Speeding-up direct implicit particle-in-cell simulations in bounded plasma by obtaining future electric field through explicitly propulsion of particles

The direct implicit particle-in-cell is a powerful kinetic method for researching plasma characteristics.However,it is time-consuming to obtain the future electromagnetic field in such a method since the field equations contain time-dependent matrix coefficients.In this work,we propose to explicitly push particles and obtain the future electromagnetic field based on the information about the particles in the future.The new method retains the form of implicit particle pusher,but the future field is obtained by solving the traditional explicit equation.Several numerical experiments,including the motion of charged particle in electromagnetic field,plasma sheath,and free diffusion of plasma into vacuum,are implemented to evaluate the performance of the method.The results demonstrate that the proposed method can suppress finite-grid-instability resulting from the coarse spatial resolution in electron Debye length through the strong damping of high-frequency plasma oscillation,while accurately describe low-frequency plasma phenomena,with the price of losing the numerical stability at large time-step.We believe that this work is helpful for people to research the bounded plasma by using particle-in-cell simulations.

On the energy conservation electrostatic particle-in-cell/Monte Carlo simulation: Benchmark and application to the radio frequency discharges

We benchmark and analyze the error of energy conservation （EC） scheme in particle-in-cell/Monte Carlo （PIC/MC） algorithms by simulating the radio frequency discharge. The plasma heating behaviors and electron distributing functions obtained by one-dimensional （1D） simulation are analyzed. Both explicit and implicit algorithms are checked. The results showed that the EC scheme can eliminated the self-heating with wide grid spacing in both cases with a small reduction of the accuracies. In typical parameters, the EC implicit scheme has higher precision than EC explicit scheme. Some ＂numerical cooling＂ behaviors are observed and analyzed. Some other errors are also analyzed. The analysis showed that the EC implicit scheme can be used to qualitative estimation of some discharge problems with much less computational resource cost without much loss of accuracies.

Implicit electrostatic particle-in-cell/Monte Carlo simulation for the magnetized plasma:Algorithms and application in gas-inductive breakdown

An implicit electrostatic particle-in-cell/Monte Carlo （PIC/MC） algorithm is developed for the magnetized discharging device simulation. The inductive driving force can be considered. The direct implicit PIC algorithm （DIPIC） and energy conservation scheme are applied together and the grid heating can be eliminated in most cases. A tensor-susceptibility Poisson equation is constructed. Its discrete form is made up by a hybrid scheme in one-dimensional （1D） and two- dimensional （2D） cylindrical systems. A semi-coarsening multigrid method is used to solve the discrete system. The algorithm is applied to simulate the cylindrical magnetized target fusion （MTF） pre-ionization process and get qualitatively correct results. The potential application of the algorithm is discussed briefly.

Spontaneous growth of the reconnection electric field during magnetic reconnection with a guide field:A theoretical model and particle-in-cell simulations

Reconnection electric field is a key element of magnetic reconnection.It quantifies the change of magnetic topology and the dissipation of magnetic energy.In this work,two-dimensional(2D)particle-in-cell(PIC)simulations are performed to study the growth of the reconnection electric field in the electron diffusion region(EDR)during magnetic reconnection with a guide field.At first,a seed electric field is produced due to the excitation of the tearing-mode instability.Then,the reconnection electric field in the EDR,which is dominated by the electron pressure tensor term,suffers a spontaneous growth stage and grows exponentially until it saturates.A theoretical model is also proposed to explain such a kind of growth.The reconnection electric field in the EDR is found to be directly proportional to the electron outflow speed.The time derivative of electron outflow speed is proportional to the reconnection electric field in the EDR because the outflow is formed after the inflow electrons are accelerated by the reconnection electric field in the EDR and then directed away along the outflow direction.This kind of reinforcing process at last leads to the exponential growth of the reconnection electric field in the EDR.

Particle-in-cell simulation for effect of anode temperature on discharge characteristics of a Hall effect thruster

Propellant gas flow has an important impact on the ionization and acceleration process of Hall effect thrusters （HETs）. In this paper, a particle-in-cell numerical method is used to study the effect of the anode temperature, i.e., the flow speed of the propellant gas, on the discharge characteristics of a HET. The simulation results show that, no matter the magnitude of the discharge voltage, the calculated variation trends of performance parameters with the anode temperature are in good agreement with the experimental ones presented in the literature. Further mechanism analysis indicates that the magnitude of the electron temperature is responsible for the two opposing variation laws found under different discharge voltages. When the discharge voltage is low, the electron temperature is low, and so is the intensity of the propellant ionization; the variation of the thruster performance with the anode temperature is thereby determined by the variation of the neutral density that affects the propellant utilization efficiency. When the discharge voltage is high, the electron temperature is large enough to guarantee a high degree of the propellant utilization no matter the magnitude of the anode temperature. The change of the thruster performance with the anode temperature is thus dominated by the change of the electron temperature and consequently the electron-neutral collisions as well as the electron cross-field mobility that affect the current utilization efficiency.

Electromagnetic Particle-in-Cell Simulations of Electron Holes Formed During the Electron Two-Stream Instability

Previous electrostatic particle-in-cell （PIC） simulations have pointed out that elec- tron phase-space holes （electron holes） can be formed during the nonlinear evolution of the electron two-stream instability. The parallel cuts of the parallel and perpendicular electric field have bipolar and unipolar structures in these electron holes, respectively. In this study, two-dimensional （2D） electromagnetic PIC simulations are performed in the x - y plane to investigate the evolution of the electron two-stream instability, with the emphasis on the magnetic structures associated with these electron holes in different plasma conditions. In the simulations, the background magnetic field （Bo = Boer） is along the x direction. In weakly magnetized plasma （Ωe 〈ωpe, where Ωe and ωpe are the electron gyrofrequency and electron plasma frequency, respectively）, several 2D electron holes are formed. In these 2D electron holes, the parallel cut of the fluctuating magnetic field δBx and δBz has unipolar structures, while the fluctuating magnetic field δBy has bipolar structures. In strongly magnetized plasma （Ωe 〉 ωpe）, several quasi-lD electron holes are formed. The electrostatic whistler waves with streaked structures of Ey are excited. The fluctuating mag- netic field δBx and δBz also have streaked structures. The fluctuating magnetic field δBx and δBy are produced by the current in the z direction due to the electric field drift of the trapped elec- trons, while the fluctuating magnetic field δBz can be explained by the Lorentz transformation of a moving quasielectrostatic structure. The influences of the initial temperature anisotropy on the magnetic structures of the electron holes are also analyzed. The electromagnetic whistler waves are found to be excited in weakly magnetized plasma. However, they do not have any significant effects on the electrostatic structures of the electron holes.

Effects of electron trapping on nonlinear electron-acoustic waves excited by an electron beam via particle-in-cell simulations

By performing one-dimensional particle-in-cell simulations, the nonlinear effects of electronacoustic(EA) waves are investigated in a multispecies plasma, whose constituents are hot electrons, cold electrons, and beam electrons with immobile neutralized positive ions. Numerical analyses have identified that EA waves with a sufficiently large amplitude tend to trap cold electrons. Because EA waves are dispersive, where the wave modes with different wavenumbers have different phase velocities, the trapping may lead to the mixing of cold electrons. The cold electrons finally get thermalized or heated. The investigation also shows that the excited EA waves give rise to a broad range of wave frequencies, which may be helpful for understanding the broadband-electrostatic-noise spectrum in the Earth’s auroral region.

Analysis of microwave propagation in a time-varying plasma slab with particle-in-cell simulations

Continuous microwave propagation through a time-varying plasma and frequency up-conversion has been demonstrated by particle-in-cell (PIC) simulation. In principle, it is possible to transform a 2.45 GHz source radiation to an arbitrary larger frequency radiation. The energy conversion is also obtained by the theoretical analysis and has been testified by PIC simulation. The source wave was propagating in a parallel plate waveguide locally filled with the ionized gas. In this paper we would discuss the effects of the rise time, the plasma length, the switching time and the collision frequency on the energy conversion, and the methods to improve the upshift wave energy are proposed. We also put forward the new concept of the critical values of the rise time and the source wave amplitude to provide a theoretical basis for the selection of parameters in the experiments.

Explicit structure-preserving geometric particle-in-cell algorithm in curvilinear orthogonal coordinate systems and its applications to whole-device 6D kinetic simulations of tokamak physics

Explicit structure-preserving geometric particle-in-cell(PIC)algorithm in curvilinear orthogonal coordinate systems is developed.The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research.The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms,discrete exterior calculus,and explicit non-canonical symplectic integration.In addition to the truncated infinitely dimensional symplectic structure,the algorithm preserves exactly many important physical symmetries and conservation laws,such as local energy conservation,gauge symmetry and the corresponding local charge conservation.As a result,the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics.The algorithm has been implemented in the Sym PIC code,which is designed for highefficiency massively-parallel PIC simulations in modern clusters.The code has been applied to carry out whole-device 6 D kinetic simulation studies of tokamak physics.A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor.The state also admits a steady-state subsonic ion flow in the range of 10 km s-1,agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated.It is shown that high-n ballooning modes have larger growth rates than low-n global modes,and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2%level by the E×B flow generated by the instability.These results are consistent with early and recent electromagnetic gyrokinetic simulations.

Particle-in-cell simulation of ion-acoustic solitary waves in a bounded plasma

We study some nonlinear waves in a viscous plasma which is confined in a finite cylinder.By averaging the physical quantities on the radial direction in some cases,we reduce this system to a simple one-dimensional model.It seems that the effects of the bounded geometry(the radius of the cylinder in this case)can be included in the damping coefficient.We notice that the amplitudes of both Korteweg–de Vries(KdV)solitary waves and dark envelope solitary waves decrease exponentially as time increases from the particle-in-cell(PIC)simulation.The dependence of damping coefficient on the cylinder radius and the viscosity coefficient is also obtained numerically and analytically.Both are in good agreement.By using a definition,we give a condition whether a solitary wave exists in a bounded plasma.Moreover,some of potential applications in laboratory experiments are suggested.

Particle-in-cell/Monte Carlo simulation of filamentary barrier discharges

The plasma behavior of filamentary barrier discharges in helium is simulated using a twodimensional（2D） particle-in-cell/Monte Carlo model. Four different phases have been suggested in terms of the development of the discharge： the Townsend phase; the space-charge dominated phase; the formation of the cathode layer, and the extinguishing phase. The spatialtemporal evolution of the particle densities, velocities of the charged particles, electric fields, and surface charges has been demonstrated. Our simulation provides insights into the underlying mechanism of the discharge and explains many dynamical behaviors of dielectric barrier discharge（DBD） filaments.

Particle-in-Cell Simulation of the Reflection of a Korteweg-de Vries Solitary Wave and an Envelope Solitary Wave at a Solid Boundary

Reflections of a Korteweg-de Vries （KdV） solitary wave and an envelope solitary wave are studied by using the particle-in-cell simulation method. Defining the phase shift of the reflected solitary wave, we notice that there is a phase shift of the reflected KdV solitary wave, while there is no phase shift for an envelope solitary wave. It is also noted that the reflection of a KdV solitary wave at a solid boundary is equivalent to the head-on collision between two identical amplitude solitary waves.

Affine particle-in-cell method for two-phase liquid simulation

Background The interaction of gas and liquid can produce many interesting phenomena,such as bubbles rising from the bottom of the liquid.The simulation of two-phase fluids is a challenging topic in computer graphics.To animate the interaction of a gas and liquid,MultiFLIP samples the two types of particles,and a Euler grid is used to track the interface of the liquid and gas.However,MultiFLIP uses the fluid implicit particle(FLIP)method to interpolate the velocities of particles into the Euler grid,which suffer from additional noise and instability.Methods To solve the problem caused by fluid implicit particles(FLIP),we present a novel velocity transport technique for two individual particles based on the affine particle-in-cell(APIC)method.First,we design a weighed coupling method for interpolating the velocities of liquid and gas particles to the Euler grid such that we can apply the APIC method to the simulation of a two-phase fluid.Second,we introduce a narrowband method to our system because MultiFLIP is a time-consuming approach owing to the large number of particles.Results Experiments show that our method is well integrated with the APIC method and provides a visually credible two-phase fluid animation.Conclusions The proposed method can successfully handle the simulation of a two phase fluid.

PIC simulation of plasma properties in the discharge channel of a pulsed plasma thruster with flared electrodes

Plasma in the discharge channel of a pulsed plasma thruster(PPT) with flared electrodes is simulated by a self-developed two-dimensional code. The fully particle-in-cell method with Monte Carlo collision is employed to model the particle movement and collisions and investigate the plasma properties and acceleration process. Temporal and spatial variations of the electron density distribution and the ion velocity between electrodes are calculated and analyzed in detail.The computational results of the electron number density, which is in the order of 1023 m-3,show good agreements with experimental results of a PPT named ADD SIMP-LEX. The ion velocity distributions along the center line of the channel lead to a comprehensive understanding of ions accelerated by electromagnetic field. The electron distributions of PPT with discharge voltages varying from 1300 to 2000 V are compared. The diffusion of electrons presents strong dependency on discharge voltage and implies higher degree of ionization for higher voltage.

基于MPI+CUDA的DSMC/PIC耦合模拟异构并行及性能优化研究

DSMC/PIC耦合模拟是一类重要的高性能计算应用,大规模DSMC/PIC耦合模拟计算量巨大,需要实现高效并行计算。由于粒子动态注入、迁移等操作,基于MPI并行的DSMC/PIC耦合模拟往往通信开销较大且难以实现负载均衡。针对自主研发的DSMC/PIC耦合模拟软件,在原有MPI并行优化版本上设计实现了高效的MPI+CUDA异构并行算法,结合GPU体系结构和DSMC/PIC计算特点,开展了GPU访存优化、GPU线程工作负载优化、CPU-GPU数据传输优化及DSMC/PIC数据冲突优化等一系列性能优化。在北京北龙超级云HPC系统的NVIDIA V100和A100 GPU上,针对数亿粒子规模的脉冲真空弧等离子体羽流应用,开展了大规模DSMC/PIC耦合异构并行模拟,相比原有纯MPI并行,GPU异构并行大幅缩短了模拟时间,两块GPU卡较192核的CPU加速比达到550%,同时具有更好的强可扩展性。

The E×B drift instability in Hall thruster using 1D PIC/MCC simulation

The E×B drift instability is studied in Hall thruster using one-dimensional particle in cell(PIC)simulation method.By using the dispersion relation,it is found that unstable modes occur only in discrete bands in k space at cyclotron harmonics.The results indicate that the number of unstable modes increases by increasing the external electric field and decreases by increasing the radial magnetic field.The ion mass does not affect the instability wavelength.Furthermore,the results confirm that there is an instability with short wavelength and high frequency.Finally,it is shown that the electron and ion distribution functions deviate from the initial state and eventually the instability is saturated by ion trapping in the azimuthal direction.Also for light mass ion,the frequency and phase velocity are very high that could lead to high electron mobility in the axial direction.

Fast weighting method for plasma PIC simulation on GPU-accelerated heterogeneous systems

Particle-in-cell (PIC) method has got much benefits from GPU-accelerated heterogeneous systems.However,the performance of PIC is constrained by the interpolation operations in the weighting process on GPU (graphic processing unit).Aiming at this problem,a fast weighting method for PIC simulation on GPU-accelerated systems was proposed to avoid the atomic memory operations during the weighting process.The method was implemented by taking advantage of GPU's thread synchronization mechanism and dividing the problem space properly.Moreover,software managed shared memory on the GPU was employed to buffer the intermediate data.The experimental results show that the method achieves speedups up to 3.5 times compared to previous works,and runs 20.08 times faster on one NVIDIA Tesla M2090 GPU compared to a single core of Intel Xeon X5670 CPU.

PIC-MCC Simulation for HPM Multipactor Discharge on Dielectric Surface in Vacuum

In order to understand the physical mechanism of multipactor discharge on dielectric window surface under high power microwave （HPM） excitation in vacuum, an electron movement simulation model based on the particle-in-cell （PIC） Monte Carlo （MC） is built in this paper. The influences of microwave electromagnetic field and electrostatic field from dielectric surface charging are simultaneously considered in this model. During the simulation, the emission velocity and angle distribution of secondary electrons from the dielectric surface are taken into account. The movement trajectories of electron clusters under complex field excitation are obtained. The influences of emergence angle and microwave electromagnetic parameters on the electron movement are analyzed. It is found that the emergence angle of electrons from the surface has significant effect on its movement, and both the impact energy and return time of electrons oscillate periodically with the phase of microwave field. The number of secondary electrons and induced electrostatic field from multipactoring are also investigated. The results reveal that both values oscillate periodically at twice the microwave frequency, which is due to the electron impact energy oscillating with microwave period. A schematic diagram is proposed to explain the periodical oscillation phenomena.

PIC/MCC Simulation of Radio Frequency Hollow Cathode Discharge in Nitrogen

A two-dimensional PIC/MCC model is developed to simulate the nitrogen radio frequency hollow cathode discharge（rf-HCD）.It is found that both the sheath oscillation heating and the secondary electron heating together play a role to maintain the rf-HCD under the simulated conditions.The mean energy of ions（N＋_2,N＋）in the negative glow region is greater than the thermal kinetic energy of the molecular gas（N2）,which is an important characteristic of rf-HCD.During the negative portion of the hollow electrode voltage cycle,electrons mainly follow pendulum movement and produce a large number of ionization collisions in the plasma region.During the positive voltage of the rf cycle,the axial electric field becomes stronger and its direction is pointing to the anode（substrate）,therefore the ions move toward the anode（substrate）via the axial electric field acceleration.Compared with dc-HCD,rf-HCD is more suitable for serving as a plasma jet nozzle at low pressure.

离子电推进器模拟中Particle-in-Cell模型的发展与应用综述

离子推进器是一种被广泛应用于各类航天器中的重要动力装置。数值模拟在揭示离子推进器的物理机理和辅助结构设计等方面发挥了重要的作用,成为离子推进器研究中必不可少的研究方法。为了推动离子电推进技术的创新发展,并确保其能够持续满足当前和未来航天任务对推进技术的要求,详细总结了Particle-in-Cell(PIC)粒子模型在离子推进器模拟中的发展和应用现状。梳理和总结了放电室、栅极系统、羽流区这三个关键区域及多区域一体化建模的全PIC和混合PIC的研究进展。针对离子推进器数值模拟研究的需求,指出了粒子模型未来的发展方向,为离子推进技术的发展提供理论支撑。