# Re: Applying Two Different Diffusion Coefficients at Single FV Face

In general, if you're going to scale things in your problem, I tend to
recommend fully non-dimensionalizing a problem with reference lengths
scales, time scales, etc. and not using different references for different
regions described by the same PDE. In other words, if you have some sort of
relevant time scale, t_ref and a chosen length scale L_ref (which could be,
e.g., L1, L2, L1+L2), I would scale everything with those values, such that
D* = D/D_ref
D_ref = L_ref^2/t_ref
and so on for other quantities in the PDE's including differential
operators and field variables.
In that case, the harmonic mean of D1* and D2* is the same as the harmonic
mean of D1 and D2 divided by D_ref. If you change the references with
position along the length of your PDE, you run into issues with things like
this.

Ray

On Fri, Sep 16, 2016 at 11:36 AM, Gopalakrishnan, Krishnakumar <
krishnaku...@imperial.ac.uk> wrote:

> Hi Raymond,
>
>
>
> Thanks a lot for your inputs. We have been working along the same lines
> that you have suggested.
>
>
>
> There is a slightly more complication that wasn’t fully discussed
> (intentionally, to keep it simple, and to find out if any other options
> existed).
>
>
>
> Assuming that we are working on a normalised co-ordinate and the two bulk
> regions within the domain have non-uniform properties.  In this situation,
> we have to normalise the PDEs too.
>
>
>
> Thus, the diffusion coefficient becomes D_region/L_region  , where region
> can be either region 1 or region 2. , and L_region denotes the original
> length of the region.
>
>
>
> Now, although it’s a perfectly straightforward idea to define D in the
> cell centers and let fiPy handle the harmonic facevalue computation,  I
> don’t know if it’s meaningful to compute harmonicfacevalue of a length
> (which appears in the denominator).
>
>
>
> What do you suggest for L_region ?  Define that at the cell-centers again,
> and use arithmeticfacevalue , or stick with harmonicfacevalue ?
>
>
>
> Best Regards
>
>
>
> Krishna & Ian.
>
>
>
>
>
>
>
> *From:* fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov] *On Behalf
> Of *Raymond Smith
> *Sent:* Friday, September 16, 2016 7:18 PM
> *To:* fipy@nist.gov
> *Subject:* Re: Applying Two Different Diffusion Coefficients at Single FV
> Face
>
>
>
> A side note, here it may actually be more convenient to think about the D
> in terms of the volumes, so you could also define it as a cell variable
> with values specified as D1 when x<=1 and D2 when x>1, then use the
> harmonicFaceValue attribute when you put it into the governing equation to
> have FiPy do this calculation for you and avoid awkward use of the mesh
> spacing in the specification of the variable values.
>
>
>
> On Fri, Sep 16, 2016 at 11:12 AM, Raymond Smith <smit...@mit.edu> wrote:
>
> Hi, Ian.
>
> I don't think there is such a thing as having two different flux
> coefficients at the same face for the same governing PDE. The flux through
> a given face is calculated by the coefficient at that face times some
> approximation of the gradient in a field variable at that face, like
> both evaluated at the face, so there is only one coefficient in that
> expression. And in the FiPy FV approach D doesn't come into play at all
> except at the faces, so it can't change immediately to the right or left of
> a face but only from one face to the next.
>
> That said, perhaps you could try following a common approach used when a
> FV interface is directly between two bulk regions with different continuum
> properties -- use some sort of a mean of the continuum transport
> coefficient in the adjacent volumes as the approximation for the
> coefficient at the face. I would suggest the harmonic mean (one simple
> reason: it preserves zero flux through the face if either of the adjacent
> cells has a zero coefficient). You could try:
> Dint = 2*D1*D2/(D1+D2)
> Coefficient.setValue(Dint, where=((1.0-dx/2 < 1D_mesh.faceCenters[0]) &
> (1D_mesh.faceCenters[0] < 1.0+dx/2)))
> If you do that, you should see a change in the slope of the field variable
> right near the face between the two bulk regions, which seems to be what
> you're missing. Of course, ideally your mesh is fine enough that the dx is
> insignificant in terms of the solution, but this is a common approach to
> get a slightly more reasonable estimate for what happens between two
> regions.
>
>
>
> Best,
>
> Ray
>
>
>
> On Fri, Sep 16, 2016 at 10:44 AM, Campbell, Ian <
> i.campbel...@imperial.ac.uk> wrote:
>
> Hello All,
>
>
>
> We have a uniformly spaced 1D mesh of length 2.0, wherein a standard
> elliptic diffusion equation (\nabla. (D \nabla \phi) = f is being solved.
>
>
>
> The diffusion coefficient, D, changes sharply at x = 1.0. Although this is
> quite similar to examples.diffusion.mesh1D,  there is a key difference:
>
>
>
> We are seeking an instantaneous change in the coefficient value at that
> face, i.e. something of the effect that on one side of the face (i.e.
> from x = 0 up to and including x = 1.0) diffusion coefficient D has a value
> D1.  Then there is an abrupt change in diffusion coefficient from x >= 1.0
> (very notable, including the face at x = 1.0) and for the remainder of the
> mesh, diffusion coefficient D has a value D2.  Currently, we have to either
> decide on whether the single face at x = 1.0 gets the value D1 or
> D2. However, when we run with either of these cases, the results are not
> accurate (when compared against a benchmark from a commercial FV solver).
> We are getting a smooth interpolation across the face , rather than the
> sharp change correctly predicted by the benchmarking code.
>
>
>
> With the code below, we have only been able to set the coefficient at that
> middle face to one of either D1 or D2. This doesn’t result in the desired
> instantaneous change in the coefficient value, but instead, the new
> coefficient value only applies from the NEXT face (which is a distance dx
> away) from this critical interior boundary.
>
>
>
> Coefficient = FaceVariable(mesh = 1D_mesh)
>
>                                                                      #
> Define facevariable on mesh
>
> Coefficient.setValue(D1, where=(1D_mesh.faceCenters[0] <= 1.0))
>
>                 # Set coefficient values in 1st half of mesh
>
> Coefficient.setValue(D2, where=((1.0 < 1D_mesh.faceCenters[0]) &
> (1D_mesh.faceCenters[0] <= 2.0)))                    # Set coefficient
> values in 2nd half of mesh
>
>
>
> An alternative with the inequalities adjusted, but that still only permits
> a single coefficient at that face:
>
> Coefficient = FaceVariable(mesh = 1D_mesh)
>
>                                                  # Define facevariable on
> mesh
>
> Coefficient.setValue(D1, where=(1D_mesh.faceCenters[0] < 1.0))
>
> # Set coefficient values in 1st half of mesh
>
> Coefficient.setValue(D2, where=((1.0 <= 1D_mesh.faceCenters[0]) &
> (1D_mesh.faceCenters[0] <= 2.0)))                  # Set coefficient
> values in 2nd half of mesh
>
>
>
> Sincerely,
>
>
>
> -          Ian
>
>
>
> P.S. Daniel, thank you very much for your previous reply on sweep speeds!
>
>
>
>
>
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