热辐射输运问题的隐式蒙特卡罗方法求解

热辐射与物质相互作用及辐射光子在物质中的传输是惯性约束聚变研究中的重要课题. 介绍了基于隐式蒙特卡罗方法的辐射输运方程,在该方程的积分-微分形式基础上,推导了利于蒙特卡罗方法模拟的等价的积分输运方程;基于积分方程设计数值模拟流程,编写三维蒙特卡罗数值模拟程序;针对热辐射输运典型问题及benchmark问题开展了数值实验,计算结果验证了方法的适应性及程序的正确性. 关键词： 热辐射 惯性约束聚变 输运方程 隐式蒙特卡罗     

Exact and approximate solutions for the electron nonsinglet structure function in QED

Two exact, valid up to infinite perturbative order, numerical solutions of the Lipatov equation for the nonsinglet electron structure function in the QED are presented. One of them is of the Monte Carlo type and another is based on the numerical inversion of the Mellin transform. They agree numerically to a very high precision (better than 0.05%). Within the leading logarithmic approximation, the exact solution is compared with the perturbative second and third order exponentiated solutions. It is shown that the perturbative second order solution inspired by the Yennie-Frautschi-Suura (exclusive) exponentiation is much closer to the exact solution than the other ones. New compact analytical formula for the third order exponentiated solution is given. It is shown to be in perfect numerical agreement with the infinite order solution of the Monte Carlo and Mellin type.     

An equation of state for the isotropic phase of linear,partially flexible and fully flexible tangent hard-sphere chain fluids

A new equation of state is developed that accurately describes the isotropic phase behaviour of linear, partially flexible and fully flexible tangent hard-sphere chain fluids and their mixtures. The equation of state is based on the equation of state of Liu and Hu [H. Liu and Y. Hu, Fluid Phase Equilibr. 122, 75 (1996)] for fully flexible chain fluids. The effect of molecular flexibility is described by a pure-component parameter that is introduced in the theory at the level of the cavity correlation function of next-to-nearest neighbour segments in a chain molecule. The equation of state contains a total of three adjustable model constants. The extension to partially flexible- and linear chain fluids is based on a refitting of the first model constant to numerical data of the second virial coefficient of partially flexible and linear tangent hard-sphere chain fluids. The numerical data were obtained from an analytical approximation for the pair-excluded volume. The other two parameters were adjusted to molecular simulation data for the pressure of linear tangent hard-sphere chain fluids. For both, pure component systems and mixtures of chains of variable flexibility, the pressure and second virial coefficient obtained from the equation of state, are in excellent agreement with the results from Monte Carlo simulations. A significant improvement to TPT1, TPT2, generalised Flory-dimer theory and scaled particle theory is observed.     

热辐射输运问题的高效蒙特卡罗模拟方法

惯性约束聚变研究中,热辐射光子在介质中的输运以及热辐射光子与介质的相互作用是重要研究课题,蒙特卡罗方法是该类问题的重要研究手段之一.隐式蒙特卡罗方法虽然能正确地模拟热辐射在介质中的输运过程,但当模拟重介质(材料的吸收系数大)问题时,该方法花费的计算时间将变得很长,导致模拟效率很低.本文以离散扩散蒙特卡罗方法为基础,开发了"离散扩散蒙特卡罗方法辐射输运模拟程序",可以较好地解决重介质区的计算效率问题,但是离散扩散蒙卡罗方法在模拟轻介质区时精度不够高.辐射输运问题中通常既有轻介质也有重介质,为了能同时解决蒙特卡罗方法模拟的效率和精度问题,本文研究了离散扩散蒙特卡罗方法与隐式蒙特卡罗方法相结合的模拟方法,并提出了新的扩散区与输运区界面处理方法,研制了混合蒙特卡罗方法的辐射输运模拟程序.典型辐射输运问题模拟显示:在模拟重介质问题时,该程序能大幅缩短模拟时间,且能取得与隐式蒙特卡罗方法一致的结果;在模拟轻重介质均存在的问题时,与隐式蒙特卡罗方法相比,混合蒙特卡罗方法的模拟精度与其相当且计算效率同样能够得到显著提升.     

A benchmark comparison of Monte Carlo particle transport algorithms for binary stochastic mixtures

We numerically investigate the accuracy of two Monte Carlo algorithms originally proposed by Zimmerman [1] and Zimmerman and Adams [2] for particle transport through binary stochastic mixtures. We assess the accuracy of these algorithms using a standard suite of planar geometry incident angular flux benchmark problems and a new suite of interior source benchmark problems. In addition to comparisons of the ensemble-averaged leakage values, we compare the ensemble-averaged material scalar flux distributions. Both Monte Carlo transport algorithms robustly produce physically realistic scalar flux distributions for the benchmark transport problems examined. The base Monte Carlo algorithm reproduces the standard Levermore-Pomraning model [3] and [4] results. The improved Monte Carlo algorithm generally produces significantly more accurate leakage values and also significantly more accurate material scalar flux distributions. We also present deterministic atomic mix solutions of the benchmark problems for comparison with the benchmark and the Monte Carlo solutions. Both Monte Carlo algorithms are generally significantly more accurate than the atomic mix approximation for the benchmark suites examined.     

Numerical modeling of thermal force in a plasma for test-ion transport simulation based on Monte Carlo Binary Collision Model

We present a new numerical model of the thermal force in a plasma, based on the Monte Carlo Binary Collision Model (BCM) [T. Takizuka, H. Abe, J. Comput.Phys. 25 (1977) 205]. This model can be applied for the transport simulation of test ions. The model consists of two steps: (i) choosing a background plasma ion velocity from a distorted Maxwell distribution under the temperature gradient, and (ii) calculating a Coulomb collision between a test particle and the above chosen ion by using the BCM. For the step (i), we developed a velocity sampling method from a distorted Maxwellian, which enables the BCM to bring the thermal force on a test particle in the step (ii).A systematic series of simulations has been performed under various conditions to examine the model. The results of these simulations have been compared with the theoretical values, and it is shown that our model simulates the thermal force correctly for important characteristic features; dependences on the temperature gradient, the test particle velocity, and the background plasma density.     

DSM-LPDF两相湍流模型及旋流两相流动的模拟

本文由流体－颗粒速度的拉氏联合概率密度函数（ＰＤＦ）输运方程出发,用Ｓｉｍｏｎｉｎ建议的Ｌａｎｇｅｖｉｎ模型封闭颗粒所遇到流体瞬时速度的条件期望项,并用Ｍｏｎｔｅ Ｃａｒｌｏ方法直接求解 ＰＤＦ输运方程,将其和求解流体雷诺应力方程模型的有限差分方法结合,建立了雷诺应力－拉氏ＰＤＦ（ＤＳＭ－ＬＰＤＦ,简称ＤＬ）两相湍流模型．用此模型模拟了旋流数为０．４７的突扩旋流气粒两相流动,并与文献中ＰＤＰＡ实验和用类似于单相流动湍流模型封闭方法的时平均统一二阶矩（ＵＳＭ）模型的预报进行了对比．     

A low diffusion particle method for simulating compressible inviscid flows

A new particle method is presented for the numerical simulation of compressible inviscid gas flows, through procedures which involve relatively small modifications to an existing direct simulation Monte Carlo (DSMC) algorithm. Implementation steps are outlined for simulations involving various grid geometries and for gas mixtures comprising an arbitrary number of species. The proposed method is compared with other numerical schemes through a series of one-dimensional and two-dimensional test cases, and is shown to provide a significant reduction in both artificial diffusion and statistical scatter effects relative to existing DSMC-based equilibrium particle methods.     

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

Phase equilibria and transport in polymer colloids

Colloidal dispersions of uniform spherical particles, achievable through special emulsion polymerization techniques, show order-disorder transitions in which the ordered phase consists of particles spaced on a close-packed lattice. This colloidal phenomenon is an example of condensation of a hard-sphere gas, the so-called Kirkwood-Alder transition. The colloid transition can be simulated by using the DLVO pair potential either in numerical integration of Newton's equations o r in a Monte Carlo method. Although suitable for equilibrium properties, these methods can not yield transport properties of colloids, due to neglect of interactions between particle and medium, Substitution of Langevin's equation for Newton's introduces both frictional and stochastic forces on the particles. Computer simulation of particle motions by integration of Langevin's equation yields correct equilibrium properties, and permits the calculation of transport properties as well. Some results of computer simulation of diffusion are presented.     

Particle in Cell/Monte Carlo Collision Method for Simulation of RF Glow Discharges: Effect of Super Particle Weighting

A parallel Particle in Cell/Monte Carlo Collision (PIC/MCC) numerical code for glow discharge plasma simulations is developed and verified. This method is based on simultaneous solution of the Lorentz equations of motion of super particles, coupled with the Poisson's equation for electric field. Collisions between the particles are modelled by the Monte Carlo method. Proper choice of particle weighting is critically important in order to perform adequate and efficient PIC simulations of plasma. Herein, effects of particle weighting on the simulations of capacitive radio‐frequency argon plasma discharges are studied in details. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonequilibrium dynamics of correlated electron transfer in molecular chains

The relaxation dynamics of correlated electron transport along molecular chains is studied based on a substantially improved numerically exact path integral Monte Carlo approach. As an archetypical model, we consider a Hubbard chain containing two interacting electrons coupled to a bosonic bath. For this generalization of the ubiquitous spin-boson model, non-Boltzmann equilibrium distributions are found for many-body states. By mapping the multiparticle dynamics onto an isomorphic single particle motion, this phenomenon is shown to be sensitive to particle statistics and, due to its robustness, allows for new control schemes in designed quantum aggregates.     

Multiple temperature model for the information preservation method and its application to nonequilibrium gas flows

The information preservation (IP) method has been successfully applied to various nonequilibrium gas flows. Comparing with the direct simulation Monte Carlo (DSMC) method, the IP method dramatically reduces the statistical scatter by preserving collective information of simulation molecules. In this paper, a multiple temperature model is proposed to extend the IP method to strongly translational nonequilibrium gas flows. The governing equations for the IP quantities have been derived from the Boltzmann equation based on an assumption that each simulation molecule represents a Gaussian distribution function with a second-order temperature tensor. According to the governing equations, the implementation of IP method is divided into three steps: molecular movement, molecular collision, and update step. With a reasonable multiple temperature collision model and the flux splitting method in the update step, the transport of IP quantities can be accurately modeled. We apply the IP method with the multiple temperature model to shear-driven Couette flow, external force-driven Poiseuille flow and thermal creep flow, respectively. In the former two cases, the separation of different temperature components is clearly observed in the transition regime, and the velocity, temperature and pressure distributions are also well captured. The thermal creep flow, resulting from the presence of temperature gradients along boundary walls, is properly simulated. All of the IP results compare well with the corresponding DSMC results, whereas the IP method uses much smaller sampling sizes than the DSMC method. This paper shows that the IP method with the multiple temperature model is an accurate and efficient tool to simulate strongly translational nonequilibrium gas flows.     

Radial Distribution Functions for Molecules in the Universal Equation of State Model for Gaseous/Fluid/Condensed Systems

Analytical expressions and a numerical method for calculation of distribution functions of hard spheres gij(r) based on inverting the Laplace transform for functions rgij(r) obtained from the Percus—Yevick equation are obtained. The method for calculation of radial distribution functions is applicable for any distances between hard spheres; it is verified by comparison of numerical results and Monte Carlo simulations. The application of the developed method for calculation of the radial distribution functions of metal atoms is demonstrated. Distribution functions are required to construct a universal theoretical model of equation of state capable of describing both dense multicomponent gas and condensed substances (liquid or solid phases) with high accuracy which is substantially faster than computer experiments (Monte Carlo and molecular dynamics methods).     

A Monte Carlo model for determination of binary diffusion coefficients in gases

A Monte Carlo method has been developed for the calculation of binary diffusion coefficients in gas mixtures. The method is based on the stochastic solution of the linear Boltzmann equation obtained for the transport of one component in a thermal bath of the second one. Anisotropic scattering is included by calculating the classical deflection angle in binary collisions under isotropic potential. Model results are compared to accurate solutions of the Chapman–Enskog equation in the first and higher orders. We have selected two different cases, H₂ in H₂ and O in O₂, assuming rigid spheres or using a model phenomenological potential. Diffusion coefficients, calculated in the proposed approach, are found in close agreement with Chapman–Enskog results in all the cases considered, the deviations being reduced using higher order approximations.     

Improved particle filters for multi-target tracking

We present a novel approach for improving particle filters for multi-target tracking. The suggested approach is based on drift homotopy for stochastic differential equations. Drift homotopy is used to design a Markov Chain Monte Carlo step which is appended to the particle filter and aims to bring the particle filter samples closer to the observations while at the same time respecting the target dynamics. We have used the proposed approach on the problem of multi-target tracking with a nonlinear observation model. The numerical results show that the suggested approach can improve significantly the performance of a particle filter.     

两相壁面射流PDF方程有限分析方法及数值模拟

提出求解位置-速度相空间中高维两相流PDF(probability density function)方程的有限分析方法,将位置-速度相空间颗粒PDF方程约化到速度空间,并解析求解,颗粒的位置PDF用轨道方法求解.对壁面射流两相流动进行数值模拟,并与颗粒雷诺应力轨道方法进行比较计算,结果优于颗粒雷诺应力轨道方法.     

非定常辐射输运问题的蒙特卡罗自适应偏倚抽样

用蒙特卡罗(MC)方法模拟高温、高压、多介质、大变形辐射输运问题时,由于网格体积悬殊,导致各网格通量的统计误差涨落很大,随着时间步的增加,误差积累甚至会导致计算结果失真.为此,发展了针对全局网格计算的源偏倚抽样技巧.用于源偏倚抽样的价值函数基于上个时间步各网格通量及误差,通过加权构造产生,它比传统MC通过解伴随方程获取价值的性价比要高得多.数值试验表明,全局源偏倚抽样通过自适应分配当前时间步各网格的粒子数,有效地降低了当前步重要网格通量误差. 关键词： 非定常 辐射输运 蒙特卡罗 源偏倚抽样     

Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations

We study a class of methods for the numerical solution of the system of stochastic differential equations (SDEs) that arises in the modeling of turbulent combustion, specifically in the Monte Carlo particle method for the solution of the model equations for the composition probability density function (PDF) and the filtered density function (FDF). This system consists of an SDE for particle position and a random differential equation for particle composition. The numerical methods considered advance the solution in time with (weak) second-order accuracy with respect to the time step size. The four primary contributions of the paper are: (i) establishing that the coefficients in the particle equations can be frozen at the mid-time (while preserving second-order accuracy), (ii) examining the performance of three existing schemes for integrating the SDEs, (iii) developing and evaluating different splitting schemes (which treat particle motion, reaction and mixing on different sub-steps), and (iv) developing the method of manufactured solutions (MMS) to assess the convergence of Monte Carlo particle methods. Tests using MMS confirm the second-order accuracy of the schemes. In general, the use of frozen coefficients reduces the numerical errors. Otherwise no significant differences are observed in the performance of the different SDE schemes and splitting schemes.