Implementation of a particle-in-cell method for the energy solver in 3D spherical geodynamic modeling

The thermal evolution of the Earth’s interior and its dynamic effects are the focus of Earth sciences.However,the commonly adopted grid-based temperature solver is usually prone to numerical oscillations,especially in the presence of sharp thermal gradients,such as when modeling subducting slabs and rising plumes.This phenomenon prohibits the correct representation of thermal evolution and may cause incorrect implications of geodynamic processes.After examining several approaches for removing these numerical oscillations,we show that the Lagrangian method provides an ideal way to solve this problem.In this study,we propose a particle-in-cell method as a strategy for improving the solution to the energy equation and demonstrate its effectiveness in both one-dimensional and three-dimensional thermal problems,as well as in a global spherical simulation with data assimilation.We have implemented this method in the open-source finite-element code CitcomS,which features a spherical coordinate system,distributed memory parallel computing,and data assimilation algorithms.

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.

Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Recent development of structure-preserving geometric particle-in-cell （PIC） algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required for these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential （or integro-differential） equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to harness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modem mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure- preserving geometric PIC algorithms in comparison with the conventional PIC methods.

Particle-in-cell simulations of EUV-induced hydrogen plasma in the vicinity of a reflective mirror

Particle-In-Cell(PIC)simulations were performed in this work to study the dynamics of the EUVinduced hydrogen plasma.The Monte-Carlo Collision(MCC)model was employed to deal with the collisions between charged particles and background gas molecules.The dynamic evolution of the plasma sheath,as well as the flux and energy distribution of ions impacting on the mirror surface,was discussed.It was found that the emission of secondary electrons under the EUV irradiation on the ruthenium mirror coating creates a positively charged wall and then prevents the ions from impacting on the mirror and therefore changes the flux and energy distribution of ions reaching the mirror.Furthermore,gas pressure has a notable effect on the plasma sheath and the characteristics of the ions impinging on the mirrors.With greater gas pressure,the sheath potential decreases more rapidly.The flux of ions received by the mirror grows approximately linearly and at the same time the energy corresponding to the peak flux decreases slightly.However,the EUV source intensity barely changes the sheath potential,and its influence on the ion impact is mainly limited to the approximate linear increase in ion flux.

Exploration of microscopic physical processes of Z-pinch by a modified electrostatic direct implicit particle-in-cell algorithm

For investigating efficiently the stagnation kinetic-process of Z-pinch,we develop a novel modified electrostatic implicit particle-in-cell algorithm in radial one-dimension for Z-pinch simulation in which a small-angle cumulative binary collision algorithm is used.In our algorithm,the electric field in z-direction is solved by a parallel electrode-plate model,the azimuthal magnetic field is obtained by Ampere’s law,and the term for charged particle gyromotion is approximated by the cross product of the averaged velocity and magnetic field.In simulation results of 2 MA deuterium plasma shell Zpinch,the mass-center implosion trajectory agrees generally with that obtained by one-dimensional MHD simulation,and the plasma current also closely aligns with the external current.The phase space diagrams and radial-velocity probability distributions of ions and electrons are obtained.The main kinetic characteristic of electron motion is thermal equilibrium and oscillation,which should be oscillated around the ions,while that of ion motion is implosion inwards.In the region of stagnation radius,the radial-velocity probability distribution of ions transits from the non-equilibrium to equilibrium state with the current increasing,while of electrons is basically the equilibrium state.When the initial ion density and current peak are not high enough,the ions may not reach their thermal equilibrium state through collisions even in its stagnation phase.

Particle residence time distribution and axial dispersion coefficient in a pressurized circulating fluidized bed by using multiphase particle-in-cell simulation

The particle residence time distribution(RTD)and axial dispersion coefficient are key parameters for the design and operation of a pressurized circulating fluidized bed(PCFB).In this study,the effects of pressure(0.1-0.6 MPa),fluidizing gas velocity(2-7 m·s^(-1)),and solid circulation rate(10-90 kg·m^(-2)·s^(-1))on particle RTD and axial dispersion coefficient in a PCFB are numerically investigated based on the multiphase particle-in-cell(MP-PIC)method.The details of the gas-solid flow behaviors of PCFB are revealed.Based on the gas-solid flow pattern,the particles tend to move more orderly under elevated pressures.With an increase in either fluidizing gas velocity or solid circulation rate,the mean residence time of particles decreases while the axial dispersion coefficient increases.With an increase in pressure,the core-annulus flow is strengthened,which leads to a wider shape of the particle RTD curve and a larger mean particle residence time.The back-mixing of particles increases with increasing pressure,resulting in an increase in the axial dispersion coefficient.

Insight Into the Separation-of-Variable Methods for the Closed-Form Solutions of Free Vibration of Rectangular Thin Plates

The separation-of-variable(SOV)methods,such as the improved SOV method,the variational SOV method,and the extended SOV method,have been proposed by the present authors and coworkers to obtain the closed-form analytical solutions for free vibration and eigenbuckling of rectangular plates and circular cylindrical shells.By taking the free vibration of rectangular thin plates as an example,this work presents the theoretical framework of the SOV methods in an instructive way,and the bisection–based solution procedures for a group of nonlinear eigenvalue equations.Besides,the explicit equations of nodal lines of the SOV methods are presented,and the relations of nodal line patterns and frequency orders are investigated.It is concluded that the highly accurate SOV methods have the same accuracy for all frequencies,the mode shapes about repeated frequencies can also be precisely captured,and the SOV methods do not have the problem of missing roots as well.

Structural Modal Parameter Recognition and Related Damage Identification Methods under Environmental Excitations: A Review

Modal parameters can accurately characterize the structural dynamic properties and assess the physical state of the structure.Therefore,it is particularly significant to identify the structural modal parameters according to the monitoring data information in the structural health monitoring(SHM)system,so as to provide a scientific basis for structural damage identification and dynamic model modification.In view of this,this paper reviews methods for identifying structural modal parameters under environmental excitation and briefly describes how to identify structural damages based on the derived modal parameters.The paper primarily introduces data-driven modal parameter recognition methods(e.g.,time-domain,frequency-domain,and time-frequency-domain methods,etc.),briefly describes damage identification methods based on the variations of modal parameters(e.g.,natural frequency,modal shapes,and curvature modal shapes,etc.)and modal validation methods(e.g.,Stability Diagram and Modal Assurance Criterion,etc.).The current status of the application of artificial intelligence(AI)methods in the direction of modal parameter recognition and damage identification is further discussed.Based on the pre-vious analysis,the main development trends of structural modal parameter recognition and damage identification methods are given to provide scientific references for the optimized design and functional upgrading of SHM systems.

5千瓦霍尔推力器磁场优化及Particle-in-Cell性能仿真

为了提升兰州空间技术物理研究所研制的一种5 k W霍尔推力器LHT—140的性能,采用ANSOFT软件进行了磁场优化设计,将磁场径向分量的轴向梯度提高了47%,相同励磁激励下放电通道中的磁场强度提高了38%。建立了一个R-Z平面内的二维particle-in-cell(PIC)等离子体模型,对磁场优化后推力器的性能进行了仿真分析,预估其在300~800 V放电电压、10~15 mg/s流率范围内,推力提升了9.6%~22.1%,效率提升了8.7%~19.3%。性能验证试验表明磁场优化后在相同放电电压与流率下,性能提升的测试值高于仿真值。

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.

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.

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.

Particle-in-cell simulation for the effect of magnetic cusp on discharge characteristics in a cylindrical Hall thruster

The cylindrical Hall thruster has the good prospect of serving as a miniaturized electric propulsion device.A 2 D-3 V particle-in-cell plus Monte Carlo（PIC-MCC） method is used to study the effect of the magnetic cusp on discharge characteristics of a cylindrical Hall thruster.The simulation results show that the main ionization region and the main potential drop of the thruster are located at the upstream of the discharge channel.When the magnetic cusp moves toward the anode side,the main ionization region is compressed and weakened,moving upstream correspondingly.The ionization near the cusp is enhanced,and the interaction between the plasma and the wall increases.The simulation results suggest that the magnetic cusp should be located near the channel exit.

Particle-in-cell simulation for different magnetic mirror effects on the plasma distribution in a cusped field thruster

Magnetic mirror used as an efficient tool to confine plasma has been widely adopted in many different areas especially in recent cusped field thrusters. In order to check the influence of magnetic mirror effect on the plasma distribution in a cusped field thruster, three different radii of the discharge channel(6 mm, 4 mm, and 2 mm) in a cusped field thruster are investigated by using Particle-in-Cell Plus Monte Carlo(PIC-MCC) simulated method, under the condition of a fixed axial length of the discharge channel and the same operating parameters. It is found that magnetic cusps inside the small radius discharge channel cannot confine electrons very well. Thus, the electric field is hard to establish. With the reduction of the discharge channel’s diameter, more electrons will escape from cusps to the centerline area near the anode due to a lower magnetic mirror ratio. Meanwhile, the leak width of the cusped magnetic field will increase at the cusp. By increasing the magnetic field strength in a small radius model of a cusped field thruster, the negative effect caused by the weak magnetic mirror effect can be partially compensated. Therefore, according to engineering design, the increase of magnetic field strength can contribute to obtaining a good performance, when the radial distance between the magnets and the inner surface of the discharge channel is relatively big.

Numerical Approach of Interactions of Proton Beams and Dense Plasmas with Quantum-Hydrodynamic/Particle-in-Cell Model

A one dimensional quantum-hydrodynamic/particle-in-cell （QHD/PIC） model is used to study the interaction process of an intense proton beam （injection density of 1017 cm-3） with a dense plasma （initial density of -10^21 cm^-3）, with the PIC method for simulating the beam particle dynamics and the QHD model for considering the quantum effects including the quantum statistical and quantum diffraction effects. By means of the QHD theory, the wake electron density and wakefields are calculated, while the proton beam density is calculated by the PIC method and compared to hydrodynamic results to justify that the PIC method is a more suitable way to simulate the beam particle dynamics. The calculation results show that the incident continuous proton beam when propagating in the plasma generates electron perturbations as well as wakefields oscillations with negative valleys and positive peaks where the proton beams are repelled by the positive wakefields and accelerated by the negative wakefields. Moreover, the quantum correction obviously hinders the electron perturbations as well as the wakefields. Therefore, it is necessary to consider the quantum effects in the interaction of a proton beam with cold dense plasmas, such as in the metal films.

A Review on Sources,Extractions and Analysis Methods of a Sustainable Biomaterial:Tannins

Condensed and hydrolysable tannins are non-toxic natural polyphenols that are a commercial commodity industrialized for tanning hides to obtain leather and for a growing number of other industrial applications mainly to substitute petroleum-based products.They are a definite class of sustainable materials of the forestry industry.They have been in operation for hundreds of years to manufacture leather and now for a growing number of applications in a variety of other industries,such as wood adhesives,metal coating,pharmaceutical/medical applications and several others.This review presents the main sources,either already or potentially commercial of this forestry by-materials,their industrial and laboratory extraction systems,their systems of analysis with their advantages and drawbacks,be these methods so simple to even appear primitive but nonetheless of proven effectiveness,or very modern and instrumental.It constitutes a basic but essential summary of what is necessary to know of these sustainable materials.In doing so,the review highlights some of the main challenges that remain to be addressed to deliver the quality and economics of tannin supply necessary to fulfill the industrial production requirements for some materials-based uses.

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.

Toward modeling of the scrape-off layer currents induced by electrode biasing using 2D particle-in-cell simulations

The property of scrape-off layer(SOL) currents induced by a biased electrode is investigated by fully kinetic collisionless two-dimensional particle-in-cell(PIC) simulations. A reduced Vlasov–Darwin model is employed, which is capable of describing the low-frequency kinetic behavior without electromagnetic vacuum modes(w^2=w_(pe)~2+ c^2k^2). A linear decay distribution of electron currents parallel to the background magnetic field is exhibited. Simulation analyses indicate that the cross field ion current is a key factor in sheath formation and global current balance. The influences of electrode area, biasing voltage and plasma source on the SOL current profile are studied, respectively.Characteristic plasma parameters in the far SOL region of the EAST tokamak are used in simulations to assess the current driving ability of the electrode biasing method. Due to the limitations of computational power, the geometrical size of the simulation domain is significantly smaller than the realistic SOL, which may lead to an absence of the quasi-neutral region in the upstream plasma.At last, a heuristic method is proposed to calculate the upper bound of the total current strength.