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This guide provides an in-depth overview of smoke simulation, including semi-Lagrangian convection, vorticity confinement, and pressure boundary conditions. Learn about compressible and incompressible equations, using an octree data structure, and rendering techniques for thick and thin smoke. Discover the Operator Splitting technique and how to maintain incompressibility in fluid simulations. Get insights on solving the system, updating velocity, density, and temperature fields, and displaying accurate smoke effects. Dive into the world of smoke simulation for a compelling visual experience.
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Motivation • Movie • Game • Engineering
Introduction • Ideally • Looks good • Fast simulation • Looks good? • Look plausible • Doesn’t need to be exactly correct • doesn’t suffer from excessive numerical dissipation
Overview • Simulation • Compressible/Imcompressible Equation • Vortex particle method • Using octree data structure • Rendering • Thick smoke: plain particles • Thin smoke: adaptive particles • Using Compensated Ray Marching
Compressible Fluids • Yngve, G. D., O'Brien, J. F., Hodgins, J. K., 2000, Animating Explosions. The proceedings of ACM SIGGRAPH 2000, New Orleans, July 23-28, pp. 29-36.
Operator Splitting • Stam used the technique in his Stable Fluids paper, 1999. • Allows us to solve the Navier-Stokes equation in Parts. • As long as each part is stable the technique will be stable.
Semi-Lagrangian Convection • Trace each point p in the field backwards in time. The new velocity at x is therefore the velocity that the particle had a time step ago at the old location.
Smoke Viscous Diffusion • In the simulation of smoke, it is reasonable to consider the fluid inviscid. Therefore this term is zero and need not be solved for. • In fact, the implicit solver that is used will take energy away from the system anyway, so there is a numerical dampening that can look like a non-zero viscosity.
Body Forces • Gravity, user forces, and object interaction. • The visible smoke is from a density field. • Areas of high density fall in the direction of gravity. • Heat moves against gravity (hot air rises).
Incompressibility • At each time step we start with a velocity field that is divergence free. We need this to be true at the end of the time step because the fluid is incompressible. • After solving for convection, and body forces the velocity field is not divergence free.
Incompressibility • Fedkiw, R., Stam, J. and Jensen, H.W.,"Visual Simulation of Smoke",SIGGRAPH 2001, 23-30 (2001).
Incompressible Euler Equations self-advection Forces incompressible (Navier-Stokes without viscosity)
Additional Equations We now know all the finite differences we need to add the forces for heat and density… smoke’s density temperature
Vorticity Confinement • The numerical dissipation, and the coarse grid size, cause the fine scale detail of turbulent swirling smoke to vanish. • Identify where the curl is highest, and add back in a rotational force there. • “Vorticity Confinement” force preserves • swirling nature of fluids.
What is vorticity? • A measure of the local rotation in a fluid flow.
Pressure Boundary Conditions • A Neumann boundary condition is a restriction on the derivative of a function.
Velocity Boundary Condition • A Dirichlet boundary condition is a restriction on the value of a function.
Solving the System • Need to calculate: • Start with initial state • Calculate new velocity fields • New state:
Smoke Simulation • While (simulating) • Get external forces (if any) from UI • Get density/heat sources (from UI or init grid cells) • Update velocity • Update density • Update temperature • Display density
Smoke Simulation • While (simulating) • Get external forces (if any) from UI • Get density/heat sources (from UI or init grid cells) • Update velocity • Update density • Update temperature • Display density
Update Velocity • Equation: • First term: • Advection • Move the fluid through its velocity field (Du/Dt=0) • Second term: • external forces • Final term: • pressure update
Update Velocity • Add external forces (fbuoy + forces from UI + fconf) • Advection (semi-lagrangian step – trace particles back) • Pressure update (solve linear system)
Smoke Simulation • While (simulating) • Get external forces (if any) from UI • Get density/heat sources (from UI or init grid cells) • Update velocity • Update density • Update temperature • Display density
Update density • Equation • First term: • Advection • Move the temperature through velocity field • Second term: • Diffusion • Can skip this term • Second term: • external sources
Update density • Add Sources (pick grid cells or from UI) • Advection (semi-lagrangian step – trace particles back) • Diffusion (solve linear system – can skip this step)
Smoke Simulation • While (simulating) • Get external forces (if any) from UI • Get density/heat sources (from UI or init grid cells) • Update velocity • Update density • Update temperature • Display density
Update temperature • Equation • First term: • Advection • Move the temperature through velocity field • Second term: • Diffusion • Can skip this term
Update Temperature • Add Sources (grid cells or objects or UI) • Advection (semi-lagrangian step – trace particles back) • Diffusion (solve linear system – can skip this step)
Smoke Simulation • While (simulating) • Get external forces (if any) from UI • Get density/heat sources (from UI or init grid cells) • Update velocity • Update density • Update temperature • Display density
Display density • Use any approach you want • Visualize the density field: • just opengl render a bunch of cubes (corresponding to grid cells) that have alpha values based on how dense the smoke is.
Numerical Dissipation • ‘Stable Fluids’ method dampens the flow • Typical with semi- Lagrangianmethods • Improve using • “Vorticity Confinement” force
Total Forces • User supplied fields • Buoyancy force • New confinement force
Other Methods • Vortex particle method • Using octree data structure
Vortex particle method-Lagrangian primitives • Curves carry the vorticity • Each local vortex induces a weighted rotation
Vortex particle method-Lagrangian primitives • Curves carry the vorticity • Each local vortex induces a weighted rotation
Method of simulation • Vortex particles (for motion) organized as curves. = tangent • Smoke particles (for visualisation) • Curves carry vortices • Vortices induce a velocity field • velocity field deforms curves & smoke At every step: • Advect the curves • Stretch • Advect the smoke
Method of simulation • Vortex particles (for motion) organized as curves. = tangent • Smoke particles (for visualisation) • Curves carry vortices • Vortices induce a velocity field • velocity field deforms curves & smoke At every step: • Advect the curves • Stretch • Advect the smoke
Method of simulation • Vortex particles (for motion) organized as curves. = tangent • Smoke particles (for visualisation) • Curves carry vortices • Vortices induce a velocity field • velocity field deforms curves & smoke At every step: • Advect the curves • Stretch • Advect the smoke
Smoke Rendering • Thick smoke: plain particles • Thin smoke: adaptive particles[AN05] • accumulate stretching
l n e Smoke Rendering • Thin smoke behaves like a surface [ William Brennan ]
Smoke Rendering • Using Compensated Ray Marching