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Development of a Massively Parallelized Fluid-Based Plasma Simulation Code With a Finite-Volume Method on an Unstructured Grid

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A massively parallelized unstructured-grid plasma simulation code based on a multifluid plasma model has been developed and validated. The code uses a collocated cell-centered finite-volume method and is designed to simulate intermediate low-pressure and atmospheric-pressure (AP) plasma sources with complex machine geometries. One of the novel features of this code is that it is implemented in a highly flexible computational platform named ultrafast massively parallel processing (ultraMPP), which allows straightforward addition and integration of different partial differential equation (PDE) solvers in a self-consistent manner. As to the numerical methods, the Scharfetter-Gummel scheme is adopted to handle the drift-diffusion flux of electrons, whereas the Harten-Lax-van Leer (HLL)-type approximate Riemann solver is used to handle the convection terms of ion momentum equations. For discretization of the diffusion terms in an unstructured grid, the Taylor expansion is used to deal with the effects of nonorthogonality of the cells, and the cell-center gradient is calculated using a least-squares method. The simulation code with a local-field approximation (LFA) was validated for two cases of AP plasmas as well as a case of an argon capacitively coupled plasma generated in a Gaseous Electronic Conference (GEC) reference cell with local mean energy approximation (LMEA) at intermediate low pressure. The simulation results were found to be in good agreement with the previously published experimental and simulation data.
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104 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
Development of a Massively Parallelized
Fluid-Based Plasma Simulation Code
With a Finite-Volume Method
on an Unstructured Grid
Kuan-Lin Chen , Meng-Fan Tseng, Ming-Chung Lo , Satoshi Hamaguchi, Meng-Hua Hu,
Yun-Ming Lee, and Jong-Shinn Wu ,Member, IEEE
Abstract— A massively parallelized unstructured-grid plasma
simulation code based on a multifluid plasma model has been
developed and validated. The code uses a collocated cell-centered
finite-volume method and is designed to simulate intermediate
low-pressure and atmospheric-pressure (AP) plasma sources
with complex machine geometries. One of the novel features
of this code is that it is implemented in a highly flexi-
ble computational platform named ultrafast massively parallel
processing (ultraMPP), which allows straightforward addition
and integration of different partial differential equation (PDE)
solvers in a self-consistent manner. As to the numerical meth-
ods, the Scharfetter–Gummel scheme is adopted to handle
the drift-diffusion flux of electrons, whereas the Harten–Lax-
van Leer (HLL)-type approximate Riemann solver is used to
handle the convection terms of ion momentum equations. For
discretization of the diffusion terms in an unstructured grid, the
Taylor expansion is used to deal with the effects of nonorthog-
onality of the cells, and the cell-center gradient is calculated
using a least-squares method. The simulation code with a local-
field approximation (LFA) was validated for two cases of AP
plasmas as well as a case of an argon capacitively coupled
plasma generated in a Gaseous Electronic Conference (GEC)
reference cell with local mean energy approximation (LMEA) at
intermediate low pressure. The simulation results were found to
be in good agreement with the previously published experimental
and simulation data.
Manuscript received April 30, 2020; revised June 29, 2020; accepted
July 29, 2020. Date of publication August 13, 2020; date of current version
January 11, 2021. This work was supported in part by the Ministry of Science
and Technology, Taiwan, under Grant 107-2221-E-009-072-MY3, Grant 108-
2811-E-009 -517, Grant 107-2622-E-009-009-CC1, Grant 108-2622-E-009-
003-CC1, and Grant 109-2622-E-009-001-CC1; in part by the Japan Society
of the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research(S)
under Grant 15H05736; and in part by the JSPS Core-to-Core Program under
Grant JPJSCCA2019002. The review of this article was arranged by Senior
Editor P. K. Chu. (Corresponding author: Jong-Shinn Wu.)
Kuan-Lin Chen, Meng-Fan Tseng, and Jong-Shinn Wu are with the Depart-
ment of Mechanical Engineering, National Chiao Tung University, Hsinchu
30010, Taiwan (e-mail: chongsin@faculty.nctu.edu.tw).
Ming-Chung Lo is with the Department of Mechanical Engineering,
National Chiao Tung University, Hsinchu 30010, Taiwan, and also with the
Department of Mechanical and Aerospace Engineering, Chung Cheng Institute
of Technology, National Defense University, Taoyuan 335, Taiwan.
Satoshi Hamaguchi is with the Center for Atomic and Molecular Tech-
nologies, Graduate School of Engineering, Osaka University, Suita 565-0871,
Japan.
Meng-Hua Hu and Yun-Ming Lee are with Plasma Taiwan Innovation
Corporation, Hsinchu 30274, Taiwan.
Color versions of one or more of the figures in this article are available
online at https://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPS.2020.3013632
Index Terms—Atmospheric-p ressure (AP) plasmas, numerical
analysis, plasma applications.
I. INTRODUCTION
LOW-TEMPERATURE gas discharges [1] play a key role
in materials processing in the modern semiconductor
industry. These include plasma-enhanced chemical vapor
deposition [2], plasma etching [3]–[6], Direct-current (dc)/
radio frequency (RF) magnetron sputtering plasmas [7], [8],
and surface cleaning [9], to name a few. A better understanding
of complicated discharge phenomena observed in these
processes is very critical in improving various qualities
of such material processing, e.g., deposition or etching
rates and process uniformity across the wafer. Gaining a
better understanding of such processes can be facilitated
through experimental observations as well as numerical
simulations. The latter has become an increasingly important
and indispensable tool since it can easily probe various plasma
properties that may be inaccessible by experimental means.
Various kinds of numerical modeling techniques have been
developed for general gas discharges under various pressure
conditions based on particle-in-cell (PIC)/Monte Carlo colli-
sion (MCC) models (see [10]–[21]) or fluid models [22]–[32].
The former can be used for highly rarefied gas discharges with
potentially high computational cost, whereas the latter is more
applicable to collisional plasmas, especially intermediate low-
to atmospheric-pressure (AP) plasmas [15]. There is an alter-
native simulation tool for capturing kinetic effects by solving
the high-dimensional Boltzmann and Fokker–Planck equa-
tions [16], [17]. However, its computing cost is much higher
than the fluid model due to high-dimensional phase spaces.
In this study, we present a massively parallelized
unstructured-grid plasma simulation code based on a multi-
fluid plasma model that we have recently developed and its
validation against a few cases given in previous publications.
In general, there are two types of plasma fluid modeling
techniques, i.e., the local field approximation (LFA) and the
local mean energy approximation (LMEA). The LFA assumes
that electrons are in equilibrium with the local electric field
(i.e., the EEDF depends only on the local electric field), which
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 105
means that the electron transport coefficients and rate constants
only depend on the reduced electric field E/N, i.e., the ratio of
the electric field strength to the number density of background
neutral species [24]–[26]. In other words, the energy gained
by the electrons from the local electric field is assumed to
be consumed locally through elastic and inelastic collision
processes of the electrons with other particles. However, if the
electric field varies rapidly in either space or time, then
the LFA becomes inaccurate or even invalid. For instance,
such a situation occurs in the head of a streamer, the sheath
region, and the RF discharges, among others. In these cases,
the electron energy density balance equation (i.e., LMEA)
should be considered to include the effect of nonlocal electron
transport. The resulting macroscopic properties depend on the
mean electron energy instead of the reduced electric field.
The use of LMEA leads to a significant improvement of the
description of electron properties [27]. Both approximations
have been implemented in our simulation code presented here,
and our simulation results based on these approximations were
validated against some published data, which will be presented
in this article.
Among the existing plasma fluid modeling simulation codes,
many use structured meshes [28]–[32]. With a structured
mesh, for example, Teunissen and Ebert [32] utilized an adap-
tive mesh refinement (AMR) method with OpenMP (shared-
memory) parallel algorithms to reduce the simulation time
while increasing accuracy where is needed. However, most
of the industrial applications need to deal with objects having
complex geometries, which may not fit well with a regular
structured mesh with finite grid sizes. To resolve this problem,
a better alternative is to use either a body-fitted approach
with structured grids (or curvilinear approach) [33], [34] or
an unstructured grid that can be set along the boundaries
of objects with complex geometries. Another alternative is
to use an immersed boundary method (IBM) on Cartesian
coordinates [35]. The limitations of the IBM are that some
significant modifications of the fluid code for different prob-
lems are required as well as it is difficult to have an arbitrarily
improved resolution near the wall, which limits its further
application in the modeling of gas discharges. In addition,
Kolobov et al. [36] used the adaptive Cartesian mesh with
the volume of fluid (VOF) and IBM for embedded bound-
aries to simulated corona discharge, in which the mesh is
automatically generated around embedded objects and can
be dynamically adapted to solution properties and moving
boundaries to resolve the problem by using IBM. Moreover,
the commercial package, COMSOL Multiphysics [37], can
also simulate the low-temperature plasmas, which uses the
finite-element method (FEM) for 2-/3-D unstructured meshes,
and has a dedicated module for plasma.
Prior to our study, some 2-D fluid-based plasma simula-
tion codes were developed with unstructured grids [38]–[40].
Among them using LFA, one employed the FEM to model
filamentary discharges in air [38], whereas another used the
flux correct transport (FCT) scheme based on a dual-cell finite-
volume (nodal FVM) method with an unstructured grid to
model AP plasmas [40]. Some codes use both FEM and FCT
with LFA [41], [42].
It is well known that the simulation of multidimensional
low-temperature plasma physics is a typical very complex mul-
tiphysics modeling problem that may include plasma physics,
electromagnetics (EM), complex chemistry, highly nonlin-
ear partial differential equations (PDEs), advanced numerical
schemes, and parallel computing, to name a few. It may take
years to develop an efficient and accurate plasma simulation
code to solve nontrivial problems in plasma science. Recently,
a parallel computing platform, named Rigorous Advanced
Plasma Integration Testbed (RAPIT), has been proposed with
both embedded-PDE and particle-solver-related objects that
can easily accommodate the development of the continuum-
and/or particle-based solvers with some proper hybridization
algorithms in a self-consistent way [43]. In addition, a com-
pact version of RAPIT, named ultrafast Massively Parallel
Processing (ultraMPP), a compact version of RAPIT, has also
been proposed, which includes various numerical features such
as single or multiple unstructured grids for different solvers
or species with automatic interpolation relations, essentially
the same source code for 2-D and 3-D problems due to
nearly operator-like programming style, and embedded parallel
implementation, to name a few. In this study, we took advan-
tage of these features of ultraMPP to implement our simulation
code efficiently.
In this article, we present details of the newly developed
plasma simulation code together with some sample simulation
results that corroborate its validity. In Section II, the numerical
methods employed in the code will be described, which
include plasma fluid modeling equations, boundary conditions,
and related numerical schemes and algorithms that were used
to discretize and solve the fluid model equations. In Section III,
some simulation results for 2-D-axisymmetric dry air positive
streamer propagation, a 2-D dry air surface dielectric barrier
discharge (SDBD), and a 2-D-axismmetric argon capacitively
coupled plasma (CCP) will be presented. The simulation
results were compared with previously published experimental
and simulation data. The major findings of this study are
summarized in Section IV.
II. NUMERICAL METHODS
A. Plasma Fluid Model
In this study, the governing equations for the plasma fluid
model are similar to those solved by Lin et al. [44] and
Chen et al. [45] and are only briefly described here for com-
pleteness. The electron, ion, and neutral transport is described
by the continuity, momentum, and energy density equations.
The momentum equations for electrons and ions are often
simplified using the drift–diffusion approximations, which are
combined with the continuity equations to form the number
density equations of electrons and ions, respectively. The
general nonlinear continuity equations with the drift–diffusion
approximation can be written as follows:
ne,i,neu
t+∇·e,i=Se,i(1)
where the subscripts e,i,andneu denote the electron, ion,
and neutral species, respectively. The variables n,S,and
are the number density, the chemical reaction source/sink, and
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106 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
the species flux vector, respectively. The expressions of the
charged particle fluxes are written as
e,i=sign(qe,ie,ine,iEDe,ine,i(2)
where μand Dare the mobility and the diffusivity of the
charged species, respectively, and Eis the local electric field.
However, in a gas discharge either driven by high-frequency
power source(s) or low-pressure background, the ion iner-
tia effect becomes nonnegligible [46]. Therefore, the ion
momentum equations should be considered directly. The ion
momentum equations without considering the magnetic force
can be written as
miniui
t+(ui·∇
)ui=qiniE−∇·Piminiυi,neuui
(3)
where miis the ion mass, uiis the ion velocity vector, Piis the
ion pressure, and υi,neu is the momentum exchange collision
frequency of the ion with the neutral species.
In addition, the neutral species particle flux without consid-
ering the convection effect can be expressed as
neu =−Dneunneu .(4)
The electron energy density equation can be written as
nε
t+∇·nε=Snε(5)
where nε=(3/2)nekbTeis the electron energy density,
kbis Boltzmann’s constant, Teis the electron temperature,
and nε=−(5/3enεE(5/3)Denεis the electron
energy density flux. The source/sink term, Snε, stands for the
summation of the energy losses due to elastic and inelastic
collisions and the electron joule heating. Furthermore, the ion
temperature is the same temperature as the background gas by
assuming thermal equilibrium between these two species.
Finally, the self-consistent electrostatic (ES) potential is
determined by solving Poisson’s equation as
∇·φ) =−
m
l=1
(qn)l(6)
where φ,ε, and mare the potential, the permittivity, and the
number of charged species, respectively. The corresponding
electric field is then obtained by the relation E=−φ.
B. Boundary Conditions
There are three different sets of boundary conditions for the
charged species that are used in the present fluid modeling
code. The first set is the simplest boundary condition, assum-
ing that the ion number density, the electron number density,
and the electron energy density are all zero at the electrode
surfaces [47], which can be written as
ni=ne=nε=0.(7)
Ions will flow out of the computational domain if the electric
field points toward the electrode surface. In addition, the zero
ion density gradient boundary condition produces the same
result as the zero density boundary condition [48], which can
be written as
ni=ne=nε=0.(8)
The second set of boundary conditions is the flux type
for the ion and the electron at the electrode surfaces can be
expressed as [49]
e·ˆ
n=aeμeE·ˆ
n+1
4uth,e(1e)neγSEEi·ˆ
n
i·ˆ
n=ai·sign(qiiE·ˆ
n+1
4uth,ini
ε·ˆ
n=ae
5
3μeE·ˆ
n+1
3uthnεγSEE εSEE i·ˆ
n(9)
where ae,i=1 if the drift velocity points toward the
electrode surface (sign(qe,ie,iE·ˆ
n0)and ae,i=0
if (sign(qe,ie,iE·ˆ
n<0).uth (=(8kbTe,ime,i)1/2)is
the mean thermal speed of ions/electron, eis the electron
reflection coefficient, and ˆ
nis the outward unit normal vector
of the electrode surface. γSEE and εSEE are the secondary
electron emission coefficient and the secondary electron emis-
sion energy, respectively. γSEE defines the average number of
electrons emitted per ion impact.
The third set of boundary conditions is proposed by
Hagelaar et al. [50], which is written as
e·ˆ
n=(2ae1eE·ˆ
n+1
2uth,ene1
2nruth,e
2(1aeSEEi·ˆ
n
i·ˆ
n=(2ai1)sign(qiiE·ˆ
n+1
2uth,ini(10)
where nris the number density of the secondary electron and
can be expressed as
nr=(1ae)γSEEi·ˆ
n
μeE·ˆ
n.(11)
Since the boundary conditions used by Hagelaar et al. [50] did
not include the electron energy density, the following boundary
condition for the electron energy density was proposed by
Lee et al. [51] as:
ε·ˆ
n=(2ae1)5
3μeE·ˆ
n+2
3uth,enε2
3nεruth,e
2(1aeSEEεSE Ei·ˆ
n(12)
where nεris the energy density of the secondary electron and
can be written as
nεr=(1ae)γSEEεSEEi·ˆ
n
μeE·ˆ
n.(13)
In general, the abovementioned boundary conditions can
only be applied for the fluid model equations based on
the drift–diffusion approximation. However, the positive ion
velocity and density at the electrode surface are assumed as
the Neumann boundary condition when solving the full ion
equations and the ion flux is set to zero for negative ions
considering the normal sheath electric field. In addition, all
the ion temperature gradients are set to zero at walls.
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 107
Finally, for the boundary conditions of neutral species,
the diffusion-type boundary condition is applied at the wall
surfaces
neu ·ˆ
n=−Dn·ˆ
n.(14)
For Poisson’s equation, the Dirichlet boundary conditions
are imposed on the potential φcorresponding to an applied
voltage φw(e.g., φw=0 for the grounded electrode) at the
solid electrode wall. Moreover, the interfacial conditions at
the plasma/dielectric interfaces and the displacement current
continuity must be enforced, which can be written as
εplasma Eplasma εdielectric Edielectric
·ˆ
n=σs(15)
where the accumulated surface charge σsvarying with time
can be evaluated by the following equation:
dσ
dt =e(iwew)·ˆ
n(16)
where iwand ewrepresent the ion and electron flux toward
the plasma/dielectric interface, respectively.
C. General Finite-Volume Method
In this article, we employed the collocated cell-centered
finite-volume method (FVM) for discretizing all the PDEs
of the plasma fluid modeling equations, in which all the
dependent variables are defined at the cell centers. The FVM
is based on the discretization of the integral forms of the
governing equations. The term “finite volume” refers to the
small volumes (control volume) surrounding the nodal points
on a mesh. The volume integral of the governing equations can
be converted into the surface integral using the well-known
divergence theorem in vector analysis. These surface terms
can be evaluated as the fluxes at the surfaces of each control
volume (i.e., cell). The general integral form of the conserva-
tion equation of any quantity using the FVM can be written
as
t
d+S
·ˆ
ndA =
Qd(17)
where is the control volume, is the dependent variable,
dA is the surrounding face area, and Qis the source term
of the conservation equation. The temporal term can be
discretized by the finite-difference method, and the source term
can be treated explicitly if not too stiff numerically. Based on
the concept of FVM, the flux integral can be approximated
by the summation of the flux vectors over the boundary faces
of the cell as
S
·ˆ
ndS =
N=k(P)
P,NAf(18)
where k(P)is the faces of the cell P,Afis the face area,
and P,Nrepresents the numerical fluxes through the interface
between cell Pand cell N, respectively. Note that the direction
of P,Nis assumed to be positive directed from the first cell
Ptoward the neighboring cell N.
D. Scharfetter–Gummel Scheme for Approximating the
Drift–Diffusion Flux
Considering the continuity equation [see (1)] without the
source term using the drift–diffusion approximation [see (2)],
the flux can be approximated by the Scharfetter–Gummel
(S-G) scheme (or exponential scheme), which originally
had been developed for the modeling the semiconductor
device [52]. This method is the discretization scheme by
applying the local 1-D exact solution for the PDE with both
the convection and diffusion terms. The drift–diffusion flux at
the interface fbetween cell Pand N, as shown in Fig. 1, can
be written as
SG ·ˆ
n=
unnN+nNnP
exp(Pe)1,if Pe <0
unnP+nPnN
exp(Pe)1,if Pe >0
(19)
where un(=sign(qEx·ˆnx+sign(q Ey·ˆny)is the normal
convective velocity at the faces and Pe(=un¯
PN/D)is the
Péclet number. It is clear that the weighting not only depends
on the ratio of mobility and diffusivity but also the potential
difference between cell Pand N. It is a good flux scheme
suitable for plasma simulation because the spatial distribution
of electron and ion densities could be both in an exponential-
like distribution.
E. HLL Approximated Riemann Solver for the Full Ion
Momentum Equations
When solving the full ion momentum equations (without
drift–diffusion approximation), the numerical flux across each
cell interface can be determined by the approximate Riemann
solver for the convection term. One of the most popular
schemes is the Harten–Lax-van Leer (HLL)-type Riemann
solver [53]. As shown in Fig. 1(a), the HLL flux function
for the convection term of the ion momentum equations in the
normal direction of each face of the cell can be expressed as
Fn=S+
NF(UP)S
PF(UN)
S+
NS
P
+S+
NS
P
S+
NS+
P
(UNUP)(20)
where U,F,S+
N,andS
Pdenote the column vector of the
conservation variables, flux function vector in the normal
direction, and the wave speed in right and left directions,
respectively. Note the abovementioned column vectors, Uand
F, are, respectively, defined as follows:
U=
ni
niui
nivi
and Fn=
niui
niu2
i+pi/mi
niuivi
·ˆnx+
nivi
niuivi
niu2
i+pi/mi
·ˆny.(21)
In addition, the wave speeds are computed by the simple
estimation, as presented in [54], as
SN=max(uN+aN,uP+aP), SP=min(uNaN,uPaP)
S+
N=max(SN,0), S
P=min(SP,0)(22)
where ais the sound speed. More details can be found in [45].
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108 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
Fig. 1. (a) Stencil used in the unstructured-grid cell-centered finite-volume discretization. (b) Control volume involved in the calculation of cell-center
gradient.
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 109
Fig. 2. Computational domain for streamer simulation.
F. Diffusion Flux and Cell Gradient Calculation
in Poisson’s Equation
The diffusion flux of the variable φin Poisson’s equation
through a cell interface fcan be approximated as
S
εφ·ˆ
ndA
N=k(P)
fφfAf)P,N.(23)
For the unstructured-grid cell-centered FVM, the simplest
second-order approximation for the diffusion flux is obtained
using auxiliary, referring to Fig. 1(a). Because the ¯
PN may
not be perpendicular to the face between the adjacent cells in
an unstructured mesh, the diffusion flux may not be accurate
due to the nonorthogonality of the grid. In this case, the term
φ·ˆ
ndA can be approximated by
f·ˆ
nfAf=∂φ
nf
AfφNφP
¯
PN·ˆ
nf
Af=φNφP
¯
PN
Af.
(24)
The variable values at the auxiliary points Nand Pare
estimated by the Taylor-series expansion, which are written
as
φp=φp+∇φP·¯
PP
N=φN+∇φN·¯
NN(25)
where ¯
PP=¯
Pf(¯
Pf·ˆ
nf)ˆ
nfand ¯
NN =¯
Nf(¯
Nf·ˆ
nf)ˆ
nf.
By substituting (25) into (24) and after some rearrangements,
one can obtain
φf·ˆ
nfAf=φNφP
|¯
PN|Af+φN·¯
NN−∇φP·¯
PP
|¯
PN|Af.
(26)
The first term on the right-hand side of (26) is the “normal
diffusion” and the second term is the “cross diffusion,” which
is a correction term for the diffusion term calculated on the
nonorthogonal grids.
From the expression in (26), one needs to accurately cal-
culate the cell gradients of the dependent variables. Here, the
least-square method is employed. Consider, e.g., a triangular
cell Pwith three neighbor cells, A,B,andC, which are shown
in Fig. 1(b). Extrapolation for the neighbor cell-center values
φA,B,Cat point (xA,B,C,yA,B,C)from φPat point (xP,yP)can
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110 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
Fig. 3. Temporal position of the dry air positive streamer head.
be written as
φA=φP+(xAxP)∂φ
xP
+(yAyP)∂φ
yP
φB=φP+(xBxP)∂φ
xP
+(yByP)∂φ
yP
φC=φP+(xCxP)∂φ
xP
+(yCyP)∂φ
yP
.(27)
Then, the above can be rewritten in a matrix form of the linear
system as
(xAxP)(yAyP)
(xBxP)(yByP)
(xCxP)(yCyP)
∂φ
x
∂φ
y
P
=
φAφP
φBφP
φCφP
.(28)
Finally, this linear system can be solved to obtain the cell
gradient of cell Ponce for all from the start because the
mesh distribution does not change during the simulation in
the current study.
G. Implementation of the Parallelized Plasma Fluid
Modeling Code
In this article, all the discretized equations by the cell-
centered, collocated, unstructured-grid finite-volume method
were implemented on a general PC cluster through the use
of the ultrafast ultraMPP platform [43], as mentioned in
Section I. Note that ultraMPP is a general-purpose paral-
lelization platform for physical problems modeled by PDEs.
Furthermore, it can deal with complex geometry using the
2-D/2-D-axisymmetric/3-D hybrid unstructured grid(s) with
parallel computing using the domain decomposition method
through the use of standard MPI (message passing inter-
face) on any PC cluster with distributed-memory architec-
ture. Domain decomposition was achieved through the use
of METIS [55] based on the criterion that approximately
the same number of cells in each processor was maintained.
No special dynamic load balancing was implemented since
no adaptive mesh was used in the current study. In addition,
the preconditioned GMRES iterative matrix solver [56] is
used throughout the study, unless otherwise specified. In brief
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 111
Fig. 4. Temporal evolution of the distribution of the magnitude of the electric field of the dry air positive streamer.
Fig. 5. Simulation domain of the SDBD plasma actuator.
summary, it can help the computational scientists greatly
shorten the development time of a large-scale multiphysics
computational code from years down to months or even weeks
for a realistic physical problem. Most important of all, there
is no need for the code developer to write any commands in
the source code about parallel computing.
H. Time-Stepping Method and Semi-implicit Poisson’s
Equation Solution Algorithm
The continuity equation with the drift–diffusion approxima-
tion is discretized by the backward Euler method. At each time
step, the resulting algebraic linear system of each equation is
solved one by one using an iterative method, e.g., parallel
preconditioned GMRES [56] which is embedded through
ultraMPP. In the case of solving the complete ion equations
(without the drift diffusion approximation), the forward Euler
method is employed because of much larger characteristic time
related to ions compared with electrons. Furthermore, it was
reported that explicit evaluation of the source term of Poisson’s
equation led to a very small computational time step due to
the restriction of dielectric relaxation time. For overcoming
this problem, the so-called semi-implicit treatment was used
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112 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
Fig. 6. Mesh distributions of the SDBD plasma actuator simulations. (a) Structured-like (84 000 cells). (b) Unstructured (9600 cells).
in the present plasma fluid modeling code, which is similar
to [44].
III. RESULTS AND DISCUSSION
In this section, three 2D or 2-D-axisymmetric gas dis-
charge problems are simulated and validated to demonstrate
the capability of the developed plasma fluid modeling code.
These problems include 2-D-axisymmetric dry air positive
streamer propagation, a 2-D dry air SDBD plasma actuator,
and 2-D-axisymmetric capacitively coupled argon plasma.
They are described in the following in turn.
A. Modeling of 2-D-Axisymmetric Dry Air Positive Streamer
Propagation
Streamer discharge is caused by quickly growing ionized
channels due to the strong electric field at their tips [57].
Streamer discharges have several applications, such as ozone
production, air purification, and plasma medicine, to name
a few. However, simulating streamer discharge proven to be
challenging for many reasons, which have been pointed out
by Teunissen and Ebert [32]. First, to simulate the phe-
nomenon of streamer branching or the interaction between
streamers needs to be considered as a 3-D description. Second,
an extremely high grid resolution for capturing the nonlinear
growth of streamers around the streamer heads is required.
Due to this reason, it is not possible to obtain an approx-
imate solution by using a coarse grid. Third, the streamers
are highly transient filamentary plasmas. Therefore, a time-
dependent simulation should be considered, and due to the
mesh spacing and high electric field at streamer tips, a very
small timestep has to be used. Due to the abovementioned
numerical difficulties, the modeling of streamer discharge
is a perfect problem to validate and test the capability of
the plasma fluid modeling code developed in the current
study.
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 113
Since the 3-D simulation is computationally too expensive
in most cases, we employ the 2-D axisymmetric model to
simulate the spatial and temporal development of individual
streamer following Bagheri et al. [58] as a validation case.
Among the six sets of the simulation data from different
groups in [58], we use the Netherlands’ CWI group’s data for
comparison because they have used the AMR method near
the streamer head to highly resolve the local large jump in
the local electric field, which we believe that its simulation
accuracy is among the best in the literature.
In this case, the positive streamers in dry air at the pressure
of 1 bar and temperature of 300 K have been considered.
A streamer discharge is simulated between planar electrodes
in an axisymmetric domain with a radius and a height of
both 0.0125 m as shown in Fig. 2. The continuity equations
of electrons and ions with the drift–diffusion approximation
[see (1) and (2)] and the Poisson equation [see q. (6)] are
solved based on the LFA. Since we are only interested in the
few hundreds of nanoseconds (or milliseconds), the ions in
the short time scale of streamer propagation are assumed to
be immobile, and the neutral species are not modeled with
the same reason. For the boundary conditions, the voltage
applied to the top electrode is φ0=18.75 kV and the bottom
electrode is grounded. The Neumann boundary conditions for
potential are applied to the symmetry axis and the right of
the computational domain. The Neumann boundary conditions
are also applied to the electron density at all boundaries. The
transport coefficients of electron and ionization rate coefficient
can also be found in [58]. The initial charged species density is
ne=ni=1013 m3, and a Gaussian seed of positive ion is set
to be ni(r,z)=(1×1018)exp((r2+(z0.01)2/0.00042))
to enhance the local electric field. In this case, a nonuni-
form structured-like unstructured grid is used to simulate the
positive streamer propagation, where the numbers of cells
are 4096–16 384 and 256–512 in the z-andr-directions,
respectively. A resulting maximum 4M cells are employed
for the simulation. The timestep size is fixed as 1 ×1012 s
for all cases. The simulation up to 20 ns with 30 processors
(Intel Xeon Gold 6148 at 2.4 GHz) takes about 24 h using the
structured grid for the 4M cell case.
Fig. 3 shows the temporal position of the air streamer head
using meshes with different levels of refinement. When the
finer mesh is used (16 384 ×256), the propagation speeds
of the streamer head are nearly the same as those in [58].
In contrast, the propagation speeds of streamer head with
coarser meshes become faster than the benchmarking case
(CWI group) in [58]. This is caused by the big jump in the
local electric field and a very high number density at the region
very close to the streamer head. In addition, the simulation
results are not sensitive to the resolution in the r-direction to
some extent with the test cases performed in the current study.
Fig. 4 shows the simulated temporal evolution of the distri-
butions of the magnitude of the electric field after discharge
ignition. As it is well known, the local electric field will
be enhanced due to the large space charge difference. The
enhanced electric field will create the electron avalanches
because of gas breakdown, which gives rise to an additional
electric field boost. This field can further increase the growth
Fig. 7. Temporal evolution of the distributions of the ion density and the ES
potential at three different times using structured-like mesh. The equipotential
contours are separated by 100 V, and the ion density levels are 0.2, 0.5, 1.0,
2.0, 5.0, and 10 ×1020 m3.
of new electron avalanches that creates enhanced ioniza-
tion, and the streamer head propagates downward along the
z-direction and accelerates with time. We have found that the
current plasma fluid model solver is capable of reproducing
essentially the same results of the CWI group as presented in
[58, Fig. 3].
B. Modeling of 2-D Dry Air SDBD Plasma Actuator
In Section III-A, the plasma dynamics of streamer dis-
charges is mostly governed by the electrons because the ions
are practically immobile. However, the ions can play a vital
role in some applications, such as aerodynamics applications.
For the flow control in aerodynamics applications, one possible
method is to apply the plasma actuator(s) installed on the flight
body to prevent the boundary layer from separation. The basic
working principle of these actuators is based on the formation
of gas discharge that ionizes the air, and the charged species
are accelerated through the electric field and exchange an
appreciable amount of momentum with the neutral air species
through collisions. This leads to a significant electrodynamics
force, which can modify the velocity profile of the background
air species within the boundary layer for laminar-turbulent
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114 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
Fig. 8. Comparison of the spatial distributions of the simulated ion density and electric potential at three different times using (a) structured-like grid and
(b) unstructured grid. The equipotential contours are separated by 100 V, and the ion density levels are 0.2, 0.5, 1.0, 2.0, 5.0, and 10 ×1020 m3.
transition control, drag reduction, or flow stabilization [59].
In this section, we will focus on the validation of the developed
fluid code, and the detailed physical mechanisms will be
reported elsewhere in the near future.
Fig. 5 shows the computational domain and related physical
configuration of the SDBD plasma actuator. The computational
domain is in rectangular geometry with the Neumann bound-
ary conditions at all boundaries except the bottom one. For
the interface of plasma and dielectric material, the interfacial
boundary condition is enforced with a secondary electron
emission coefficient equal to 0.05 [see (9)], and the dis-
placement current continuity across the interface is enforced
[see(15)]. The relative permittivity of the dielectric layer is
set to be 10. The dry air gas at a pressure of 760 torr
and a temperature of 300 K is considered with an initial
homogenous charged number density as 1013 m3. A constant
voltage pulse of 1.2 kV is applied between the electrodes. The
transport coefficients and chemical reaction rate coefficients
can be found in [59]. In addition, the fluid modeling equations
include the continuity equations of electrons and ions with
the drift–diffusion approximation [see (1) and (2)], and the
Poisson equation [see (6)].
Fig. 6 shows two different sets of mesh distributions used
for the plasma fluid modeling. First, a structured-like fine grid
with 84 000 cells is used and the results are compared with the
simulation data in [59]. The timestep size is kept as a constant
with t=1×1012 s. On the other hand, another unstructured
coarse grid with 9600 cells is used for the same simulation.
Fig. 7 shows the temporal evolution of the distributions of
the ion density and the ES potential at three different times
using a structured-like mesh with those in [59, Fig. 2]. Before
t=30 ns, The ion space charge builds up due to the electron
avalanches generated by the ion-induced secondary electron
emission at the interface of plasma and dielectric material.
It is also known as a Townsend breakdown mechanism. At t=
30 ns, the plasma has formed near the edge of the power elec-
trode surface due to the continuous charging of the dielectric
surface nearby. The potential progressively reaches the applied
dc voltage. After the potential rises to the applied dc voltage,
the region of potential gradient will move further downstream
because the plasma plays the role of a virtual power electrode
expanding along the surface, as shown in Fig. 7 at t=50 ns
and t=70 ns. The propagation speed of the ion that shows
a good agreement with [59] is about 2500 m/s.
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 115
Fig. 9. Schematic of the 2-D-axisymmetric GEC CCP discharge chamber.
Fig. 8 shows the comparison of the temporal evolution of the
instantaneous spatial distributions of the ion densities and the
ES potentials at three different times between structured-like
and unstructured grids. The results of the different types of
grids show a similar trend, even though the spatial distribution
of the ion density and the electric potential is slightly different.
Nevertheless, the computational time is greatly reduced by
eight times by using the unstructured grid compared with the
very fine structured grid. For example, the simulation up to
100 ns with 30 processors (Intel Xeon Gold 6148 at 2.4 GHz)
takes only 15 min using the unstructured grid compared with
2 h using a structured-like grid.
C. 2-D-Axisymmetric Modeling of an Argon CCP
The advantage of CCP discharge is low cost, simple config-
uration, and easy to produce uniform plasma over a larger area
compared to the other types of plasma sources [60]. In gen-
eral, the basic configuration of low-pressure CCP discharge
consists of two parallel electrodes, in which one electrode is
driven by the RF power source, and the other is grounded or
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116 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 49, NO. 1, JANUARY 2021
Fig. 10. Mesh distribution of CCP reference cell discharge chamber.
intentionally biased. The gap between the electrodes is about a
few centimeters. As a result, the plasma is generated between
the electrodes, and the sheath regions are close to the two
electrodes. The most critical issue in addition to the growth
rate or etching rate for the application of CCP in material
processing is the plasma uniformity, which can be controlled
by applying the dual frequencies at the power electrode or
at different electrodes. Furthermore, the benefits of using dual
frequency are control of the ion energy (control by a lower fre-
quency power source) and plasma density (control by a higher
frequency power source) independently [61]. However, when
the frequency increases to a very high frequency (100 MHz)
in a large-scale reactor, the standing-wave, ES, and EM edge
effects and skin effects become more significant and can result
in a highly nonuniform plasma [62]. However, those effects
could be mitigated by using a Gaussian-shaped electrode to
suppress, which has been proposed in [63]. To simulation
this type of gas discharge, the use of the unstructured-grid
fluid modeling solver is indispensable. Thus, in this section,
we would like to demonstrate the simulation of a typi-
cal CCP discharge using the unstructured-grid fluid model
code.
Fig. 9 shows the schematic of the Gaseous Electronic
Conference (GEC) reference cell, in which the gap between
two electrodes is 25.4 mm with a dielectric material (insulator)
and a conductor (both 1.5 mm in thickness) covering the outer
edges of both powered and grounded electrodes. A power
source with a peak-to-peak voltage of 200 V and a frequency
of 13.56 MHz is applied to the powered electrode at a pressure
of 250 mtorr with a uniform gas temperature of 300 K.
Fig. 10 shows the grid distribution with 8979 quadrilateral
and triangle cells in total, in which 50 cells are used across
the electrode gap. The number of timesteps per cycle is set
to be 400 and 2000 cycles is performed using ten cores of
processors. It takes 18 h to complete the run. In this case, all
the transport coefficient and model equations are the same as
in [64]. The rate constant coefficients are obtained from the
BOLSIG+[65], and we employ two different sets of argon
collision cross section, Phepls [66] and [67].
Fig. 11 shows the comparison of simulated and measured
number densities of charged species at various radial positions
in the GEC reference cell at a distance of 0.0125 m from the
powered electrode [64], [68]. We have found that the electron–
neutral collision cross section plays an important rule for the
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CHEN et al.: DEVELOPMENT OF A MASSIVELY PARALLELIZED FLUID-BASED PLASMA SIMULATION CODE 117
Fig. 11. Comparison of simulated and measured radial number densities of charged species in a GEC conference cell at a distance of 0.0125 m from the
powered electrode. Test conditions: f=13.56 MHz, Vpp =100 V, and Pc=250 mtorr.
simulated plasma density, as shown in Fig. 11. The results
show that our simulated data are in reasonable agreement with
the measured data and other simulation data, considering the
intrusive measurement by the Langmuir probe and unclear
documentation of the transport coefficients and rate constants.
IV. CONCLUSION
In this study, we have presented the development and vali-
dation of a parallelized 2-D rectangular or 2-D-axisymmetric
plasma simulation code based on a fluid plasma model with
LFA or LMEA and an unstructured grid. A few gas discharges
at intermediate low or atmospheric pressure were simulated
with this code and compared with existing simulation results
and experimental data. The simulated results were found to
be in good agreement with such data, which corroborated the
validity of this code. However, some gas discharge problems
with strong electron kinetic effects cannot be properly modeled
by a pure fluid model because, under strong nonequilibrium
conditions, the electron energy distribution strongly deviates
from Maxwellian and its high energy tails can strongly
affect the ionization processes. This is typically the case for
inductively coupled plasma and magnetron sputtering plasma
at a very low pressure. To solve such problems with high
computational efficiency, a hybrid particle-fluid model is need.
Inclusion of kinetic electrons in this code is currently in
progress.
ACKNOWLEDGMENT
The authors would like to thank the National Center for
High-Performance Computing of Taiwan for providing com-
puting resources.
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