Issue about Dielectric Boundary Condition

[garlan2018]

Hi all,

I'm trying to build an Argon plasma simulation with ZAPDOS dielectric boundary condition in a 2D square domain (uniform structured mesh).
However, I encountered convergence issues in the solution when I use EconomouDielectricBC.
My simulation have no chemical reactions at this moment.
It has no issues when I apply Dirichlet BC to all boundaries in the Poissson solver.
When I change the BC for the top boundary to EconomouDielectricBC, it fails to converge for the first iteration.

Does anyone have experience in applying dielectric boundary condition in plasma models?

Below are the things I've tried, but still still not giving success.

1. Start the simulation with using the solution from the simulation with all Dirichlet BC.
2. I came across with a discussion on the Structured Mesh vs Unstructured Mesh some years ago and followed the Github from that discussion to implement the fluid flux variation of dielectric BC.
   Discussion: https://groups.google.com/g/zapdos-users/c/7nKaOAaV8w8
   https://github.com/csdechant/zapdos/tree/structured/debug
3. Add chemical reaction back to the simulation.

I've attached the input code for reference.
test.txt

[csdechant]

Hi!
I will take your input file and see what I can do. In the meantime, I did have a few questions and comments for you:

• We have two types of dielectric BC, DielectricBCWithEffEfield and EconomouDielectricBC. Physical, these are the some BC. Numerical, however, DielectricBCWithEffEfield is based on a Neumann formulation, while EconomouDielectricBC is based on a Penalty Dirichlet formulation. In most situations, the Neumann formulation is more numerical stable, so I recommend trying that one out.
• I don't see a supply voltage in your simulations in your input file and you start with zero potential. I never tried that with the dielectric BC before. The EconomouDielectricBC calculates time derivative of the electric field on the boundary. I am assuming something might be zeroing out on the first time step. That is the exact error message your received? Is it a ZERO_PIVOT error?
• Your time step seems good, but you have a really coarse mesh (uniform mesh of 25 X 25). Do you see any different behavior with you refine the mesh?
• This is more of a side note, depend on your needs. The dielectric BC in Zapdos assumes an ideal and grounded dielectric. If you plan to have a supply voltage on the other side of the dielectric, I recommend looking at the surface charge tests. This simulations dielectrics was MOOSE's Interface systems.

[garlan2018]

Hi!

Thanks for your advice!
I'm new to the plasma modelling and ZAPDOS.

1. I'll try DielectricBCWithEffEfield and post the result later. Is there any recommendation on the mesh requirement for this BC?
2. Both error messages of using EconomouDielectricBC and the flux variation of this BC are DIVERGED_FNORM_NAN.
   EconomouDielectricBC gives Nonlinear |R| = -nan when the system cuts half to dt while the flux variation gives a very large nonlinear |R| (10e17).
   Both can't converge even when dt is cut to minimum. Do I need to provide time derivative for this type of BC?
3. The solution still can't be converged when I refine the mesh to 250 elements per side for both BC. The error message is still DIVERGED_FNORM_NAN.
   Time Step 1, time = 2.5e-11, dt = 2.5e-11

• Nonlinear |R| = 3.162278e+17 (Before preset BCs, predictors, correctors, and constraints)

Performing automatic scaling calculation

Computing Automatic Scaling
  Computing Jacobian.                                                                [ 12.10 s] [  783 MB]
Finished Computing Automatic Scaling                                                 [ 12.21 s] [  783 MB]

0 Nonlinear |R| = 3.162278e+17
Computing Jacobian... [ 22.63 s] [ 801 MB]
0 Linear |R| = 3.162278e+17
1 Linear |R| = 3.469447e+02
1 Nonlinear |R| = 1.163356e+17

Computing Jacobian...                                                                [ 22.87 s] [ 3969 MB]
  0 Linear |R| = 1.163356e+17
  1 Linear |R| = 7.658276e+02

2 Nonlinear |R| = 6.793140e+16

Computing Jacobian...                                                                [ 22.75 s] [ 3969 MB]
  0 Linear |R| = 6.793140e+16
  1 Linear |R| = 1.670514e+02

3 Nonlinear |R| = 6.154200e+18

4. I'm trying to build a ICP simulation. Voltage may be supplied at the electrode later, but maybe not on the dielectric material.
   To begin, I want to keep it simple. Thus, I first start with the box domain and no voltage is supplied to it.
   I've also tried to add most of chemical reactions involving Ar just like the example Lymberopoulos_rf_discharge, but still got the similar behaviour.
   That's why I moved back to experiment with the one without the reactions.

[csdechant]

1. Yes, please let me know how that goes. For plasma, the mesh is usually refined near the boundaries due to the high gradient profiles that can appear in the plasma sheath. There is the hard limit of the Courant–Friedrichs–Lewy (CFL) condition to get an idea of the absolute limit of time step and grid size, but we often use second order implicit time integration to use larger time steps and grid sizes then the CFL condition. If you are new to plasma fluid simulations, it is still a good start, though.

2. DIVERGED_FNORM_NAN is an other type of zeroing out error. When you say "the flux variation" I am assuming you did use the DielectricBCWithEffEfield already, or is there there something else you used? If both dielectric BC is failing, I am assuming the error due to the initial conditions of the potential. This is because of the formulation for the dielectric equations, which I will give in detail to your other question, but in short, these boundary conditions equate the electric fields at the dielectric to the surface charge (i.e., ion and electron surface flux). The electric field is zero (because of the zero IC and surrounding Dirichlet BC), the ion flux is zero (because it is assumed to be purely advection driven, $\Gamma_i = \mu n_i \nabla V$), and the electric flux is non-zero (because it is assumed to be the thermal velocity driven at the surface $\Gamma_e = v_{th} n_e$). This lead to $v_{th} n_e = 0$ at the boundary, which can't be done, both physically or numerically in Zapdos. The reason being, in order for $v_{th} n_e = 0$ either the electron energy or density needs to be zero, and in Zapdos, we use logarithmic forms for energy and densities to avoid unrealistic zeros. I have two suggestion:

   1. Add an applied voltage to one of you other boundaries. Having an applied voltage on one boundary while grounding the other boundary (expect of the dielectric) should induce a potential gradient on the first time step and removing the unrealistic condition of $v_{th} n_e = 0$ at the dielectric boundary. (NOTE: This could still be numerically ill-condition and the simulation might still fail, but hopefully this removes the DIVERGED_FNORM_NAN error).
   2. Look at the potential profile in your "Dirichlet-Only" case. I am assuming you has some type of non-zero potential profile due to the charge separation in the plasma sheath (if a plasma sheath didn't form in your previous simulation, that is something else to look into). If you have a non-zero potential profile, your can use that output file as the initial conditions for your dielectric simulation (link here for detail, but let us know if you know help).

3. Refining the mesh won't help if my assumption of the error being $v_{th} n_e = 0$ is correct. Since mesh refinement didn't help here, I am more confident in my assumption.

4. We can talk more about modeling a ICP in a separate discussion. Zapdos hasn't be used to model ICP, yet (Zapdos has mainly modeled CCP and microwave-driven discharges, with current development focused on magnetized plasmas), but it should be doable. I usually recommend people starting with Zapdos to model a CCP , because computationally, they are usually simpler than ICP. Once you can model a CCP correctly, then more to a ICP.

[garlan2018]

The flux variation is the one that computes the electron flux with drift-diffusion at the boundary, rather than the electron temperature.
As you explained, it certainly fails to produce converged solutions.

[garlan2018]

As you pointed out that EconomouDielectricBC calculates time derivative of the electric field on the boundary.
For a unchanging electric field, it is better to remove the time derivative terms in this type of BC?

[csdechant]

No, I wouldn't recommend removing the time derivative because that captures the time response of the surface charge of the dielectric. The equation for the electric field at a dielectric surface is:
$\varepsilon_1 \vec{E}_1 - \varepsilon_2 \vec{E}_2 = \sigma$, where $1$ and $2$ denotes the perpendicular electric field on opposite sides of the dielectric surface, and $\sigma$ is the surface charge. For a plasma, the surface charge is:
$\int e \left( \vec{\Gamma}_i - \vec{\Gamma}_e \right) dt$ where the time integration accounts for the surface charge buildup over time. (NOTE, for simplicity this formulation assumes only positive ions, but Zapdos can simulation both positive and negative ions).

This is where the two Zapdos dielectric BC diverge, where DielectricBCWithEffEfield solves for the integrated form and EconomouDielectricBC solver for the time derivative form (i.e., take the time derivative of the equation to remove the integral of the plasma flux and include the time derivative of the electric field). Removing the time derivative of the electric field removes the time dependent surface charge.

[garlan2018]

Hi @csdechant,

I've used DielectricBCWithEffEfield you suggested in the testing case.
It provides converged solution and can run until it reaches some physical limits of the case setup.
Thanks a lot for your suggestions and explanations!
I learnt a lot from these.