# Non-linear 4rth order Diffusion

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February 21, 2019, 09:34
Non-linear 4rth order Diffusion
#1
New Member

Thomas N.
Join Date: Nov 2018
Posts: 11
Rep Power: 7
Dear OpenFoamers,Foamers and HopeFoamers,

I am struggling lately with the problem of how to solve a non linear 4rth order diffusion equation(see below). Of course, it can be solved full explicitly but on this way someone has to use small time-steps. One idea is to split this equation into two 2nd order equations by defining the brackets as the chemical potential. When someone does this, it will come up with the problem of OpenFOAM and general most of the softwares/toolkits can't solve equations for two variables simultaneously. With OpenFOAM words: FATAL ERROR incompatible fields. Nevertheless, I have attached a picture that I was thinking on this way maybe I could solve my problem but it doesn't work. The idea was to calculate first explicit the chemical potential in order to find the correct values of C and then, to calculate implicity the chemical potential with the new correct values of C that have been calculated in the previous step. After this, you solve the diffusion equation. This idea doesn't work. Therefore, I would like to ask if someone has any idea of how to deal with this problem. Last, of course the ideal case would be to define an implicity laplacian^2 with an implicit scheme but I guess this over my programming skills. Thanks in advance!

Thomas

Attached Images
 bam.png (53.5 KB, 41 views)

February 22, 2019, 07:01
#2
Senior Member

Andrew Somorjai
Join Date: May 2013
Posts: 175
Rep Power: 13
Quote:
 Originally Posted by physicosm Dear OpenFoamers,Foamers and HopeFoamers, " can't solve equations for two variables simultaneously. With OpenFOAM words: FATAL ERROR incompatible fields. "
I'd love to help you with this Thomas but this is a daunting problem, something on the level of a thesis. The best I can do is this link.

Incompatible fields for operation

Is there some way to end up with just one variable and one matrix? What are the variables by the way in your equation?

February 22, 2019, 07:26
#3
New Member

Thomas N.
Join Date: Nov 2018
Posts: 11
Rep Power: 7
Quote:
 Originally Posted by massive_turbulence I'd love to help you with this Thomas but this is a daunting problem, something on the level of a thesis. The best I can do is this link. Incompatible fields for operation Is there some way to end up with just one variable and one matrix? What are the variables by the way in your equation?

Dear Andrew,

first of all, thanks for your answer and you are right this one kind of thesis range problem. Regarding to the link that you have attached: I have seen it and in this case, it used chain rule and you treat the half of the term explicitly. In my case, when someone introduces different methods to solve the Cahn-Hilliard in order to increase the time-step, it turns out that someone has to treat the laplacian^2 term implicitly. Therefore, the idea of splitting the 4rth order equation into two 2nd order equations holds and you can define implicitly the laplacians. By the way, this is a common approach in all of the phase field softwares (FiPy, Moose, PRISMS PF, ... ). However, OpenFOAM can't solve for two variables in the same time.

Regarding to your second question, the two variables are the order parameter C and the chemical potential chemPot in the code respectively. So, there boundary conditions for each field separately.

Thomas

February 22, 2019, 16:10
#4
Senior Member

Andrew Somorjai
Join Date: May 2013
Posts: 175
Rep Power: 13
Quote:
 Originally Posted by physicosm Dear Andrew, first of all, thanks for your answer and you are right this one kind of thesis range problem. Regarding to the link that you have attached: I have seen it and in this case, it used chain rule and you treat the half of the term explicitly. In my case, when someone introduces different methods to solve the Cahn-Hilliard in order to increase the time-step, it turns out that someone has to treat the laplacian^2 term implicitly. Therefore, the idea of splitting the 4rth order equation into two 2nd order equations holds and you can define implicitly the laplacians. By the way, this is a common approach in all of the phase field softwares (FiPy, Moose, PRISMS PF, ... ). However, OpenFOAM can't solve for two variables in the same time. Regarding to your second question, the two variables are the order parameter C and the chemical potential chemPot in the code respectively. So, there boundary conditions for each field separately. Thomas
One of the things I've noticed about semi-implicit methods in openfoam is that there are some calculations with matrices such as the velocity matrix

volScalarField rUA = 1.0/UEqn().A();

in order to use it for the pressure equation e.g.
Code:
``` fvScalarMatrix pEqn
(
fvm::laplacian(rUA, p) == fvc::div(phi)
);
pEqn.setReference(pRefCell, pRefValue);
pEqn.solve();```
Wouldn't your code need something like this similarly (not in terms of the actual variables but in terms of the methodology), if not why not? I know that this sounds like a hack but I'm very new to PISO and SIMPLE.

 February 25, 2019, 10:11 #5 New Member   Thomas N. Join Date: Nov 2018 Posts: 11 Rep Power: 7 Dear Andrew, Unfortunately, it can't. For more details regarding to the PISO loop I am encouraging you to check the wonderful summary of interFoam solver. Download the pdf file from the section 1.2 Useful links. https://openfoamwiki.net/index.php/InterFoam Thomas

February 25, 2019, 17:14
#6
Senior Member

Arjun
Join Date: Mar 2009
Location: Nurenberg, Germany
Posts: 1,278
Rep Power: 34
Quote:
 Originally Posted by physicosm Dear Andrew, Unfortunately, it can't. For more details regarding to the PISO loop I am encouraging you to check the wonderful summary of interFoam solver. Download the pdf file from the section 1.2 Useful links. https://openfoamwiki.net/index.php/InterFoam Thomas

I think if you are open to using other solvers then you can (possibly) do with Wildkatze solver using user coding where you supply the descretization and solver solves the linear system for you (also does other tasks like creating variables, and providing them for post processing etc).

March 1, 2019, 10:49
Progress...
#7
New Member

Thomas N.
Join Date: Nov 2018
Posts: 11
Rep Power: 7
If anyone is still interested on the above issue, there was a little bit of progress. What is more, I am treating the 4th order non linear Cahn-Hilliard equation by splitting into two 2nd order equations. So, the equations are coupled and on this way have to be solved both of them for every time step. The solution to this, it can be given by using the block coupled matrix of foam-extend 4.0. Moreover, I have found on an OpenFOAM workshop 2015 that Holger Marschall is doing this in order to solve the system of coupled equations (He said that it will be released in foam-extend but I haven't found something). Unfortunately, I couldn't find more details of how he is doing it and so, I decided to try to implement it on my own. Therefore, I have attached 2 files, one for the details of the Cahn-Hilliard equation and one of how this implementation could be done. Unfortunately, the way that I have implemented the solver, it doesn't work. It would be nice if you have any ideas or if you know a different way to handle these equations.

Thomas
Attached Images
 scheme.png (63.6 KB, 41 views)
Attached Files
 scheme.pdf (51.6 KB, 27 views)

 August 10, 2020, 05:40 Non-linear 4rth order Diffusion #8 New Member   Azad Join Date: Nov 2019 Posts: 6 Rep Power: 6 Hello Foamers, Have you been able to solve the 4th order Diffusion equation using OpenFOAM. Please help me with the same. Thanks

 Tags c++, cahn hilliard, diffusion, non linear, openfoam