Hi Krishina and Ray,
Thanks for the interesting discussion. I'm not 100% sure about everything
that Krishina is asking for in the latter part of the discussion so I'm
just going to address the code that Ray has developed below (my code is
below). I think there is a way to handle the right hand side boundary
condition both implicitly and with second order accuracy using an
ImplicitSourceTerm boundary condition trick. Sort of similar to what is
here,
http://www.ctcms.nist.gov/fipy/documentation/USAGE.html#applying-fixed-flux-boundary-conditions
.
Like I said, I think the code below is both second order accurate (for
fixed dx) and implicit. Extending this to 2D might raise a few issues. The
grid spacing needs to be in the coefficient so, obviously, dx needs to be
fixed in both x and y directions. Also, I'm not sure if I'm missing a
factor of dx in the source term in 1D as I'm using both a divergence and an
ImplicitSourceTerm so there is some question about volumes and face areas
in 2D as well. I'm also confused about the signs, I had to flip the sign in
front of the source to make it work. It seems to look right though in 1D
and you can just take one massive time step to get to the answer. This will
only work if k is negative otherwise it's unstable, right?
The way I came up with the source was the following
n.grad(phi) = k * (phi_P + n.grad(phi) * dx / 2)
and then solve for n.grad(phi) which gives
n.grad(phi) = k * phi_P / (1 - dx * k / 2)
where phi_P is the value of phi at the cell next to the boundary with the
implicit boundary condition. Then we fake the outbound flux to be the
expression on the right of the equation.
Just as a general note it would be great in FiPy if we could come up with a
nice way to write boundary conditions in scripts that did all these tricks
implicitly without having to know all these background details about FV and
how FiPy works.
~~~~
import fipy as fp
nx = 50
dx = 1.
mesh = fp.Grid1D(nx=nx, dx=dx)
phi = fp.CellVariable(name="field variable", mesh=mesh, value=1.0)
D = 1.
k = -1.
diffCoeff = fp.FaceVariable(mesh=mesh, value=D)
diffCoeff.constrain(0., mesh.facesRight)
valueLeft = 0.0
phi.constrain(valueLeft, mesh.facesLeft)
#phi.faceGrad.constrain([phi], mesh.facesRight) # This is the problematic BC
#phi.faceGrad.constrain(phi.harmonicFaceValue, mesh.facesRight) # This is
the problematic BC
#phi.faceGrad.constrain([k * phi.harmonicFaceValue], mesh.facesRight) #
This is the problematic BC
implicitCoeff = -D * k / (1. - k * dx / 2.)
eq = (fp.TransientTerm() == fp.DiffusionTerm(diffCoeff) - \
fp.ImplicitSourceTerm((mesh.faceNormals * implicitCoeff *
mesh.facesRight).divergence))
timeStep = 0.9 * dx**2 / (2 * D)
timeStep = 10.0
steps = 800
viewer = fp.Viewer(vars=phi, datamax=1., datamin=0.)
for step in range(steps):
eq.solve(var=phi, dt=timeStep)
viewer.plot()
~~~~
On Thu, Jun 9, 2016 at 12:02 PM, Raymond Smith <smit...@mit.edu> wrote:
> Hi, Krishna.
>
> Perhaps I'm misunderstanding something, but I'm still not convinced the
> second version you suggested -- c.faceGrad.constrain([-(j_at_c_star +
> partial_j_at_op_point*(c.faceValue - c_star))], mesh.facesTop) -- isn't
> working like you want. Could you look at the example I suggested to see if
> that behaves differently than you expect?
>
> Here's the code I used. To me it looks very similar in form to c
> constraint above and at first glance it seems to behave exactly like we
> want -- that is, throughout the time stepping, n*grad(phi) is constrained
> to the value, -phi at the surface. Correct me if I'm wrong, but my
> impression is that this is the behavior you desire.
>
> from fipy import *
>
> nx = 50
> dx = 1.
> mesh = Grid1D(nx=nx, dx=dx)
>
> phi = CellVariable(name="field variable", mesh=mesh, value=1.0)
> D = 1.
>
> valueLeft = 0.0
> phi.constrain(valueLeft, mesh.facesLeft)
> #phi.faceGrad.constrain([phi], mesh.facesRight) # This is the problematic
> BC
> #phi.faceGrad.constrain(phi.harmonicFaceValue, mesh.facesRight) # This is
> the problematic BC
> phi.faceGrad.constrain([-phi.harmonicFaceValue], mesh.facesRight) # This
> is the problematic BC
>
> eq = TransientTerm() == DiffusionTerm(coeff=D)
>
> timeStep = 0.9 * dx**2 / (2 * D)
> steps = 800
>
> viewer = Viewer(vars=phi, datamax=1., datamin=0.)
>
> for step in range(steps):
> eq.solve(var=phi, dt=timeStep)
> viewer.plot()
>
> Cheers,
> Ray
>
> On Thu, Jun 9, 2016 at 11:44 AM, Gopalakrishnan, Krishnakumar <
> k.gopalakrishna...@imperial.ac.uk> wrote:
>
>> Hi Ray,
>>
>>
>>
>> Yes. You make a good point. I see that the analytical solution to the
>> particular problem I have posted is also zero.
>>
>>
>>
>> The reason I posted is because I wanted to present an (oversimplified)
>> analogous problem when posting to the group, retaining the generality,
>> since many other subject experts might have faced similar situations.
>>
>>
>>
>> The actual problem I am solving is the solid diffusion PDE (only 1
>> equation) in a Li-ion battery. I am solving this PDE in a pseudo-2D
>> domain. i.e. I have defined a Cartesian 2D space, wherein the y-coordinate
>> corresponds to the radial direction. The bottom face corresponds to
>> particle centres, and the top face corresponding to surface of each
>> spherical particle. The x-axis co-ordinate corresponds to particles along
>> the width (or thickness) of the positive electrode domain. Diffusion of Li
>> is restricted to be within the solid particle (i.e. y-direction only), by
>> defining a suitable tensor diffusion coefficient as described in the
>> Anisotropic diffusion example and FAQ in FiPy. I have normalised my x and
>> y dimensions to have a length of unity.
>>
>>
>>
>> Now, the boundary condition along the top face is
>>
>>
>>
>> Now, j is non-linear (Butler-Volmer), and I am using a Taylor-expanded
>> linear version for this boundary condition. All other field variables
>> are assumed as constants. The idea is to set up the infrastructure and
>> solve this problem independently, before worrying upon the rubrics of
>> setting up the coupled system. In a similar fashion, I have built up and
>> solved the solid phase potential PDE (thanks to your help for pointing out
>> about the implicit source term). Thus, the idea is to build up the coupled
>> P2D Newman model piecemeal.
>>
>>
>>
>> The linearised version of my BC’s RHS at a given operating point ()is
>>
>>
>>
>> As you can see, the linearised Boundary condition, is cast in terms of
>> the field variable, . Hence, we need it in an implicit form
>> corresponding to (pseudocode: c.faceGrad.constrain([-( (j_at_c_star -
>> partial_j_at_op_point*c_star) + coeff =
>> partial_j_at_op_point)],mesh.facesTop) , or something of this
>> form/meaning. (just like the very useful ImplicitSourceterm method)
>>
>>
>>
>> If I instead apply the c.faceValue method,i.e. using it in setting the
>> BC as
>>
>>
>>
>> c.faceGrad.constrain([-(j_at_c_star + partial_j_at_op_point*(c.faceValue
>> - c_star))], mesh.facesTop), then c.faceValue gets immediately
>> evaluated at the operating point, c_star, and we are left with 0
>> multiplying the first-order derivative.
>>
>>
>>
>> ie. the Boundary conditions becomes,
>>
>>
>>
>> Leading to huge loss of accuracy.
>>
>>
>>
>> Is there any hope at all in this situation ? J . Cheers and thanks for
>> your help thus far.
>>
>>
>>
>>
>>
>> Krishna
>>
>>
>>
>> *From:* fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov] *On Behalf
>> Of *Raymond Smith
>> *Sent:* 09 June 2016 16:06
>>
>> *To:* fipy@nist.gov
>> *Subject:* Re: casting implicit Boundary Conditions in FiPy
>>
>>
>>
>> Hi, Krishna.
>>
>> Could you give a bit more detail and/or an example about how you know
>> it's doing the wrong thing throughout the solution process? In the example
>> you sent, the correct solution is the same (c(x, t) = 0) whether you set
>> n*grad(phi) to zero or to phi at the boundary, so it's not a good example
>> for concluding that it's not behaving as you'd expect. It's helpful here to
>> find a situation in which you know that analytical solution to confirm one
>> way or the other. For example, you should be able to get the solution to
>> the following problem using a Fourier series expansion:
>>
>> dc/dt = Laplacian(c)
>>
>> c(t=0) = 1
>>
>> x=0: c = 0
>>
>> x=1: c - dc/dx = 0
>>
>> Ray
>>
>>
>>
>> On Thu, Jun 9, 2016 at 10:52 AM, Gopalakrishnan, Krishnakumar <
>> k.gopalakrishna...@imperial.ac.uk> wrote:
>>
>> Hi Ray,
>>
>>
>>
>> Thanks for your help.
>>
>>
>>
>> But when I apply phi.harmonicFaceValue , it is immediately evaluated to
>> a numerical result (a zero vector in this case, since initial value = 0,
>> the data type is
>> fipy.variables.harmonicCellToFaceVariable._HarmonicCellToFaceVariable',
>> i.e. the boundary condition is not remaining implicit.
>>
>>
>>
>> The same is the case with the examples.convection.robin example. Here,
>> the phi.faceValue method is used. However, this also results in an
>> immediate numerical evaluation.
>>
>>
>>
>> However, what is actually required is that, the BC must remain implicit
>> (in variable form, without getting numerically evaluated), being cast in
>> terms of the field variable being solved for. Then the solver needs to
>> solve the PDE on the domain to yield the solution of the field variable.
>>
>>
>>
>>
>>
>> I think we need to solve for the PDE, keeping this implicit BC, rather
>> than immediately evaluating the term , since is the field variable to be
>> solved for, i.e. there ought to be some way to cast the Boundary condition
>> as implicit.
>>
>>
>>
>> In FiPy, I have previously set up an implicit source term, by using
>> the following code snippet, ImplicitSourceTerm(coeff=k) . Perhaps there
>> might be an equivalent method in FiPy to set up the implicit BC, I think ?
>>
>>
>>
>>
>>
>> Krishna
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> *From:* fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov] *On Behalf
>> Of *Raymond Smith
>> *Sent:* 09 June 2016 14:23
>>
>>
>> *To:* fipy@nist.gov
>> *Subject:* Re: casting implicit Boundary Conditions in FiPy
>>
>>
>>
>> Oh, right, the boundary condition is applied on a face, so you need the
>> facevalue of phi:
>>
>> phi.faceGrad.constrain([phi.harmonicFaceValue])
>>
>> Ray
>>
>>
>>
>> On Thu, Jun 9, 2016 at 7:28 AM, Gopalakrishnan, Krishnakumar <
>> k.gopalakrishna...@imperial.ac.uk> wrote:
>>
>> Hi ray,
>>
>>
>>
>> Casting the implicit PDE does not work for my problem. FiPy throws up a
>> ton of errors.
>>
>> I am attaching a minimal example (based off example1.mesh.1D)
>>
>>
>>
>> *from *fipy *import **
>>
>> nx = 50
>> dx = 1.
>> mesh = Grid1D(nx=nx, dx=dx)
>>
>> phi = CellVariable(name=*"field variable"*, mesh=mesh, value=0.0)
>> D = 1.
>>
>> valueLeft = 0.0
>> phi.constrain(valueLeft, mesh.facesLeft)
>> phi.faceGrad.constrain([phi], mesh.facesRight)
>>
>> *# This is the problematic BC *eq = TransientTerm() == DiffusionTerm(
>> coeff=D)
>>
>> timeStep = 0.9 * dx**2 / (2 * D)
>> steps = 100
>>
>> viewer = Viewer(vars=phi)
>>
>> *for *step *in *range(steps):
>> eq.solve(var=phi, dt=timeStep)
>> viewer.plot()
>>
>>
>>
>> The errors are as follows:
>>
>>
>>
>> line 22, in <module>
>>
>> eq.solve(var=phi, dt=timeStep)
>>
>> \fipy\terms\term.py", line 211, in solve
>>
>> solver = self._prepareLinearSystem(var, solver, boundaryConditions,
>> dt)
>>
>> \fipy\terms\term.py", line 170, in _prepareLinearSystem
>>
>> buildExplicitIfOther=self._buildExplcitIfOther)
>>
>> \fipy\terms\binaryTerm.py", line 68, in _buildAndAddMatrices
>>
>> buildExplicitIfOther=buildExplicitIfOther)
>>
>> \fipy\terms\unaryTerm.py", line 99, in _buildAndAddMatrices
>>
>> diffusionGeomCoeff=diffusionGeomCoeff)
>>
>> \fipy\terms\abstractDiffusionTerm.py", line 337, in _buildMatrix
>>
>> nthCoeffFaceGrad = coeff[numerix.newaxis] *
>> var.faceGrad[:,numerix.newaxis]
>>
>> \fipy\variables\variable.py", line 1575, in __getitem__
>>
>> unit=self.unit,
>>
>> \fipy\variables\variable.py", line 255, in _getUnit
>>
>> return self._extractUnit(self.value)
>>
>> \fipy\variables\variable.py", line 561, in _getValue
>>
>> value[..., mask] = numerix.array(constraint.value)[..., mask]
>>
>> IndexError: index 50 is out of bounds for axis 1 with size 50
>>
>>
>>
>> I have tried including the implicit BC within the time-stepper loop, but
>> that does not still help.
>>
>>
>>
>>
>>
>> Best Regards
>>
>>
>>
>> Krishna
>>
>>
>>
>>
>>
>>
>>
>> *From:* fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov] *On Behalf
>> Of *Gopalakrishnan, Krishnakumar
>> *Sent:* 08 June 2016 23:42
>> *To:* fipy@nist.gov
>> *Subject:* RE: casting implicit Boundary Conditions in FiPy
>>
>>
>>
>> Hi Raymond,
>>
>>
>>
>> Sorry, it was a typo.
>>
>>
>>
>> Yes, It is indeed d (phi)/dx, the spatial derivative BC. I shall try
>> setting phi.faceGrad.constrain([k*phi], mesh.facesRight), and see if it
>> will work.
>>
>>
>>
>> Thanks for pointing this out.
>>
>>
>>
>>
>>
>> Krishna
>>
>>
>>
>> *From:* fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov
>> <fipy-boun...@nist.gov>] *On Behalf Of *Raymond Smith
>> *Sent:* 08 June 2016 23:36
>> *To:* fipy@nist.gov
>> *Subject:* Re: casting implicit Boundary Conditions in FiPy
>>
>>
>>
>> Hi, Krishna.
>>
>> Just to make sure, do you mean that the boundary condition is a
>> derivative with respect to the spatial variable or with respect to time
>> as-written? If you mean spatial, such that d\phi/dx = k*phi, have you tried
>>
>> phi.faceGrad.constrain(k*phi) and that didn't work?
>>
>> If you mean that its value is prescribed by its rate of change, then I'm
>> not sure the best way to do it. Could you maybe do it explicitly?
>> - Store the values from the last time step with hasOld set to True in
>> the creation of the cell variable
>> - In each time step, calculate the backward-Euler time derivative
>> manually and then set the value of phi with the phi.constrain method.
>>
>>
>>
>> Ray
>>
>>
>>
>> On Wed, Jun 8, 2016 at 6:26 PM, Gopalakrishnan, Krishnakumar <
>> k.gopalakrishna...@imperial.ac.uk> wrote:
>>
>> I am trying to solve the standard fickean diffusion equation on a 1D
>> uniform mesh in (0,1)
>>
>>
>>
>> $$\frac{\partial \phi}{\partial t} = \nabla.(D \nabla\phi)$$
>>
>>
>>
>> with a suitable initial value for $\phi(x,t)$.
>>
>>
>>
>> The problem is that, one of my boundary conditions is *implicit*, i.e.
>> is a function of the field variable being solved for.
>>
>>
>>
>> $ \frac{\partial\phi}{\partial t} = k \phi $ , at the right boundary
>> edge, k = constant
>>
>>
>>
>> The left BC is not a problem, it is just a standard no-flux BC.
>>
>>
>>
>> How do I cast this *implicit BC* in FiPy ? Any help/pointers will be
>> much appreciated.
>>
>>
>>
>>
>>
>> Best regards
>>
>>
>>
>> Krishna
>>
>> Imperial College London
>>
>>
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>>
>>
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