Particle-in-cell Schemes Research Articles

Acceleration of particle-in-cell simulations using sparse grid algorithms. I. Application to dual frequency capacitive discharges

The use of sparse particle-in-cell (PIC) algorithms to accelerate the standard explicit PIC scheme has recently been successfully applied in the context of single-frequency capacitively coupled plasma discharges [Garrigues et al., J. Appl. Phys. 129, 153303 (2021)]. We have extended the sparse PIC scheme to model dual-frequency capacitive discharges. Comparisons between standard and sparse PIC algorithms show that the plasma properties as well as the electron and ion distribution functions can be retrieved with a maximum error of 2%. This work opens the interest of using the sparse PIC algorithm to perform 2D and 3D simulations under real operating conditions of capacitively coupled plasma discharges.

Dynamic mode decomposition for data-driven analysis and reduced-order modeling of E × B plasmas: I. Extraction of spatiotemporally coherent patterns

The advent of data-driven/machine-learning based methods and the increase in data available from high-fidelity simulations and experiments has opened new pathways toward realizing reduced-order models for plasma systems that can aid in explaining the complex, multi-dimensional phenomena and enable forecasting and prediction of the systems’ behavior. In this two-part article, we evaluate the utility and the generalizability of the dynamic mode decomposition (DMD) algorithm for data-driven analysis and reduced-order modeling of plasma dynamics in cross-field E × B configurations. The DMD algorithm is an interpretable data-driven method that finds a best-fit linear model describing the time evolution of spatiotemporally coherent structures (patterns) in data. We have applied the DMD to extensive high-fidelity datasets generated using a particle-in-cell (PIC) code based on the cost-efficient reduced-order PIC scheme. In this part, we first provide an overview of the concept of DMD and its underpinning proper orthogonal and singular value decomposition methods. Two of the main DMD variants are next introduced. We then present and discuss the results of the DMD application in terms of the identification and extraction of the dominant spatiotemporal modes from high-fidelity data over a range of simulation conditions. We demonstrate that the DMD variant based on variable projection optimization (OPT-DMD) outperforms the basic DMD method in identification of the modes underlying the data, leading to notably more reliable reconstruction of the ground-truth. Furthermore, we show in multiple test cases that the discrete frequency spectrum of OPT-DMD-extracted modes is consistent with the temporal spectrum from the fast Fourier transform of the data. This observation implies that the OPT-DMD augments the conventional spectral analyses by being able to uniquely reveal the spatial structure of the dominant modes in the frequency spectra, thus, yielding more accessible, comprehensive information on the spatiotemporal characteristics of the plasma phenomena.

Reduced-order particle-in-cell simulations of a high-power magnetically shielded Hall thruster

High-power magnetically shielded Hall thrusters have emerged in recent years to meet the needs of the next-generation on-orbit servicing and exploration missions. Even though a few such thrusters are currently undergoing their late-stage development and qualification campaigns, many unanswered questions yet exist concerning the behavior and evolution of the plasma in these large-size thrusters that feature an unconventional magnetic field topology. Noting the complex, multi-dimensional nature of plasma processes in Hall thrusters, high-fidelity particle-in-cell (PIC) simulations are optimal tools to study the intricate plasma behavior. Nonetheless, the significant computational cost of traditional multi-dimensional PIC schemes renders simulating the high-power thrusters without any physics-altering speed-up factors unfeasible. The novel reduced-order ‘quasi-2D’ PIC scheme enables a significant reduction in the computational cost requirement of the PIC simulations. Thus, in this article, we demonstrate the applicability of the reduced-order PIC for a cost-efficient, self-consistent study of the physics in high-power Hall thrusters by performing simulations of a 20 kW-class magnetically shielded Hall thruster along the axial-azimuthal and radial-azimuthal coordinates. The axial-azimuthal quasi-2D simulations are performed for three operating conditions in a rather simplified representation of the thruster’s inherently 3D configuration. Nevertheless, we have resolved self-consistently an unprecedented 650 µs of the discharge evolution without any ad-hoc electron mobility model, capturing several breathing cycles and approximating the experimental performance parameters with an accuracy of 70%–80% across the operating conditions. The radial-azimuthal simulations, carried out at three cross-sections corresponding to different axial locations within the discharge channel, have casted further light on the evolution of the azimuthal instabilities and the resulting variations in the electrons’ cross-field mobility and the plasma-wall interactions. Particularly, we observed the development of a long-wavelength, relatively low-frequency wave mode near the exit plane of the thruster’s channel that induces a notable electron transport and a significant ion heating.

An asymptotic-preserving and energy-conserving particle-in-cell method for Vlasov–Maxwell equations

In this paper, we develop an asymptotic-preserving and energy-conserving (APEC) Particle-In-Cell (PIC) algorithm for the Vlasov–Maxwell system. This algorithm not only guarantees that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model but also preserves Gauss’s law and energy conservation at the same time; therefore, it is promising to provide stable simulations of complex plasma systems even in the quasi-neutral regime. The key ingredients for achieving these properties include the generalization of Ohm’s law for electric fields such that asymptotic-preserving discretization can be achieved and a proper decomposition of the effects of the electromagnetic fields such that a Lagrange multiplier method can be appropriately employed for correcting the kinetic energy. We investigate the performance of the APEC method with three benchmark tests in one dimension, including the linear Landau damping, the bump-on-tail problem, and the two-stream instability. Detailed comparisons are conducted by including the results from the classical explicit leapfrog and the previously developed asymptotic-preserving PIC schemes. Our numerical experiments show that the proposed APEC scheme can give accurate and stable simulations of both kinetic and quasi-neutral regimes, demonstrating the attractive properties of the method across scales.

Concept of the generalized reduced-order particle-in-cell scheme and verification in an axial-azimuthal Hall thruster configuration

The reduced-order particle-in-cell (PIC) scheme is a novel modeling approach that enables computationally efficient electrostatic kinetic simulations of plasma. In our previous publications, we demonstrated that a proof-of-concept implementation of this novel PIC scheme resolves the multi-dimensional plasma processes and their interactions in a Hall thruster in a manner close to traditional electrostatic PIC codes. In this work, we extend our efforts on this topic and present a mathematically mature formulation for the dimensionality reduction of Poisson’s equation in the Vlasov–Poisson system, which enables the generalized reduced-order ‘quasi-multi-dimensional’ PIC scheme. The applicability of the dimensionality-reduction approach to solve general 2D Poisson problems is numerically verified. Next, we present several reduced-order quasi-2D (Q2D) simulations of a well-defined axial-azimuthal simulation case from the literature using approximation orders of the 2D problem whose computational costs are 2%–15% of a full-2D simulation. It is shown that these reduced-order simulations allow us to recover the same characteristics, behaviors and effects reported in the literature regarding the azimuthal instabilities in Hall thrusters. Moreover, in terms of the time-averaged plasma properties, it was found that, when increasing the approximation order, the error associated with the Q2D simulations’ predictions decreases from 15% to 4% for the electric field and from 20% to 2% for the ion number density. We have additionally discussed a series of sensitivity analysis results, including the influence of the initial number of macroparticles per cell on the predictions of the Q2D simulations. According to the detailed results and analyses presented, we conclude that the generalized reduced-order PIC scheme serves as a rigorous foundation for eventual cost-effective and comprehensive three-dimensional kinetic studies of the physics in Hall thrusters and similar electrostatic plasma technologies.

Verification of the generalized reduced-order particle-in-cell scheme in a radial–azimuthal E × B plasma configuration

In this article, we present an in-depth verification of the generalized electrostatic reduced-order particle-in-cell (PIC) scheme in a cross electric and magnetic field configuration representative of a radial–azimuthal section of a Hall thruster. The setup of the simulations follows a well-established benchmark case. The main purpose of this effort is to demonstrate that our novel PIC scheme can reliably resolve the complex two-dimensional dynamics and interactions of the plasma instabilities in the radial–azimuthal coordinates of a Hall thruster at a fraction of the computational cost compared to full-2D PIC codes. To this end, we first present the benchmarking of our newly developed full-2D PIC code. Next, we provide an overview of the reduced-order PIC scheme and the resulting “quasi-2D” code, specifying that the degree of order reduction in the quasi-2D PIC is defined in terms of the number of “regions” along the simulation’s directions used to divide the computational domain. We compare the predictions of the quasi-2D simulation in various approximation degrees of the 2D problem against our full-2D simulation results. We show that, with an increase in the number of regions in the Q2D simulations, the quasi-2D results converge to the 2D ones. Nonetheless, we also highlight that a quasi-2D simulation that provides a factor of 5 reduction in the computational cost resolves the underlying physical processes in an almost indistinguishable manner with respect to the full-2D simulation and incurs an L2-norm error of only about 2% in the ion number density and below 1% in the electron temperature.

Resolving multi-dimensional plasma phenomena in Hall thrusters using the reduced-order particle-in-cell scheme

Plasma in Hall thrusters exhibits a complex behavior, characterized by the interplay between various dominant processes in each of the thruster’s coordinates. The emergence of high-power Hall thrusters in the recent years and the design modifications intended to extend the lifetime of these devices have further amplified the three-dimensional nature of the plasma behavior. In this regard, the novel reduced-order particle-in-cell (PIC) scheme provides the possibility to resolve the multi-dimensional interactions in a Hall thruster at a computational cost up to two orders of magnitude lower than current multi-dimensional PIC simulations. To demonstrate this point, we present in this article the results from a series of pseudo-two-dimensional simulations we performed in three configurations: axial-azimuthal, azimuthal-radial, and axial-radial. We show that, in each configuration, the pseudo-2D PIC scheme provides a significantly improved picture of the involved physics compared to a one-dimensional PIC simulation and captures self-consistently the coupling between the plasma processes in different directions, notably similar to the observations from full-2D kinetic simulations.

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 hybrid nodal-staggered pseudo-spectral electromagnetic particle-in-cell method with finite-order centering

Electromagnetic particle-in-cell (PIC) codes are widely used to perform computer simulations of a variety of physical systems, including fusion plasmas, astrophysical plasmas, plasma wakefield particle accelerators, and secondary photon sources driven by ultra-intense lasers. In a PIC code, Maxwell's equations are solved on a grid with a numerical method of choice. This article focuses on pseudo-spectral analytical time-domain (PSATD) algorithms and presents a novel hybrid PSATD PIC scheme that combines the respective advantages of standard nodal and staggered methods. The novelty of the hybrid scheme consists in using finite-order centering of grid quantities between nodal and staggered grids, in order to combine the solution of Maxwell's equations on a staggered grid with the deposition of charges and currents and the gathering of electromagnetic forces on a nodal grid. The correctness and performance of the novel hybrid scheme are assessed by means of numerical tests that employ different classes of PSATD equations in a variety of physical scenarios, ranging from the modeling of electron-positron pair creation in vacuum to the simulation of laser-driven and particle beam-driven plasma wakefield acceleration. It is shown that the novel hybrid scheme offers significant numerical and computational advantages, compared to purely nodal or staggered methods, for all the test cases presented.

Enhancing one-dimensional particle-in-cell simulations to self-consistently resolve instability-induced electron transport in Hall thrusters

The advent of high-power Hall thrusters and the increasing interest toward their use as a primary propulsion system for various missions have given a new boost to the efforts aiming at self-consistent predictive modeling of this thruster technology. In this article, we present a novel approach, which allows enhancing the predictive capability of one-dimensional particle-in-cell (PIC) simulations to self-consistently capture the wave-induced electron transport due to the azimuthal instabilities in Hall thrusters. The so-called “pseudo-2D” PIC scheme resulting from this approach is extensively tested in several operating conditions. The results are compared against a well-established 2D3V axial–azimuthal reference case in terms of the axial profiles of the time-averaged plasma properties, the azimuthal electric field fluctuations and their dispersion features, and the contributions of the force terms in the electron azimuthal momentum equation to the cross-field mobility. We have demonstrated that the pseudo-2D PIC provides a prediction of the above aspects that compares very closely in almost all conditions with those from the full-2D simulation. In addition, the sensitivity of the pseudo-2D simulation results to the numerical parameters associated with our approach is assessed in detail. The outcomes of these analyses have casted light on the next steps to further improve the approach.

Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes

We propose a sparse grid-based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto sparse grids we reduce the high-frequency particle noise. Thus, we exploit the ability of sparse grids to act as a multidimensional low-pass filter in our approach. Thanks to the truncated combination technique [1–3], we can reduce the larger grid-based error of the standard sparse grid approach for non-aligned and non-smooth functions. The truncated approach also provides a natural framework for minimizing the sum of grid-based and particle-based errors in the charge density. We show that our approach is, in fact, a filtering perspective for the noise reduction obtained with the sparse PIC schemes first introduced in [4]. This enables us to propose a heuristic based on the formal error analysis in [4] for selecting the optimal truncation parameter that minimizes the total error in charge density at each time step. Hence, unlike the physical and Fourier domain filters typically used in PIC codes for noise reduction, our approach automatically adapts to the mesh size, number of particles per cell, smoothness of the density profile and the initial sampling technique. It can also be easily integrated into high performance large-scale PIC code bases, because we only use sparse grids for filtering the charge density. All other operations remain on the regular grid, as in typical PIC codes. We demonstrate the efficiency and performance of our approach with two test cases: the diocotron instability in two dimensions and the three-dimensional electron dynamics in a Penning trap. Our run-time performance studies indicate that our approach can provide significant speedup and memory reduction to PIC simulations for achieving comparable accuracy in the charge density.

A parallel particle-in-cell code for spacecraft charging problems

This paper reports the recent development of a full-scale particle-in-cell (PIC) simulation package highlighting an efficient electro-statics (ES)-PIC implementation for spacecraft charging problems. Numerical simulations are crucial in studying plasma flows because analytical solutions are rare, and experiments are expensive. There are many types of plasma flow that need various numerical methods for efficient and accurate simulations; as such, how to implement and organize those different methods into one comprehensive simulation package is challenging. This work adopted several modern software design patterns and developed a versatile package that includes various PIC schemes. This package has an open architecture, clean interfaces with both serial and parallel simulation capabilities. Two benchmark test cases are included to demonstrate the capabilities of this package. Then two plasma flows around a positively charged probe are simulated, and the results are discussed. The simulation results are consistent with past simulation results, and new insights are obtained. This work can lead to the development and organization of more sophisticated plasma simulation solvers in the future.

Mitigation of the self-force effect in unstructured PIC codes using generalized moving least squares

Electrostatic particle-in-cell (PIC) methods use point clouds to represent a collection of charged particles and a mesh to compute the electric field generated by these particles. Such methods involve a data transfer of charge from the point cloud to the mesh and another transfer of the electric field from the mesh to the point cloud. The approximation errors introduced during these data transfers manifest as an artificial self-force acting on particles, causing a loss of momentum conservation. There exist PIC schemes on uniform Cartesian grids that provably avoid this effect. However, these schemes rely on a translation invariance property of a discrete Green’s function and cannot be extended to arbitrary unstructured grids. We present a new meshless data transfer approach that uses generalized moving least squares (GMLS) to reconstruct the electric field at particles from local mesh data on non-uniform meshes. The approach enriches the GMLS approximation space with singular functions to recover the singularity introduced at particles. This reduces the strength of the self-force by several orders of magnitude. We present results reconstructing both from linear finite elements and from Raviart–Thomas elements, demonstrating that the approach is generally applicable to PIC codes having degrees of freedom of arbitrary type.

Multiple-scale kinetic simulations with the energy conserving semi-implicit particle in cell method

The recently developed energy conserving semi-implicit method (ECsim) for particle-in-cell (PIC) simulation is applied to multiple-scale problems where the electron-scale physics needs to be only partially retained and the interest is on the macroscopic or ion-scale processes. Unlike hybrid methods, the ECsim is capable of providing kinetic electron information, such as wave–electron interaction (Landau damping or cyclotron resonance) and non-Maxwellian electron velocity distributions. However, like hybrid methods, the ECsim does not need to resolve all electron scales, allowing time steps and grid spacings orders of magnitude larger than in explicit PIC schemes. The additional advantage of the ECsim is that the stability at large scale is obtained while conserving energy exactly. Three examples are presented: ion acoustic waves, electron acoustic instability and reconnection processes.

Sparse grid techniques for particle-in-cell schemes

We propose the use of sparse grids to accelerate particle-in-cell (PIC) schemes. By using the so-called ‘combination technique’ from the sparse grids literature, we are able to dramatically increase the size of the spatial cells in multi-dimensional PIC schemes while paying only a slight penalty in grid-based error. The resulting increase in cell size allows us to reduce the statistical noise in the simulation without increasing total particle number. We present initial proof-of-principle results from test cases in two and three dimensions that demonstrate the new scheme’s efficiency, both in terms of computation time and memory usage.

Asymptotically Stable Particle-In-Cell Methods for the Vlasov--Poisson System with a Strong External Magnetic Field

This paper deals with the numerical resolution of the Vlasov--Poisson system with a strong external magnetic field by particle-in-cell (PIC) methods. In this regime, classical PIC methods are subject to stability constraints on the time and space steps related to the small Larmor radius and plasma frequency. Here we propose an asymptotic-preserving PIC scheme which is not subjected to these limitations. Our approach is based on first- and higher-order semi-implicit numerical schemes already validated on dissipative systems [S. Boscarino, F. Filbet, and G. Russo, J. Sci. Comput., 2016, doi:10.1007/s10915-016-0168-y]. Additionally, when the magnitude of the external magnetic field becomes large, this method provides a consistent PIC discretization of the guiding-center equation, that is, an incompressible Euler equation in vorticity form. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts.

Finite time step and spatial grid effects in δf simulation of warm plasmas

This paper introduces a technique for analyzing time integration methods used with the particle weight equations in δf method particle-in-cell (PIC) schemes. The analysis applies to the simulation of warm, uniform, periodic or infinite plasmas in the linear regime and considers the collective behavior similar to the analysis performed by Langdon for full-f PIC schemes [1] , [2] . We perform both a time integration analysis and spatial grid analysis for a kinetic ion, adiabatic electron model of ion acoustic waves. An implicit time integration scheme is studied in detail for δf simulations using our weight equation analysis and for full- f simulations using the method of Langdon. It is found that the δf method exhibits a CFL-like stability condition for low temperature ions, which is independent of the parameter characterizing the implicitness of the scheme. The accuracy of the real frequency and damping rate due to the discrete time and spatial schemes is also derived using a perturbative method. The theoretical analysis of numerical error presented here may be useful for the verification of simulations and for providing intuition for the design of new implicit time integration schemes for the δf method, as well as understanding differences between δf and full- f approaches to plasma simulation.

Efficient Parallelization of a Three-Dimensional High-Order Particle-in-Cell Method for the Simulation of a 170 GHz Gyrotron Resonator

We present the simulation of the transient excitation and evolution of a TE34, 19 mode in the resonant cavity of the 170 GHz gyrotron. This gyrotron is planned for electron cyclotron resonance heating in the ITER tokamak fusion reactor. The numerical computation of a state-of-the-art gyrotron resonant cavity with a transient 3-D full wave particle-in-cell (PIC) method is a computationally demanding task. It was enabled by a highly scalable PIC scheme. To allow the numerical simulation of the high-frequency electromagnetic waves, we use a high-order discontinuous Galerkin method.

Modeling and simulations of high-power microwave devices using the CHIPIC code

AbstractThe CHIPIC code, a fully electromagnetic particle-in-cell (PIC) code for modeling and simulations of high-power microwave (HPM) devices, is introduced in this paper. It consists of a two-dimensional (2D) code and a three-dimensional (3D) code. The 2D code can model and simulate HPM devices with symmetric structure on 2D Cartesian, cylindrical and polar grids, while the 3D code can model and simulate HPM devices on 3D Cartesian and cylindrical grids. The fields are calculated using the finite-difference time-domain scheme, and the particles are described by the PIC scheme. Various types of boundary conditions have also been implemented for different kinds of applications. In addition, the 3D code is specifically designed for high-performance modeling and computing. It uses the message passing interface and the open specifications for multiprocessing (OpenMP) for parallelization. Its parallel design ensures that it is capable of efficiently executing on a variety of architectures. In order to allow efficient use of parallel architectures, it provides automated partitioning and dynamic load balancing. Even though this code is still in development, it has successfully simulated various real-world HPM experimental devices. Simulation results on some typical HPM devices by using the CHIPIC code are given, which agree well with those obtained from some well-known PIC codes. Direction for future work is also presented.

LEADS-DC: A computer code for intense dc beam nonlinear transport simulation

An intense dc beam nonlinear transport code has been developed. The code is written in Visual FORTRAN 6.6 and has ∼13000 lines. The particle distribution in the transverse cross section is uniform or Gaussian. The space charge forces are calculated by the PIC (particle in cell) scheme, and the effects of the applied fields on the particle motion are calculated with the Lie algebraic method through the third order approximation. Obviously, the solutions to the equations of particle motion are self-consistent. The results obtained from the theoretical analysis have been put in the computer code. Many optical beam elements are contained in the code. So, the code can simulate the intense dc particle motions in the beam transport lines, high voltage dc accelerators and ion implanters.