0$} \\
0 & \text{when $\phi \le 0$}
\end{cases}
The velocity of the interface generally has a linear dependence on ion
concentration. The following boundary condition applies at the zero
level set,
.. math::
D \hat{n} \cdot \nabla c = \frac{v(c)}{\Omega} \qquad \text{at $phi = 0$}
where
.. math::
v(c) = c V_0
The test case below is for a 1D steady state problem. The solution is
given by:
.. math::
c(x) = \frac{c^{\infty}}{\Omega D / V_0 + L}\left(x - L\right) + c^{\infty}
This is the test case,
>>> from fipy.meshes import Grid1D
>>> nx = 11
>>> dx = 1.
>>> from fipy.tools import serialComm
>>> mesh = Grid1D(nx = nx, dx = dx, communicator=serialComm)
>>> x, = mesh.cellCenters
>>> from fipy.variables.cellVariable import CellVariable
>>> ionVar = CellVariable(mesh = mesh, value = 1.)
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> disVar = DistanceVariable(mesh = mesh,
... value = (x - 0.5) - 0.99,
... hasOld = 1)
>>> v = 1.
>>> diffusion = 1.
>>> omega = 1.
>>> cinf = 1.
>>> eqn = buildMetalIonDiffusionEquation(ionVar = ionVar,
... distanceVar = disVar,
... depositionRate = v * ionVar,
... diffusionCoeff = diffusion,
... metalIonMolarVolume = omega)
>>> ionVar.constrain(cinf, mesh.facesRight)
>>> from builtins import range
>>> for i in range(10):
... eqn.solve(ionVar, dt = 1000)
>>> L = (nx - 1) * dx - dx / 2
>>> gradient = cinf / (omega * diffusion / v + L)
>>> answer = gradient * (x - L - dx * 3 / 2) + cinf
>>> answer[x < dx] = 1
>>> print(ionVar.allclose(answer))
1
Testing the interface source term
>>> from fipy.meshes import Grid2D
>>> from fipy import numerix, serialComm
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2, communicator=serialComm)
>>> from fipy.variables.distanceVariable import DistanceVariable
>>> distance = DistanceVariable(mesh = mesh, value = (-.5, .5, .5, 1.5))
>>> ionVar = CellVariable(mesh = mesh, value = (1, 1, 1, 1))
>>> depositionRate = CellVariable(mesh=mesh, value=(1, 1, 1, 1))
>>> source = depositionRate * distance.cellInterfaceAreas / mesh.cellVolumes / ionVar
>>> sqrt = numerix.sqrt(2)
>>> ans = CellVariable(mesh=mesh, value=(0, 1 / sqrt, 1 / sqrt, 0))
>>> print(numerix.allclose(source, ans))
True
>>> distance[:] = (-1.5, -0.5, -0.5, 0.5)
>>> print(numerix.allclose(source, (0, 0, 0, sqrt)))
True
:Parameters:
- `ionVar`: The metal ion concentration variable.
- `distanceVar`: A `DistanceVariable` object.
- `depositionRate`: A float or a `CellVariable` representing the interface deposition rate.
- `transientCoeff`: The transient coefficient.
- `diffusionCoeff`: The diffusion coefficient
- `metalIonMolarVolume`: Molar volume of the metal ions.
"""
diffusionCoeff = _LevelSetDiffusionVariable(distanceVar, diffusionCoeff)
eq = TransientTerm(transientCoeff) - DiffusionTermNoCorrection(
diffusionCoeff)
mesh = distanceVar.mesh
coeff = depositionRate * distanceVar.cellInterfaceAreas / (
mesh.cellVolumes * metalIonMolarVolume) / ionVar
return eq + ImplicitSourceTerm(coeff)
def buildSurfactantBulkDiffusionEquation(bulkVar=None,
distanceVar=None,
surfactantVar=None,
otherSurfactantVar=None,
diffusionCoeff=None,
transientCoeff=1.,
rateConstant=None):
r"""
The `buildSurfactantBulkDiffusionEquation` function returns a bulk diffusion of a
species with a source term for the jump from the bulk to an interface.
The governing equation is given by,
.. math::
\frac{\partial c}{\partial t} = \nabla \cdot D \nabla c
where,
.. math::
D = \begin{cases}
D_c & \text{when $\phi > 0$} \\
0 & \text{when $\phi \le 0$}
\end{cases}
The jump condition at the interface is defined by Langmuir
adsorption. Langmuir adsorption essentially states that the ability for
a species to jump from an electrolyte to an interface is proportional to
the concentration in the electrolyte, available site density and a
jump coefficient. The boundary condition at the interface is given by
.. math::
D \hat{n} \cdot \nabla c = -k c (1 - \theta) \qquad \text{at $\phi = 0$}.
Parameters
----------
bulkVar : ~fipy.variables.cellVariable.CellVariable
The bulk surfactant concentration variable.
distanceVar : ~fipy.variables.distanceVariable.DistanceVariable
surfactantVar : ~fipy.variables.surfactantVariable.SurfactantVariable
otherSurfactantVar : ~fipy.variables.surfactantVariable.SurfactantVariable
Any other surfactants that may remove this one.
diffusionCoeff : float or ~fipy.variables.faceVariable.FaceVariable
transientCoeff : float
In general 1 is used.
rateConstant : float
The adsorption coefficient.
"""
spCoeff = rateConstant * distanceVar.cellInterfaceAreas / bulkVar.mesh.cellVolumes
spSourceTerm = ImplicitSourceTerm(spCoeff)
bulkSpCoeff = spCoeff * bulkVar
coeff = bulkSpCoeff * surfactantVar.interfaceVar
diffusionCoeff = _LevelSetDiffusionVariable(distanceVar, diffusionCoeff)
eq = TransientTerm(transientCoeff) - DiffusionTermNoCorrection(
diffusionCoeff)
if otherSurfactantVar is not None:
otherCoeff = bulkSpCoeff * otherSurfactantVar.interfaceVar
else:
otherCoeff = 0
return eq - coeff + spSourceTerm - otherCoeff
phaseY = phase.getFaceGrad().dot((0, 1))
phaseX = phase.getFaceGrad().dot((1, 0))
psi = theta + numerix.arctan2(phaseY, phaseX)
Phi = numerix.tan(N * psi / 2)
PhiSq = Phi**2
beta = (1. - PhiSq) / (1. + PhiSq)
betaPsi = -N * 2 * Phi / (1 + PhiSq)
A = alpha**2 * c * (1. + c * beta) * betaPsi
D = alpha**2 * (1. + c * beta)**2
dxi = phase.getFaceGrad()._take((1, 0), axis=1) * (-1, 1)
anisotropySource = (A * dxi).getDivergence()
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.explicitDiffusionTerm import ExplicitDiffusionTerm
from fipy.terms.implicitSourceTerm import ImplicitSourceTerm
phaseEq = TransientTerm(tau) == ExplicitDiffusionTerm(D) + \
ImplicitSourceTerm(mVar * ((mVar < 0) - phase)) + \
((mVar > 0.) * mVar * phase + anisotropySource)
from fipy.terms.implicitDiffusionTerm import ImplicitDiffusionTerm
temperatureEq = TransientTerm() == \
ImplicitDiffusionTerm(tempDiffusionCoeff) + \
(phase - phase.getOld()) / timeStepDuration
bench.stop('terms')
phase.updateOld()
temperature.updateOld()
phaseEq.solve(phase, dt=timeStepDuration)
temperatureEq.solve(temperature, dt=timeStepDuration)
steps = 10
def __init__(self,
surfactantVar=None,
distanceVar=None,
bulkVar=None,
rateConstant=None,
otherVar=None,
otherBulkVar=None,
otherRateConstant=None,
consumptionCoeff=None):
"""
Create a `AdsorbingSurfactantEquation` object.
:Parameters:
- `surfactantVar`: The `SurfactantVariable` to be solved for.
- `distanceVar`: The `DistanceVariable` that marks the interface.
- `bulkVar`: The value of the `surfactantVar` in the bulk.
- `rateConstant`: The adsorption rate of the `surfactantVar`.
- `otherVar`: Another `SurfactantVariable` with more surface affinity.
- `otherBulkVar`: The value of the `otherVar` in the bulk.
- `otherRateConstant`: The adsorption rate of the `otherVar`.
- `consumptionCoeff`: The rate that the `surfactantVar` is consumed during deposition.
"""
self.eq = TransientTerm(coeff=1) - ExplicitUpwindConvectionTerm(
SurfactantConvectionVariable(distanceVar))
self.dt = Variable(0.)
mesh = distanceVar.mesh
adsorptionCoeff = self.dt * bulkVar * rateConstant
spCoeff = adsorptionCoeff * distanceVar._cellInterfaceFlag
scCoeff = adsorptionCoeff * distanceVar.cellInterfaceAreas / mesh.cellVolumes
self.eq += ImplicitSourceTerm(spCoeff) - scCoeff
if otherVar is not None:
otherSpCoeff = self.dt * otherBulkVar * otherRateConstant * distanceVar._cellInterfaceFlag
otherScCoeff = -otherVar.interfaceVar * scCoeff
self.eq += ImplicitSourceTerm(otherSpCoeff) - otherScCoeff
vars = (surfactantVar, otherVar)
else:
vars = (surfactantVar, )
total = 0
for var in vars:
total += var.interfaceVar
maxVar = (total > 1) * distanceVar._cellInterfaceFlag
val = distanceVar.cellInterfaceAreas / mesh.cellVolumes
for var in vars[1:]:
val -= distanceVar._cellInterfaceFlag * var
spMaxCoeff = 1e20 * maxVar
scMaxCoeff = spMaxCoeff * val * (val > 0)
self.eq += ImplicitSourceTerm(spMaxCoeff) - scMaxCoeff - 1e-40
if consumptionCoeff is not None:
self.eq += ImplicitSourceTerm(consumptionCoeff)