Large-scale simulations on multiple Graphics Processing Units (GPUs) for the direct simulation Monte Carlo method

In this study, the application of the two-dimensional direct simulation Monte Carlo (DSMC) method using an MPI-CUDA parallelization paradigm on Graphics Processing Units (GPUs) clusters is presented. An all-device (i.e. GPU) computational approach is adopted where the entire computation is performed on the GPU device, leaving the CPU idle during all stages of the computation, including particle moving, indexing, particle collisions and state sampling. Communication between the GPU and host is only performed to enable multiple-GPU computation. Results show that the computational expense can be reduced by 15 and 185 times when using a single GPU and 16 GPUs respectively when compared to a single core of an Intel Xeon X5670 CPU. The demonstrated parallel efficiency is 75% when using 16 GPUs as compared to a single GPU for simulations using 30 million simulated particles. Finally, several very large-scale simulations in the near-continuum regime are employed to demonstrate the excellent capability of the current parallel DSMC method.     

On the lattice Boltzmann method for phonon transport

The lattice Boltzmann method is a discrete representation of the Boltzmann transport equation that has been employed for modeling transport of particles of different nature. In the present work, we describe the lattice Boltzmann methodology and implementation techniques for the phonon transport modeling in crystalline materials. We show that some phonon physical properties, e.g., mean free path and group velocity, should be corrected to their effective values for one- and two-dimensional simulations, if one uses the isotropic approximation. We find that use of the D2Q9 lattice for phonon transport leads to erroneous results in transient ballistic simulations, and the D2Q7 lattice should be employed for two-dimensional simulations. Furthermore, we show that at the ballistic regime, the effect of direction discretization becomes apparent in two dimensions, regardless of the lattice used. Numerical methodology, lattice structure, and implementation of initial and different boundary conditions for the D2Q7 lattice are discussed in detail.     

Application of the modified differential approximation for radiative transfer to arbitrary geometry

The first-order spherical harmonics method (or P₁ approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P₁ approximation deteriorates as the optical thickness of the medium is decreased. The modified differential approximation (MDA) was originally proposed to remove the shortcomings of the P₁ approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry—in particular, geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results. The accuracy of MDA was found to be superior to the P₁ approximation for all optical thicknesses. Its accuracy, when compared with the discrete ordinates method (both S₆ and S₈), was found to be clearly superior in optically thin situations, but problem dependent in optically intermediate and thick situations. For 3D geometries, calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint. Several strategies to reduce both memory and computational time are discussed and demonstrated.     

A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering

We describe a semi-analytical numerical method for coherent isotropic scattering time-dependent radiative transfer problems in slab geometry. This numerical method is based on a combination of two classes of numerical methods: the spectral methods and the Laplace transform (LTS_N) methods applied to the radiative transfer equation in the discrete ordinates (S_N) formulation. The basic idea is to use the essence of the spectral methods and expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable and then solve recursively the resulting set of “time-independent” S_N problems by using the LTS_N method. We show some numerical experiments for a typical model problem.     

An efficient mixed-precision,hybrid CPU–GPU implementation of a nonlinearly implicit one-dimensional particle-in-cell algorithm

Recently, an implicit, nonlinearly consistent, energy- and charge-conserving one-dimensional (1D) particle-in-cell method has been proposed for multi-scale, full-f kinetic simulations [G. Chen et al., J. Comput. Phys. 230 (18) (2011)]. The method employs a Jacobian-free Newton–Krylov (JFNK) solver, capable of using very large timesteps without loss of numerical stability or accuracy. A fundamental feature of the method is the segregation of particle-orbit computations from the field solver, while remaining fully self-consistent. This paper describes a very efficient, mixed-precision hybrid CPU–GPU implementation of the 1D implicit PIC algorithm exploiting this feature. The JFNK solver is kept on the CPU in double precision (DP), while the implicit, charge-conserving, and adaptive particle mover is implemented on a GPU (graphics processing unit) using CUDA in single-precision (SP). Performance-oriented optimizations are introduced with the aid of the roofline model. The implicit particle mover algorithm is shown to achieve up to 400 GOp/s on a Nvidia GeForce GTX580. This corresponds to 25% absolute GPU efficiency against the peak theoretical performance, and is about 100 times faster than an equivalent single-core CPU (Intel Xeon X5460) compiler-optimized execution. For the test case chosen, the mixed-precision hybrid CPU–GPU solver is shown to over-perform the DP CPU-only serial version by a factor of ～100, without apparent loss of robustness or accuracy in a challenging long-timescale ion acoustic wave simulation.     

Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements

We propose the P_N approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P_N approximation is compared against Monte Carlo simulations and other state-of-the-art methods.     

图形处理器并行计算用于离子发动机粒子模拟

 为了研究离子发动机羽流对航天器的影响,采用质点网格-蒙特卡罗碰撞方法对离子发动机羽流中的交换电荷离子进行了模拟。利用计算设备统一架构技术,开发出一套基于图形处理器的并行粒子模拟程序。随机数生成采用并行MT19937伪随机数生成器算法,电场方程使用完全近似存储格式的代数多重网格法求解。r-z轴对称坐标系中,在z=0 m处获得的电流密度均值为4.5×10^-5 A/m²,图形处理器所得结果与中央处理器模拟结果吻合。在16核心的NVIDIA GeForce 9400 GT图形显示卡上,取得相对于Intel Core 2 E6300中央处理器4.5~10.0倍的加速比。     

Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

CORRECTIONS TO THE COLLISION TERM IN THE BGK BOLTZMANN EQUATION

With the discrete method of the hexagonal cell and three different velocities of particle population in each cell, a two-dimensional lattice Boltzmann model is developed in this paper.^[1,2] The collision operator in the Boltzmann equation is expanded to fourth order using the Taylor expansion.^[3,4] With this model, good results have been obtained from the numerical simulation of the reflection phenomenon of the shock wave on the surface of an obstacle, and the numerical stability is also good. Thus the applicability of the D2Q 19 model is verified.     

A finite element-spherical harmonics radiation transport model for photon migration in turbid media

In this paper, we solve the steady-state form of the Boltzmann transport equation in homogeneous and heterogeneous tissue-like media with a finite element-spherical harmonics (FE-P_N) radiation transport method. We compare FE-transport and diffusion solutions in terms of the ratio of absorption to reduced scattering coefficient, (μ_a/μ_s′) and the anisotropy factor g. Two different scattering phase function formulas are employed to model anisotropic scattering in the slab media with high g-value. Influence of void-like heterogeneities, and of their boundaries with the surrounding medium on the transport of photons are also examined.     

Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes

Standard and goal-oriented adaptive mesh refinement (AMR) techniques are presented for the linear Boltzmann transport equation. A posteriori error estimates are employed to drive the AMR process and are based on angular-moment information rather than on directional information, leading to direction-independent adapted meshes. An error estimate based on a two-mesh approach and a jump-based error indicator are compared for various test problems. In addition to the standard AMR approach, where the global error in the solution is diminished, a goal-oriented AMR procedure is devised and aims at reducing the error in user-specified quantities of interest. The quantities of interest are functionals of the solution and may include, for instance, point-wise flux values or average reaction rates in a subdomain. A high-order (up to order 4) Discontinuous Galerkin technique with standard upwinding is employed for the spatial discretization; the discrete ordinates method is used to treat the angular variable.     

Transport Coefficients for Inelastic Maxwell Mixtures

The Boltzmann equation for inelastic Maxwell models (IMM) is used to determine the Navier–Stokes transport coefficients of a granular binary mixture in d-dimensions. The Chapman–Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are exactly obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres (IHS) obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux     

Low temperature preparation and superconductivity of F-doped SmFeAsO

A low temperature (1100 °C) process of preparing F-doped SmFeAsO samples has been developed using SmF₃ with nanometer scale as the source of fluorine. A series of the SmFeAsO_1−xF_x (x = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3) samples have been prepared using the present method. Compared with previous reports, the present SmF₃ is more effective to introduce F into SmFeAsO system in which a transition temperature of 39 K can be observed when x = 0.05. The superconductivity is definitely enhanced with the increasing F-doping level. All the samples presented to be layered structure and the crystal particle size is about three times larger with sintering time increasing from 36 h to 48 h. Except for the nanometer scale of SmF₃, the flux effect of SmF₃ is recognized to be another reason for the decrease of the sintering temperature. Further more, a relatively large amount of SmF₃ was also employed in the raw materials to introduce excessive F and this has induced higher T_c (55 K) in SmFeAsO_0.8F_0.2+δ system.     

Time-dependent diffusion and transport calculations using a finite-element-spherical harmonics method

In this paper, the finite-element-spherical harmonics (FE-P_N) method is applied to the solution of transient Boltzmann transport equation. Firstly, transport and diffusion calculations are obtained for homogeneous and inhomogeneous circular regions. Results are compared in order to show the effects of different absorption coefficient values on the propagation of photons. Significant differences between two theories are shown to occur especially in cases when the absorption is increased. Secondly, to validate the FE-P_N method, results from this method are compared with Monte Carlo calculations for different cases. Comparisons show good agreements between FE-transport and Monte Carlo solutions and demonstrate the correctness of the results obtained.     

Solving the Boltzmann equation in a 2-D-configuration and2-D-velocity space for capacitively coupled RF discharges

A new kinetic scheme, the generalized Monte Carlo flux (GMCF) method, provides the electron particle distribution function in phase space, f(ν, μ, r, z, t) (ν: speed, μ: velocity angle, r: radial position, z: axial position, and t: time), for solving the Boltzmann equation in modeling capacitively coupled RP discharges. For a simulation with spatial- and temporal-varying fields in RF discharges, the GMCF method handles the collision terms of the Boltzmann equation by using one transition matrix to compute the collision transition between velocity space cells. An anti-diffusion flux transport scheme is developed to overcome the numerical diffusion in the velocity and configuration spaces. The major advantages of the GMCF method are the increase in resolution in the tail of distribution functions and the decrease of computation time. The GMCF calculation results in terms of microscopic electron distribution function and macroscopic quantities of density, electric field and ionization rate, are presented for RF discharges and compared with other kinetic and fluid simulation and experimental results. The effects of the induced radial electric field in the sheath close to the radial wall in a cylindrically symmetric parallel-plate geometry are discussed     

Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method

The Schur-decomposition for three-dimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3D Schur-decomposition solver is the least, and almost only one thirtieth to one fiftieth CPU time is needed when using the spectral method with 3D Schur-decomposition solver compared with the standard discrete ordinates method.     

Immiscible multicomponent lattice Boltzmann model for fluids with high relaxation time ratio

An immiscible multicomponent lattice Boltzmann model is developed for fluids with high relaxation time ratios, which is based on the model proposed by Shan and Chen (SC). In the SC model, an interaction potential between particles is incorporated into the discrete lattice Boltzmann equation through the equilibrium velocity. Compared to the SC model, external forces in our model are discretized directly into the discrete lattice Boltzmann equation, as proposed by Guo et al. We develop it into a new multicomponent lattice Boltzmann (LB) model which has the ability to simulate immiscible multicomponent fluids with relaxation time ratio as large as 29.0 and to reduce ‘spurious velocity’. In this work, the improved model is validated and studied using the central bubble case and the rising bubble case. It finds good applications in both static and dynamic cases for multicomponent simulations with different relaxation time ratios.     

Entropy considerations in numerical simulations of non-equilibrium rarefied flows

Non-equilibrium rarefied flows are encountered frequently in supersonic flight at high altitudes, vacuum technology and in microscale devices. Prediction of the onset of non-equilibrium is important for accurate numerical simulation of such flows. We formulate and apply the discrete version of Boltzmann’s H-theorem for analysis of non-equilibrium onset and accuracy of numerical modeling of rarefied gas flows. The numerical modeling approach is based on the deterministic solution of kinetic model equations. The numerical solution approach comprises the discrete velocity method in the velocity space and the finite volume method in the physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter and a third-order WENO schemes. The use of entropy considerations in rarefied flow simulations is illustrated for the normal shock, the Riemann and the two-dimensional shock tube problems. The entropy generation rate based on kinetic theory is shown to be a powerful indicator of the onset of non-equilibrium, accuracy of numerical solution as well as the compatibility of boundary conditions for both steady and unsteady problems.     

Rotation-vibration energy cluster formation in XH2D and XHD2 molecules (X = Bi, P, and Sb)

We investigate theoretically the energy cluster formation in highly excited rotational states of several pyramidal XH₂D and XHD₂ molecules (X = Bi, P, and Sb) by calculating, in a variational approach, the rotational energy levels in the vibrational ground states of these species for J?70. We show that at high J the calculated energy levels of the di-deuterated species XHD₂ exhibit distinct fourfold cluster patterns highly similar to those observed for H₂X molecules. We conclude from eigenfunction analysis that in the energy cluster states, the XHD₂ molecule rotates about a so-called localization axis which is approximately parallel to one of the X-D bonds. For the mono-deuterated XH₂D isotopologues, the rotational spectra are found to have a simple rigid-rotor structure with twofold clusters.     

Hybrid discrete ordinates—spherical harmonics solution to the Boltzmann Transport Equation for phonons for non-equilibrium heat conduction

The Boltzmann Transport Equation (BTE) for phonons has found prolific use for the prediction of non-equilibrium heat conduction phenomena in semiconductor materials. This article presents a new hybrid formulation and associated numerical procedures for solution of the BTE for phonons. In this formulation, the phonon intensity is first split into two components: ballistic and diffusive. The governing equation for the ballistic component is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). The diffusive component, on the other hand, is determined by invoking the first-order spherical harmonics (or P₁) approximation, which results in a Helmholtz equation with Robin boundary conditions. Both governing equations, referred to commonly as the ballistic-diffusive equations (BDE), are solved using the unstructured finite-volume procedure. Results of the hybrid method are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two two-dimensional transient heat conduction problems at various Knudsen numbers. Subsequently, the method is explored for a large-scale three-dimensional geometry in order to assess convergence and computational cost. It is found that the proposed hybrid method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method is found to be superior to direct solution of the BTE both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.