Plasma Simulations with

Adaptive Cartesian Mesh

Vladimir Kolobov and Robert Arslanbekov,

6th International Conference on

Numerical Modeling of Space Plasma Flows

ASTRONUM-2011

Valencia, Spain

June 15, 2011

Plasma Types & Physical Models

• Space Plasmas

• Laboratory & Atmospheric Plasmas

• Hypersonic Reentry Plasmas

• Energy Range

• Collisions, chemical reactions & electron kinetics

• Geometry & Computational Grid

• Rarefied-continuum regimes: Kinetic-Fluid models

• Magnetic Fields

• Radiation Transport (Radiation Trapping for Atomic lines)

• Turbulence

Commonalities & Differences

Agenda

• Body-Fitted and Cartesian Meshes

• Adaptive Mesh Refinement (AMR) using OctreeCartesian mesh

• GFS framework for solving Partial Differential Equations with AMR

• Minimal Plasma model for streamer development in Corona Discharges

• Extended Plasma model for simulations of Glow and Corona Discharges

• Radial and Axial Structure of DC Glow Discharges

• Effects of Electron Thermal Conductivity

• Dynamics of pulsed breakdown and micro-plasma formation

• Conclusions

Body-Fitted and Cartesian Meshes

• Partial Differential Equations governing Plasma Dynamics are solved numerically using computational mesh

• The choice of mesh geometry and boundary treatment define accuracy and efficiency of simulations

Unstructured body-fitted meshStructured body-fitted rectangular mesh

Combination of structured Cartesian mesh and hybrid unstructured body-fitted

mesh near the wall

Structured Cartesian mesh with embedded boundary

Comparison of Mesh Types

• Advantages of body-fitted meshes

• place nodes optimally to resolve geometrical features of the problem

• increase mesh density towards the wall

• Disadvantages of body-fitted meshes

• approximation errors caused by spread of inter-nodal distances (LTE)

• loss of robustness caused by skewed faces

• difficulties with automated mesh generation and adaptation

•Advantages of Cartesian meshes

• Local Truncation Error (LTE) minimization

• robustness of differencing scheme

• simplicity and robustness of mesh generation algorithm

• simplicity of navigation on mesh

• dynamic mesh adaptation

• multi-grid solvers

• Disadvantages of Cartesian meshes

• more complicated boundary treatment: cut cells versus Immersed Boundary Methods

Plasma Solvers and Mesh Techniques

Fluid models have been widely used for

simulations of weakly ionized plasmas

using traditional body-fitted mesh

techniques.

These methods lack efficiency for

simulations of dynamically evolving

features requiring high resolution in

selected areas of computational

domain (shock waves, ionization fronts,

streamers, etc).

We have selected a Gerris Flow Solver

(GFS) as a framework for the

development of new plasma solver

with adaptive Cartesian mesh:An example of 3D fluid simulations of an

industrial ICP source performed with CFD-ACE+ software using body-fitted

mesh techniqueshttp://gfs.sourceforge.net

Highlights of GFS

• Fully Threaded Three for navigation on Mesh

• Multi-Grid Poisson Solver

• Poisson equation is solved using a relaxation operator

• Multilevel acceleration is used with a classical “V-cycle”

• Advection Tracer Solver

• Conservative formulation with Godunov scheme for

the advection term in the tracer solver

• The diffusion term is treated using the multigrid Poisson solver

• GFS is a free software for solving PDEs with AMR

• Complex boundaries embedded into computational domain

• Cut-cell methods to define BC

• Some moving boundary capabilities

• The computational grid is generated by subsequent division of square boxes into smaller boxes with linear dimensions equal to half of the initial dimension.

• This procedure of creating computational mesh is represented by a tree. The root of the tree (0th level) corresponds to initial cube; the first level corresponds to 4 (in 2D) or 8 (in 3D) cubes obtained by division of the initial cube.

• Computational cells correspond to leaves of the tree. All computation procedures are called

only for the leaves.

Quad/Octree Cartesian Mesh

Cell Accessing Procedure

level 0

level 1

level 2

level 3

Conventional octree:

connectivity info between a cell

and its neighbors is stored

explicitly, which requires 19 words

of computer memory

(PARAMESH)

Fully threaded tree (FTT):

cells are organized into groups

called octs. Each oct stores

pointers to eight child cells and to

a parent cell, requiring words

per cell

Hash tables: cell traversal is

reduced to key manipulations,

5/8 words per cells

32

8

• To perform computations for a part of the domain (for instance, the sub-domain shown by blue color), one introduces a flag for each leaf.

• After introducing flags, the procedure of cell traversing is modified to visit only the cells belonging to the selected sub-domain. Only the branches connected by solid lines are traversed.

• To specify boundary conditions, additional cells marked by dashed lines are used. The branches corresponding to boundary cells are shown by dashed lines. The boundary cells are marked using a

different flag, and can be traversed separately by the code.

Specifying Boundary Conditions

Simulations of a part of the domain

level 0

level 1

level 2

level3

• During sequential traversing of cells by natural order, the physical space is filled with curves in N-order (Morton ordering). After this ordering of cells, all cells can be considered as a one-dimensional array.

• A weight is assigned to each cell, proportional to CPU time required for computations in this cell. The array modified with corresponding weights, is subdivided into sub-arrays equal to the number of processors.

• This method allows most efficient domain decomposition between different processors.

Space Filling Curves & Forest of Octrees

The procedure of domain decomposition is performed using SFC. Using SFC, 2D or 3D spatial data can be stored in a single dimensional array and domain decomposition is easily achieved by chopping this array into even pieces.

Minimal Plasma Model

ee e

ndiv n

t

ie

nn

t

4 e in n

e eEn A simplest set of equations for the electron

and ion densities, and Poisson equation for

electrostatic potential.

Electron diffusion and ion transport are

neglected

00 exp

E

E

10 10 10 1020 3 log ( ) log ( ) log ( ) log ( )max( ) max( )

e ee i i

e e

n nn n E

n n

Mesh Adaptation criteria

When gradient of exceeded a fixed value in a cell,

this particular cell was refined

When in a cell, this cell was coarsened

1

1/ 4

The procedure of mesh adaptation performed every 10 time steps

Surface Treatment & Boundary Conditions

Cut cell Embedded Boundaries

• Treatment of surface cells is done using cut-cell approach

• Cell merging for small cells (volume < 0.5)

• Time step is calculated using the classical CFL condition

Standard set of boundary conditions

• On electrodes thermal flux and secondary electron emission is accounted for.

• On dielectrics surface charge is calculated

/ n

Streamer development in negative corona

2D axi-symmetric simulation of streamer development near a needle-like

elliptic cathode and a flat anode: p=760 Torr

The plasma channel is formed on the axis of the discharge.

A constant electron density 105 cm-3 is assumed at the cathode surface. The streamer

propagates with a velocity by an order of magnitude higher than the electron drift velocity .

The calculated streamer velocity agrees with the classical estimate for the fast streamer

velocity.

Streamer development in positive corona

The computational grid (left), electrostatic potential (center), and the electron

density (right) for a streamer developing near a rod-shape anode.

2D axisymmetric simulations of positive corona discharge

An off-axis toroidal channel was formed in our simulations.

This behavior was observed with different physical models for different

electrode shapes.

Streamer development in positive corona

3D simulations revealed that the 2D axi-symmetric channels break into separate

streamers. The streamers form near the anode with an almost periodic pattern and

propagate in slightly different directions with different speeds. Only two streamers survived

at a long distance from the anode. The code generates about 1.5 million cells during

parallel simulations on 8 cores.

Computational grid, electron density contours at 107 cm -3, and the electric field

strength (color, maximum value 7 106 V/m ) at 15 ns (left) and 20 ns (right)

Extended Plasma Model

ee e

ndiv n

t

( )ii e

ndiv n

t

4 e in n

e e e e eEn D n

• A set of equations for the electron and ion

densities, and Poisson equation for

electrostatic potential

• Ion transport is included

• Multiple species ions are allowed by

introducing species specific mass,

diffusion coefficients, and ionization rates

• Local field approximation for ionization

rate

• Electron Energy Transport with account

for electron thermal conductivity

• CANTERA link implementation is

underway for complex reaction

mechanisms

00 exp

E

E

i i i i iEn D n

nS

t

5 5

3 3e eEn D n

en n

External Circuit

• External circuit solved analytically using function calls from simulation scripts. Simple circuits can be treated

• RC external circuit used in the present studies

• External circuit can be attached to any computation domain boundary patch

potential

DC Discharges: Normal Current Density

Electron density and computational mesh (on the left), ion density and

equipotential contours (on the right) in a normal DC discharge. The

maximal values of electron and ion densities are 1.8 108 cm-3 and 9.5 108

cm-3, correspondingly

• Normal current density: discharge current occupies only a part of the cathode (near the axis).

• Mesh adaptation based on gradient of electron density.

• Discharge conditions: pressure 3 Torr, gap length 1 cm, radius 2 cm, voltage 400 V, resistor 1 MOhm, and capacitor 0.1 pF, g = 0.1

Normal Current Density in CCP

Electron density and computational mesh (on

the left), ion density and equipotential

contours (on the right) in a normal CCP

discharge. The maximal values of electron

and ion densities are 3.5 108 cm-3 and 1.4 109

cm-3, correspondingly.

Applied voltage and gap voltage (red)

6 KHz, 3 Torr, gap length 1 cm, radius 2 cm, voltage 400 V,

resistor 0.5 MOhm,

capacitor 0.1 pF,

g = 0.1

Classical DC discharge: local ionization

Ion density and equipotentials,ni max = 1 1010 cm-3 , U = 195 V, 2 MO, 0.01 pF

Electron density and computational meshne max = 1.3 109 cm-3

No Faraday Dark Space, No field reversals in the cathode region

2D axi-symmetric simulations

Effect of Electron Energy Transport

Spatial distribution of electron density

Axial distributions of electron and ion densities (left), electrostatic potential and electron temperature (right) with account of electron thermal conductivity

The model reproduces basic structure of classical DC glow discharges

2D axi-symmetric simulations

Short DC Glow Discharge

Argon, 3 Torr,

radius R = 1 cm

gap L = 1 cm

An RC external circuit is attached to the left electrode (anode), cathode is grounded.

For a discharge current of 34 µA, we obtained a normal discharge with a well pronounced cathode spot.

The electron thermal conductivity defines the axial structure of the discharge. Due to nonlocal ionization, a potential well is formed in the plasma region.

Ion Density

Electron Density

Potential

Temperature

Discharges with Rod Shape Cathode

Axial profiles of the electron and ion density (left), the electron temperature and the x-component of the electric field (zoom on the plasma region)

Computational grid and the electrostatic potential (left), and electron density contours (right) for a glow discharge between a rod cathode and a planar anode.

15 Torr

32 µA

Dynamics of pulsed breakdown

Computational mesh and electrostatic potential (top), and electron density (bottow) at three time moments during gas breakdown between two wires.

Gas breakdown between two wires of radius 100 nm separated by a 1 m distance. A voltage of amplitude 0.2-4 kV applied between the wires to initiate the gas breakdown induced by field emission. Gas pressure 1-100 atm.

Micro Plasma is formed at a sup-picosecond time scale

Cage Discharge

Computational grid and electric potential and log10(ne) (right) for a cage discharge

Schematic of cage discharge and discharge

cross section illustrating high-density

plasma generation process (right).

Conclusions & Future Plans

• We have shown that fluid plasma models previously developed for the traditional body-fitted mesh techniques can be extended for the adaptive Cartesian mesh.

• We have illustrated the benefits of AMR for simulations of corona and streamer discharges over a wide range of gas pressures.

• The calculated fields near the streamer tips were of the order of Bp, indicating that non-local ionization should play an important role in these regions.

• Using kinetic description of electrons in dynamically evolving ionization fronts and fluid description in the high density streamer channel resembles shock wave propagation in gas dynamics previously studied by UFS.

• Development of adaptive kinetic-fluid simulations of plasmas is a subject of our current research.

Closing Notes

• Financial support provided by NASA SBIR Project, AFOSR STTR Project, and DoE Office of Fusion Sciences.

• Further details can be found in the Invited Review paper “Towards Adaptive Kinetic-Fluid Models of Weakly Ionized Plasmas” accepted for publication in the Journal of Computational Physics.