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.

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.

Fe-Cr二元合金微观组织演化的质量密度场耦合动力学Monte-Carlo模拟研究

本文建立了一种全新的将动力学Monte-Carlo粒子模拟与基于归一化Gauss函数基组的质量密度场空间粗粒化模型耦合的杂化模拟算法.采用该杂化模拟算法,系统对比研究了4种Cr原子含量分别为12.8%,20.0%,30.0%和40.0%的Fe-Cr合金中Cr相在温度为673 K下的时效析出动力学机制,及其时效不同阶段微观组织形貌的演变规律.研究得出Fe-Cr(12.8%)合金富Cr相时效组织形貌呈现孤立颗粒状空间分布形态,时效机制属于形核-长大(NG)机制;对于Fe-Cr(30.0%)和Fe-Cr(40.0%),富Cr相时效形貌在形核-生长及熟化阶段均呈现为三维蠕虫状空间分布特征,时效机制属于条幅分解(SD)机制;对于Fe-Cr(20.0%)合金,其富Cr相组织演化特征介于NG和SD机制之间.研究进一步发现Cr原子短程序参量可用来分析富Cr相形核-生长阶段Fe-Cr合金原子尺度结构的演变,但对于时效熟化阶段微观结构组织变化不敏感.基于空间粗粒化后Fe-Cr合金微观组织形貌,进一步分析了4种Cr原子含量下Fe-Cr合金相变动力学参数如富Cr相体积分数、平均粒径及相颗粒数密度随时效时间演变.本文建立的质量密度场耦合动力学Monte-Carlo模拟方法,为开发多尺度算法模拟合金时效动力学机制及微观组织形貌演变提供了新的思路和研究基础.

基于Monte-Carlo迭代求解策略的局部社区发现算法

针对现有的局部社区发现算法因采用贪心策略进行社区扩张而导致的过早收敛和查全率低的问题,提出一种基于Monte-Carlo迭代求解策略的局部社区发现算法。首先,在每轮迭代的社区扩张阶段,根据节点对社区紧密度增益的贡献比例为所有邻接候选节点赋予选择概率,并结合此概率,再随机选择一个节点加入社区。然后,为避免随机选择导致扩张方向偏离目标社区,根据社区质量变化情况判断本轮迭代中是否触发节点淘汰机制。若触发,计算各个已加入社区节点与社区内其他节点的相似度和,根据相似度和的倒数赋予淘汰概率,并结合此概率,再随机淘汰一个节点。最后,在给定数量的最近迭代轮次中,根据社区规模是否增加判断是否继续迭代。在三个真实的网络数据集上进行实验,相较于局部紧密度扩展(LTE)算法、Clauset算法、加权共同邻居节点(CNWNN)算法和模糊相似关系(FSR)算法,所提算法的局部社区发现结果的F-score值分别提升了32.75、17.31、20.66和25.51个百分点,且能够有效避免查询节点在社区中所处位置对局部社区发现结果的影响。

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。

具有下界约束混料试验的Monte-Carlo渐近最优设计

在具有下界约束的混料试验设计中,由于难以求得含有未知变量的信息矩阵的行列式的显式表达,所以难以得到各类最优准则下的精确最优设计解。文章使用Monte-Carlo方法构造随机试验点的样本,进而可以通过模拟得出D-准则和MV-准则下的渐近最优设计。

基于Monte-Carlo模拟的小样本下齿轮疲劳极限计算方法及软件开发

齿轮疲劳极限评估Dixon-Mood(D-M)法的试验样本需求量较大,标准差估计值存在较大偏差,因此基于Monte-Carlo模拟提出小样本的齿轮疲劳极限分析方法(CQUboot),并结合大量弯曲疲劳极限试验对D-M法和CQUboot法进行了对比分析。与GB/T 14230—2021《齿轮弯曲疲劳强度试验方法》推荐的20~22个样本计算结果相比,样本量降至12时,D-M法的最大误差为15.76%~24.99%,CQUboot法为8.02%~12.98%;以12.66%为疲劳极限预估允许的最大误差时,D-M法至少需要10~18个样本点,CQUboot法只需8个样本点。

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.

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.

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.

Development of multi-group Monte-Carlo transport and depletion coupling calculation method and verification with metal-fueled fast reactor

The accurate modeling of depletion,intricately tied to the solution of the neutron transport equation,is crucial for the design,analysis,and licensing of nuclear reactors and their fuel cycles.This paper introduces a novel multi-group Monte-Carlo depletion calculation approach.Multi-group cross-sections(MGXS)are derived from both 3D whole-core model and 2D fuel subassembly model using the continuous-energy Monte-Carlo method.Core calculations employ the multi-group Monte-Carlo method,accommodating both homogeneous and specific local heterogeneous geometries.The proposed method has been validated against the MET-1000 metal-fueled fast reactors,using both the OECD/NEA benchmark and a new refueling benchmark introduced in this paper.Our findings suggest that microscopic MGXS,produced via the Monte-Carlo method,are viable for fast reactor depletion analyses.Furthermore,the locally heterogeneous model with angular-dependent MGXS offers robust predictions for core reactivity,control rod value,sodium void value,Doppler constants,power distribution,and concentration levels.

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.

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

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

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

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

A Particle-in-Cell Simulation of Double Layers and Ion-Acoustic Waves

Double layers and ion-acoustic waves are investigated by using a one-dimensional electrostatic particle-in-cell simulation code. Our results show that double layers can be formed even when the drift velocity between electrons and ions is less than the electron thermal velocity. Electron and ion density depressions were clearly seen. Electrons gradually developed a distribu- tion comprising both background and beam components. In fact, as the initial electron-ion drift velocity was less than the electron thermal velocity, intense ion-acoustic waves could be found only at the places where the electron beam was located, suggesting that they are excited by the self-consistently developed electron beam. Besides the Langmuir waves and ion-acoustic waves, the beam mode excited by electron beams produced in our simulation has been clearly found.

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.

Particle-in-cell investigation on the resonant absorption of a plasma surface wave

The resonant absorption of a plasma surface wave is supposed to be an important and efficient mechanism of power deposition for a surface wave plasma source. In this paper, by using the particle-in-cell method and Monte Carlo simulation, the resonance absorption mechanism is investigated. Simulation results demonstrate the existence of surface wave resonance and show the high efficiency of heating electrons. The positions of resonant points, the resonance width and the spatio-temporal evolution of the resonant electric field are presented, which accord well with the theoretical results. The paper also discusses the effect of pressure on the resonance electric field and the plasma density.

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

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