==================================================== The Lattice-Boltzmann Method for Gaseous Phenomena ==================================================== Summary This paper presents physically-based method for simulating gaseous phenomena by simulating the microscopic movement of fluid particles by linear and local rules on a grid of cells, so that the macroscopic averaged properties obey the desired Navier-Stokes equations. The model is defined on a discrete lattice. The problem this paper addresses is devising a way to render fluid dynamics satisfying the conditions of the NS equations while still achieving real-time computation and rendering speeds. Key Ideas 1. The lattice grid is based on cellular automata. Microscopic movements and collisions of a fluid are simulated in order to get the continuum macroscopic equations of fluid dynamics. 2. Updating the density packet distributions at each node involves two types of rules: propagation and collision. Propagation means microscopic particles will move to the nearest neighbor along their velocity direction. Collision governs how a particle moves along the lattice edges and must satisfy the conservation of mass and momentum. 3. Each cell has a sublattice of velocity vectors that point in the direction of 19 of each node's 26 3-space neighbors. The packet distributions are associated with these edges and get propagated to the nearest neighboring cell along that velocity edge at each time step. 4. The packet distribution value is a function of the average distribution of microscopic particles at that node. 5. Boundary conditions are handled using bounce-back, sending boundary-oriented packet distributions back in the direction from which they originated. 6. The real-time rendering speeds are achieved by mapping the packet distributions and velocity maps to the graphics hardware. 7. Texture splatting is used in conjunction with the hardware mapping to visually output the results. Contributions (a) Using a D3Q19 sublattice to satisfy the isotropic requirement of fluid properties (b) A new collision operator for calculating the movement of microscopic fluid particles along the lattice structure. (c) Achieving real-time computational speeds by mapping the LBM to the graphics hardware ==================================================== Equations of the Lattice-Boltzmann Method ==================================================== Summary This paper attempts to give a better explanation of the equations governing the Lattice-Boltzmann method. It discusses the local collision operator in detail and breaks down the "approximation" operator that is implemented in place of the "exact" collision operator that is unnecessarily complex and numerically inefficient. Key Ideas 1. For the Lattice-Boltzmann method, the particle distribution does not have an upper bound. The packet distribution must be greater than zero, however. 2. Lattice-Boltzmann avoids solving incompressible flows because the incompressibility requirement has been relaxed and the effects of pressure changes are controlled by an equation of state rather than a Poisson equation. Contributions (a) Fully-extended version of the collision operator function used in the LBM. ============================================================= Texture Splats for 3D Scalar and Vector Field Visualization ============================================================= Summary This paper discusses a new technique called texture splatting. This is an extension of a technique used for volume rendering known as splatting. Texture splats take advantage of the texture mapping and compositing features in graphics hardware in order to solve the problem of volume visualization with a substantial gain in performance. The paper describes methods for creating and using texture splats, and also discusses how texture splats can be extended to also represent vector fields. Key Ideas 1. Color and opacity values for a master splat are inserted into a texture map, from which the texture hardware interpolates sampled values. 2. 3D optimization is a 3D version of the 2D optimization, which is explicitly solved for. 3. Vector fields can be integrated into the scalar reconstruction function by adding a slight disturbance in the function, such as vector particles or "scratch marks." Contributions (a) A reconstruction function for 3D splats that is accurate from all viewing directions (b) Method for mathematically optimizing splats to give a smooth overlap ============================================================= Lattice-Gas Automata for the Navier-Stokes Equation ============================================================= Summary This paper explores the construction, use, and analysis of a lattice-gas automata model. The purpose of deriving such a structure is to asymptotically go over to the incompressible 2D and 3D Navier-Stokes equations. The lattice-gas automata are a grid-like structure of hexagonally connected nodes where each cell represents a microscopic "Boolean" molecule. The paper describes how such a structure can be used to model features of fluid dynamics that are described in the Navier-Stokes equations. Key Features 1. The hexagonal 2D-lattice is defined. At each vertex, there are up to four molecules of equal mass, withunit speed, whose velocities point in oneo f the four link directions. Only one molecule can occupy a single vertex at a time. The lattice is updated by having each molecule move one like, to the nearest vertex to which its velocity was point. A collision/propagation system handles interactions when two molecules with opposing velocity vectors make contact, typically sending them in directions 60 degrees off of their original direction. The important thing about this interaction is that collision operator attempts to maintain equilibrium in momentum and mass. 2. In lattice-gas models, as in general cellular automata, boundary conditions are very easy to implement.