晶格波茲曼法為一較新且適合作高速計算解流體力學的數值方法。本研究中為利用Lee 的多相流模型並加以實做在多張圖形顯示卡叢集上以模擬複雜且具高密度差之流體並得到高效率的結果。其中驗證的例子有消除靜止液珠介面上因力量不平衡所造成的假性速度、液珠的融合震盪、二相的分離現象以及不同的液珠碰撞結果。而在碰撞過程中主要為調查韋伯數與碰撞參數對於結果的影響，液珠的碰撞融合與碰撞分離現象也與理論實驗相符合。此外，在本研究中對於多圖形顯示卡施作部分也提出了有效率的資料傳遞型態，結果顯示即便是使用多張圖形顯示卡計算，仍舊能維持高平行效率。

Lattice Boltzmann method (LBM) is a novel and relative new approach in the field of hydrodynamics and well-suited to efficient high-performance computing implementation. In this thesis, a three-dimensional two-phase lattice Boltzmann model is adopted based on Lee et al. on multi graphic processing unit (GPU) cluster. Such a complex binary system at high density ratio is achieved easily and yields high performance. Here, several numerical validation are examed. First, the spurious velocity appearing near the two-phase interface caused by the force imbalance is successfully eliminated in a stationary droplet case. Droplets oscillation is validated and the results are in good agreement with analytical solutions. The behaviours of two droplets collisions are also investigated. "Coalescence" and "stretching separation" phenomena are observed and studied in terms of two dimensionless variables, Weber number We and impact parameter B, showing good consistency to benchmark. Further, an efficient multi-GPU cluster implementation is proposed and achieves good scalability even in the case with high GPU number.

[1] U. Frisch, B. Hasslacher, and Y. Pomeau, \Lattice-gas automata for the Navier-Stokes equation," Phys. Rev. Lett. 56, 1505, (1986).
[2] S. Wolfram, \Cellular automaton fluids 1: Basic theory," J. Stat. Phys. 45, 471, (1986).
[3] F. J. Higuera, S. Sussi, and R. Benzi, \3-dimensional flows in complex geometries with the lattice Boltzmann method," Europhys. Lett. 9, 345, (1989).
[4] F. J. Higuera, and J. Jemenez, \Boltzmann approach to lattice gas simulations," Europhys. Lett. 9, 663, (1989).
[5] P. L. Bhatnagar, E. P. Gross, and M. Grook, \A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems," Phys. Rev. E 94, 511, (1954).
[6] S. Harris, \An introduction to the theory of the Boltzmann equation," Holt, Rinehart and Winston, New York, (1971).
[7] U. Frisch, D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. P. Rivet, \Lattice gas hydrodynamics in two and three dimensions," Complex Syst. 1, 649, (1987).
[8] D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, \Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics," Phys. Fluids. 6, 1285, (1994).
[9] R. Scardovelli and S. Zaleski, \Direct numerical simulation of free-surface and intercal flow," Annu.Rev.Fluid Mech. 31, 567, (1999).
[10] S. Osher and R. P. Fedkiw, \Level set method: An overview and some recent results," J. Comput. Phys. 169, 463, (2001).
[11] T. Y. Hou, J. S. Lowengrub, M. J. Shelley, \Boundary integral methods for multicomponent fluids and multiphase materials," J. Comput.Phys. 169, 302, (2001).
[12] Andrew K. Gunstensen and Daniel H. Rothman, \Lattice Boltzmann model of immiscible fluids," Phys. Rev. 43, 4320-4327, (1991).
[13] Daniel H. Rothman and Jeffrey M. Keller, \Immiscible cellular-automaton fluids," J. Stat. Phys. 52(3), 1119-1127, (1988).
[14] X. Shan and H. Chen, \Lattice Boltzmann model for simulating flows with multiple phases and components," Phys. Rev. E. 47, 1815-1819, (1993).
[15] X. Shan and H. Chen, \Simulation of Nonideal Gases and Liquid-GasPhase Transitions by the Lattice Boltzmann Equation," Phys. Rev. E. 49, 2941-2948, (1994).
[16] X. Shan, and G. D. Doolen, \Multicomponent Lattice-Boltzmann Model With Interparticle Interaction," J. Stat. Phys. 52, 379-393, (1995).
[17] M. R. Swift, W. R. Osborn, J. M. Yeomans \Lattice Boltzmann simulation of nonideal fluids," Phys. Rev. Lett. 75(5), 830-833, (1995).
[18] M. R. Swift, W. R. Osborn, J. M. Yeomans \Lattice Boltzmann simulations of liquid-gas and binary-fluid systems," Phys. Rev. E,54, 5041-5052, (1996).
[19] X. He. Shan, G.D. Doolen, \A discrete Boltzmann equation model for non-ideal gases," Phys. Rev. 57, R13, (1998).
[20] X. He, S. Chen, R. Zhang, \ A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of RayleighTaylor instability," J. Comput. Phys. 152, 642, (1999).
[21] T. Inamuro, T. Ogata, S. Tajima, N. Konishi, \A lattice Boltzmann method for incompressible two-phase flows with large density differences," J. Comput. Phys. 198, 628, (2004).
[22] T. Lee and P. F. Fischer, \Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases," Phys. Rev.E. 74, 046709, (2006).
[23] D. Jacqmin, \Calculation of two-phase Navier-Stokes flows using phase-field modeling," J. Comput. Phys. 155, 96-127, (1999).
[24] T. Lee and C. L. Lin, \A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio," J. Comput. Phys. 206, 16-47, (2005).
[25] X. He, S. Chen, R. Zhang, \A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability," J. Comput. Phys. 152(2), 642-663, (1999).
[26] T. Lee, \Effects of incompressibility on the elimination of parasitic currents in the lattice Boltzmann equation method for binary fluids ," Comput. Math. Appl.58, 987-994, (2010).
[27] T. Lee and C. L. Lin, \Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces," J. Comput. Phys. 229, 8045-8063, (2010).
[28] J. Bolz, I. Farmer, E. Grinspun, and P. Schroder, \Sparse matrix solvers on the GPU: Conjugate gradients and multigrid," ACM Trans. Graph. (SIGGRAPH) 22, 917, (2003).
[29] F. A. Kuo, M. R. Smith, C. W. Hsieh, C. Y. Chou, and J. S. Wu, \GPU acceleration for general conservation equations and its application to several engineering problems," Comput. Fluids 45, 147, (2011).
[30] J. Tolke, \Implementation of a lattice Boltzmann kernel using the compute unied device architecture developed by nVIDIA," Comput. Visual Sci. 13, 29, (2008).
[31] J. Tolke, and M. Krafczyk, \TeraFLOP computing on a desktop PC with GPUs for 3D CFD," Int. J. Comput. Fluid D. 22, 443, (2008).
[32] E. Riegel, T. Indinger, and N. A. Adams, \Implementation of a Lattice-Boltzmann method for numerical fluid mechanics using the nVIDIA CUDA technology," CSRD 23, 241, (2009).
[33] F. Kuznik, C. Obrecht, G. Rusaouen, and J. J. Roux, \LBM based flow simulation using GPU computing processor," Comput. Math. Appl. 59, 2380, (2010).
[34] J. Habich, T. Zeiser, G. Hager, and G. Wellein, \Performance analysis and optimization strategies for a D3Q19 lattice Boltzmann kernel on nVIDIA GPUs using CUDA," Adv. Eng. Softw. 42, 266, (2011).
[35] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, \A new approach to the lattice Boltzmann method for graphics processing units," Comput. Math. Appl. 61, 3628, (2011).
[36] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, \Multi-GPU implementation of a hybrid thermal lattice Boltzmann solver using the TheLMA framework," Comput. Fluids. 80, 269, (2013).