Particle In Cell Method Research Articles

A numerical approach for nonlinear transmission line analysis with bidirectional coupling to lumped-element and particle-in-cell models

In this article, an iterative method for nonlinear transmission lines (TLs) based on the Lax-Wendroff method has been established to describe a bidirectional coupling model of nonlinear lumped-element models, nonlinear transmission line models, and particle-in-cell (PIC) models. The proposed self-consistent global circuit model is achieved through the introduction of the charge conservation equation at the upper plate as the Robin boundary condition and its coupling with the PIC method used for plasma simulation. The effectiveness of the proposed model is validated through five numerical experiments, including nonlinear transmission line coupling to lumped circuits, and the most general case of plasma-TLs-lumped elements model (LEM) coupling systems. This non-linear global circuit coupling model provides a powerful tool for simulating the complex behavior of various physical systems, including but not limited to capacitively coupled plasmas, vacuum electronic devices, and Z-pinches.

Diffusive shock acceleration in relativistic, oblique shocks

Cosmic rays are charged particles that are accelerated to relativistic speeds by astrophysical shocks. Numerical models have been successful in confirming the acceleration process for (quasi-)parallel shocks, which have the magnetic field aligned with the direction of the shock motion. However, the process is less clear when it comes to (quasi-)perpendicular shocks, where the field makes a large angle with the shock-normal. For such shocks, the angle between the magnetic field and flow ensures that only highly energetic particles can travel upstream at all, reducing the upstream current. This process is further inhibited for relativistic shocks, since the shock can become superluminal when the required particle velocity exceeds the speed of light, effectively inhibiting any upstream particle flow. In order to determine whether such shocks can accelerate particles, we use the particle-in-cell (PIC) method to determine what fraction of particles gets reflected initially at the shock. We then use this as input for a new simulation that combines the PIC method with grid-based magnetohydrodynamics to follow the acceleration (if any) of the particles over a larger time-period in a two-dimensional grid. We find that quasi-perpendicular, relativistic shocks are capable of accelerating particles through the DSA process, provided that the shock has a sufficiently high Alfvénic Mach number.

Recent development of fully kinetic particle-in-cell method and its application to fusion plasma instability study

This paper reviews the recent advancements of the algorithm and application to fusion plasma instability study of the fully kinetic Particle-in-Cell (PIC) method. The strengths and limitations of both explicit and implicit PIC methods are described and compared. Additionally, the semi-implicit PIC method and the code ECsim used in our research are introduced. Furthermore, the application of PIC methods in fusion plasma instabilities is delved into. A detailed account of the recent progress achieved in the realm of tokamak plasma simulation through fully kinetic PIC simulations is also provided. Finally the prospective future development and application of PIC methods are discussed as well.

RelSIM: A Relativistic Semi-implicit Method for Particle-in-cell Simulations

We present a novel Relativistic Semi-Implicit Method (RelSIM) for particle-in-cell (PIC) simulations of astrophysical plasmas, implemented in a code framework ready for production runs. While explicit PIC methods have gained widespread recognition in the astrophysical community as a reliable tool to simulate plasma phenomena, implicit methods have been seldom explored. This is partly due to the lack of a reliable relativistic implicit PIC formulation that is applicable to state-of-the-art simulations. We propose the RelSIM to fill this gap: our new method is relatively simple, being free of nonlinear iterations and only requiring a global linear solve of the field equations. With a set of one- and two-dimensional tests, we demonstrate that the RelSIM produces more accurate results with much smaller numerical errors in the total energy than standard explicit PIC, in particular when characteristic plasma scales (skin depth and plasma frequency) are heavily underresolved on the numerical grid. By construction, the RelSIM also performs much better than the relativistic implicit-moment method, originally proposed for semi-implicit PIC simulations in the relativistic regime. Our results are promising to conduct large-scale (in terms of duration and domain size) PIC simulations of astrophysical plasmas, potentially reaching physical regimes inaccessible by standard explicit PIC codes.

INCLUSION OF CONTACT FRICTION FOR PARTICLE-BASED SIMULATION OF SEDIMENT TRANSPORT OVER MOBILE BED

The particle based approach, including the particle resolving method, such as CFD-DEM, e.g. Drake and Calantoni (2001), Schmeeckle (2014), and the Particle-In-Cell (PIC) method, e.g. Patankar and Joseph (2001); Finn, M. Li, and Apte (2016); Y. Li et al. (2014), has become important tool for simulation of sediment transport in recent years. The latter is advantageous in the required computing resources when large amount of particles are involved and hence is more suitable for simulation of sediment transport over mobile bed. However, unlike that in CFD-DEM, special treatment is needed in the PIC method in order to prevent overlap and over-packing of sediment particles in a computational cell. Most models so far ignore the contact friction force between particles that hinders relative movement but often is essential to maintain particles in static position, especially in the seabed where the contact forces between particles are the largest. An new friction force is proposed to simulate the particle interactions, similar to the collision used in previous studies, so that the kinetic energy driving particles motion can be effectively dissipated and over-packing can be minimised under either static or dynamic stages of the particle motion.

Investigating the effect of changing parameters in the IEC device in comparative study

Kinetic simulations have been performed on an Inertial Electrostatic Confinement Fusion (IECF) device. These simulations were performed using the particle-in-cell (PIC) method to analyze the behavior of ions in an IEC device and the effects of some parameters on the Confinement Time (CT). CT is an essential factor that significantly contributes to the IEC's performance as a nuclear fusion device. Using the PIC method, the geometry of a two-grided device with variable grid radius, the number of cathode grid rings, variable pressure and different dielectric thickness for the feed stalk was simulated. In this research, with the development of previous works, the interaction of particles was simulated and compared with previous results. The simulation results are in good agreement with the previous results. In these simulations, it was found that with the increase of the dielectric thickness of the feed stalk, the electric field was weakened and as a result, the confinement time was reduced. On the other hand, with the increase of the cathode radius, the confinement time increased. Using the results, an IEC device can be designed with higher efficiency and more optimal CT for ions.

Accurate simulation of direct laser acceleration in a laser wakefield accelerator

In a laser wakefield accelerator (LWFA), an intense laser pulse excites a plasma wave that traps and accelerates electrons to relativistic energies. When the pulse overlaps the accelerated electrons, it can enhance the energy gain through direct laser acceleration (DLA) by resonantly driving the betatron oscillations of the electrons in the plasma wave. The traditional particle-in-cell (PIC) algorithm, although often the tool of choice to study DLA, contains inherent errors due to numerical dispersion and the time staggering of the electric and magnetic fields. Furthermore, conventional PIC implementations cannot reliably disentangle the fields of the plasma wave and laser pulse, which obscures interpretation of the dominant acceleration mechanism. Here, a customized field solver that reduces errors from both numerical dispersion and time staggering is used in conjunction with a field decomposition into azimuthal modes to perform PIC simulations of DLA in an LWFA. Comparisons with traditional PIC methods, model equations, and experimental data show improved accuracy with the customized solver and convergence with an order-of-magnitude fewer cells. The azimuthal-mode decomposition reveals that the most energetic electrons receive comparable energy from DLA and LWFA.

A Necessarily Incomplete Review of Electromagnetic Finite Element Particle-in-Cell Methods

While finite-difference time-domain methods have long been the foundational basis for particle-in-cell (PIC) codes, there has been increasing momentum in developing a suite of finite element-based PIC methods. The beauty of finite difference-based methods is that it is easily cast within the correct mathematical framework to represent fields, fluxes, currents, and charges. However, more importantly, these methods are cost-effective. In the intervening years, since finite difference methods were developed, the state of the art of field modeling has shifted rather dramatically. Indeed, the most popular and trusted field simulators are based on the finite element method (FEM), thanks, in large part, to the discovery of the correct function spaces for quantities of interest for Maxwell’s equations, but also the flexibility that it brings to modeling geometry with the ability to refine in space and numerical order to better capture the underlying physics. Together with time-stepping schemes that are unconditionally stable, these methods provide the framework necessary to correctly capture the nuance of the physical evolution with high fidelity. The intent of this article is to review advances in electromagnetic finite element PIC (EM-FEMPIC). We will address the progress made in fundamental challenges in such a method for charge conservation to more programmatic ones, such as computational complexity.

Modeling of very high frequency large-electrode capacitively coupled plasmas with a fully electromagnetic particle-in-cell code

Phenomena taking place in capacitively coupled plasmas with large electrodes and driven at very high frequencies are studied numerically utilizing a novel energy- and charge-conserving implicit fully electromagnetic particle-in-cell (PIC)/Monte Carlo code ECCOPIC2M. The code is verified with three model problems and is validated with results obtained in an earlier experimental work (Sawada et al 2014 Japan. J. Appl. Phys. 53 03DB01). The code shows a good agreement with the experimental data in four cases with various collisionality and absorbed power. It is demonstrated that under the considered parameters, the discharge produces radially uniform ion energy distribution functions for the ions hitting both electrodes. In contrast, ion fluxes exhibit a strong radial nonuniformity, which, however, can be different at the powered and grounded electrodes at increased pressure. It is found that this nonuniformity stems from the nonuniformity of the ionization source, which is in turn shaped by mechanisms leading to the generation of energetic electrons. The mechanisms are caused by the interaction of electrons with the surface waves of two axial electric field symmetry types with respect to the reactor midplane. The asymmetric modes dominate electron heating in the radial direction and produce energetic electrons via the relatively inefficient Ohmic heating mechanism. In the axial direction, the electron energization occurs mainly through an efficient collisionless mechanism caused by the interaction of electrons in the vicinity of an expanding sheath with the sheath motion, which is affected by the excitation of the surface modes of both types. The generation of energetic electron populations as a result of such mechanisms is shown directly. Although some aspects of the underlying physics were demonstrated in the previous literature with other models, the PIC method is advantageous for the predictive modeling due to a complex interplay between the surface mode excitations and the nonlocal physics of the corresponding type of plasma discharges operated at low pressures, which is hard to reproduce in other models realistically.

Particle-in-cell method for plasmas in the one-dimensional electrostatic limit

We discuss the particle-in-cell (PIC) method, which is one of the most widely used approaches for the kinetic description of plasmas. The positions and velocities of the charged particles take continuous values in phase space, and spatial macroscopic quantities, such as the charge density and self-generated electric fields, are calculated at discrete spatial points of a grid. We discuss the computer implementation of the PIC method for one-dimensional plasmas in the electrostatic regime and discuss a desktop application (PlasmAPP), which includes the implementation of different numerical and interpolation methods and diagnostics in a graphical user interface. To illustrate its functionality, the electron-electron two-stream instability is discussed. Readers can use PlasmAPP to explore advanced numerical methods and simulate different phenomena of interest.

Optimization of Ion Energy for the Treatment of Cancerous Tumors

Background & Objectives: For over 60 years, proton beams and heavy ions have been a powerful ion therapy method for treating cancerous tumors. It is an inherent ability of high-energy ions to discharge their energy at a certain depth with high doses, which is impossible for other beams such as X-rays, gamma rays, and electron beams. Moreover, it is an excellent way to protect healthy tissues in ion therapy.
 Materials & Methods: Scientists have used radio frequency (RF) accelerators to generate high-energy ions. However, due to the high price and large devices, laser-plasma accelerators have received much attention. This paper has studied the best conditions for producing high-energy argon ions. For this purpose, the interaction of high-intensity laser pulses with argon nanoclusters is simulated. This simulation is based on the nanoplasma model by the particle in cell method (PIC).
 Results: The simulation results show that the ion energy is dependent on the parameters of the irradiated laser and the parameters of the target cluster.
 Conclusions: The energy of the ions increases with the intensity of the laser and the duration of the laser pulse. Access to higher energy ions is also possible by changing the radius of the cluster. The density of the initial atoms of the cluster is also an important parameter that influences the energy of ions.

MPI Parallelization of Numerical Simulations for Pulsed Vacuum Arc Plasma Plumes Based on a Hybrid DSMC/PIC Algorithm

The transport characteristics of the unsteady flow field in rarefied plasma plumes is crucial for a pulsed vacuum arc in which the particle distribution varies from 1016 to 1022 m−3. The direct simulation Monte Carlo (DSMC) method and particle-in-cell (PIC) method are generally combined to study this kind of flow field. The DSMC method simulates the motion of neutral particles, while the PIC method simulates the motion of charged ions. A hybrid DSMC/PIC algorithm is investigated here to determine the unsteady axisymmetric flow characteristics of vacuum arc plasma plume expansion. Numerical simulations are found to be consistent with the experiments performed in the plasma mass and energy analyzer (EQP). The electric field is solved by Poisson’s equation, which is usually computationally expensive. The compressed sparse row (CSR) format is used to store the huge diluted matrix and PETSc library to solve Poisson’s equation through parallel calculations. Double weight factors and two timesteps under two grid sets are investigated using the hybrid DSMC/PIC algorithm. The fine PIC grid is nested in the coarse DSMC grid. Therefore, METIS is used to divide the much smaller coarse DSMC grid when dynamic load imbalances arise. Two parameters are employed to evaluate and distribute the computational load of each process. Due to the self-adaption of the dynamic-load-balancing parameters, millions of grids and more than 150 million particles are employed to predict the transport characteristics of the rarefied plasma plume. Atomic Ti and Ti2+ are injected into the small cylinders. The comparative analysis shows that the diffusion rate of Ti2+ is faster than that of atomic Ti under the electric field, especially in the z-direction. The fully diffuse reflection wall model is adopted, showing that neutral particles accumulate on the wall, while charged ions do not—due to their self-consistent electric field. The maximum acceleration ratio is about 17.94.

Sparse grid reconstructions for Particle-In-Cell methods

In this article, we propose and analyse Particle-In-Cell (PIC) methods embedding sparse grid reconstructions such as those introduced in Ricketson and Cerfon [Plasma Phys. Control. Fusion 59 (2017) 024002] and Muralikrishnan et al. [J. Comput. Phys. X 11 (2021) 100094]. The sparse grid reconstructions offer a significant improvement on the statistical error of PIC schemes as well as a reduction in the complexity of the problem providing the electric field. Main results on the convergence of the electric field interpolant and conservation properties are provided in this paper. Besides, tailored sparse grid reconstructions, in the frame of the offset combination technique, are proposed to introduce PIC methods with improved efficiency. The methods are assessed numerically and compared to existing PIC schemes thanks to classical benchmarks with remarkable prospects for three dimensional computations.

A charge-momentum-energy-conserving 1D3V hybrid Lagrangian–Eulerian method for Vlasov–Maxwell system

Here we propose hybrid Lagrangian–Eulerian (HLE) method for kinetic simulations of plasmas. The HLE method solves advection equations of shape functions unlike particle-in-cell (PIC) method. Although the PIC method cannot preserve the conservation laws of momentum and energy simultaneously, the HLE method can resolve this issue while the particle velocity is formulated by the Lagrangian description. The HLE method discretizes the advection equations and Maxwell's equations with the same scheme, so the momentum and energy exchange between charged particles and electromagnetic field exactly balances in discrete level. Numerical experiments show that linear growth rates of two-stream and Weibel instabilities are reproduced well, and global conservation is exactly preserved except for round-off errors although the velocity space is discretized by the Lagrangian description like the PIC method.

The Multipactor Simulation and Thresholds Analysis for an Impedance Transformer by an Improved 3-D PIC Method Combined With Particle Merging

In this study, a secondary electron emission (SEE) module is updated to Alioth, a 3-D electromagnetic (EM) particle-in-cell (PIC) software for simulating the multipactor. To solve the problems of low computational efficiency and overburden caused by the considerable number of secondary electrons, a particle merging scheme based on the Russian roulette is introduced. The multipactor process of an impedance transformer is investigated from the generation of seed electrons to the saturation through the 3-D PIC method combined with the particle merging scheme. The time evolution characteristics of the space charge during the multipactor are calculated to analyze the multipactor threshold of the impedance transformer. Based on the evolution of the electrons and space charge, the changes of S₁₁, and the generation of the third harmonic, the multipactor threshold is researched thoroughly. The error of the threshold results between the simulation and experiment is 0.296 dB, which shows the good agreement of Alioth with the measures.

Two‐dimensional analysis for the transition from nonlocal to local electron kinetics and its effect on the spatial uniformity of a capacitively coupled plasma

AbstractIn capacitively coupled plasmas (CCPs) operated under various pressure conditions in semiconductor manufacturing processes, the transition in discharge characteristics is observed depending on local or nonlocal electron kinetics. A two‐dimensional (2D) particle‐in‐cell (PIC) simulation is required because the one‐dimensional PIC method cannot capture the effect of device structures and because fluid models cannot treat the self‐consistent change in the energy distributions. Using a 2D PIC simulation parallelized with a graphics processing unit, we investigated the transition of electron kinetics for the variation of the driving voltage, gas pressure, the gap distance between upper and lower electrodes, and the sidewall condition. Moreover, the mechanism for electron energy relaxation in a CCP is elucidated to develop a method to improve the process uniformity.

Diffusive Shock Acceleration at Oblique High Mach Number Shocks

Abstract The current paradigm of cosmic-ray (CR) origin states that the greater part of galactic CRs is produced by supernova remnants. The interaction of supernova ejecta with the interstellar medium after a supernova's explosions results in shocks responsible for CR acceleration via diffusive shock acceleration (DSA). We use particle-in-cell (PIC) simulations and a combined PIC-magnetohydrodynamic (PIC-MHD) technique to investigate whether DSA can occur in oblique high Mach number shocks. Using the PIC method, we follow the formation of the shock and determine the fraction of the particles that gets involved in DSA. With this result, we use PIC-MHD simulations to model the large-scale structure of the plasma and the magnetic field surrounding the shock and find out whether or not the reflected particles can generate upstream turbulence and trigger DSA. We find that the feasibility of this process in oblique shocks depends strongly on the Alfvénic Mach number, and the DSA process is more likely to be triggered at high Mach number shocks.

An Energy Conserving Vlasov Solver That Tolerates Coarse Velocity Space Resolutions: Simulation of MMS Reconnection Events

AbstractVlasov solvers that operate on a phase‐space grid are highly accurate but also numerically demanding. Coarse velocity space resolutions, which are largely unproblematic in particle‐in‐cell (PIC) simulations, can lead to numerical heating or oscillations in continuum Vlasov methods. To address this issue, we present a new dual Vlasov solver which is based on an established positivity preserving advection scheme for the update of the distribution function and an energy conserving partial differential equation solver for the kinetic update of mean velocity and temperature. The solvers work together via moment fitting during which the maximum entropy part of the distribution function is replaced by the solution from the partial differential equation solver. This numerical scheme makes continuum Vlasov methods competitive with PIC methods concerning computational cost and enables us to model large scale reconnection in Earth's magnetosphere with a fully kinetic continuum method. The simulation results agree well with measurements by the MMS spacecraft.

Accelerated steady-state electrostatic particle-in-cell simulation of Langmuir probes

First-principles particle-in-cell (PIC) simulation is a powerful tool for understanding plasma behavior, but this power often comes at great computational expense. Artificially reducing the ion/electron mass ratio is a time-honored practice to reduce simulation costs. Usually, this is a severe approximation. However, for steady-state collisionless, electrostatic (Vlasov–Poisson) systems, the solution with reduced mass ratio can be scaled to the solution for the real mass ratio, with no approximation. This “scaled mass” method, which works with already-existing PIC codes, can reduce the computation time for a large class of electrostatic PIC simulations by the square root of the mass ratio. The particle distributions of the resulting steady state must be trivially rescaled to yield the true distributions, but the self-consistent electrostatic field is independent of the mass ratio. This method is equivalent to “numerical timestepping,” an approach that evolves electron and ion populations with different time steps. Numerical timestepping can be viewed as a special case of the speed-limited PIC (SLPIC) method, which is not restricted to steady-state phenomena. Although the scaled-mass approach is simplest, numerical timestepping and SLPIC more easily generalize to include other effects, such as collisions. The equivalence of these new approaches is demonstrated by applying them to simulate a cylindrical Langmuir probe in electron–argon plasma, speeding up simulation by two orders of magnitude. Methods such as SLPIC can therefore play an invaluable role in interpreting probe measurements by including geometric effects, collisions, secondary emission, and non-Maxwellian distributions.

Ion Current Screening Modeling of the Ion–Weibel Instability

Abstract Ion–Weibel instability (IWI) is an important mechanism of generating a magnetic field in supernova remnants; it plays a key role in the generation of high-energy cosmic rays. Computational efficiency has been a bottleneck in numerical exploration of the large-scale evolution of IWI. Here I report a new hybrid particle-in-cell (PIC) method that can quickly simulate IWI. The method is based on a new model that describes the relation of the ion current and its magnetic field under the electron screening. The new method’s computational efficiency is nearly two orders of magnitude higher than that of the PIC method. This method is suitable for the full-scale simulation of the IWI in laser-plasma experiments and supernova remnants.