Physically Based Simulations (on the GPU)

Transcription

1 Physically Based Simulations (on the GPU) (some material from slides of Mark Harris) CS535 Fall 2014 Daniel G. Aliaga Department of Computer Science Purdue University

2 Simulating the world Floating point arithmetic on GPUs and their speed enable us to simulate a wide variety of phenomena using PDEs

3 Some Basics Operators (on images/lattices) Diffusion Bouyancy

4 Operators Given an image: Gradient (vector) f x, y Laplacian (scalar) 2 f x, y = f x x + f y y = 2 f x f y 2

5 Discrete Laplacian 2 f x, y = f x 1, y + f x + 1, y + f x, y 1 + f x, y + 1 4f x, y Matrix form =??

6 Discrete Laplacian 2 f x, y = f x 1, y + f x + 1, y + f x, y 1 + f x, y + 1 4f x, y Matrix form =

7 Heat Equation f t = 2 f

8 Diffusion Equation [Weisstein 1999] f x, y = f x, y + c d 4 2 f(x, y) where c d is the coefficient of diffusion

9 (Anisotropic) Diffusion

10 Buoyancy Used in convection, cloud formations, etc. Given a temperature state T: a vertical buoyancy velocity is upwards if a node is hotter than its neighbors and a vertical buoyancy velocity is downwards if a node is cooler than its neighbors

11 Buoyancy v x, y = v x, y + c b (2f x, y f x + 1, y f(x 1, y)) 2 where c b is the buoyancy strength

12 Bouyancy (considering neighbors) f x, y = f x, y σ f x, y 2 [ρ(f x, y + 1 ρ(f x, y 1 ] where ρ f = tanh α f f c (an approx. of density relative to temperature f) and σ is buoyancy strength and α and f c are constants

13 Euler Method (for ODE) Given: Do: y t = f t, y t with y t 0 = y 0 y n+1 = y n + hf(t n, y n )

14 Classical Runge Kutta Method Given: y t = f t, y t with y t 0 = y 0 Do: y n+1 = y n + h/6(k 1 + 2k 2 + 2k 3 + k 4 ) t n+1 = t n + h where

15 Example: (Water) Boiling Based on [Harris et al. 2002] State = Temperature Three operations: Diffusion, buoyancy, & latent heat 3D Simulation Stack of 2D texture slices

16 Turing: Morphogenesis and Reaction-Diffusion (1952) Alan Turing in 1952 describing the way in which non-uniformity (stripes, spots, spirals, etc.) may arise naturally out of a homogeneous, uniform state. The theory (which can be called a reaction diffusion theory of morphogenesis), has served as a basic model in theoretical biology, and is seen by some as the very beginning of chaos theory. U t = D U 2 U k UV 16 V t = D V 2 V + k UV 12 V

17 Gray-Scott Reaction-Diffusion State = two scalar chemical concentrations Simple: just Diffusion and Reaction ops U t V t D u 2 U UV 2 F(1 U ), D v 2 V UV 2 (F k)v U, V are chemical concentrations, F, k, D u, D v are constants

18 Some research usion/reaction_diffusion.html

19 Navier-Stokes Equations Describe flow of an incompressible fluid u t (u )u 1 p 2 u f Advection Pressure Gradient Diffusion (viscosity) External Force u 0 Velocity is divergence-free

20 Fluid Dynamics Solution of Navier-Stokes flow eqs. Stable for arbitrary time steps (=fast!) [Stam 1999], [Fedkiw et al. 2001] Can be implemented on latest GPUs Quite a bit more complex than R-D or boiling See Fast Fluid Dynamics Simulation on the GPU (Harris, GPU Gems, 2004)

21 Fluid Simulations

22 Thermodynamics Temperature affected by Heat sources Advection Latent heat released / absorbed during condensation / evaporation temperature = advection + latent heat release + temperature input

23 Cloud Dynamics 3 components 7 unknowns Fluid dynamics Motion of the air Thermodynamics Temperature changes Water continuity Evaporation, condensation Velocity: u (u,v,w) Pressure: p Potential temperature: q (see dissertation) Water vapor mixing ratio: q v Liquid water mixing ratio: q c

24 Cloud Dynamics

25 Wave Equation Remember heat equation: Rate of change of value proportional to Laplacian Wave equation: Rate of change of the rate of change is also proportional to the Laplacian

26 Wave Equation 2 u t 2 = c2 2 u where u models the displacement and c is the propagation speed

27 Water Simulation: Wave Equation U = value, V = rate of change U t = b k + d 2 U V t = k 2 U

28 Water Simulation: Wave Equation Demo

29 Water Simulation: Sine Waves Asin(ωx + t)

30 Water Simulation: Sine Waves A 1 sin(ω 1 x + t 1 ) + A 2 sin ω 2 x + t 2 +

31 Water Simulation: Sine Waves Using sine-wave summations: H x, y, t = A i sin(d i x, y ω i + tφ i ) [use H as height or a pixel intensity] Pixel values over time are: P x, y, t = (x, y, H x, y, t )

32 Water Simulation: Sine Waves (here, pixel normals are computed as well for reflections)

33 Water: Surface Normals Use binormal and tangent: B x, y, t = dx, dy dx dx, dh x,y,t dx T x, y, t = = 0,1, Normal is: N x, y, t = B T N x, y, t dh x, y, t = ( dx, = (1,0, dh x, y, t dy dh x, y, t dy dh x,y,t dx, 1) )

34 Water Simulation: Gerstner Waves These waves also change the x, y of the wave imitating how points at top of wave are squished together and points at bottom are separated

35 Water Simulation: Gerstner Waves P x, y, t x + Q i A i D i. x cos (ω i D i x, y + φ i t) = y + Q i A i D i. y cos (ω i D i x, y + φ i t) A i sin (ω i D i x, y + φ i t) where Q i =sharpness

36 Water Simulation: Gerstner Waves

37 Video i4s

38 Simulation Algorithm Advect quantities Similar to [Stam, 1999] Compute and apply accelerations Buoyancy Compute condensation, evaporation, and temperature changes Enforce momentum conservation Otherwise velocity dissipates, loses swirls Projection step of Stable Fluids [Stam, 1999]

39 Simulation Algorithm Most steps are simple Most use one fragment program, one pass Programs come directly from equations Tricky parts: Staggered grid discretization Stable Fluids projection step Boundary conditions 3D Simulation

40 Flat 3D Textures

41 Flat 3D Textures Advantages One texture update per operation Better use of GPU parallelism Non-power-of-two Textures Quick simulation preview Disadvantage Must compute texture offsets