Fluids in Games. Jim Van Verth Insomniac Games
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1 Fluids in Games Jim Van Verth Insomniac Games
2 Introductory Bits General summary with some details Not a fluids expert Theory and examples
3 What is a Fluid? Deformable Flowing Examples Smoke Fire Water
4 What is a Fluid?
5 What is a Fluid?
6 What is a Fluid?
7 Fluid Concepts Fluids have variable density (Density field)
8 Fluid Concepts Fluids flow (Vector field)
9 Fluid Concepts Need way to represent Density (x) Velocity (u) Sometimes temperature
10 Fluid Concepts Our heroes:
11 Fluid Concepts Our heroes: Navier
12 Fluid Concepts Our heroes: Navier Stokes
13 Fluid Concepts Their creation: " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
14 Fluid Concepts Their creation: " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f THE END!
15 Fluid Concepts Their creation: " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f SERIOUSLY -- WHAT DOES THIS MEAN?
16 Fluid Concepts Want change in velocity field
17 Fluid Concepts Want change in velocity field
18 Fluid Concepts Want change in velocity field
19 Fluid Concepts Want change in velocity field
20 Fluid Concepts What affects it?
21 Fluid Concepts What affects it? External Forces
22 Fluid Concepts What affects it? Viscosity
23 Fluid Concepts What affects it? Advection
24 Fluid Concepts What affects it? Pressure
25 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
26 Fluid Concepts Brief notational diversion " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
27 Fluid Concepts Brief notational diversion " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Gradient (vector along partial derivative)
28 Fluid Concepts Brief notational diversion " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Divergence (real derivative of vec. field)
29 Fluid Concepts Brief notational diversion " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Laplacian (divergence of gradient)
30 Fluid Concepts Brief notational diversion " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Advection operator (transport of flow)
31 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Change in Velocity
32 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Advection
33 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Pressure
34 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Viscosity
35 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f External Forces
36 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f HOLD ON THERE BUCKO
37 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f WHAT S THIS ONE?
38 Fluid Concepts Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Mass Conservation
39 Fluid Concepts In principle then, Navier-Stokes is " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
40 Fluid Concepts In principle then, Navier-Stokes is " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f THE END!
41 Fluid Concepts But not really, of course
42 Fluid Concepts But not really, of course Little tiny detail of implementation
43 Computational Fluid Types Grid-based/Eulerian (Stable Fluids) Particle-based/Lagrangian (Smoothed Particle Hydrodynamics) Surface-based (wave composition)
44 Grid-Based Store density, temp in grid centers
45 Grid-Based Velocity (flow) from centers as well Could also do edges
46 Grid-Based How do we use this? " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
47 Grid-Based Jos Stam devised stable approximation: Stable Fluids, SIGGRAPH 99
48 Grid-Based How do we use this? " u = 0 Must maintain #u #t = $(u ")u $ 1 % "p + &" 2 u + f
49 Grid-Based How do we use this? " u = 0 Idea: compute #u #t = $(u ")u $ 1 % "p + &" 2 u + f
50 Grid-Based How do we use this? " u = 0 Idea: compute #u #t = $(u ")u $ 1 % "p + &" 2 u + f
51 Grid-Based How do we use this? " u = 0 Idea: compute #u #t = $(u ")u $ 1 % "p + &" 2 u + f then project to 0 div field
52 Grid-Based How do we use this? End up with "u "t = P(#(u $)u + %$ 2 u + f)
53 Grid-Based How do we use this? Easy "u "t = P(#(u $)u + %$ 2 u + f)
54 Grid-Based How do we use this? Sparse linear system "u "t = P(#(u $)u + %$ 2 u + f)
55 Grid-Based How do we use this? Sparse linear system "u "t = P(#(u $)u + %$ 2 u + f)
56 Grid-Based How do we use this? Non-linear ugh "u "t = P(#(u $)u + %$ 2 u + f)
57 Grid-Based Updating advection General idea:
58 Grid-Based Updating advection General idea: look at current position
59 Grid-Based Updating advection General idea: follow flow to prev. position
60 Grid-Based Updating advection General idea: get velocity there
61 Grid-Based Updating advection General idea: assign to current position
62 Grid-Based Overview Update velocities based on Forces, then Advection, then Viscosity Project velocities to zero divergence Update densities based on Input sources Velocity Diffusion (similar to viscosity, sometimes not used) Draw it
63 Rendering Grid-Based Build level surface
64 Rendering Grid-Based Determining color, transparency
65 Issues Limited space Water splashes get lost Can be computationally expensive Dampens down But stable
66 Implementation Little Big Planet Death smoke Bubble pop Other smoke effects Hellgate: London GDC09 NVIDIA demo
67 Smoothed Particle Hydrodynamics Approximate fluid with small(er) set of particles
68 SPH Velocities at particles provide flow
69 SPH Idea: treat as particle system
70 SPH Idea: treat as particle system Determine forces
71 SPH Idea: treat as particle system Determine forces Update velocities, positions
72 SPH Weighted average gives density (smoothing kernel)
73 SPH Can also use kernel to get general velocity
74 SPH Usually center at particle
75 SPH Specify width by h h
76 SPH Specify vector from other particles by r r h
77 SPH Example kernel h
78 SPH Common kernel W poly 6 (r,h) = 315 $ (h 2 # r 2 ) 3 % 64"h 9 & 0 0 ' r ' h otherwise
79 SPH Common kernel W poly 6 (r,h) = 315 $ (h 2 # r 2 ) 3 % 64"h 9 & 0 0 ' r ' h otherwise Clamps to zero at boundary
80 SPH Common kernel W poly 6 (r,h) = 315 $ (h 2 # r 2 ) 3 % 64"h 9 & 0 0 ' r ' h otherwise Clamps to zero at boundary Uses length squared
81 SPH General SPH rule A S (x) = " m A j j j # j W (r $ r j,h)
82 SPH General SPH rule A S (x) = " m A j j j # j W (r $ r j,h) Particle mass
83 SPH General SPH rule A S (x) = " m A j j j # j W (r $ r j,h) Quantity at particle j
84 SPH General SPH rule A S (x) = " m A j j j # j W (r $ r j,h) Density at particle j
85 SPH General SPH rule A S (x) = " m A j j j # j W (r $ r j,h) Weighting function
86 SPH Computing density " S (x) = # m " j j W (r $ r j,h) " j j
87 SPH Computing density " S (x) = # m j W (r $ r j,h) j
88 SPH Local pressure p i = k" i k is gas constant
89 SPH Local pressure p i = k" i k is gas constant Can be unstable, so
90 SPH Local pressure (alternative) p i = k(" i # " 0 ) No effect on gradient, more stable
91 SPH Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
92 SPH Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Have fixed # particles and mass, so
93 SPH Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Have fixed # particles and mass, so mass is automatically conserved
94 SPH Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Advection automagically handled by particle update, so
95 SPH Back to Navier-Stokes " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f Advection automagically handled by particle update, so
96 SPH Simplifies to " u = 0 #u #t = $(u ")u $ 1 % "p + &" 2 u + f
97 SPH Simplifies to " dv dt = #$p + µ$ 2 u + "f
98 SPH Functional breakdown " du dt = #$p + µ$ 2 u + "f
99 SPH Functional breakdown " du dt = #$p + µ$ 2 u + "f Change in velocity
100 SPH Functional breakdown " du dt = #$p + µ$ 2 u + "f Pressure
101 SPH Functional breakdown " du dt = #$p + µ$ 2 u + "f Viscosity
102 SPH Functional breakdown " du dt = #$p + µ$ 2 u + "f External forces
103 SPH Compute densities, local pressure Generate forces on particles External Pressure Viscosity Update velocities, positions Render
104 SPH Pressure f i pressure = "#p(r i )
105 SPH Pressure f i pressure = " m p j j j # j $W (r i % r j,h)
106 SPH Pressure f i pressure = " m p j j j # j $W (r i % r j,h) Asymmetric, so
107 SPH Pressure f i pressure = " m p i + p j j j 2# j $W (r i % r j,h) Ensures 2-particle interaction equal
108 SPH Pressure kernel Problem: gradient of poly6 kernel is zero at origin
109 SPH Pressure kernel Problem: gradient of poly6 kernel is zero at origin
110 SPH Pressure kernel W spiky (r,h) = 15 $ (h # r) 3 % "h 6 & 0 0 ' r ' h otherwise
111 SPH Viscosity f i viscosity = µ" 2 v(r i )
112 SPH Viscosity f viscosity i = µ " m v j j $ 2 W (r i % r j,h) # j j
113 SPH Viscosity f viscosity i = µ " m v j j $ 2 W (r i % r j,h) # j Also asymmetric, so j
114 SPH Viscosity f viscosity i = µ " m v j # v i j % 2 W (r i # r j,h) $ j j Ensures 2-particle interaction opposite
115 SPH Viscosity kernel W viscosity (r,h) = 15 $ # r 3 & 2"h 3 2h + r2 3 h + h % 2 2r #1 '& 0 0 ( r ( h otherwise
116 SPH Rendering Marching cubes (using smoothing kernel) Blobs around particles/splatting
117 SPH Implementations Takahiro Harada Kees van Kooten (Playlogic) NVIDIA PhysX Rama Hoetzlein* (SPH Fluids 2.0) Takashi AMADA* * Source code available
118 SPH Issues Need a lot of particles Computing level surface can be a pain Can be difficult to get stable simulation
119 SPH Improvements Spatial hashing Variable kernel width CFD/SPH Hybrid CFD manages general flow SPH splashes
120 Surface Simulation Idea: for water, all we care about is the airwater boundary (level surface) Why simulate the rest? This is what Insomniac R20 system does
121 R20 Done by Mike Day, based on Titanic water Basic idea: convolve sinusoids procedurally * = Much cheaper to multiply in frequency domain and do FFT (assuming periodic)
122 R2O Review Sinusoid in spatial domain
123 R2O Review Can represent as magnitude+phase in frequency slot
124 R2O Review Requires periodic function FFT
125 R2O Review Multiple sinusoids end up at multiple entries FFT
126 R2O Wave speed dependant on wavelength
127 R2O Wave speed dependant on wavelength
128 R2O Wave speed dependant on wavelength
129 R2O Wave speed dependant on wavelength
130 R2O Wave speed dependant on wavelength I.e. phases update at different rates AKA dispersion
131 R2O General procedure Start with convolved data in (r,!) form Update phase angles for each sinusoid Angular velocity*dt Dependent on frequency Do inverse FFT to get spatial result
132 R2O FFT kernel limited to 32x32 Combine multiple levels via LOD height field scheme Gives high detail close to camera
133 R2O Interactive waves Just adding in splashes looks fake Instead, do some more FFT trickery so all our work occurs in the same domain Non-periodic, so have to manage edges Gives nice dispersion effects
134 R2O Rendering Rendered as height field mesh Add normal map for detail Cube map/frame buffer map for reflections Distortion effect for refractions
135 R2O Nifty video
136 References Jos Stam, Stable Fluids, SIGGRAPH 1999 Mattias Müller, et. al, Particle-Based Fluid Simulation for Interactive Applications, SIGGRAPH Symposium on Computer Animation 2003 Jerry Tessendorf, Simulating Ocean Water, SIGGRAPH Course Notes.
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