startingArray = numerix.zeros(nx, 'd')
startingArray[50:90] = 1.
var = CellVariable(
name = "advection variable",
mesh = mesh,
value = startingArray)
boundaryConditions = (
FixedValue(mesh.getFacesLeft(), valueLeft),
FixedValue(mesh.getFacesRight(), valueRight)
)
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.explicitUpwindConvectionTerm import ExplicitUpwindConvectionTerm
eq = TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (velocity,))
if __name__ == '__main__':
viewer = fipy.viewers.make(vars=(var,))
for step in range(steps):
eq.solve(var,
dt = timeStepDuration,
boundaryConditions = boundaryConditions,
solver = LinearCGSSolver(tolerance = 1.e-15, steps = 2000))
viewer.plot()
viewer.plot()
raw_input('finished')
startingArray[50:90] = 1.
var = CellVariable(
name = "advection variable",
mesh = mesh,
value = startingArray)
boundaryConditions = (
FixedValue(mesh.getFacesLeft(), valueLeft),
FixedValue(mesh.getFacesRight(), valueRight)
)
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.powerLawConvectionTerm import PowerLawConvectionTerm
eq = TransientTerm() - PowerLawConvectionTerm(coeff = (velocity,))
if __name__ == '__main__':
viewer = fipy.viewers.make(vars=(var,))
viewer.plot()
raw_input("press key to continue")
for step in range(steps):
eq.solve(var,
dt = timeStepDuration,
boundaryConditions = boundaryConditions,
solver = LinearLUSolver(tolerance = 1.e-15))
viewer.plot()
viewer.plot()
raw_input('finished')
from fipy.variables.cellVariable import CellVariable
phi = CellVariable(name="solution variable", mesh=mesh, value=0)
D = 1
valueLeft = 1
valueRight = 0
from fipy.boundaryConditions.fixedValue import FixedValue
BCs = (FixedValue(faces=mesh.getFacesRight(), value=valueRight),
FixedValue(faces=mesh.getFacesLeft(), value=valueLeft))
from fipy.terms.explicitDiffusionTerm import ExplicitDiffusionTerm
from fipy.terms.transientTerm import TransientTerm
eqX = TransientTerm() == ExplicitDiffusionTerm(coeff=D)
timeStepDuration = 0.9 * dx**2 / (2 * D)
steps = 100
from fipy import viewers
viewer = viewers.make(vars=(phi), limits={'datamin': 0., 'datamax': 1.})
def hit_continue(Prompt='Hit any key to continue'):
raw_input(Prompt)
for step in range(steps):
eqX.solve(var=phi, boundaryConditions=BCs, dt=timeStepDuration)
viewer.plot()
hit_continue()
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.vanLeerConvectionTerm import VanLeerConvectionTerm
eq1 = TransientTerm() - VanLeerConvectionTerm(coeff=(-velocity, ))
eq2 = TransientTerm() - VanLeerConvectionTerm(coeff=(-velocity, ))
if __name__ == '__main__':
import fipy.viewers
viewer1 = fipy.viewers.make(vars=var1)
viewer2 = fipy.viewers.make(vars=var2)
viewer1.plot()
viewer2.plot()
from fipy.solvers.linearLUSolver import LinearLUSolver
newVar2 = var2.copy()
for step in range(steps):
eq1.solve(var=var1, dt=dt, solver=LinearLUSolver())
eq2.solve(var=var2, dt=dt, solver=LinearLUSolver())
viewer1.plot()
viewer2.plot()
newVar2[:nx / 4] = var2[nx / 4:]
newVar2[nx / 4:] = var2[:nx / 4]
print 'maximum absolute difference between periodic and non-periodic grids:', numerix.max(
abs(var1[nx / 4:3 * nx / 4] - newVar2))
raw_input('finished')
from fipy.boundaryConditions.nthOrderBoundaryCondition import NthOrderBoundaryCondition
BCs = (FixedFlux(mesh.getFacesRight(), 0),
FixedFlux(mesh.getFacesLeft(), 0),
NthOrderBoundaryCondition(mesh.getFacesLeft(), 0, 3),
NthOrderBoundaryCondition(mesh.getFacesRight(), 0, 3),
NthOrderBoundaryCondition(mesh.getFacesTop(), 0, 3),
NthOrderBoundaryCondition(mesh.getFacesBottom(), 0, 3))
if __name__ == '__main__':
import fipy.viewers
viewer = fipy.viewers.make(vars = var, limits = {'datamin': 0., 'datamax': 1.0})
viewer.plot()
dexp=-5
for step in range(steps):
dt = numerix.exp(dexp)
dt = min(100, dt)
dexp += 0.01
var.updateOld()
eqch.solve(var, boundaryConditions = BCs, solver = solver, dt = dt)
if __name__ == '__main__':
viewer.plot()
print 'step',step,'dt',dt
def _run():
pass
class AdsorbingSurfactantEquation():
r"""
The `AdsorbingSurfactantEquation` object solves the
`SurfactantEquation` but with an adsorbing species from some bulk
value. The equation that describes the surfactant adsorbing is
given by,
.. math::
\dot{\theta} = J v \theta + k c (1 - \theta - \theta_{\text{other}}) - \theta c_{\text{other}} k_{\text{other}} - k^- \theta
where :math:`\theta`, :math:`J`, :math:`v`, :math:`k`, :math:`c`,
:math:`k^-` and :math:`n` represent the surfactant coverage, the curvature,
the interface normal velocity, the adsorption rate, the concentration in the
bulk at the interface, the consumption rate and an exponent of consumption,
respectively. The :math:`\text{other}` subscript refers to another
surfactant with greater surface affinity.
The terms on the RHS of the above equation represent conservation of
surfactant on a non-uniform surface, Langmuir adsorption, removal of
surfactant due to adsorption of the other surfactant onto non-vacant sites
and consumption of the surfactant respectively. The adsorption term is added
to the source by setting :math:` S_c = k c (1 - \theta_{\text{other}})` and
:math:`S_p = -k c`. The other terms are added to the source in a similar
way.
The following is a test case:
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> from fipy.tools import numerix
>>> from fipy.variables.cellVariable import CellVariable
>>> dx = .5
>>> dy = 2.3
>>> dt = 0.25
>>> k = 0.56
>>> initialValue = 0.1
>>> c = 0.2
>>> from fipy.meshes import Grid2D
>>> from fipy import serialComm
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = (-dx*3/2, -dx/2, dx/2,
... 3*dx/2, 5*dx/2),
... hasOld = 1)
>>> surfactantVar = SurfactantVariable(value = (0, 0, initialValue, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar = CellVariable(mesh = mesh, value = (c , c, c, c, c))
>>> eqn = AdsorbingSurfactantEquation(surfactantVar = surfactantVar,
... distanceVar = distanceVar,
... bulkVar = bulkVar,
... rateConstant = k)
>>> eqn.solve(surfactantVar, dt = dt)
>>> answer = (initialValue + dt * k * c) / (1 + dt * k * c)
>>> print numerix.allclose(surfactantVar.interfaceVar,
... numerix.array((0, 0, answer, 0, 0)))
1
The following test case is for two surfactant variables. One has more
surface affinity than the other.
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> dx = 0.5
>>> dy = 2.73
>>> dt = 0.001
>>> k0 = 1.
>>> k1 = 10.
>>> theta0 = 0.
>>> theta1 = 0.
>>> c0 = 1.
>>> c1 = 1.
>>> totalSteps = 10
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = dx * (numerix.arange(5) - 1.5),
... hasOld = 1)
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> var1 = SurfactantVariable(value = (0, 0, theta1, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> bulkVar1 = CellVariable(mesh = mesh, value = (c1, c1, c1, c1, c1))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = k0)
>>> eqn1 = AdsorbingSurfactantEquation(surfactantVar = var1,
... distanceVar = distanceVar,
... bulkVar = bulkVar1,
... rateConstant = k1,
... otherVar = var0,
... otherBulkVar = bulkVar0,
... otherRateConstant = k0)
>>> for step in range(totalSteps):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> answer0 = 1 - numerix.exp(-k0 * c0 * dt * totalSteps)
>>> answer1 = (1 - numerix.exp(-k1 * c1 * dt * totalSteps)) * (1 - answer0)
>>> print numerix.allclose(var0.interfaceVar,
... numerix.array((0, 0, answer0, 0, 0)), rtol = 1e-2)
1
>>> print numerix.allclose(var1.interfaceVar,
... numerix.array((0, 0, answer1, 0, 0)), rtol = 1e-2)
1
>>> dt = 0.1
>>> for step in range(10):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> x, y = mesh.cellCenters
>>> check = var0.interfaceVar + var1.interfaceVar
>>> answer = CellVariable(mesh=mesh, value=check)
>>> answer[x==1.25] = 1.
>>> print check.allequal(answer)
True
The following test case is to fix a bug where setting the adosrbtion
coefficient to zero leads to the solver not converging and an eventual
failure.
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = 0)
>>> eqn0.solve(var0, dt = dt)
>>> eqn0.solve(var0, dt = dt)
>>> answer = CellVariable(mesh=mesh, value=var0.interfaceVar)
>>> answer[x==1.25] = 0.
>>> print var0.interfaceVar.allclose(answer)
True
The following test case is to fix a bug that allows the accelerator to
become negative.
>>> nx = 5
>>> ny = 5
>>> dx = 1.
>>> dy = 1.
>>> mesh = Grid2D(dx=dx, dy=dy, nx = nx, ny = ny, communicator=serialComm)
>>> x, y = mesh.cellCenters
>>> disVar = DistanceVariable(mesh=mesh, value=1., hasOld=True)
>>> disVar[y < dy] = -1
>>> disVar[x < dx] = -1
>>> disVar.calcDistanceFunction() #doctest: +LSM
>>> levVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> accVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> levEq = AdsorbingSurfactantEquation(levVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0)
>>> accEq = AdsorbingSurfactantEquation(accVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0,
... otherVar = levVar,
... otherBulkVar = 0,
... otherRateConstant = 0)
>>> extVar = CellVariable(mesh = mesh, value = accVar.interfaceVar)
>>> from fipy import TransientTerm, AdvectionTerm
>>> advEq = TransientTerm() + AdvectionTerm(extVar)
>>> dt = 0.1
>>> for i in range(50):
... disVar.calcDistanceFunction()
... extVar.value = (numerix.array(accVar.interfaceVar))
... disVar.extendVariable(extVar)
... disVar.updateOld()
... advEq.solve(disVar, dt = dt)
... levEq.solve(levVar, dt = dt)
... accEq.solve(accVar, dt = dt) #doctest: +LSM
>>> print (accVar >= -1e-10).all()
True
"""
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)
def solve(self, var, boundaryConditions=(), solver=None, dt=None):
"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear system once.
:Parameters:
- `var`: A `SurfactantVariable` to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
"""
self.dt.setValue(dt)
if solver is None:
import fipy.solvers.solver
if fipy.solvers.solver == 'pyamg':
from fipy.solvers.pyAMG.linearGeneralSolver import LinearGeneralSolver
solver = LinearGeneralSolver(tolerance=1e-15, iterations=2000)
else:
from fipy.solvers import LinearPCGSolver
solver = LinearPCGSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions,)
var.constrain(0, var.mesh.exteriorFaces)
self.eq.solve(var,
boundaryConditions=boundaryConditions,
solver = solver,
dt=1.)
def sweep(self, var, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None):
r"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear
system once. This method also recalculates and returns the
residual as well as applying under-relaxation.
:Parameters:
- `var`: The variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
- `underRelaxation`: Usually a value between `0` and `1` or `None` in the case of no under-relaxation
"""
self.dt.setValue(dt)
if solver is None:
from fipy.solvers import DefaultAsymmetricSolver
solver = DefaultAsymmetricSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions,)
var.constrain(0, var.mesh.exteriorFaces)
return self.eq.sweep(var, solver=solver, boundaryConditions=boundaryConditions, underRelaxation=underRelaxation, residualFn=residualFn, dt=1.)
from fipy.boundaryConditions.fixedValue import FixedValue
BCs = (FixedValue(faces=mesh.getFacesRight(), value=valueRight),
FixedValue(faces=mesh.getFacesLeft(), value=valueLeft))
from fipy.terms.explicitDiffusionTerm import ExplicitDiffusionTerm
from fipy.terms.transientTerm import TransientTerm
eqX = TransientTerm() == ExplicitDiffusionTerm(coeff=D)
timeStepDuration = 0.9 * dx**2 / (2 * D)
steps=100
from fipy import viewers
viewer = viewers.make(vars=(phi),
limits={'datamin': 0., 'datamax': 1.})
def hit_continue(Prompt='Hit any key to continue'):
raw_input(Prompt)
for step in range(steps):
eqX.solve(var=phi,
boundaryConditions=BCs,
dt=timeStepDuration)
viewer.plot()
hit_continue()
bench.start()
D = 1.
from fipy.terms.implicitDiffusionTerm import ImplicitDiffusionTerm
from fipy.terms.transientTerm import TransientTerm
eq = TransientTerm() == ImplicitDiffusionTerm(coeff = D)
bench.stop('terms')
## from fipy import viewers
## viewer = viewers.make(vars = C, limits = {'datamin': 0, 'datamax': 1})
## viewer.plot()
## raw_input("initial")
bench.start()
dt = 1e0
steps = 1
for step in range(steps):
eq.solve(var = C, dt = dt)
## viewer.plot()
bench.stop('solve')
print bench.report(numberOfElements=N, steps=steps)
## raw_input("finished")
value = startingArray[:nx / 2])
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.vanLeerConvectionTerm import VanLeerConvectionTerm
eq1 = TransientTerm() - VanLeerConvectionTerm(coeff = (-velocity,))
eq2 = TransientTerm() - VanLeerConvectionTerm(coeff = (-velocity,))
if __name__ == '__main__':
import fipy.viewers
viewer1 = fipy.viewers.make(vars=var1)
viewer2 = fipy.viewers.make(vars=var2)
viewer1.plot()
viewer2.plot()
from fipy.solvers.linearLUSolver import LinearLUSolver
newVar2 = var2.copy()
for step in range(steps):
eq1.solve(var = var1, dt = dt, solver = LinearLUSolver())
eq2.solve(var = var2, dt = dt, solver = LinearLUSolver())
viewer1.plot()
viewer2.plot()
newVar2[:nx / 4] = var2[nx / 4:]
newVar2[nx / 4:] = var2[:nx / 4]
print 'maximum absolute difference between periodic and non-periodic grids:',numerix.max(abs(var1[nx / 4:3 * nx / 4] - newVar2))
raw_input('finished')
phase = CellVariable(name = 'PhaseField', mesh = mesh, value = 1.)
from fipy.variables.modularVariable import ModularVariable
theta = ModularVariable(name = 'Theta', mesh = mesh, value = 1.)
x, y = mesh.getCellCenters()[...,0], mesh.getCellCenters()[...,1]
theta.setValue(0., where=(x - L / 2.)**2 + (y - L / 2.)**2 < (L / 4.)**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):
phase.updateOld()
phaseEq.solve(phase, dt = timeStepDuration)
phaseViewer.plot()
raw_input('finished')
5.39556276e-14, 1.55308505e-11, 2.94461712e-09, 3.63798469e-07,
2.74326174e-05, 1.01935828e-03, 9.76562500e-24, 1.92578125e-20,
1.70937109e-17, 8.99433979e-15, 3.10726059e-12, 7.36603377e-10,
1.21397338e-07, 1.37456643e-05, 1.02532568e-03, 4.57589878e-02,
2.63278194e-07, 5.70863224e-12, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 1.51165440e-13, 1.23805218e-07,
1.51873310e-03, 5.87457842e-01, 3.78270971e-06, 2.41898556e-10,
2.62440000e-16, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 1.55453500e-09, 6.18653630e-05, 8.85109369e-02,
7.24354518e-05, 1.32738123e-08, 8.11158300e-14, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 2.27755930e-11,
3.31776157e-06, 1.39387353e-02, 3.78270971e-06, 2.41898556e-10,
2.62440000e-16, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 1.55453500e-09, 6.18653630e-05, 8.85109369e-02
])
if __name__ == '__main__':
viewer = fipy.viewers.make(vars=var)
for step in range(steps):
var.updateOld()
eqn.solve(var,
boundaryConditions=boundaryConditions,
dt=timeStepDuration)
if (not (step % 100)):
print(step / 100)
print var
theMask = numerix.array([[10, 1, 20, 2]])
viewer.plot()
## viewer.plot(mask = theMask, graphwidth = 15, graphheight = 3)
raw_input('finished')
# create a field variable, set the initial conditions:
from fipy.variables.cellVariable import CellVariable
var = CellVariable(mesh=mesh, value=0)
def centerCells(cell):
return abs(cell.getCenter()[0] - L / 2.0) < L / 10
var.setValue(value=1.0, cells=mesh.getCells(filter=centerCells))
# create the equation:
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.implicitDiffusionTerm import ImplicitDiffusionTerm
eq = TransientTerm() - ImplicitDiffusionTerm(coeff=1) == 0
# create a viewer:
from fipy.viewers.gist2DViewer import Gist1DViewer
viewer = Gist1DViewer(vars=(var,), limits=('e', 'e', 0, 1))
viwer.plot()
# solve
for i in range(steps):
var.updateOld()
eq.solve()
viewer.plot()
## Define boundary condition values
leftValue = 1.0
rightValue = 0
## Creation of boundary conditions
from fipy.boundaryConditions.fixedValue import FixedValue
BCs = (FixedValue(faces = mesh_example.getFacesRight(), value=rightValue),
FixedValue(faces=mesh_example.getFacesLeft(),value=leftValue))
D = 1.0 ## diffusivity (m^2/s)
## Transient diffusion equation is defined next
from fipy.terms.explicitDiffusionTerm import ExplicitDiffusionTerm
from fipy.terms.transientTerm import TransientTerm
eqX = TransientTerm() == ExplicitDiffusionTerm(coeff = D)
timeStep = 0.09*deltax**2/(2*D) ## size of time step
steps = 9000 ## number of time-steps
## create a GUI
from fipy import viewers
viewer = viewers.make(vars = phi, limits={'datamin':0.0, 'datamax':1.0})
## iterate until you get tired
for step in range(steps):
eqX.solve(var = phi, boundaryConditions = BCs, dt = timeStep) ## solving the equation
viewer.plot() ## update the GUI
steps = 1000
mesh = Grid1D(dx=dx, nx=nx)
startingArray = numerix.zeros(nx, 'd')
startingArray[50:90] = 1.
var = CellVariable(name="advection variable", mesh=mesh, value=startingArray)
boundaryConditions = (FixedValue(mesh.getFacesLeft(), valueLeft),
FixedValue(mesh.getFacesRight(), valueRight))
from fipy.terms.transientTerm import TransientTerm
from fipy.terms.powerLawConvectionTerm import PowerLawConvectionTerm
eq = TransientTerm() - PowerLawConvectionTerm(coeff=(velocity, ))
if __name__ == '__main__':
viewer = fipy.viewers.make(vars=(var, ))
viewer.plot()
raw_input("press key to continue")
for step in range(steps):
eq.solve(var,
dt=timeStepDuration,
boundaryConditions=boundaryConditions,
solver=LinearLUSolver(tolerance=1.e-15))
viewer.plot()
viewer.plot()
raw_input('finished')
class AdsorbingSurfactantEquation():
r"""
The `AdsorbingSurfactantEquation` object solves the
`SurfactantEquation` but with an adsorbing species from some bulk
value. The equation that describes the surfactant adsorbing is
given by,
.. math::
\dot{\theta} = J v \theta + k c (1 - \theta - \theta_{\text{other}}) - \theta c_{\text{other}} k_{\text{other}} - k^- \theta
where :math:`\theta`, :math:`J`, :math:`v`, :math:`k`, :math:`c`,
:math:`k^-` and :math:`n` represent the surfactant coverage, the curvature,
the interface normal velocity, the adsorption rate, the concentration in the
bulk at the interface, the consumption rate and an exponent of consumption,
respectively. The :math:`\text{other}` subscript refers to another
surfactant with greater surface affinity.
The terms on the RHS of the above equation represent conservation of
surfactant on a non-uniform surface, Langmuir adsorption, removal of
surfactant due to adsorption of the other surfactant onto non-vacant sites
and consumption of the surfactant respectively. The adsorption term is added
to the source by setting :math:` S_c = k c (1 - \theta_{\text{other}})` and
:math:`S_p = -k c`. The other terms are added to the source in a similar
way.
The following is a test case:
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> from fipy.tools import numerix
>>> from fipy.variables.cellVariable import CellVariable
>>> dx = .5
>>> dy = 2.3
>>> dt = 0.25
>>> k = 0.56
>>> initialValue = 0.1
>>> c = 0.2
>>> from fipy.meshes import Grid2D
>>> from fipy import serialComm
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = (-dx*3/2, -dx/2, dx/2,
... 3*dx/2, 5*dx/2),
... hasOld = 1)
>>> surfactantVar = SurfactantVariable(value = (0, 0, initialValue, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar = CellVariable(mesh = mesh, value = (c , c, c, c, c))
>>> eqn = AdsorbingSurfactantEquation(surfactantVar = surfactantVar,
... distanceVar = distanceVar,
... bulkVar = bulkVar,
... rateConstant = k)
>>> eqn.solve(surfactantVar, dt = dt)
>>> answer = (initialValue + dt * k * c) / (1 + dt * k * c)
>>> print numerix.allclose(surfactantVar.interfaceVar,
... numerix.array((0, 0, answer, 0, 0)))
1
The following test case is for two surfactant variables. One has more
surface affinity than the other.
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> dx = 0.5
>>> dy = 2.73
>>> dt = 0.001
>>> k0 = 1.
>>> k1 = 10.
>>> theta0 = 0.
>>> theta1 = 0.
>>> c0 = 1.
>>> c1 = 1.
>>> totalSteps = 10
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = dx * (numerix.arange(5) - 1.5),
... hasOld = 1)
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> var1 = SurfactantVariable(value = (0, 0, theta1, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> bulkVar1 = CellVariable(mesh = mesh, value = (c1, c1, c1, c1, c1))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = k0)
>>> eqn1 = AdsorbingSurfactantEquation(surfactantVar = var1,
... distanceVar = distanceVar,
... bulkVar = bulkVar1,
... rateConstant = k1,
... otherVar = var0,
... otherBulkVar = bulkVar0,
... otherRateConstant = k0)
>>> for step in range(totalSteps):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> answer0 = 1 - numerix.exp(-k0 * c0 * dt * totalSteps)
>>> answer1 = (1 - numerix.exp(-k1 * c1 * dt * totalSteps)) * (1 - answer0)
>>> print numerix.allclose(var0.interfaceVar,
... numerix.array((0, 0, answer0, 0, 0)), rtol = 1e-2)
1
>>> print numerix.allclose(var1.interfaceVar,
... numerix.array((0, 0, answer1, 0, 0)), rtol = 1e-2)
1
>>> dt = 0.1
>>> for step in range(10):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> x, y = mesh.cellCenters
>>> check = var0.interfaceVar + var1.interfaceVar
>>> answer = CellVariable(mesh=mesh, value=check)
>>> answer[x==1.25] = 1.
>>> print check.allequal(answer)
True
The following test case is to fix a bug where setting the adsorption
coefficient to zero leads to the solver not converging and an eventual
failure.
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = 0)
>>> eqn0.solve(var0, dt = dt)
>>> eqn0.solve(var0, dt = dt)
>>> answer = CellVariable(mesh=mesh, value=var0.interfaceVar)
>>> answer[x==1.25] = 0.
>>> print var0.interfaceVar.allclose(answer)
True
The following test case is to fix a bug that allows the accelerator to
become negative.
>>> nx = 5
>>> ny = 5
>>> dx = 1.
>>> dy = 1.
>>> mesh = Grid2D(dx=dx, dy=dy, nx = nx, ny = ny, communicator=serialComm)
>>> x, y = mesh.cellCenters
>>> disVar = DistanceVariable(mesh=mesh, value=1., hasOld=True)
>>> disVar[y < dy] = -1
>>> disVar[x < dx] = -1
>>> disVar.calcDistanceFunction() #doctest: +LSM
>>> levVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> accVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> levEq = AdsorbingSurfactantEquation(levVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0)
>>> accEq = AdsorbingSurfactantEquation(accVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0,
... otherVar = levVar,
... otherBulkVar = 0,
... otherRateConstant = 0)
>>> extVar = CellVariable(mesh = mesh, value = accVar.interfaceVar)
>>> from fipy import TransientTerm, AdvectionTerm
>>> advEq = TransientTerm() + AdvectionTerm(extVar)
>>> dt = 0.1
>>> for i in range(50):
... disVar.calcDistanceFunction()
... extVar.value = (numerix.array(accVar.interfaceVar))
... disVar.extendVariable(extVar)
... disVar.updateOld()
... advEq.solve(disVar, dt = dt)
... levEq.solve(levVar, dt = dt)
... accEq.solve(accVar, dt = dt) #doctest: +LSM
>>> # The following test fails sometimes on linux with scipy solvers
>>> # See issue #575. We ignore for now.
>>> print (accVar >= -1e-10).all() #doctest: +NOTLINUXSCIPY
True
"""
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)
def solve(self, var, boundaryConditions=(), solver=None, dt=None):
"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear system once.
:Parameters:
- `var`: A `SurfactantVariable` to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
"""
self.dt.setValue(dt)
if solver is None:
import fipy.solvers.solver
if fipy.solvers.solver == 'pyamg':
from fipy.solvers.pyAMG.linearGeneralSolver import LinearGeneralSolver
solver = LinearGeneralSolver(tolerance=1e-15, iterations=2000)
else:
from fipy.solvers import LinearPCGSolver
solver = LinearPCGSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions, )
var.constrain(0, var.mesh.exteriorFaces)
self.eq.solve(var,
boundaryConditions=boundaryConditions,
solver=solver,
dt=1.)
def sweep(self,
var,
solver=None,
boundaryConditions=(),
dt=None,
underRelaxation=None,
residualFn=None):
r"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear
system once. This method also recalculates and returns the
residual as well as applying under-relaxation.
:Parameters:
- `var`: The variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
- `underRelaxation`: Usually a value between `0` and `1` or `None` in the case of no under-relaxation
"""
self.dt.setValue(dt)
if solver is None:
from fipy.solvers import DefaultAsymmetricSolver
solver = DefaultAsymmetricSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions, )
var.constrain(0, var.mesh.exteriorFaces)
return self.eq.sweep(var,
solver=solver,
boundaryConditions=boundaryConditions,
underRelaxation=underRelaxation,
residualFn=residualFn,
dt=1.)
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')
NthOrderBoundaryCondition(mesh.getFacesRight(), 0, 3),
NthOrderBoundaryCondition(mesh.getFacesTop(), 0, 3),
NthOrderBoundaryCondition(mesh.getFacesBottom(), 0, 3))
if __name__ == '__main__':
import fipy.viewers
viewer = fipy.viewers.make(vars=var,
limits={
'datamin': 0.,
'datamax': 1.0
})
viewer.plot()
dexp = -5
for step in range(steps):
dt = numerix.exp(dexp)
dt = min(100, dt)
dexp += 0.01
var.updateOld()
eqch.solve(var, boundaryConditions=BCs, solver=solver, dt=dt)
if __name__ == '__main__':
viewer.plot()
print 'step', step, 'dt', dt
def _run():
pass
value = valueLeft)
eq = TransientTerm() == ExplicitDiffusionTerm()
solver = LinearLUSolver(tolerance = 1.e-6, iterations = 100)
boundaryConditions=(FixedValue(mesh.getFacesLeft(),valueLeft),
FixedValue(mesh.getFacesRight(),valueRight))
answer = numerix.array([ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 1.58508452e-07, 6.84325019e-04,
7.05111362e-02, 7.81376523e-01, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 4.99169535e-05, 1.49682805e-02, 3.82262622e-01,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 4.06838361e-06,
3.67632029e-03, 1.82227062e-01, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 4.99169535e-05, 1.49682805e-02, 3.82262622e-01])
if __name__ == '__main__':
steps = 1000
for step in range(steps):
eq.solve(var, solver = solver, boundaryConditions = boundaryConditions, dt = timeStepDuration)
print var
viewer = fipy.viewers.make(vars = var)
viewer.plot()
raw_input('finished')
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
bench.start()
for i in range(steps):
phase.updateOld()
temperature.updateOld()
phaseEq.solve(phase, dt=timeStepDuration)
temperatureEq.solve(temperature, dt=timeStepDuration)
bench.stop('solve')
print bench.report(numberOfElements=numberOfElements, steps=steps)
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
bench.start()
for i in range(steps):
phase.updateOld()
temperature.updateOld()
phaseEq.solve(phase, dt=timeStepDuration)
temperatureEq.solve(temperature, dt=timeStepDuration)
bench.stop('solve')
print bench.report(numberOfElements=numberOfElements, steps=steps)
bench.stop('variables')
bench.start()
D = 1.
from fipy.terms.implicitDiffusionTerm import ImplicitDiffusionTerm
from fipy.terms.transientTerm import TransientTerm
eq = TransientTerm() == ImplicitDiffusionTerm(coeff=D)
bench.stop('terms')
## from fipy import viewers
## viewer = viewers.make(vars = C, limits = {'datamin': 0, 'datamax': 1})
## viewer.plot()
## raw_input("initial")
bench.start()
dt = 1e0
steps = 1
for step in range(steps):
eq.solve(var=C, dt=dt)
## viewer.plot()
bench.stop('solve')
print bench.report(numberOfElements=N, steps=steps)
## raw_input("finished")
