Physics-Based Animation

Transcription

1 CSCI 5980/8980: Special Topics in Computer Science Physics-Based Animation 13 Fluid simulation with grids October 20, 2015

2 Today Presentation schedule Fluid simulation with grids Course feedback survey (again)

3 (Tentative) presentation schedule Oct Nov Dec Tue Thu 20 Fluids on grids Liquid surfaces Elasticity Finite element method 29 3 Sound (Bryan) Cloth and hair 5 10 Cloth (Alex & Morgan) Fluids (Ran & Jingying) Fire (Devin & Lei) TBD Solids (George & Jie) Thanksgiving 26 1 Real-time stuff (Zahra) Real-time stuff (Dan & Matt) 3 8 Wrap-up day Project presentations Project presentations

4 Course feedback survey

5 Representing continua

6 Particles (SPH) Values on moving particles Approximate continuous field by weighted averaging A(x) = X A i m i i W (x x i,h) Derivatives by differentiating weighting kernel

7 Grids Values on nodes of rectilinear grid Easy to interpolate using only 4 (in 2D) or 8 (in 3D) nearest values

8 Grids class Grid { int m, n, o; Vec3d origin; double dx; Type *values; Type get(int i, int j, int k);... };

9 Grids 11 c 21 c 11 x c 02 c 12 c 22 c 01 c 11 c 21 c 00 c c 11 c 01 x c 21 c 01 2 x ½ 0 ½

10 Grids (finite differences) c 02 c 12 c 22 c 01 c 11 c 21 r 2 c 1 h c 00 c 10 c 20

11 Meshes (FEM) Values on nodes of (not necessarily regular) mesh Evaluate derivatives on elements (triangles/ tetrahedra/etc.)

12 Fluids on grids

13 Grids vs. particles Particles Mass, velocity, pressure, etc. stored on particles Particles move with fluid Grids Velocity, pressure, etc. stored on grid cells Grid doesn t move

14 Advection Grid doesn t move, fluid flows through grid Lagrangian Dc =0 6=0 Eulerian

15 Advection Dc + u rc Lagrangian derivative Change due to movement of fluid ( advection ) Eulerian derivative Proof: differentiate c(x(t), t) with respect to t

16 The fluid equations Du Dt = f ext rp + µr 2 u Du Dt = f ext 1 rp + + u ru = f ext 1 rp + r2 u

17 + u ru = f ext 1 rp + r2 u Lots of different terms, hard to integrate safely in one step Deal with one term at a time, ignoring all the others (e.g. IMEX)

18 u n Operator + 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

19 Advection

20 Advection Goal: Integrate c/ t + u c = 0 for time Δt Interpretation: c moves with speed u for time Δt Solution: To get the value of c n+1 at any point, figure out where it came from and take the value of c n from there

21 Advection Trace backwards through u and look up values c n u n c n+1 Semi-Lagrangian advection

22 Advection Input: initial grid c n, velocity field u n Output: final grid c n+1 For each grid cell x i Backtrace position, e.g. x back = x i u i Δt Set output c i n+1 = interpolate c n at x back To advect velocities, just use u n as the initial grid too.

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24 Pressure

25 Pressure In SPH, density ρ was used for two things: 1. Normalizing for uneven particle distribution 2. Computing pressure forces With grids, we don t need to track ρ. But what about pressure? Treat it as a constraint force: ρ = const dρ/dt = 0

26 Incompressibility Flow Fixed surface n u Net flow ZZinto/out of region = u ˆn da ZZZ = r u dv [Divergence theorem] We want net flow to be 0 for all possible regions, so u = 0 everywhere

27 Pressure u new = u t rp r u new =0 Constraint: u new should be divergence-free, find the constraint response p that makes it so (analogous to computing λ so that Jv = 0) r 2 p = t r u Ignore the /Δt (just a rescaling)

28 Staggered 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 p i,j+1 p v u i,j v i,j+1 u i+1,j p i 1,j p i,j p i+1,j v u u v i,j u u v v p p i,j 1 p

29 Boundary conditions Solid obstacles: u n = 0 Fluid cannot flow into or out of obstacles Free surface: p = 0 p = 0 Air applies negligible force on water

30 Pressure solve 2 p = u Build a big linear system Ap = b and solve it Rows of A contain stencil for 2 (except at boundaries, be careful!) b contains values of u Then update u u p

31 The other bits External forces e.g. f = β(t T 0 )g Gravity, buoyancy, user interaction, Explicit integration is fine Viscosity Often ignored because numerical damping is enough For really thick fluids, use implicit integration

32 u n Operator + 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

33 Further reading Bridson and Müller-Fischer, Fluid Simulation for Computer Animation (SIGGRAPH 2007 course notes) Bridson, Fluid Simulation for Computer Graphics (2008 book)

34 Course feedback survey

35 Next class Even more fluids Hybrid (particles + grid) techniques for advection Liquid surface reconstruction