In this thesis, the implementation of GPU with CUDA architecture on lattice Boltzmann model is presented. Simulations of 3D lid-driven cavity flow is conducted as a test case with multi-GPU. The optimization of multi-GPU with two-dimensional domain decomposition is also discussed here. The numerical results are validated with benchmark solutions and the performance of the GPU implementation is also discussed. In the present work, we can achieve 17664.23 MLUPS for 384X384X384 grids with 96 nVIDIAr Tesla M2070 GPU cards.

[1] U. Firsch, B. Hasslacher, and Y. Pomeau, \Lattice-gas automata for the NavierStokes equation," Phys. Rev. Lett. 56, 1505, (1986).
[2] S. Wolfram, \Cellular automata
uids 1: Basic theory," J. Stat. Phys. 45, 471
(1986).
[3] P.L. Bhatnagar, E. P. Gross, and M. Krook, \A model for collision processes
in gases. I. Small amplitude processes in charged and neutral one-component
systems," Phys. Rev. 94, 511, (1954).
[4] S. Harris, \An introduction to the theory of the Boltzmann equation," Holt,
Rinehart and Winston, New York, (1971).
[5] F. J. Higuera, S. Succi, and R. Benzi, \Lattice gas dynamics with enhanced
collisions," Europhys. Lett. 9, 345, (1989).
[6] F. J. Higuera, and J. Jimenez, \Boltzmann approach to lattice gas
simulations," Europhys. Lett. 9, 663, (1989).
[7] Y. H. Qian, D. d'Humieres, and P. Lallemand, \Lattice BGK models for NavierStokes equation," Europhys. Lett. 17, 479, (1992).
[8] H. Chen, S. Chen, and W. H. Matthaeus, \Recovery of the Navier-Stokes
equations using a lattice-gas Boltzmann method," Phys. Rev. A. 45, 5339,
(1992).
[9] D. V. Patil, K. N. Lakshmisha, and B. Rogg, \Lattice Boltzmann simulations
of lid-driven
ow in deep cavities," Comput. Fluids 35, 1116, (2006).
[10] G. R. McNamara, and G. Zanetti, \Use of the Boltzmann equation to simulate
lattice-gas automata," Phys. Rev. Lett. 61, 2332, (1988).
[11] S. Chen, H. Chen, D. O. Martinez, and W. H. Matthaeus, \Lattice Boltzmann
model for simulation of magnethydrodynamics," Phys. Rev. Lett. 67, 3776,
(1991).
[12] X. He, and L. S. Luo, \Theory of the lattice Boltzmann method: From the
Boltzmann equation to the lattice Boltzmann equation," Phys. Rev. E 56,
6811, (1997).
[13] X. He, and L. S. Luo, \A priori derivation of the lattice Boltzmann equation,"
Phys. Rev. E 55, 6333, (1997).
[14] K. Kono, T. Ishizuka, H. Tsuda, and A. Kurosawa, \Application of lattice
Boltzmann model to multiphase
ows with phase transition," Comput. Phys.
Commun. 129, 110, (2000).
[15] S. Hou, X. Shan, Q. Zou, G. D. Doolen, and W. E. Soll, \Evaluation of two
lattice Boltzmann models for multiphase
ows," J. Comput. Phys. 138, 695,
(1997).
[16] X. He, S. Chen, and R. Zhang, \A lattice Boltzmann scheme for
incompressible multiphase
ow and its application in simulation of RayleighTaylor instability," J. Comput. Phys. 152, 642, (1999).
[17] C. H. Shih, C. L. Wu, L. C. Chang, and C. A. Lin, \Lattice Boltzmann
simulations of incompressible liguid-gas system on partial wetting surface,"
Phil. Trans. R. Soc. A 369, 2510, (2011).
[18] M. Krafczyk, M. Schulz, and E. Rank, \Lattice-gas simulations of two-phase

ow in porous media," Commun. Numer. Meth. Engng 14, 709, (1998).
[19] J. Bernsdorf, G. Brenner, and F. Durst, \Numerical analysis of the pressure
drop in porous media
ow with lattice Boltzmann (BGK) automata," Comput.
Phys. Commun. 129, 247, (2000).
[20] D. M. Freed, \Lattice-Boltzmann method for macroscopic porous media
modeling," Int. J. Mod. Phys. C 9, 1491, (1998).
[21] Y. Hashimoto, and H. Ohashi, \Droplet dynamics using the lattice-gas
method," Int. J. Mod. Phys. 8, 977, (1997).
[22] H. Xi, and C. Duncan, \Lattice Boltzmann simulations of three-dimensional
single droplet deformation and breakup under simple shear
ow," Phys. Rev.
E 59, 3022, (1999).
[23] R. A. Brownlee, A.N Gorban, J.Levesley,"Stabilisation of the latticeBoltzmann method using the Ehrenfests' coarse-graining", (2008)
[24] D. d'Humieres, \Generalized lattice Boltzmann equation," In Rareed gas
dymanics - Theory and simulations, Progress in Astronautics and Aeronautics,
vol. 159, Shizgal BD, Weaver DP(eds). AIAA: Washington, DC, 45-458, (1992).
[25] P. Lallemand, and L. S. Luo, \Theory of the lattice Boltzmann method:
dispersion, dissipation, isotropy, Galilean invariance, and stability," Phys. Rev.
E 61, 6546, (2000).
[26] D. d'Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, and L. S. Luo, \Multirelaxation-time lattice Boltzmann models in three dimensions," Phil. Trans. R.
Soc. Lond. A 360, 437, (2002).
[27] J. S. Wu, and Y. L. Shao, \Simulation of lid-driven cavity
ows by parallel
lattice Boltzmann method using multi-relaxation-time scheme," Int. J. Numer.
Meth. Fluids 46, 921, (2004).
[28] H. Yu, L. S. Luo, and S. S. Girimaji, \LES of turbulent square jet
ow using
an MRT lattice Boltzmann model," Comput. Fluids 35, 957, (2006).
[29] L. S. Lin, Y. C. Chen, and C. A. Lin, \Multi relaxation time lattice Boltzmann
simulations of deep lid driven cavity
ows at different aspect ratios," Comput.
Fluids 45, 233, (2011).
[30] X. He, Q. Zou, L. S. Luo, and M. Dembo, \Analytic solutions of simple
ows
and analysis of nonslip boundary conditions for the lattice Boltzmann BGK
model," J. Stat. Phys. 87, 115, (1997).
[31] P. A. Skordos, \Initial and boundary conditions for the lattice Boltzmann
method," Phys. Rev. E 48, 4823, (1993).
[32] D. R. Noble, S. Chen, J. G. Georgiadis, and R. O. Buckius, \A consistent
hydrodynamics boundary condition for the lattice Boltzmann method," Phys.
Fluids 7, 203, (1995).
[33] T. Inamuro, M. Yoshino, and F. Ogino, \A non-slip boundary condition for
lattice Boltzmann simulations," Phys. Fluids 7, 2928, (1995).
[34] S. Chen, D. Martinez, and R. Mei, \On boundary conditions in lattice
Boltzmann methods," Phys. Fluids 8, 2527, (1996).
[35] Q. Zou, and X. He, \On pressure and velocity boundary conditions for the
lattice Boltzmann BGK model," Phys. Fluids 9, 1591, (1997).
[36] C. F. Ho, C. Chang, K. H. Lin, and C. A. Lin, \Consistent boundary conditions
for 2D and 3D lattice Boltzmann simulations," CMES 44, 137, (2009).
[37] S. Hou, Q. Zou, S. Chen, G. Doolean, A. C. Cogley, Simulation of cavity
ow
by the lattice Boltzmannmethod, J. Comput Phys,118, 329, (1995).
[38] P. N. Shankar, and M. D. Deshpande, \Fluid mechanics in the driven cavity,"
Annu. Rev. Fluid Mech. 32, 93, (2000).
[39] S. Albensoeder, H. C. Kuhlmann, \Accurate three-dimensional lid-driven
cavity
ow," J. Comput. Phys., 206, 536, (2005).
[40] 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).
[41] 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).
[42] J. Tolke, \Implementation of a lattice Boltzmann kernel using the compute
unied device architecture developed by nVIDIA," Comput. Visual Sci. 13,
29, (2008).
[43] J. Tolke, and M. Krafczyk, \TeraFLOP computing on a desktop PC with GPUs
for 3D CFD," Int. J. Comput. Fluid D. 22, 443, (2008).
[44] E. Riegel, T. Indinger, and N. A. Adams, \Implementation of a LatticeBoltzmann method for numerical
uid mechanics using the nVIDIA CUDA
technology," CSRD 23, 241, (2009).
[45] F. Kuznik, C. Obrecht, G. Rusaouen, and J. J. Roux, \LBM based
ow
simulation using GPU computing processor," Comput. Math. Appl. 59, 2380,
(2010).
[46] 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).
[47] 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).
[48] 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).
[49] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, \Multi-GPU
implementation of the lattice Boltzmann method," Comput. Math. Appl. 65,
252, (2013).
[50] X. Wang, T. Aoki, \Multi-GPU performance of incompressible
ow
computation by lattice Boltzmann method on GPU cluster," Parallel.
Computing. 37, 521, (2011).
[51] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, \Scalable lattice
Boltzmann solvers for CUDA GPU clusters," Comput. Fluids. 39, 259, (2013).
[52] H. W. Chang, P.Y. Hong, L.S. Lin and C.A. Lin, \Simulations of threedimensional cavity
ows with multi relaxation time lattice Boltzmann method
and graphic processing units," Procedia Engineering. 61, 94, (2013)
[53] P. Y. Hong\Lattice Boltzmann model with multi-GPU implementation,"
Master's thesis, National Tsing Hua University, (2013)
[54] T. I. Gombosi, \Gas kinetic theory," Cambridge University Press, (1994).
[55] R. Cornubert, D. d'Humieres, and D Levermore, \A Knudsen layer theory for
lattice gases," Physica D 47, 241, (1991)
[56] I. C. Kim \Second Order Bounce Back Boundary Condition for the Latice
Boltzmann Fluid Simulation," KSME 14, 84, (2000)
[57] A. J. C. Ladd, \Numerical simulations of particulate suspensions via a
discretized Boltzmann equation. Part 1. Theoretical foundation, J. Fluid Mech,
271, 285, (1994).
[58] A. J. C. Ladd, \Numerical simulations of particulate suspensions via a
discretized Boltzmann equation. Part 2. Numerical results, J. Fluid Mech, 271,
311, (1994).
[59] M. Bouzidi, M. Firdaouss and P. Lallemand \Momentum transfer of a
Boltzmann-lattice
uid with boundaries," Phys. Fluids, 13, 3452, (2001)
[60] S. Albensoeder, H. C. Kuhlmann, H. J. Rath, \Three-dimensional centrifugal-
ow instabilities in the lid-driven-cavity problem," Phys. Fluids, 13, 121,
(2001).
[61] F. Auteri, N. Parolini, L. Quartapelle, \Numerical investigation on the stability
of singular driven cavity
ow," J. Comput. Phys., 183, 1, (2002).
[62] M. Sahin, R. G. Owens, \A novel fully-implicit nite volume method applied to
the lid-driven cavity problem. Part II. Linear stability analysis," Int. J. Numer.
Meth. Fluids, 42, 79, (2003).
[63] L. S. Lin, H. W. Chang, C. A. Lin, \Multi relaxation time lattice Boltzmann
simulations of transition in deep 2D lid driven cavity using GPU," Comput.
Fluids, 80, 381, (2013).
[64] R. Iwatsu, K. Ishii, T. Kawamura, K. Kuwahara, J. M. Hyun, \Numerical
simulation of three-dimensional
ow structure in a driven cavity," Fluid Dyn.
Res., 5, 173, (1989).
[65] G. Guj, F. Stella, \A vorticity-velocity method for the numerical solution of
3D incompressible
ows," J. Comput. Phys., 106, 286, (1993).
[66] Y. Feldman, A. Yu. Gelfgat, \Oscillatory instability of a three-dimensional
lid-driven
ow in a cube," Phys. Fluids, 22, 093602, (2010).
[67] A. Liberzon, Yu. Feldman , A. Yu. Gelfgat, \Experimental observation of the
steady-oscillatory transition in a cubic lid-driven cavity," Phys. Fluids, 23,
084106, (2011).
[68] C. Obrecht, F. Kuznik, B. Tourancheau, and J. J. Roux, \The TheLMA
project: A thermal lattice Boltzmann solver for the GPU," Comput. Fluids.
54, 118, (2012).