Fluid Simulation http kucg korea ac kr Graphics

Fluid Simulation http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Contents n n n KUCG Introduction Diffusion Semi-Lagrangian advection Poisson Equation Implementation http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Motivation KUCG Fedkiw et al. 2001 http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Fluids in Computer Graphics KUCG Fast n Looks good n Easy to code n http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Fluid Mechanics n Natural framework for fluid modeling n n Full Navier-Stokes Equations Has a long history n n KUCG Reuse code/algorithms Equations are hard to solve n Non-linear http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Application n KUCG Use velocity to move densities http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Equations n Density field n Velocity field http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University

Finite Difference Scheme http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University

Equations n KUCG Evolution of density (assume velocity known) Over a time step. . . http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Equations n KUCG Evolution of density (assume velocity known) Density changes in the direction of the flow http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Equations n KUCG Evolution of density (assume velocity known) Density diffuses over time http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Equations n KUCG Evolution of density (assume velocity known) Increases due to sources from the UI http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Algorithm KUCG Subdivide space into voxels Velocity + density defined in the center of each voxel http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Algorithm add source http: //kucg. korea. ac. kr KUCG diffuse move Graphics Lab @ Korea University

Diffusing Densities KUCG dt Dn http: //kucg. korea. ac. kr Dn+1 Graphics Lab @ Korea University

Diffusing Densities KUCG Exchange of density between neighbors http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Diffusing Densities KUCG Exchange of density between neighbors http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Diffusing Densities KUCG i, j+1 i-1, j i+1, j i, j-1 Dn+1 i, j = Dn i, j + k dt (Dn i-1, j + Dn i+1, j + Dn i, j-1 + Dn i, j+1 - 4 Dn i, j)/h 2 http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Algorithm add source http: //kucg. korea. ac. kr KUCG diffuse move Graphics Lab @ Korea University

Moving Densities KUCG dt Velocity known http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Moving Densities n KUCG Finite differences Transfer only between neighbors n Unstable with large time steps n http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Semi-Lagrangian Advection http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University

Semi-Lagrangian Advection http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University

Semi-Lagrangian Advection http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University

Semi-Lagrangian Advection KUCG Interpolate the density at new location http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Semi-Lagrangian Advection KUCG Set interpolated density at grid location Requires two grids http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Computing Velocities KUCG Use same algorithms as for density Velocity is moved by itself http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Moving Velocity KUCG Trace particle backwards in time http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Moving Velocity KUCG Interpolate the velocity at new location http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Moving Velocity KUCG Set interpolated velocity at grid location Requires two grids http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Conservation of Mass KUCG Inflow = Outflow Ui+1, j - Ui-1, j + Vi, j+1 - Vi, j-1 = 0 http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Poisson Equation n KUCG Poisson Equation n Step 1 n Step 2 http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Implementation KUCG /* Running */ while(simulating) { velocityfield. advect(dt); velocityfield += gravityfield*(densityfield*(dt*10. 0 f)); fvelocityfield = velocityfield; fvelocityfield. remove. Divergence(); velocityfield = fvelocityfield; densityfield. advect(velocityfield, dt); } http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Implicit Diffusion http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Implicit Diffusion n Explicit diffusion n n KUCG Dn+1 i, j = Dn i, j + k dt (Dn i-1, j + Dn i+1, j + Dn i, j-1 + Dn i, j+1 - 4 Dn i, j)/h 2 Implicit diffusion n Dn+1 i, j – k dt (Dn+1 i-1, j+Dn+1 i+1, j+Dn+1 i, j-1+Dn+1 i, j+1 -4 Dn+1 i, j)/h 2 = Dni, j Ax=b http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Diffusing Densities KUCG Linear solvers: Name Cost Comments Gaussian elimination N 3 Use only for very small N (test code) Jacobi/SOR relaxation N 2 Easy to code but slow FFT/cyclical reduction N log. N Use when no internal boundaries Conjugate gradient N 1. 5 Use when internal boundaries Multi-grid N http: //kucg. korea. ac. kr Slower than FFT in practice. Hard to code when internal boundaries present Graphics Lab @ Korea University

Mass Conservation - Numerical Example- http: //kucg. korea. ac. kr Graphics Lab @ Korea University

Given Velocity Field KUCG P 0 P 1 P 2 P 3 P 4 P 5 10. 0 f P 6 http: //kucg. korea. ac. kr P 7 P 8 Graphics Lab @ Korea University

Poisson Equation http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University

Solution http: //kucg. korea. ac. kr KUCG 0. 0 0. 41 0. 0 -0. 41 1. 25 -0. 41 -2. 50 -4. 58 -2. 50 Graphics Lab @ Korea University

Superposition KUCG + 5. 83 http: //kucg. korea. ac. kr 10. 0 f Graphics Lab @ Korea University

Divergence-Free Velocity Field http: //kucg. korea. ac. kr KUCG Graphics Lab @ Korea University