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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,i)μe,ine,iE−De,i∇ne,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

mini∂ui

∂t+(ui·∇

)ui=qiniE−∇·Pi−miniυi,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 =−Dneu∇nneu .(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=−(5/3)μenεE−(5/3)De∇nεis 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

∇·(ε∇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(1−e)ne−γSEEi·ˆ

n

i·ˆ

n=ai·sign(qi)μiE·ˆ

n+1

4uth,ini

ε·ˆ

n=−ae

5

3μeE·ˆ

n+1

3uthnε−γSEE εSEE i·ˆ

n(9)

where ae,i=1 if the drift velocity points toward the

electrode surface (sign(qe,i)μe,iE·ˆ

n≥0)and ae,i=0

if (sign(qe,i)μe,iE·ˆ

n<0).uth (=(8kbTe,i/πme,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=−(2ae−1)μeE·ˆ

n+1

2uth,ene−1

2nruth,e

−2(1−ae)γSEEi·ˆ

n

i·ˆ

n=(2ai−1)sign(qi)μiE·ˆ

n+1

2uth,ini(10)

where nris the number density of the secondary electron and

can be expressed as

nr=(1−ae)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=−(2ae−1)5

3μeE·ˆ

n+2

3uth,enε−2

3nεruth,e

−2(1−ae)γSEEεSE Ei·ˆ

n(12)

where nεris the energy density of the secondary electron and

can be written as

nεr=(1−ae)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=−D∇n·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(iw−ew)·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+nN−nP

exp(Pe)−1,if Pe <0

unnP+nP−nN

exp(−Pe)−1,if Pe >0

(19)

where un(=sign(q)μEx·ˆnx+sign(q)μ 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+

N−S−

P

+S+

NS−

P

S+

N−S+

P

(UN−UP)(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(uN−aN,uP−aP)

S+

N=max(SN,0), S−

P=min(SP,0)(22)

where ais the sound speed. More details can be found in [45].

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.

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

ε∇φ·ˆ

ndA ≈

N=k(P)

(ε f∇φfAf)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

Af≈φN−φP

¯

PN·ˆ

nf

Af=φN−φP

¯

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,φ

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=φN−φP

|¯

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

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+(xA−xP)∂φ

∂xP

+(yA−yP)∂φ

∂yP

φB=φP+(xB−xP)∂φ

∂xP

+(yB−yP)∂φ

∂yP

φC=φP+(xC−xP)∂φ

∂xP

+(yC−yP)∂φ

∂yP

.(27)

Then, the above can be rewritten in a matrix form of the linear

system as

⎡

⎣

(xA−xP)(yA−yP)

(xB−xP)(yB−yP)

(xC−xP)(yC−yP)

⎤

⎦⎡

⎢

⎢

⎣

∂φ

∂x

∂φ

∂y

⎤

⎥

⎥

⎦P

=⎡

⎣

φA−φP

φB−φP

φC−φP

⎤

⎦.(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

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

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.

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 m−3, and a Gaussian seed of positive ion is set

to be ni(r,z)=(1×1018)exp(−(r2+(z−0.01)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 ×10−12 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 m−3.

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

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 m−3.

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 m−3. 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=1×10−12 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.

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

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

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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