This Article 
 Bibliographic References 
 Add to: 
The Lattice-Boltzmann Method on Optimal Sampling Lattices
July/August 2009 (vol. 15 no. 4)
pp. 630-641
Usman R. Alim, Simon Fraser University, Burnaby
Alireza Entezari, University of Florida, Gainesville
Torsten Möller, Simon Fraser University, Burnaby
In this paper, we extend the single relaxation time Lattice-Boltzmann Method (LBM) to the 3D body-centered cubic (BCC) lattice. We show that the D3bQ15 lattice defined by a 15 neighborhood connectivity of the BCC lattice is not only capable of more accurately discretizing the velocity space of the continuous Boltzmann equation as compared to the D3Q15 Cartesian lattice, it also achieves a comparable spatial discretization with 30 percent less samples. We validate the accuracy of our proposed lattice by investigating its performance on the 3D lid-driven cavity flow problem and show that the D3bQ15 lattice offers significant cost savings while maintaining a comparable accuracy. We demonstrate the efficiency of our method and the impact on graphics and visualization techniques via the application of line-integral convolution on 2D slices as well as the extraction of streamlines of the 3D flow. We further study the benefits of our proposed lattice by applying it to the problem of simulating smoke and show that the D3bQ15 lattice yields more detail and turbulence at a reduced computational cost.

[1] S. Albensoeder and H.C. Kuhlmann, “Accurate Three-Dimensional Lid-Driven Cavity Flow,” J. Computational Physics, vol. 206, no. 2, pp. 536-558, 2005.
[2] B. Cabral and L.C. Leedom, “Imaging Vector Fields using Line Integral Convolution,” Proc. ACM SIGGRAPH '93, pp. 263-270, 1993.
[3] J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, thirded. Springer, 1999.
[4] A. Cortes and J. Miller, “Numerical Experiments with the Lid Driven Cavity Flow Problem,” Computers and Fluids, vol. 23, no. 8, pp. 1005-1027, 1994.
[5] C. de Boor, K. Höllig, and S. Riemenschneider, Box Splines. Springer Verlag, 1993.
[6] P. Dellar, “Incompressible Limits of Lattice Boltzmann Equations Using Multiple Relaxation Times,” J. Computational Physics, vol. 190, no. 2, pp. 351-370, 2003.
[7] D. d'Humières, M. Bouzidi, and P. Lallemand, “Thirteen-Velocity Three-Dimensional Lattice Boltzmann Model,” Physical Rev. E, vol. 63, no. 6, p. 66702, 2001.
[8] Y. Dobashi, K. Kaneda, H. Yamashita, T. Okita, and T. Nishita, “A Simple, Efficient Method for Realistic Animation of Clouds,” Proc. 27th Ann. Conf. Computer Graphics and Interactive Techniques, pp. 19-28, 2000.
[9] D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, first ed. Prentice-Hall, Inc., 1984.
[10] D. Ebert and R. Parent, “Rendering and Animation of Gaseous Phenomena by Combining Fast Volume and Scanline A-Buffer Techniques,” Proc. 17th Ann. Conf. Computer Graphics and Interactive Techniques, pp. 357-366, 1990.
[11] A. Entezari, “Optimal Sampling Lattices and Trivariate Box Splines,” PhD dissertation, Simon Fraser Univ., Vancouver, Canada, July 2007.
[12] A. Entezari, R. Dyer, and T. Möller, “Linear and Cubic Box Splines for the Body Centered Cubic Lattice,” Proc. IEEE Conf. Visualization, pp. 11-18, Oct. 2004.
[13] A. Entezari, D. Van De Ville, and T. Möller, “Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice,” IEEE Tran. Visualization and Computer Graphics, vol. 14, no. 2, pp. 313-328, Mar./Apr. 2008.
[14] R. Fedkiw, J. Stam, and H.W. Jensen, “Visual Simulation of Smoke,” Proc. ACM SIGGRAPH '01, pp. 15-22, 2001.
[15] N. Foster and D. Metaxas, “Modeling the Motion of a Hot, Turbulent Gas,” Proc. 24th Ann. Conf. Computer Graphics and Interactive Techniques, pp. 181-188, 1997.
[16] S. Hou, Q. Zou, S. Chen, G. Doolen, and A. Cogley, “Simulation of Cavity Flow by the Lattice Boltzmann Method,” J. Computational Physics, vol. 118, no. 2, pp. 329-347, 1995.
[17] M. Junk, A. Klar, and L. Luo, “Asymptotic Analysis of the Lattice Boltzmann Equation,” J. Computational Physics, vol. 210, no. 2, pp.676-704, 2005.
[18] M. Junk and Z. Yang, “One-point Boundary Condition for the Lattice Boltzmann Method,” Physical Rev. E, vol. 72, no. 6, p.66701, 2005.
[19] H. Ku, R. Hirsh, and T. Taylor, “A Pseudospectral Method for Solution of the Three-Dimensional Incompressible Navier-Stokes Equations,” J. Computational Physics, vol. 70, no. 2, pp. 439-462, 1987.
[20] R. Mei, D. Yu, and L. Luo, “Lattice Boltzmann Method for 3D Flows with Curved Boundary,” J. Computational Physics, vol. 161, no. 2, pp. 680-699, 2000.
[21] T. Meng, B. Smith, A. Entezari, A.E. Kirkpatrick, D. Weiskopf, L. Kalantari, and T. Möller, “On Visual Quality of Optimal 3D Sampling and Reconstruction,” Proc. Graphics Interface Conf. '07, pp.265-272, May 2007.
[22] K. Morton and D. Mayers, Numerical Solution of Partial Differential Equations. Cambridge Univ. Press, 1994.
[23] D.P. Petersen and D. Middleton, “Sampling and Reconstruction of Wave-Number-Limited Functions in + N-Dimensional Euclidean Spaces,” Information and Control, vol. 5, no. 4, pp. 279-323, Dec. 1962.
[24] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation. Morgan Kaufmann, 2004.
[25] Y. Qian, D. D'Humières, and P. Lallemand, “Lattice BGK Models for Navier-Stokes Equation,” Europhysics Letters, vol. 17, 479, 1992.
[26] F. Qiu, Y. Zhao, Z. Fan, X. Wei, H. Lorenz, J. Wang, S. Yoakum-Stover, A. Kaufman, and K. Mueller, “Dispersion Simulation and Visualization for Urban Security,” Proc. IEEE Conf. Visualization '04, pp. 553-560, 2004.
[27] D. Stalling and H.-C. Hege, “Fast and Resolution Independent Line Integral Convolution,” Proc. ACM SIGGRAPH '95, pp.249-256, 1995.
[28] J. Stam, “Stable Fluids,” Proc. ACM SIGGRAPH '99, pp. 121-128, 1999.
[29] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford Univ. Press, 2001.
[30] T. Theußl, T. Möller, and E. Gröller, “Optimal Regular Volume Sampling,” Proc. IEEE Visualization Conf. '01, pp. 91-98, Oct. 2001.
[31] D. Van De Ville, T. Blu, M. Unser, W. Philips, I. Lemahieu, and R. Van de Walle, “Hex-Splines: A Novel Spline Family for Hexagonal Lattices,” IEEE Trans. Image Processing, vol. 13, no. 6, pp. 758-772, June 2004.
[32] X. Wei, W. Li, K. Mueller, and A. Kaufman, “Simulating Fire with Texture Splats,” Proc. IEEE Visualization Conf. '02, pp. 227-235, 2002.
[33] X. Wei, W. Li, K. Mueller, and A. Kaufman, “The Lattice-Boltzmann Method for Simulating Gaseous Phenomena,” IEEE Trans. Visualization and Computer Graphics, vol. 10, no. 2, pp. 164-176, Mar./Apr. 2004.
[34] X. Wei, Y. Zhao, Z. Fan, W. Li, S. Yoakum-Stover, and A. Kaufman, “Blowing in the Wind,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, pp. 75-85, 2003.
[35] D.A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer-Verlag Telos, 2000.

Index Terms:
Visual simulation, animation, physically based modeling, BCC, volume modeling, vector field data, flow visualization, optimal regular sampling.
Usman R. Alim, Alireza Entezari, Torsten Möller, "The Lattice-Boltzmann Method on Optimal Sampling Lattices," IEEE Transactions on Visualization and Computer Graphics, vol. 15, no. 4, pp. 630-641, July-Aug. 2009, doi:10.1109/TVCG.2008.201
Usage of this product signifies your acceptance of the Terms of Use.