Ah, you make a good point: I was using terms incorrectly. I don't actually know if anyone has studied that PDE with the non-linear Butler-Volmer and show existence/uniqueness. However, it is true that without phi dependence on the RHS, it admits infinite solutions. This is demonstrated by noting that phi only appears as grad(phi) within the equation. Thus, the system could only ever be determined to an additive constant (the electric _field_ is the actually relevant quantity). However, because the BV equation is written in terms of phi directly, then there is explicit dependence not on the field but on the actual value of the electric potential. This is done by introducing a reference state somewhere else within the equations such that the absolute value of phi can be given some meaning. To be honest, I've never done simulations of the solid phase without coupling it to the electrolyte phase, so I can't comment much more about that. The reaction term should of course also depend on various other field variables in the system such as the electric potential in the electrolyte, but I don't immediately see that assuming that's a uniform constant would cause troubles.
In terms of implementation, I'm not sure exactly what you're asking. Specifying a flux can be done indirectly by specifying a gradient phi.faceGrad.constrain([value], mesh.facesRight) Ray On Thu, May 26, 2016 at 3:43 PM, Campbell, Ian <i.campbel...@imperial.ac.uk> wrote: > Thank you for your descriptive reply Ray. > > > > Can you please provide an explanation (or links to an explanation) of the > connection between the implicit nature of the equation (phi_s on both sides > of the PDE) and the well-posedness / existence and uniqueness of the > solution? > > > > For clarity for the rest of the users on the list, the RHS of the PDE in > question is strongly non-linear (the j term expands to the Butler-Volmer > equation). > > > > Since the right hand side of the PDE is non-linear, theoretically there is > no guarantee of a unique solution. In this context, we would like to better > understand your concluding statement that the solution gets automatically > pinned uniquely for this PDE. The implementation strategy – i.e. how to > represent these boundary conditions in FiPy – still isn’t clear to us – can > you kindly elaborate? > > > > With best regards, > > > > - Ian & Krishna > > > > *From:* fipy-boun...@nist.gov [mailto:fipy-boun...@nist.gov] *On Behalf > Of *Raymond Smith > *Sent:* 26 May 2016 19:42 > *To:* fipy@nist.gov > > *Subject:* Re: Pure Neumann Boundary Conditions; Elliptic PDE > > > > At least for now, why don't we keep this on the list. > > A couple of thoughts. First, I have actually already implemented a > Newman-like model in Python. I didn't use Fipy, but I did use the finite > volume method. It's missing a few features compared to dualfoil, but has > several extra things. If you are interested in using it, please be in touch > off the list. I plan to publish it, but I haven't quite gotten around to > that yet. Hopefully quite soon. > > To your questions/comments about the PDE, you raise a couple of critical > differences. First, your source term, j, is a function of phi (which you > didn't actually state but is true) and second, the specified flux boundary > condition at the positive current collector changes the well-posed-ness > entirely. In this case, the integral of the source term has to match the > outlet flux into the current collector. That allows for a steady state. > Then, the fact that j depends on phi is what actually "pins" the solution > rather than allowing any vertical shift. > > Best, > > Ray > > > > On Thu, May 26, 2016 at 2:25 PM, Gopalakrishnan, Krishnakumar < > k.gopalakrishna...@imperial.ac.uk> wrote: > > Hi Ray, > > > > Many thanks for your quick reply. I am Krishna, Ian’s colleague here at > Imperial. Hope you remember me from our earlier conversations about access > to Martin’s 10.626 materials. > > > > Ian & I are currently working on implementing the basic pseudo-2D porous > electrode Newman model using FiPy. > > > > The issue is that all of the PDE Boundary conditions are cast as Neumann. > However, when we try to apply your recommendation in FiPy (i.e. omit the > BC definitions/revert to default), it still does not converge. > > > > · The PDE that we are currently solving is the solid-potential in > the positive electrode. Sorry, there was an typo in the last email. What we > have is: > > o at the positive/separator boundary, there is no potential flux. (a > no-flux BC) > > o at the positive current collector, we have a fixed flux ( ) BC > > > > · Meaning of the source term components: > > o specific interfacial surface area > > o faraday’s constant > > o = Li-molar flux density ( ) > > > > · The integral of the source term is , where =Solid Phase Porosity, = > particle radius and = the thickness of the positive electrode, > respectively. > > o This above integral is clearly non-zero for a non-zero applied > current. > > o However, the Newman model potential equation is considered to be > well-posed and solution plots are found in papers. All other PDEs of the > model also have pure Neumann BCs. Hence, we don’t understand the > inconsistency that arises from having no steady-state solution. > > > > Any help will be much appreciated. > > > > > > Best Regards > > > > Krishna & Ian > > > > > > > > > > *From:* Campbell, Ian > *Sent:* 26 May 2016 18:07 > *To:* Gopalakrishnan, Krishnakumar <k.gopalakrishna...@imperial.ac.uk> > *Subject:* Fwd: Re: Pure Neumann Boundary Conditions; Elliptic PDE > > > > ---------- Forwarded message ---------- > From: Raymond Smith <smit...@mit.edu> > Date: 26 May 2016 3:49 p.m. > Subject: Re: Pure Neumann Boundary Conditions; Elliptic PDE > To: fipy@nist.gov > Cc: > > Hi, Ian. > > Some thoughts. Are you able to give us any more detail about your PDE? For > example, the form/meaning of the things in the source term on the right > hand side? Presumably the flux is defined by > > \sigma \nabla \phi > > If so, then not specifying any boundary conditions would lead to no-flux > conditions because FiPy defaults to assuming zero-gradients for field > variables on the boundary. However, you could do it explicitly with > something like > > phi.faceGrad.constrain([0.], mesh.exteriorFaces) > > However, the form of the source is important here because if the integral > of the source is non-zero, then there is no steady solution. If the > integral _is_ zero and the source is not a function of \phi, then the > system admits an infinite family of solutions, all shifted by a constant. > There's some discussion of this issue in the 1D diffusion example: > > http://www.ctcms.nist.gov/fipy/examples/diffusion/generated/examples.diffusion.mesh1D.html#module-examples.diffusion.mesh1D > > Search for "Fully implicit solutions are not without their pitfalls" > > Best, > > Ray > > > > On Thu, May 26, 2016 at 6:30 AM, Campbell, Ian < > i.campbel...@imperial.ac.uk> wrote: > > Dear All, > > > > Thanks for the great work on FiPy so far. > > > > We have a 2nd order elliptic PDE* defined in the domain (0, 1). At each > boundary, we have a no-flux boundary condition. > > > > We have tried to represent this in FiPy, but specifying such conditions > results only in incorrect results with large error. We have searched this > page (below) in depth but can’t see which function call is appropriate, if > any. > > > > http://www.ctcms.nist.gov/fipy/fipy/generated/fipy.boundaryConditions.html > > > > Do you have a recommendation for how we may implement two no-flux boundary > conditions on our PDE? > > > > Best regards, > > > > - Ian > > > > * $$\nabla.(\sigma_{eff} \nabla \bar{\phi}_s) = a_s F \bar{j}$$ > > > > Ian Campbell | PhD Candidate > > Electrochemical Science & Engineering Group > > Room 506, City & Guilds Building, Imperial College London, SW7 2AZ, United > Kingdom > > Phone: +44 (0)7449 815 520 | E-mail: i.campbel...@imperial.ac.uk > > > > > _______________________________________________ > fipy mailing list > fipy@nist.gov > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] > > > > > > _______________________________________________ > fipy mailing list > fipy@nist.gov > http://www.ctcms.nist.gov/fipy > [ NIST internal ONLY: https://email.nist.gov/mailman/listinfo/fipy ] > >
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