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Hello Ahmed Kasdi

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Hi Ahmed and others, Did you ever get your model completed? I'm trying to do the same and got the velocities and fields looking reasonably correct. The only problem is the space charge rho looks WRONG: I'm doing a positive corona and yet I get the space charge to be negative in some areas, especially close to the ground electrodes. The space charge distribution is also very patchy with highly positive and negative patches next to each other. My explanation is that maybe the nonlinear solvers cannot handle large (electric field) and small (rho) numbers simultaneously? Did you have the same problem and can you help me with my problem, since rho is what I'm interested in, which is then used to track a charge particle. Thanks in advance for your help.

Hello Peter, Ahmed,

the system of equation Ahmed described (Corona with so-called Peek's boundary condition) can be solved efficiently by rewriting the equation in an equivalent form that fits the lack of boundary condition for rho.
You can read some détails of the equations here :
www.comsol.fr/papers/3285/

With the current version of COMSOL, the weak form is not required any longer and the process is even simpler : You can use instead a Point ODE equation that you'll find under the Mathematics part of the Physics Library.

Of course this is an approximation of the plasma physics that occurs near the electrode. If you are interested in what happens near the charged electrode, you should have a look to the Plasma Module.

I hope this helps.
Regards,
Eric Favre - COMSOL France

Hello everyone,

I have a convergence problem with modelling of Electrostatics equation coupled with PDE( for charge transport equation). I attached the equations and .mph files.

MODEL : two electrodes placed in domain of air. voltage difference applied. I have to calculate resultant electric field, charge density values and use them to add a body force to laminar flow to simulate plasma.

1. Geometry is simple. Mesh is fine.
2. I tried fully coupled( various damping factors), segregated, parametric sweep of the source term in charge transport equation( which makes it nonlinear) but couldn't get it to converge.
3. I tried using PDEs alone instead of electrostatics,but again same problem ( Plasma Actuator Mod.mph)

Could anyone please help me with way to obtain convergence or any other way to solve the equations attached.

Hello,

this classic Corona problem cannot be solved the way you tried. It is simply not well posed mathematically and won't converge.
Instead, you should use the approach described in the pdf attached, that reformulates the problem.
I add the corresponding 3D model for convenience. 2D or 1D is straightforward. Adding Fluid Flow with particle tracing should be the easy part.
I suggest as well to read the reference mentionned earlier.

Good luck!
Eric Favre
COMSOL France

Thank you for your reply ,Mr.Favre . I am sorry that I couldn't respond earlier as I had some technical problems.

I understand that you are saying only one BC is needed for charge density as first order equation is solved .But I just tried to follow Mr.Martins who solved in his paper "Modelling of an Improved corona disharge and positive corona thruster (2012)" in which he used COMSOL and does the same as I had described in my previous mail.

Now, I tried to do exactly the same way that you suggested, with my problem. I just have two electrodes as setup. But it still doesnt converge. Again, like it was given in the pdf that you provided, I solved for phi alone and then used it as initial condition for solving three equations. I tried different damp factors also. But yet the error oscillates between 1 and 10. Could you please see the file and help me in getting a converged solution. Please note that I just changed rho to rhoc. Rest is all same.

Error is same as tolerance, isnt it? error should be less than or equal to tolerance for certain iterations consecutively to get the "converged" solution. Isnt that the case?

Thank you
Raghuvir

- As you know in these systems there is a theshold in electric potentiel under which the plasma does not form. So increase the potential at the electrode PHI_A by a factor 5. I suspect your parameters are not in the domain of validity of this crude Peek's plasma model.
- There is another question of importance : which boundary conditions at the exterior boundaries?
You cannot leave "zero flux" which impose the electric field to be zero, which is probably not physical. You probably want to impose PHI = 0 as in the other electrode? (usually these systems have a finite size). So adjust PHI = 0 at the external boundaries.
- mesh with default
- Solve with constant (Newton) using Fully coupled approach.
- there is no need to solve in 2 steps : a single one should be enough if your initial condition is variable in space (which is ok in your file).
- You should get a nice convergence in a few iterations within seconds.

Good luck!
Hope this helps.
Eric

Mr. Favre , Thankfully equations are converging now. I just had to change the PHI at outer enclosure to zero, instead of flux.

Now, I seem to have to problem with Laminar flow module . I added volume force as rhoc*PHIx. Error oscillates between 1 and 0.1 but doesnt converge. I tried segregated approach and parametric sweep(ramping load) also. That didnt help either.

Could you please check the file once and see if I am making any more mistakes. Is it because that I am imposing "no slp" condition at the electrode surface even though the body force is not zero. I tried "slip" condition also.

I want to be in control of the body force that I am giving and I dont have information about particle diameter/ No. of particles. Hence I am not using Particle tracer Module.

Thank you
Raghuvir

Dear Raghuvir,

This is a bit difficult to guess the precise conditions you want to simulate.
Please allow me to stress out somehow the method that needs to be used in such cases where the solution does not converge but you don't know exactly why.

1. you absolutely need to go back with difficulties, in a region that is (or should be) trivial to solve. Even if far from your real problem. In your case, choose a laminar flow, impose rho and viscosity (pay attention rho = density is probably different from rho = charge density in a electric potential equation) such that you get a laminar flow (calculate Reynolds number). Solve and play a bit with the solution, investigate how the parameters make the solution change, compare to what you think it should be (some kind of common sense validation)...

2. Based on this working (technically speaking) case, change one and only one parameter so that your case is a bit closer to the final case that you want to solve. Get it to converge, play a bit with the parameters (...). If it does not converge, you know which parameter you have changed. Is it expected or not? Try to answer these questions by yourself (you learn a lot here). If it's impossible to sort out, perhaps you should try another parameter to play with. Go back to step 1 and investigate other directions. Or if you get stuck you might contact the technical support saying : here is my working case, change this parameter, it does not work any longer (whereas it should becasue of...) : any hint?

3. iterate until you get closer and closer to the reality with step 2 above.

This is the only methodology that is proven to work to solve such a (possibly quite complicated) problem like yours.

Good luck,
Eric

Hello Eric Favre
The link you gave www.comsol.fr/papers/3285/
is no longer available.
Can you give us a new one or a pdf document?
Thanks
Best Regards,
Tu Gongming

Dear Tu Gongming,

the document you refer to is now a bit old.
For updated and superior way of solving this kind of coupled problem, please ask your Customer Representative at COMSOL, he/she should be able to send the updated relevant information.

Best regards,
Eric Favre
COMSOL France

Thanks for sharing the documents, Dr. Favre.

Originally, I tried to solve the fully coupled equations using weak PDE form for both the electric potential and space charge density. For the space charge density weak PDE, I used a weak contribution along with an auxiliary variable as the Lagrange Multiplier. Though, I was able to obtain converged solution. The solution is nothing like the analytical solution for a typical cylindrical plasma generator.

Now, I am implementing the method used in your COMSOL paper with three equations. However, I still cannot obtain the correct solution. I am solving an 1-D Axisymmetric problem by using a line as the geometry with two point boundaries. I have tried to match the settings you used in your 3D model. Any suggestions?

Thanks.

Hello,
please send details to technical support, we will be gald to help.
As a sidenote, compared to the (old) paper you refer to, the implementation in latest versions of COMSOL has improved and is much easier now.

Best regards,
Eric Favre
COMSOL France

Thanks Eric. I have my technical problem fixed.

The method you proposed by splitting rho into rho0 + drho works really well. I assumed the integration operator plus the point ODE (pode) serves as a way to impose a integral constraint on a point on the emitting electrode (Am I right?)

However, for me so far, the method works only for geometry with two concentric electrodes (2D). I have been trying to implement this new method to 1D Axis and 2D Axis, but no converging results yet.

Thanks for the help.

Hello Eric,

1D cartesian should not be different from 2D cartesian.
In 1D or 2D axisymetric case, there is this subtility in the case you use a mathematical PDE : those mathematical equations should be slightly adjusted to the coordinate system in use.
The model example "Spherically symmetric Transport" (www.comsol.fr/model/spherically-symmetric-transport-8481) demonstrates how to multiply some coefficients of the PDE by r^2 in 1D spherical coordinate. The source term of Neumann-like boundary conditions should be adjusted as well.
In axisymmetric case (r in 1D or (r,z) in 2D ), the same approach leads to multiplication by r.

I hope this corresponds to your case.
Best regards,
Eric Favre
COMSOL France

Thanks Eric. Your method is working perfectly with my geometry now.

I omitted the fact that when using general PDE, the gradient for cylindrical or spherical coordinate is not compensated. With the gradient compensated, a converged solution can be obtained efficiently.

Hi Eric,

I am trying to model the effect of dust layer on space charge density (rhoc) and electric potential (PHI). Therefore, I am modifying the boundary condition on the collection electrode to become a non-linear one.

Old BC:
PHI = 0

New BC:
PHI = - constant * rhoc * PHIr, where rhoc = rho0p + drhoc

Both the constant Newton and automatic Newton solvers lead to singular matrix. The double dogleg seems to be converging after 1000 iterations, but I am not sure if this is the correct way to pursue convergence.

I would like to know if implementing such a non-linear BC is possible?

Thanks in advance.

Eric -- Thank you for your explanations, methodology, and examples regarding Peek's Law problems. Very nice.

Hi Eric,

convergence should be straightforward. If not, this is likely that the problem is not well posed. The information you give is not detailed enough to locate the possible source of the problem. I encourage you to contact the technical support with more details for more advanced assistance.

Good luck,
Eric Favre
COMSOL France

Hi -- Is it possible to prescribe the total current on the electrode rather than V0? Something like integral(rho1*En) = I0 on the electrode? I'm a fairly new to Comsol . . .

BTW: This constraint method works well for me, though I did need to adjust the solver configs for more complex geometries. Parametric sweeps are also useful for convergence (and getting at initiation voltage, if you are interested in that).

Dear Richard,

many different options are possible with COMSOL, including integral boundary conditions. The integral is easily defined using ready to use "coupling operators". The integral can in some cases simply be re-used in another boundary condition.
It is often straigthforward to enter the equation as a constraint. What is often less straightforward is to be sure your problem is well-posed : the number of equations should balance the number of unknowns for such a direct problem (as opposed to some inverse problems).

I take the opportunity to extend the question to the integro-differential equations (not only on a boundary, and adding differential).
There is a recent blog about this :
www.comsol.com/blogs/integrals-with-moving-limits-and-solving-integro-differential-equations/

In case of problem entering your own equation into COMSOL Multiphysics (providing the fact that you have checked out it is well-posed), don't hesitate to contact technical support.

Best regards,

Eric Favre
COMSOL France

Eric,

Thanks for the suggestion. I have the issue resolved. The electric potential at the collection electrode needs to be a Dirichlet BC to make it well-posed. The variable BC can be applied with the point ODE.

I have another question about the integral constrain using the point ODE. I am solving a 2D Axisymmetric ESP (see attachment). Instead of using a fixed point to constrain the electric potential to V0 at the discharge electrode, is it possible to move the point ODE along the z-axis during each iteration step to obtain a 2D distribution of the electric potential?

Thanks again.

Hello Eric,

What is possible is to have a fixed potential at the electrode together with a non constant charge density over the electrode. I don't think the location of the point matters. I might not fully understand your request though. We can help you regarding these questions through the technical support. I encourage you to contact support@comsol.com regarding this.

Best regards,
Eric Favre
COMSOL France