基于SPH和DSMC的多尺度耦合算法的研究及应用
摘要
为了能连续、高效地提取抗新型突发性病毒和增强人体免疫力的生化制品，需要开发一种结合凝胶色谱分离特点和微流控分离技术相结合的新型微分离器件。在这类微分离器件中，被分离流体需要完成入口进样、微通道分离、再到出口取样等的通道流动过程，在整个流动过程中会经历从宏观连续流动—微观稀薄流动—宏观连续流动的跨尺度流动。因此，其复杂的流道分离特性研究是此类新型分离器件设计的关键。随着并行计算技术的飞速发展，数值模拟方法越来越多地被用于解决跨尺度问题。针对此类流动过程中的跨尺度问题，若单一采用基于连续假设的方法，不能凑效，而采用单一的微观流动方法，计算量又十分巨大。本文的研究是发展一种将连续流动算法和稀薄流动算法相结合的跨尺度耦合方法，尝试了将适用于连续流动的SPH方法和基于粒子模型的DSMC方法相耦合，以实现对以上新型分离器件中的跨尺度通道流动的分析计算。
首先，本文根据分离装置中通道流动的出入口边界条件的特点，提出了基于缓冲区的SPH入口流动边界条件和基于单元格压力线性假设的DSMC压力边界实现方法。采用自行编写的SPH计算程序和改进的DSMC计算程序对二维通道流动模型分别进行了计算分析，论证了改进后两种方法各自的计算有效性。
其次，对以前研究中提出的耦合方法中存在的，物理参数的匹配、交界面处的连接方式及数据传递和耦合时间步长的确定等问题进行了完善，给出了这些技术的改进要点、解决方法和程序中的实现技巧。在此基础上，通过大量的编程改进和运行调试，尝试将两种方法的计算程序结合起来实现了二维通道流动模型的耦合计算，并将计算结果和单独采用DSMC计算的结果进行了比较。结果显示两者有较好的一致性，表明了耦合方法及程序的正确性和有效性，也验证了耦合理论的可行性。
最后，探讨了SPH方法和DSMC方法中各自的参数设置，对耦合算法的计算效率和计算精度的影响，并提出了相应有效的解决办法，这为以后耦合算法的进一步研究提供了改进思路和方向。
A new micro-separation device combined characteristics of gel chromatogram method and technologies of microfluidic chips is developed, in order to produce biochemical drug for resisting new virus and enhancing human immunity consecutively and effectively. In micro-separation device, the separated fluid flows from the input region to the micro-channel for separation, then samples at the output region. In the whole separation process, multiscale flows which go through continuum flow region, rarefied flow region and continuum flow region, will be encountered. So Investigation of the separate characteristic of the complicated micro-channels is the key for the design and development of the new micro-separation device. Numerical simulations have been used extensively for the simulation of multiscale flows based on the rapid development of the computer technology.
Simulation of the complete flow path is, however, very complicated. On one hand, the use of continuum theories for the entire device can produce inaccurate results because of the breakdown of the continuum theories in the rarefied region, on the other hand, the simulation of such multiscale problem by molecular methods alone has low computational efficiency. An efficient approach for such multiple length scale problem is to develop a multiscale approach combining continuum theories with molecular approaches. In this paper, we investigate the application of coupling smoothed particle hydrodynamics (SPH) for continuum region with direct simulation Monte Carlo (DSMC) for rarefied region.
Firstly, setting buffer cells, a new treatment of flow boundary conditions is proposed in SPH method, and the method based on the assumption of the linear pressure distribution in the cells is used to implement pressure boundary conditions in DSMC method. The validation of the improved SPH method and DSMC method is testified by simulating 2-D channel flow separately.
Secondly, issues of the coupled method, such as the matching of parameters, the connect mode and data transfer at the interface, and the choice of the couple time, are
presented. Improvements and solutions of these issues, techniques in couple code were concretely given. Then the coupled method was used to calculate the 2-D channel flow model. The good agreement of the simulation results with the DSMC solution shows the correctness and the efficiency of the coupled SPH/DSMC method
and its code, and the feasibility of the coupled theories is also proved.
Finally, the setting of SPH and DSMC parameters and their influence for the coupled algorithmic are respectively discussed. Furthermore, effective measurements are presented, which are available for further research.
首先，本文根据分离装置中通道流动的出入口边界条件的特点，提出了基于缓冲区的SPH入口流动边界条件和基于单元格压力线性假设的DSMC压力边界实现方法。采用自行编写的SPH计算程序和改进的DSMC计算程序对二维通道流动模型分别进行了计算分析，论证了改进后两种方法各自的计算有效性。
其次，对以前研究中提出的耦合方法中存在的，物理参数的匹配、交界面处的连接方式及数据传递和耦合时间步长的确定等问题进行了完善，给出了这些技术的改进要点、解决方法和程序中的实现技巧。在此基础上，通过大量的编程改进和运行调试，尝试将两种方法的计算程序结合起来实现了二维通道流动模型的耦合计算，并将计算结果和单独采用DSMC计算的结果进行了比较。结果显示两者有较好的一致性，表明了耦合方法及程序的正确性和有效性，也验证了耦合理论的可行性。
最后，探讨了SPH方法和DSMC方法中各自的参数设置，对耦合算法的计算效率和计算精度的影响，并提出了相应有效的解决办法，这为以后耦合算法的进一步研究提供了改进思路和方向。
A new micro-separation device combined characteristics of gel chromatogram method and technologies of microfluidic chips is developed, in order to produce biochemical drug for resisting new virus and enhancing human immunity consecutively and effectively. In micro-separation device, the separated fluid flows from the input region to the micro-channel for separation, then samples at the output region. In the whole separation process, multiscale flows which go through continuum flow region, rarefied flow region and continuum flow region, will be encountered. So Investigation of the separate characteristic of the complicated micro-channels is the key for the design and development of the new micro-separation device. Numerical simulations have been used extensively for the simulation of multiscale flows based on the rapid development of the computer technology.
Simulation of the complete flow path is, however, very complicated. On one hand, the use of continuum theories for the entire device can produce inaccurate results because of the breakdown of the continuum theories in the rarefied region, on the other hand, the simulation of such multiscale problem by molecular methods alone has low computational efficiency. An efficient approach for such multiple length scale problem is to develop a multiscale approach combining continuum theories with molecular approaches. In this paper, we investigate the application of coupling smoothed particle hydrodynamics (SPH) for continuum region with direct simulation Monte Carlo (DSMC) for rarefied region.
Firstly, setting buffer cells, a new treatment of flow boundary conditions is proposed in SPH method, and the method based on the assumption of the linear pressure distribution in the cells is used to implement pressure boundary conditions in DSMC method. The validation of the improved SPH method and DSMC method is testified by simulating 2-D channel flow separately.
Secondly, issues of the coupled method, such as the matching of parameters, the connect mode and data transfer at the interface, and the choice of the couple time, are
presented. Improvements and solutions of these issues, techniques in couple code were concretely given. Then the coupled method was used to calculate the 2-D channel flow model. The good agreement of the simulation results with the DSMC solution shows the correctness and the efficiency of the coupled SPH/DSMC method
and its code, and the feasibility of the coupled theories is also proved.
Finally, the setting of SPH and DSMC parameters and their influence for the coupled algorithmic are respectively discussed. Furthermore, effective measurements are presented, which are available for further research.
引文
[1] 李淑芬，姜忠义编．高等制药分离工程．北京：化学工业出版社，2004．
[2] 李校堃，袁辉编．药物蛋白质分离纯化技术．北京：化学工业出版社，2005．
[3] 高崐玉编．色谱法在精细化工中的应用．北京：中国石化出版社，1997．
[4] Humble P H, Kelly R T, Woolley A T, Tolley H D. Electric Field Gradient Focusing of Proteins Based on Shaped Ionically Conductive Acrylic Polymer[J]. Analytical Chemistry, 2004(76): 5641-5648
[5] Jacobson S C et al. Fused Quartz Substrates for Microchip Electrophoresis[J]. Analytical Chemistry, 1995 (67): 2059-2063.
[6] Seiler K, et al. Electroosmotic Pumping and Valveless Control of Fluid Flow Within a Manifold of Capillaries on a Glass Chip[J]. Anal. Chem, 1994 (66): 3485-3491.
[7] 王立文，高殿荣．微流动系统的研究现状与趋势[J]．液压与气动，2005(10)：4-7．
[8] [美]Beskok A，Kamiadakis G E，微流动—基础与模拟．中国科学院过程工程研究所多相反应重点实验室多相复杂系统与多尺度方法课题组译，北京：化学工业出版社，2002．
[9] Rajesh Khare, Pawel Keblinskib, Arun Yethiraj. Molecular dynamics simulations of heat and momentum transfer at asolid-fluid interface: Relationship between thermal and velocity slip[J]. International Journal of Heat and Mass Transfer, 2006(49): 3401—3407.
[10] Bourgat J F, Tallec P L, Tidriri M D. Coupling Boltzmann and Navier-Stokes equations by friction[J]. Journal of Computational Physics, 1996(127): 227-245.
[11] Tallec P L, Mallinger F. Coupling Boltzmann and Navier-Stokes equations by half fluxes. Journal of Computationai Physics, 1997(136): 51-67.
[12] Tiwari S. Coupling of the Boltzmann and Euler Equations with Automatic Domain Decomposition[J]. Journal of Computer Physics, 1998(144): 710-726.
[13] Wadsworth D C, Erwin D A. One-Dimensional hybrid continuum/particle simulation approach for rarefied hypersonic flows[R], AIAA paper 90-1690(1990).
[14] Wadsworth D C, Erwin D A. Two-dimensional hybrid continuum/particle approach for rarefied flows[R], AIAA paper 92-2975(1992).
[15] Hash D B, Hassan H A. Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods[J]. Journal of Thermophysics and Heat Transfer,1996(10): 242-249.
[16] Hash D B, Hassan H A. A decoupled DSMC/Navier-Stokes analysis of a transitional flow experiment[R]. AIAA Paper 96-0353, January 1996, Presented at the 34th AIAA Aerospace Sciences Meeting and Exhibit.
[17] Hash D B , Hassan H A . Two-dimensional coupling issues of hybrid DSMC/Navier-Stokes solvers[R]. AIAA Paper 97-2507, June 1997, presented at the 32nd AIAA Thermophysics Conference, Atlanta, GA.
[18] Glass C E, Gnoffo PA. A 3-D coupled CFD-DSMC solution method with application to the mars sample return orbiter. NASA Report TM-2000-210322, 2000.
[19] Wu J S, et al. Development and verification of a coupled DSMC-NS scheme using unstructured mesh[J]. Journal of Computational Physics, 2006(219): 579-607.
[20] Fan J, Shen C. Statistical simulation of low-speed unidirectional flows in transition regime[J]. Rarefied Gas Dynamics, 1998(2): 245-252.
[21] Fan J, Shen C. Statistical simulation of low-speed rarefied gas flows[J]. Journal of Computational Physics, 2001(167): 393-412.
[22] Wang W L, Sun Q H, Boyd I D. Towards development of a hybrid DSMC-CFD method for simulating hypersonic interacting flows[R]. AIAA Paper 2002-3099,2002.
[23] Sun Q, Boyd ID, Candler GV. A hybrid continuum/particle approach for modeling subsonic, rarefied gas flows[J]. Journal of Computational Physics, 2002(179):400-425.
[24] Wijesinghe H S , Hornung R D , Garcia A L , Hadjiconstantinou N G . Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics[J]. Journal of Fluids Engineering, 2004(126): 768-777.
[25] Garcia A L, Bell J B, Crutchfield W Y, Alder B J. Adaptive mesh and algorithm refinement using direct simulation Monte Carlo[J]. Journal of Computational Physics, 1999(154): 134-155.
[26] Wijesinghe H S, Hornung R D, Garcia A L, Hadjiconstantinou N G. Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics[J]. Journal of Fluids Engineering, 2004(126): 768-777.
[27] Garcia A L, Bell J B, Crutchfield W Y, Alder B J. Adaptive mesh and algorithm refinement using direct simulation Monte Carlo[J]. Journal of Computational Physics 1999(154): 134-155.
[28] Ozgur Aktas, Aluml N R. A Combined Continuum/DSMC Technique for Multiscale Analysis of Microfluidic Filters[J]. Journal of Computational Physics, 2002(178): 342-372.
[29] Hadjiconstantinou N G. Hybrid atomistic-continuum formulations and the moving contact-line problem. Journal of Computational Physics, 1999(154): 245-265.
[30] R. Roveda, Goldstein D B, Varghese P L. Hybrid Euler/direct simulation Monte Carlo calculation of unsteady slit flow[J]. Journal of Spacecraft and Rockets 2000(37): 753-760.
[31] 王福军编．计算流体动力学分析—CFD软件原理与应用．北京：清华大学出版社，2004．
[32] 陈伟芳，吴明巧，任兵．DSMC／EPSM混合算法研究[J]．计算力学学报，2003，20(3)：274-278．
[33] Liu M B, Liu G R, Lam K Y. Coupling meshfree particle method with molecular dynamics-a novel approach for multi-scale simulations[J]. Advances in meshfree and X-FEM methods, pp, 2002: 211-216.
[34] Rapaport D C. Microscale hydrodynamics: discrete-particle simulation of evolving flow patterns[J]. Physical Review A, 1987(36): 3288-3299.
[35] Koumoutsakos P. Multiscale flow simulation using particles. Annual Review of Fluid Mechanics, 2005(37): 457-487.
[36] 毛益明．光滑粒子流体动力学数值模拟方法研究[D]．长沙：国防科技大学，2000
[37] 董师舜，王邦荣．三维耦合光滑粒子动力学方法[J]．爆炸与冲击，1999，(增刊)：188．
[38] Monaghan J J, Kocharyan A. SPH Simulation of Multi-phase flow[J]. Computer Physics Communications, 1995(87): 225-235.
[39] Liu G R，Liu M B著．光滑粒子流体动力学—一种无网格粒子法．韩旭等译，长沙：湖南大学出版社，2005．
[40] 邓嘉胤，化工分离流动模拟中的SPH方法及跨尺度耦合算法研究，浙江大学硕士学位论文．
[41] Gingold R A, Monaghan J J. Smoothed particle hydrodynamics: theory and application to non-spherial stars. Mon. Not. R. Astron. Sot., 1997(181): 375-389.
[42] Monaghan J J. Simulating free surface flows with SPH. J. Comput. Phys, 1994, 110: 399-406.
[43] Alexander V P, Melany L H, Charles S C. Liquid-solid flows using smoothed particle hydrodynamics and the discrete element method. Powder Technology, 2001(116): 204-213.
[44] Monaghan J J, Kocharyan A. SPH simulation of multi-phase flow. Computer Physics Communications, 87(1995): 225-235.
[45] Jin Hongbin, Ding Xin. On criterions for smoothed particle hydrodynamics kernel in stable field[J]. Journal of Computational Physics, 2005(202): 699-709.
[46] Jin Hongbin, Ding Xin. On criterions for smoothed particle hydrodynamics kernels in stable field[J]. Journal of Computational Physics, 2005(202): 699-709.
[47] Morris J P, Fox P J, Zhu Y. Modeling Low Reynolds Number Incompressible Flows Using SPH[[J]. Journal of Computational Physics, 1997(136): 214-226.
[48] 吴其芬，陈伟芳．高温稀薄气体热化学非平衡流动DSMC方法．长沙：国防科技大学出版社，1999．
[49] Bird G A. Direct simulation of the Boltzmann equation[J]. Physics of Fluids, 1970(13): 2676-2681.
[50] Skorov Y V, Rickman H. Simulation of gas flow in a cometary Knudsen layer[J]. Plant Space scieuce, 1998(46): 975-996.
[51] Tsuji Y T, Yonemura S. Cluster patterns in circulating fluidized bed sp re dictedb yn umericalsi mulation[J]. Powder Technology, 1998,95(3): 254-264.
[52] Kaplan C R, Oran E S. Nonlinear filtering for low-velocity gaseous microflows[J]. AIAA Journal, 2002(40): 82-90.
[53] Nance R P, Hash D B, Hassan H A. Role of Boundary conditions in Monte Carlo simulation of MEMS devices[J]. Journal of Thermalphysics and Heat Transfer, 1998(12): 447-449.
[54] 贺群武，王秋旺等．压力边界条件下微通道内气体流动换热特性研究[J]．工程热物理学报，2004(25)：837-839．
[55] LIOU W W, FANG Y. Implicit boundary conditions for direct simulation Monte Carlo method in MEMS flow predictions [J]. Computer Modeling in Engineering & Science, 2000(1): 119-128.
[56] WANG M R, LI Z X. Simulation for gas flows in microgeometries using the direct simulation Monte Carlo method [J]. International Journal of Heat and Fluid Flow, 2004(25): 975-985.
[57] Liu M. B, Liu G. R., Lam K. Y. Coupling meshfree particle method with molecular dynamics-a novel approach for multi-scale simulations. Advances in Meshfree and X-FEM Methods, 2002: 211-216.
[58] Wu J S, et al. Development and verification of a coupled DSMC-NS scheme using unstructured mesh[J]. Journal of Computational Physics, 2006(219): 579-607.
[59] Bazilevs Y, Hughes T J R. Weak imposition of Dirichlet boundary conditions in fluid mechanics. Computers and Fluids, 2005(36): 12-26.
[60] Xing Y, Joon M Y, Tai Y C. Micromachined membrane particle filters[J]. Sensors and Actuators, 1999(73): 184-191.
[2] 李校堃，袁辉编．药物蛋白质分离纯化技术．北京：化学工业出版社，2005．
[3] 高崐玉编．色谱法在精细化工中的应用．北京：中国石化出版社，1997．
[4] Humble P H, Kelly R T, Woolley A T, Tolley H D. Electric Field Gradient Focusing of Proteins Based on Shaped Ionically Conductive Acrylic Polymer[J]. Analytical Chemistry, 2004(76): 5641-5648
[5] Jacobson S C et al. Fused Quartz Substrates for Microchip Electrophoresis[J]. Analytical Chemistry, 1995 (67): 2059-2063.
[6] Seiler K, et al. Electroosmotic Pumping and Valveless Control of Fluid Flow Within a Manifold of Capillaries on a Glass Chip[J]. Anal. Chem, 1994 (66): 3485-3491.
[7] 王立文，高殿荣．微流动系统的研究现状与趋势[J]．液压与气动，2005(10)：4-7．
[8] [美]Beskok A，Kamiadakis G E，微流动—基础与模拟．中国科学院过程工程研究所多相反应重点实验室多相复杂系统与多尺度方法课题组译，北京：化学工业出版社，2002．
[9] Rajesh Khare, Pawel Keblinskib, Arun Yethiraj. Molecular dynamics simulations of heat and momentum transfer at asolid-fluid interface: Relationship between thermal and velocity slip[J]. International Journal of Heat and Mass Transfer, 2006(49): 3401—3407.
[10] Bourgat J F, Tallec P L, Tidriri M D. Coupling Boltzmann and Navier-Stokes equations by friction[J]. Journal of Computational Physics, 1996(127): 227-245.
[11] Tallec P L, Mallinger F. Coupling Boltzmann and Navier-Stokes equations by half fluxes. Journal of Computationai Physics, 1997(136): 51-67.
[12] Tiwari S. Coupling of the Boltzmann and Euler Equations with Automatic Domain Decomposition[J]. Journal of Computer Physics, 1998(144): 710-726.
[13] Wadsworth D C, Erwin D A. One-Dimensional hybrid continuum/particle simulation approach for rarefied hypersonic flows[R], AIAA paper 90-1690(1990).
[14] Wadsworth D C, Erwin D A. Two-dimensional hybrid continuum/particle approach for rarefied flows[R], AIAA paper 92-2975(1992).
[15] Hash D B, Hassan H A. Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods[J]. Journal of Thermophysics and Heat Transfer,1996(10): 242-249.
[16] Hash D B, Hassan H A. A decoupled DSMC/Navier-Stokes analysis of a transitional flow experiment[R]. AIAA Paper 96-0353, January 1996, Presented at the 34th AIAA Aerospace Sciences Meeting and Exhibit.
[17] Hash D B , Hassan H A . Two-dimensional coupling issues of hybrid DSMC/Navier-Stokes solvers[R]. AIAA Paper 97-2507, June 1997, presented at the 32nd AIAA Thermophysics Conference, Atlanta, GA.
[18] Glass C E, Gnoffo PA. A 3-D coupled CFD-DSMC solution method with application to the mars sample return orbiter. NASA Report TM-2000-210322, 2000.
[19] Wu J S, et al. Development and verification of a coupled DSMC-NS scheme using unstructured mesh[J]. Journal of Computational Physics, 2006(219): 579-607.
[20] Fan J, Shen C. Statistical simulation of low-speed unidirectional flows in transition regime[J]. Rarefied Gas Dynamics, 1998(2): 245-252.
[21] Fan J, Shen C. Statistical simulation of low-speed rarefied gas flows[J]. Journal of Computational Physics, 2001(167): 393-412.
[22] Wang W L, Sun Q H, Boyd I D. Towards development of a hybrid DSMC-CFD method for simulating hypersonic interacting flows[R]. AIAA Paper 2002-3099,2002.
[23] Sun Q, Boyd ID, Candler GV. A hybrid continuum/particle approach for modeling subsonic, rarefied gas flows[J]. Journal of Computational Physics, 2002(179):400-425.
[24] Wijesinghe H S , Hornung R D , Garcia A L , Hadjiconstantinou N G . Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics[J]. Journal of Fluids Engineering, 2004(126): 768-777.
[25] Garcia A L, Bell J B, Crutchfield W Y, Alder B J. Adaptive mesh and algorithm refinement using direct simulation Monte Carlo[J]. Journal of Computational Physics, 1999(154): 134-155.
[26] Wijesinghe H S, Hornung R D, Garcia A L, Hadjiconstantinou N G. Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics[J]. Journal of Fluids Engineering, 2004(126): 768-777.
[27] Garcia A L, Bell J B, Crutchfield W Y, Alder B J. Adaptive mesh and algorithm refinement using direct simulation Monte Carlo[J]. Journal of Computational Physics 1999(154): 134-155.
[28] Ozgur Aktas, Aluml N R. A Combined Continuum/DSMC Technique for Multiscale Analysis of Microfluidic Filters[J]. Journal of Computational Physics, 2002(178): 342-372.
[29] Hadjiconstantinou N G. Hybrid atomistic-continuum formulations and the moving contact-line problem. Journal of Computational Physics, 1999(154): 245-265.
[30] R. Roveda, Goldstein D B, Varghese P L. Hybrid Euler/direct simulation Monte Carlo calculation of unsteady slit flow[J]. Journal of Spacecraft and Rockets 2000(37): 753-760.
[31] 王福军编．计算流体动力学分析—CFD软件原理与应用．北京：清华大学出版社，2004．
[32] 陈伟芳，吴明巧，任兵．DSMC／EPSM混合算法研究[J]．计算力学学报，2003，20(3)：274-278．
[33] Liu M B, Liu G R, Lam K Y. Coupling meshfree particle method with molecular dynamics-a novel approach for multi-scale simulations[J]. Advances in meshfree and X-FEM methods, pp, 2002: 211-216.
[34] Rapaport D C. Microscale hydrodynamics: discrete-particle simulation of evolving flow patterns[J]. Physical Review A, 1987(36): 3288-3299.
[35] Koumoutsakos P. Multiscale flow simulation using particles. Annual Review of Fluid Mechanics, 2005(37): 457-487.
[36] 毛益明．光滑粒子流体动力学数值模拟方法研究[D]．长沙：国防科技大学，2000
[37] 董师舜，王邦荣．三维耦合光滑粒子动力学方法[J]．爆炸与冲击，1999，(增刊)：188．
[38] Monaghan J J, Kocharyan A. SPH Simulation of Multi-phase flow[J]. Computer Physics Communications, 1995(87): 225-235.
[39] Liu G R，Liu M B著．光滑粒子流体动力学—一种无网格粒子法．韩旭等译，长沙：湖南大学出版社，2005．
[40] 邓嘉胤，化工分离流动模拟中的SPH方法及跨尺度耦合算法研究，浙江大学硕士学位论文．
[41] Gingold R A, Monaghan J J. Smoothed particle hydrodynamics: theory and application to non-spherial stars. Mon. Not. R. Astron. Sot., 1997(181): 375-389.
[42] Monaghan J J. Simulating free surface flows with SPH. J. Comput. Phys, 1994, 110: 399-406.
[43] Alexander V P, Melany L H, Charles S C. Liquid-solid flows using smoothed particle hydrodynamics and the discrete element method. Powder Technology, 2001(116): 204-213.
[44] Monaghan J J, Kocharyan A. SPH simulation of multi-phase flow. Computer Physics Communications, 87(1995): 225-235.
[45] Jin Hongbin, Ding Xin. On criterions for smoothed particle hydrodynamics kernel in stable field[J]. Journal of Computational Physics, 2005(202): 699-709.
[46] Jin Hongbin, Ding Xin. On criterions for smoothed particle hydrodynamics kernels in stable field[J]. Journal of Computational Physics, 2005(202): 699-709.
[47] Morris J P, Fox P J, Zhu Y. Modeling Low Reynolds Number Incompressible Flows Using SPH[[J]. Journal of Computational Physics, 1997(136): 214-226.
[48] 吴其芬，陈伟芳．高温稀薄气体热化学非平衡流动DSMC方法．长沙：国防科技大学出版社，1999．
[49] Bird G A. Direct simulation of the Boltzmann equation[J]. Physics of Fluids, 1970(13): 2676-2681.
[50] Skorov Y V, Rickman H. Simulation of gas flow in a cometary Knudsen layer[J]. Plant Space scieuce, 1998(46): 975-996.
[51] Tsuji Y T, Yonemura S. Cluster patterns in circulating fluidized bed sp re dictedb yn umericalsi mulation[J]. Powder Technology, 1998,95(3): 254-264.
[52] Kaplan C R, Oran E S. Nonlinear filtering for low-velocity gaseous microflows[J]. AIAA Journal, 2002(40): 82-90.
[53] Nance R P, Hash D B, Hassan H A. Role of Boundary conditions in Monte Carlo simulation of MEMS devices[J]. Journal of Thermalphysics and Heat Transfer, 1998(12): 447-449.
[54] 贺群武，王秋旺等．压力边界条件下微通道内气体流动换热特性研究[J]．工程热物理学报，2004(25)：837-839．
[55] LIOU W W, FANG Y. Implicit boundary conditions for direct simulation Monte Carlo method in MEMS flow predictions [J]. Computer Modeling in Engineering & Science, 2000(1): 119-128.
[56] WANG M R, LI Z X. Simulation for gas flows in microgeometries using the direct simulation Monte Carlo method [J]. International Journal of Heat and Fluid Flow, 2004(25): 975-985.
[57] Liu M. B, Liu G. R., Lam K. Y. Coupling meshfree particle method with molecular dynamics-a novel approach for multi-scale simulations. Advances in Meshfree and X-FEM Methods, 2002: 211-216.
[58] Wu J S, et al. Development and verification of a coupled DSMC-NS scheme using unstructured mesh[J]. Journal of Computational Physics, 2006(219): 579-607.
[59] Bazilevs Y, Hughes T J R. Weak imposition of Dirichlet boundary conditions in fluid mechanics. Computers and Fluids, 2005(36): 12-26.
[60] Xing Y, Joon M Y, Tai Y C. Micromachined membrane particle filters[J]. Sensors and Actuators, 1999(73): 184-191.