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Introduction: Lattice Boltzmann Method for Non-fluid Applications

Introduction: Lattice Boltzmann Method for Non-fluid Applications. Ye Zhao. Lattice Boltzmann Method. Microscopic numerical solver initially desinged for fluid dynamics Simple, explicit and parallel scheme Parallel scheme for hardware acceleration Graphics hardware: GPU, GPU cluster.

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Introduction: Lattice Boltzmann Method for Non-fluid Applications

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  1. Introduction:Lattice Boltzmann Method for Non-fluid Applications Ye Zhao

  2. Lattice Boltzmann Method • Microscopic numerical solver initially desinged for fluid dynamics • Simple, explicit and parallel scheme • Parallel scheme for hardware acceleration • Graphics hardware: GPU, GPU cluster

  3. LBM in Computer Graphics • Natural phenomena with fluids • Fluid Flows • Smoke and fire • Ink dispersion • Snowing • Liquid mixture (Wang et al. 2006) • More … • Non-fluids • Lighting (Geist et al. 2004) • Image Processing

  4. Navier-Stokes Equation • Model the macroscopic behavior of fluid

  5. Fictitious particles moving along lattice links Microscopic particles inside fluids Microscopic Dynamics

  6. Lattice Boltzmann Equation • Discretize Boltzmann equation on discrete moving directions i(1988) • f : probability distribution function of particle populations on each link • Recover the Navier-Stokes equation • At the limit of low Mach number fluids

  7. Lattice Structure

  8. Collision • BGK model (1954) for equilibrium • Streaming along a link to a neighbor Single-relaxation-time LBM

  9. Macroscopic Properties • Density, velocity and viscosity

  10. Body Forces

  11. Graphics Hardware • Graphics processing units(GPU) • Low price • Inherently parallel • Programmable • Booming growth rate on speed • 3.0 GHz dual-core Pentium4: 24.6 GFLOPs • NVIDIA GeForce FX 7800: 165 GFLOPs • Ideal general-purpose applications • Large data sets • High parallelism • Minimal dependencies between data elements

  12. Graphics: Computation: Fragments Fragments Texture Images Data Fragment Processing Numerical Calculation Fragment w/ Colors Results Computation on GPU

  13. LBM Acceleration on GPU • Computation on cluster • Large-scale simulation • Domain decomposition • Simple addition, subtraction and multiplication

  14. LBM for Non-fluids Applications • Actually LBM is a special numerical solver for partial differential equations (PDE) • PDEs are widely used in graphics and visualization applications • Image processing • Surface processing • Volume graphics and visualization • Computer vision …

  15. Use LBM in these fields? • Pros: • LBM is simple to implement • both CPU and GPU • Flexible and easy to modify its scheme • Computational speed very fast with hardware acceleration • Cons: • Difficult understanding at the beginning • Memory usage • Worth a try!

  16. Research Topics • How to modify LBM scheme in order to theoretically recover the preferred PDEs • How to improve hardware acceleration performance • New-generation GPU architecture • Memory use optimization • How to incorporate with particular applications

  17. BGK model (1954) for equilibrium Modify LBM Equilibrium Equation for Diffusion • For fluid dynamics • For diffusion only

  18. Recover Diffusion Equation • Chapman-Enskog expansion, ε represents a small expansion parameter (Knudsen number) • Use this and Taylor expansion on LBM equation

  19. Recover Diffusion Equation (Continue) • We get expanded equation : • Constraints • Summation on expanded equation over i

  20. Recover Diffusion Equation (Continue) • Diffusion tensor defined by lattice structure • For a particular lattice structure we obtain • For D3Q19

  21. Applications • Volume Smoothing

  22. Volume Smoothing

  23. Volume Smoothing

  24. Surface Smoothing

  25. Surface Smoothing

  26. Image De-noising

  27. Poisson Image Editing • Poisson Equation with Dirichlet boundary • is a computed guidance field computed from origin image

  28. Performance • GPU NVidia 8800GTX with 768MB memory

  29. More Applications • Apply to others beyond diffusion • Use level set equation as a pivot example • It has many application fields • Combine several different types of PDEs

  30. Level Set • Propagation interface as zero level set • d is distance to interface • Surface motion F = Signed speed in direction of normal

  31. F Speed Function • Curvature flow  diffusion on distance field • Advected propagation by a external field • Constant speed expansion

  32. LBM for Level Set • Use distance field as the computational primitive in LBM • Play the role as density in fluids • Design appropriate equlibrium function for PDEs

  33. Thanks

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