CS-184: Computer Graphics Lecture #21: Fluid Simulation II
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1 CS-184: Computer Graphics Lecture #21: Fluid Simulation II Rahul Narain University of California, Berkeley Nov , 2013
2 Grid-based fluid simulation Recap: Eulerian viewpoint Grid is fixed, fluid moves through it How does the velocity at a grid cell change over time?
3 Eulerian and Lagrangian time derivatives Consider a weather balloon moving with the wind, measuring air temperature T(x, t) x(t0) x(t1) x(tnow)
4 Eulerian and Lagrangian time derivatives Consider a weather balloon moving with the wind. It measures air temperature as T(x(t), t) d dt Temperature as seen by balloon T dx dt + rt + u rt dt [Chain rule] Velocity of balloon = velocity of air
5 Eulerian and Lagrangian time derivatives DT + u rt Lagrangian derivative: change seen by a point moving with the fluid Change due to movement of fluid ( advection ) Eulerian derivative: change at a fixed point
6 The fluid equations Newton s second law a = f/ Forces: f = f ext rp + µr 2 u Acceleration is Lagrangian: Reminder: Velocity is u now a = Du + u ru Acceleration of fluid molecule Change at a grid cell Flow advects itself
7 The + u ru = 1 f ext p =? rp + µr 2 u The Navier-Stokes equations Millenium prize: $1,000,000 to prove (or disprove) existence & smoothness of solutions C.-L. Navier G.G. Stokes
8 + u ru = 1 f ext rp + µr 2 u Lots of different terms; hard to integrate in one go Deal with one term at a time (ignoring all the others)
9 Operator splitting u + u ru =0 for time Δt = 1 f ext for time Δt External forces = µ r2 u for time Δt = 1 rp for time Δt u n+1
10 Advection + u ru =0 Transport a quantity (in this case, u) via the velocity field u Confusing Let s transport something + u ra =0 e.g. colour, temperature, concentration of ink in water / smoke in air / etc.
11 Advection + u ra =0 A n u n A n+1 Transport A by tracing backwards and looking up its value
12 Advection Input: initial grid A n, velocity field u n Output: final grid A n+1 For each grid cell xi Backtrace position, e.g. x back = x i u i t Set output A n+1 i = interpolate A n at x back To advect velocities, just feed in u n as the initial grid too.
13 External forces Gravity Buoyancy Buoyancy strength f ext = (T T amb )g User interaction Temperature of fluid Ambient temperature
14 External forces Lentine, Zheng, and Fedkiw, 2010
15 Pressure Becker and Teschner, 2007
16 Incompressibility We will prohibit compressibility from our simulation. Net flow ZZinto/out of region Flow = u ˆn da Fixed surface n u ZZZ = r u dv [Divergence theorem] We want net flow to be 0 for all possible regions, so u = 0 everywhere
17 Pressure = p =? 1 rp Let s just do forward Euler
18 Pressure u new = u r u new =0 t rp Let s just do forward Euler and choose the p which makes u new divergence-free t r u r2 p =0 r 2 p = t r u Ignore the /Δt (just a rescaling)
19 The Helmholtz-Hodge decomposition Equivalently: separate u into curl-free and divergencefree components then throw away the curl-free part Input Curl-free Divergence-free = + u p u new =
20 Pressure We just have to solve r 2 p = r u How? Q1: How to evaluate and ² on a grid Q2: How to store p and u on the grid in the first place
21 Finite differences in 1D A A 0 (x) = lim h0 A(x + h) A(x) h x x+h On a grid, we only have samples at grid spacing Δx A 0 i A i+1 x A i Ai+1 or A 0 i A i A i 1 x Ai 1 Ai A 0 i A i+1 A i 1 or xi 1 xi xi+1 2 x Δx Δx
22 Finite differences in 2D r 2 A Apply forward and backward differences i,j A i 1,j 2A i,j + A i+1,j x 2 A i,j 1 2A i,j + A i,j+1 x Five point stencil for the Laplacian
23 Staggered (marker-and-cell) grids Store pressure at cell centers, but velocity at cell faces (r u) i,j u i+1,j u i,j x Finite differences line up u and p at cell centers Components of p and u at cell faces + v i,j+1 v i,j x p u pi,j+1 u p v ui,j vi,j+1 ui+1,j pi 1,j pi,j pi+1,j v vi,j p u pi,j 1 u p v v
24 Boundary conditions At solid obstacles, u ˆn =0 Fluid cannot flow into or out of obstacles At free surface, p =0 p=0 Air applies negligible force on water
25 Pressure solve Build a linear system with one equation per cell (r 2 p) i,j =(r u) i,j Whole system: Ax = b, where x is a vector containing all the pi,j b is a vector containing ( u)i,j Rows of A contain stencil for ² Be careful about boundaries
26 Pressure solve Solve Ax = b to get pressure values p A is large, sparse, symmetric, positive (semi)definite Use e.g. preconditioned conjugate gradient method Update velocities: u new = u rp Refer to Bridson & Müller-Fischer 2007 for full details
27 Viscosity = µ r2 u Often ignored: advection causes enough diffusion already For high-viscosity fluids, just use implicit integration u new = u old + µ r2 u new t
28 Viscosity Carlson, Mucha, Van Horn, and Turk, 2002
29 Smoke simulation Lentine, Zheng, and Fedkiw, 2010
30 Surface tracking How to represent the surface of a liquid? Option 1: Level set method Store the signed distance to surface φ(x) on grid cells Advect forward at each time step Surface is the level set (isosurface) at φ = 0
31 Surface tracking How to represent the surface of a liquid? Option 2: Particles (easier) Keep lots of particles in the fluid, passively advected with the flow Reconstruct surface as in SPH More options: level set + particles, volume-of-fluid, meshes, Zhu and Bridson, 2005
32 Liquid simulation English, Qiu, Yu, and Fedkiw, 2013
33 Real-time liquid simulation Chentanez and Müller, 2011
34 References Bridson and Müller-Fischer, Fluid Simulation for Computer Animation, SIGGRAPH 2007 course notes Stam, Stable Fluids, 1999 Enright, Marschner, and Fedkiw, Animation and Rendering of Complex Water Surfaces, 2002 Zhu and Bridson, Animating Sand as a Fluid, 2005 (for particle-based surface tracking, and more)
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