Particle In Cell Method Research Articles

A fast convergence fourth-order Vlasov model for Hall thruster ionization oscillation analyses

A 1D1V hybrid Vlasov-fluid model was developed for this study to elucidate discharge current oscillations of Hall thrusters (HTs). The Vlasov equation for ions velocity distribution function with ionization source term is solved using a constrained interpolation profile conservative semi-Lagrangian method. The fourth-order weighted essentially non-oscillatory (4th WENO) limiter is applied to the first derivative value to minimize numerical oscillation in the discharge oscillation analyses. The fourth-order accuracy is verified through a 1D scalar test case. Nonoscillatory and high-resolution features of the Vlasov model are confirmed by simulating the test cases of the Vlasov–Poisson system and by comparing the results with a particle-in-cell (PIC) method. A 1D1V HT simulation is performed through the hybrid Vlasov model. The ionization oscillation is analyzed. The oscillation amplitude and plasma density are compared with those obtained from a hybrid PIC method. The comparison indicates that the hybrid Vlasov-fluid model yields noiseless results and that the steady-state waveform is calculable in a short time period.

Rubrics for Charge Conserving Current Mapping in Finite Element Electromagnetic Particle in Cell Methods

Modeling of kinetic plasmas using electromagnetic particle in cell (EM-PIC) methods is a well worn problem, in that methods developed have been used extensively both understanding physics and exploiting them for device design. EM-PIC tools have largely relied on finite difference methods coupled with particle representations of the distribution function. Refinements to ensure consistency and charge conservation have largely been <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ad hoc</i> efforts specific to finite difference methods. Meanwhile, solution methods for field solver have grown by leaps and bounds with significant performance metrics compared to finite difference methods. Developing new EM-PIC computational schemes that leverage modern field solver technology means re-examining analysis framework necessary for self-consistent EM-PIC solution. In this article, we prescribe general rubrics for charge conservation, demonstrate how these are satisfied in conventional finite difference PIC as well as finite element PIC, and prescribe a novel charge conserving finite element PIC. Our effort leverages proper mappings on to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">de-Rham</i> sequences and lays a groundwork for understanding conditions that must be satisfied for consistency. Several numerical results demonstrate the applicability of these rubrics.

A continuum approach for multipactor using Vlasov–Poisson analysis

Multipactor is studied based on the coupled Vlasov–Poisson equation set and applied to a parallel plate geometry. This approach can be considered complementary to the particle-in-cell (PIC) methods that have provided excellent insight into multipactor behavior. However, PIC methods have limitations in terms of ‘particle noise,’ which can affect electron energy distribution functions and create re-scaling issues under conditions of charge growth. Utilizing our continuum Vlasov–Poisson approach yields susceptibility curves that are in line with reports in the literature, Spark3D PIC simulations, and experimental data. Playing to the strength of the Vlasov–Poisson approach, the differences between various multipactor orders are elucidated as they are observed in phase-space, revealing electron density dynamics without requiring increased computational resources due to electron growth. The method presented is general and can be extended to multi-input excitations and higher-dimensional phase-space.

Magnetohydrodynamic with Adaptively Embedded Particle-in-Cell model: MHD-AEPIC

Space plasma simulations have seen an increase in the use of magnetohydrodynamic (MHD) with embedded Particle-in-Cell (PIC) models. This combined MHD-EPIC algorithm simulates some regions of interest using the kinetic PIC method while employing the MHD description in the rest of the domain. The MHD models are highly efficient and their fluid descriptions are valid for most part of the computational domain, thus making large-scale global simulations feasible.However, in practical applications, the regions where the kinetic effects are critical can be changing, appearing, disappearing and moving in the computational domain. If a static PIC region is used, this requires a much larger PIC domain than actually needed, which can increase the computational cost dramatically.To address the problem, we have developed a new method that is able to dynamically change the region of the computational domain where a PIC model is applied. We have implemented this new MHD with Adaptively Embedded PIC (MHD-AEPIC) algorithm using the BATS-R-US Hall MHD and the Adaptive Mesh Particle Simulator (AMPS) as the semi-implicit PIC models. We describe the algorithm and present a test case of two merging flux ropes to demonstrate its accuracy. The implementation uses dynamic allocation/deallocation of memory and load balancing for efficient parallel execution. We evaluate the performance of MHD-AEPIC compared to MHD-EPIC and the scaling properties of the model to large number of computational cores.

Geometric electrostatic particle-in-cell algorithm on unstructured meshes

We present a geometric particle-in-cell (PIC) algorithm on unstructured meshes for studying electrostatic perturbations with frequency lower than electron gyrofrequency in magnetized plasmas. In this method, ions are treated as fully kinetic particles and electrons are described by the adiabatic response. The PIC method is derived from a discrete variational principle on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that the electric field is interpolated using Whitney 1-forms, the charge is deposited using Whitney 0-forms and the electric field is computed by discrete exterior calculus. The algorithm has been applied to study the ion Bernstein wave (IBW) in two-dimensional magnetized plasmas. The simulated dispersion relations of the IBW in a rectangular region agree well with theoretical results. In a two-dimensional circular region with fixed boundary condition, the spectrum and eigenmode structures of the IBW are obtained from simulations. We compare the energy conservation property of the geometric PIC algorithm derived from the discrete variational principle with that of previous PIC methods on unstructured meshes. The comparison shows that the new PIC algorithm significantly improves the energy conservation property.

Assessment of model parameters in MFiX particle-in-cell approach

The limitations in numerical treatment of solids-phase in conventional methods like Discrete Element Model and Two-Fluid Model have facilitated the development of alternative techniques such as Particle-In-Cell (PIC). However, a number of parameters are involved in PIC due to its empiricism. In this work, global sensitivity analysis of PIC model parameters is performed under three distinct operating regimes common in chemical engineering applications, viz. settling bed, bubbling fluidized bed and circulating fluidized bed. Simulations were performed using the PIC method in Multiphase Flow with Interphase eXchanges (MFiX) developed by National Energy Technology Laboratory (NETL). A non-intrusive uncertainty quantication (UQ) based approach is applied using Nodeworks to first construct an adequate surrogate model and then identify the most influential parameters in each case. This knowledge will aid in developing an effective design of experiments and determine optimal parameters through techniques such as deterministic or statistical calibration.

Particle-in-Cell Simulation of Phase-Shift and Faraday Rotation in Interferometry-Polarimetry Diagnostic Device of ITER Fusion Plasma

Kinetic particle-in-cell (PIC) method is a reliable technique for the laser–plasma interactions simulation based on particles’ statistical behavior. This method can be applied in simulating physical phenomena involved in optical diagnostics tools, such as the interferometry-polarimetry (IP) system. IP is one of the significant laser diagnostic methods suggested in the large tokamaks, such as international thermonuclear experimental reactor (ITER), as to improve the accuracy of density and magnetic field measurements. In this article, two separate simulations of the poloidal IP system are run by applying a 2-D/3-V XOOPIC code to observe the phase-shift and Faraday effect in a magnetized plasma with input density and poloidal magnetic field of ${n}_{e} =3.00\times 10^{19}\,\,\mathrm {m}^{-3}$ and ${B}_{p}=1.00$ T, respectively. For this purpose, a gaseous $\mathrm {(CO}_{2})$ linear-polarized far-infrared laser beam of $\boldsymbol {\lambda }_{i}=118\,\,\mu \text{m}$ wavelength, and about ${30}~{\text {W}}/ {\text {m}}^{2}$ intensity is passed through the plasma, parallel to the magnetic field. The density and poloidal magnetic field are computed from laser phase-shift and Faraday rotation theories, at ${n}_{e,\text {com}}=2.99\times {10}^{19}\,\,\text {m}^{-3}$ and ${B}_{p,\text {com}}=0.99$ T, which correspond to the input values of these simulations. The obtained results indicate the competence and potency of the PIC method in simulating role-playing phenomena in ITER diagnostic devices like the IP system.

A Set of Benchmark Tests for Validation of 3-D Particle in Cell Methods

While the particle-in-cell (PIC) method is quite mature, verification and validation of both newly developed methods and individual codes has largely focused on an idiosyncratic choice of a few test cases. Many of these test cases involve either one- or two-dimensional simulations. This is either due to availability of (quasi) analytic solutions or historical reasons. Additionally, tests often focus on investigation of particular physics problems, such as particle emission or collisions, and do not necessarily study the combined impact of the suite of algorithms necessary for a full featured PIC code. As three dimensional (3D) codes become the norm, there is a lack of benchmarks test that can establish the validity of these codes; existing papers either do not delve into the details of the numerical experiment or provide other measurable numeric metrics (such as noise) that are outcomes of the simulation. This paper seeks to provide several test cases that can be used for validation and bench-marking of particle in cell codes in 3D. We focus on examples that are collisionless, and can be run with a reasonable amount of computational power. Four test cases are presented in significant detail; these include, basic particle motion, beam expansion, adiabatic expansion of plasma, and two stream instability. All presented cases are compared either against existing analytical data or other codes. We anticipate that these cases should help fill the void of bench-marking and validation problems and help the development of new particle in cell codes.

Numerical simulation of the effect of aircraft engines impact on the nuclear power plant containment

A new version of the governing equations for concrete is proposed. A numerical algorithm for solving the governing equations proposed has been developed. The developed algorithm is implemented as part of numerical 3D-code for the finite-size particle in cell method. Using the code the simulation of an airliner engine impact on the concrete containment was performed. It was obtained that engines with the speed of 100 m/s significantly damage the containment structure, but without full penetration (the engines stop right inside the protective shield). Nonetheless the total collapse of the central part of spherical dome onto the inner metal shell of the containment is probable when the engine impacts in downward direction.

Application of sparse grid combination techniques to low temperature plasmas Particle-In-Cell simulations. II. Electron drift instability in a Hall thruster

Three-dimensional simulations of partially magnetized plasma are real challenges that actually limit the understanding of the discharge operations such as the role of kinetic instabilities using explicit Particle-In-Cell (PIC) schemes. The transition to high performance computing cannot overcome all the limits inherent to very high plasma densities and thin mesh sizes employed to avoid numerical heating. We have applied a recent method proposed in the literature [L. F. Ricketson and A. J. Cerfon, Plasma Phys. Controlled Fusion 59, 024002 (2017)] to model low temperature plasmas. This new approach, namely, the sparse grid combination technique, offers a gain in computational time by solving the problem on a reduced number of grid cells, hence allowing also the reduction of the total number of macroparticles in the system. We have modeled the example of the two-dimensional electron drift instability, which was extensively studied in the literature to explain the anomalous electron transport in a Hall thruster. Comparisons between standard and sparse grid PIC methods show an encouraging gain in the computational time with an acceptable level of error. This method offers a unique opportunity for future three-dimensional simulations of instabilities in partially magnetized low temperature plasmas.

Space charge limited current with distributed velocity of initial electrons in planar diode

The research on the space-charge limited current is of great importance for developing insight into the high current diodes. The Child-Langmuir law presents one-dimensional (1D) space-charge limited current density by considering a full electrostatic solution. Jaffe investigated the 1D space-charge limited current density modified by the uniform initial velocity of electrons emitted from the cathode by using the equation of motion for the individual electron. Luginsland et al. presented the two-dimensional (2D) space-charge limited current density by using the particle-in-cell (PIC) codes neglecting initial velocity distribution of the emitted electrons. In this Letter, the effects of initial electrons with the uniform energy distribution and thermal equilibrium distribution on the space charge limited current density are studied by using the full electromagnetic PIC method; the 2D modification factors to the space charge limited current density for the two cases are presented and discussed.

Material point method simulations using an approximate full mass matrix inverse

All material point method (MPM) codes approximate the full mass matrix with a lumped mass matrix. Because this approach causes dissipation, most MPM simulations rely on so-called FLIP methods to limit dissipation. Recent work to deal with noise caused by those FLIP methods derived the XPIC method (for extended particle in cell method) that filters null space noise from particle velocities using a projection operator. This paper shows that the XPIC projection operator is equivalent to doing grid calculations using an asymptotic expansion of the full mass matrix inverse. From that insight, we derived FMPM(k) for full mass matrix MPM of order k (where the mass matrix inverse is expanded to k terms). Compared to prior MPM algorithms, FMPM changes the methods used to update particle velocities, positions, stresses, and strains. Several examples show that FMPM is more stable and accurate, has less dissipation than XPIC, and with high enough k has less dissipation than FLIP methods. One challenge in MPM is imposing velocity conditions on the grid and those challenges are amplified in FMPM. This paper describes a new moving-wall approach that improves grid boundary conditions and is beneficial to both FMPM and prior MPM. Finally, the use of FMPM for multimaterial mode MPM, with explicit cracks, and with affine extrapolation options is each discussed.

PicFoam: An OpenFOAM based electrostatic Particle-in-Cell solver

picFoam is a fully kinetic electrostatic Particle-in-Cell (PIC) solver, including Monte Carlo Collisions (MCC), for non-equilibrium plasma research in the open-source framework of OpenFOAM. The solver’s modular design, based on the same principles used in OpenFOAM, makes it highly flexible, by allowing the user to choose different methods at run time, and extendable, by building upon templated modular classes. The implementation of the PIC method employing the finite volume method, allows it to simulate on arbitrary geometries in one to three dimensions. OpenFOAM’s barycentric particle tracking is used effectively to perform charge and field weighting from the Lagrangian particle based description to the Eulerian field description and backwards without computational expensive particle searching algorithm. picFoam also includes open and general circuit boundary models for the description of real plasma devices. Program summaryProgram Title: picFoamCPC Library link to program files:http://dx.doi.org/10.17632/bbsm8tjgjy.1Developer’s repository link:https://github.com/TFDzarm/picFoamLicensing provisions: GPLv3Programming language: C++Nature of problem: The description of equilibrium and non-equilibrium plasma with its complex collective behavior between charged species. As well as their interaction with circuits connected to the boundaries of the plasma. The applications are versatile and lie e.g. in the description of plasma phenomena like discharges in arcjet electric propulsion systems, as well as a general study of plasma phenomena.Solution method: Implementation of the Particle-in-Cell method with Monte Carlo Collisions [1] in the open-source numerical toolbox OpenFOAM [2]. By this the newly implemented solver gains access to OpenFOAM’s powerful tool sets and an easy to set up simulation case structure. Fields are solved by employing OpenFOAM’s cell-centered finite volume solvers, this allows for the simulation on arbitrary structured meshes. The leapfrog scheme [1] is employed for the integration of the particle’s equation of motion combined with relativistic and non-relativistic integration schemes for Newton’s second law of motion [3]. Monte Carlo Collision methods include models for the collision of electrons and neutral species, considered collision events are elastic collision events and inelastic excitation and ionization events. Additional models include Coulomb collisions and isotropic scattering of neutral species [4]. The solver is able to bind circuit models, include open circuit, ideal voltage and current sources, as well as general purpose circuits with an ideal voltage source and a resistance, an impedance and a capacity in series, to the plasma domain [5].Additional comments including restrictions and unusual features: The finite volume method is a relatively rarely used method for implementations of the PIC method, here by employing this method we gain the ability to use arbitrary structured meshes in one to three dimensions. OpenFOAM’s barycentric particle tracking is a novel approach for moving the particles through the mesh. With this method we are able to interpolate (weight) the charges efficiently to the mesh without using computational expensive algorithms to locate the particle’s positions beforehand.

Numerical research on ion extraction with resonant charge exchange process

The electric ion extraction, which plays an important role in productivity and abundance of product, is a critical step in laser isotope separation (LIS) technology. Several collision processes happen during the electric ion extraction, such as resonant and non-resonant charge exchange between ion and atom, the secondary ionization process and the ion-electron recombination. The resonant charge exchange process between target isotope ions and no-target background atoms is one of the major reasons of product contamination. As a result, the study of ion extraction with the consideration of resonant charge exchange process is essential. However, the resonant charge exchange process in ion extraction has not received enough attention. Besides, contradictory findings have been deduced in published studies. Therefore, it is necessary to clarify the effect of resonant charge exchange process in the electric ion extraction. In this article, the particle-in-cell (PIC) method and preprocessing hybrid-PIC method are adopted in both one- and two-dimensional numerical simulation. The preprocessing hybrid-PIC method is a calculation scheme by which accurate results can be obtained with less computational consumption. In this calculation scheme, the PIC method and hybrid-PIC method are used sequentially in different stages of ion extraction process. One-dimensional parallel type simulation cases are carried out under the circumstances of different initial plasma densities, applied voltages and background atom densities. The results show that the resonant charge exchange process happens in both shield layer and sheath layer. The ionic resonant charge exchange proportion in shield layer and sheath layer are related to the ion extraction time and average travel length in background vapor, respectively. Besides, they are proportional to the resonant charge exchange cross section and background atom density. And an empirical formula for deriving the resonant charge exchange ratio roughly is proposed. Two-dimensional simulations are carried out in four electrode configurations: parallel type, alternately biased parallel type, Π-type, and M-type. The extraction mechanisms are discussed and compared with each other. The simulation results show that M-type electrode configuration has the minimum resonant charge exchange ratio and extraction time among the configurations above. The results and conclusions provide an important reference for designing the LIS device.

Effect of particle concentration on turbulent modulation inside hydrocyclone using coupled MPPIC-VOF method

We discuss the implementation of a coupled Multi-phase particle in cell method (MPPIC) and Volume of Fluid (VOF) flow solver to simulate particles in a fluid flow field having a free surface. Using the solver developed using OpenFOAM 4.1 library, we study the flow of particles inside a highly anisotropic turbulent flow field of a hydrocyclone. The free surface from the air-core inside the hydrocyclone is resolved using the VOF method coupled with the LES turbulence model. We test multiple sub-grid scale model and find the dynamic k-equation model to have the best-predicting capability of the mean and turbulent flow field as well as the air-core shape. Using the established flow field, we study the particle flow properties and its effect on the fluid flow field using a four-way coupled description between MPPIC and VOF. Important flow properties such as the particle flow field velocity and turbulence modulation with varying particle concentration were studied. It was found that even with low solid concentration (0.3–1.5% by volume) there is a quite substantial increase in the fluctuating flow field (around 12%) and with a further increase (1.8%) we observe a decrease in the turbulent levels by approximately 6%.

A quasi-static particle-in-cell algorithm based on an azimuthal Fourier decomposition for highly efficient simulations of plasma-based acceleration: QPAD

The three-dimensional (3D) quasi-static particle-in-cell (PIC) algorithm is a very efficient method for modeling short-pulse laser or relativistic charged particle beam–plasma interactions. In this algorithm, the plasma response, i.e., plasma wave wake, to a non-evolving laser or particle beam is calculated using a set of Maxwell’s equations based on the quasi-static approximate equations that exclude radiation. The plasma fields are then used to advance the laser or beam forward using a large time step. The algorithm is many orders of magnitude faster than a 3D fully explicit relativistic electromagnetic PIC algorithm. It has been shown to be capable to accurately model the evolution of lasers and particle beams in a variety of scenarios. At the same time, an algorithm in which the fields, currents and Maxwell equations are decomposed into azimuthal harmonics has been shown to reduce the algorithmic complexity of a 3D explicit PIC algorithm to that of a 2D algorithm when the expansion is truncated while maintaining accuracy for problems with near azimuthal symmetry. This hybrid algorithm uses a PIC description in r–z and a gridless description in ϕ. We describe a novel method that combines the quasi-static and hybrid PIC methods. This algorithm expands the fields, charge and current density into azimuthal harmonics. A set of the quasi-static field equations is derived for each harmonic. The complex amplitudes of the fields are then solved using the finite difference method. The beam and plasma particles are advanced in Cartesian coordinates using the total fields. Details on how this algorithm was implemented using a similar workflow to an existing quasi-static code, QuickPIC, are presented. The new code is called QPAD for QuickPIC with Azimuthal Decomposition. Benchmarks and comparisons between a fully 3D explicit PIC code (OSIRIS), a full 3D quasi-static code (QuickPIC), and the new quasi-static PIC code with azimuthal decomposition (QPAD) are also presented.

PiCOPIC: 2.5-D PARTICLE-IN-CELL CODE, OPTIMIZED FOR SIMULATION OF BEAM-PLASMA INTERACTIONS

Original code optimized forsimulation of interactions between plasma and charged particles beams and bunches, based on particle in cell method described. The code is electromagnetic and fully relativistic with 2.5D axial symmetric geometry. Binary coulomb particle collisions are taken into account. The code is fully parallelized and designed for computer systems with shared memory. Ability to extend supported platforms to systems with distributed memory (like computer clusters or grids) is embedded into code architecture.

Analysis of influencing parameters in ion thruster plume simulation

Effects from ion thruster plume have long raised concerns. However, little work has systematically analyzed the influencing parameters in the ion thruster plume simulation. This paper analyzes LIPS 200 ion thruster plume simulations about the influencing parameters. The numerical simulations are carried out by a hybrid particle-in-cell (PIC) method and the direct simulation Monte Carlo (DSMC) method. The PIC method is employed for the plasma dynamics, and the DSMC method is used for collisions. Simulation results were compared in detail to obtain the variation of the results with the parameters. Besides, experimental data were compared to simulation results to optimize the parameters. Finally, with these researches, the optimal parameters are obtained.

A generalized particle in cell method for explicit solid dynamics

Within the framework of the material point method, a generalized particle in cell method for explicit solid dynamics is presented. In this method, the solids are discretized by finite element meshes, and these meshes are embedded in a background Eulerian grid. The external and internal forces of the solids are calculated on their respective finite element meshes and are mapped to the background grid for solving the momentum equation. Two formulations for computing these forces are presented: one using the original configuration of the solids and the other employs the current configuration. Two and three dimensional numerical examples, covering massive tensile, compressive elastic and plastic deformation, demonstrate that the former formulation performs much better for extremely large deformation problems. Compared with previous material point methods, the proposed method offers these advantages: (i) better representation of solid boundaries, (ii) seamless enforcement of Dirichlet and Neumann boundary conditions, (iii) seamless treatment of material interfaces and (iv) higher efficiency. All of these are achieved with the introduction of a finite element mesh in a supposedly meshfree method.

The geometric theory of charge conservation in particle-in-cell simulations

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.