Physically-Based Physically-Based Simulation on the GPUSimulation on the GPU

Mark Harris NVIDIA Developer Technology

Simulating the worldSimulating the world

• Simulate a wide variety of phenomena on GPUs– Anything we can describe with discrete PDEs or

approximations of PDEs– I will talk about boiling, reaction-diffusion,

fluids, and clouds

Approximate MethodsApproximate Methods

• Several approximations for PDEs– Cellular Automata (CA)– Coupled Map Lattice (CML)– Lattice-Boltzmann Methods (LBM)

• I’ll talk briefly about CML

Coupled Map LatticeCoupled Map Lattice

• Mapping:– Continuous state lattice nodes

• Coupling:– Nodes interact with each other to produce new state

according to specified rules

• Extension of CA:– Continuous state at cells instead of discrete

Example: BoilingExample: Boiling

• State = Temperature

• Three operations:– Diffusion, buoyancy, & latent heat

• Based on [Yanagita 1992]

• 3D Simulation– Stack of 2D texture slices– Sample from neighboring slices

• DEMO

PDE SimulationsPDE Simulations

• Floating-point GPUs open up new possibilities– Less Ad Hoc methods: real PDEs– Must be able to discretize in space and time

• I’ll discuss three examples:– Reaction-Diffusion– Fluid Dynamics– Cloud Dynamics

Reaction-DiffusionReaction-Diffusion

• Gray-Scott reaction-diffusion model [Pearson 1993]

• State = two scalar chemical concentrations

• Simple: just Diffusion and Reaction ops

U

tD

u2U UV 2 F(1 U ),

V

tD

v2V UV 2 (F k)V

U, V are chemical concentrations, F, k, Du, Dv are constants

Demo: “Disease”Demo: “Disease”

Fluid DynamicsFluid Dynamics

• Solution of Navier-Stokes flow eqs.– Stable for arbitrary time steps

• Means you can run it 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

Navier-Stokes EquationsNavier-Stokes Equations

• Describe flow of an incompressible fluid

u

t (u)u

1

p 2u f

Advection PressureGradient

Diffusion (viscosity)

External Force

u 0 Velocity is divergence-free

Fluid Simulator DemoFluid Simulator Demo

Cloud Dynamics OverviewCloud Dynamics Overview

• 3 components

– 7 unknowns

• Fluid dynamics

– Motion of the air

• Thermodynamics

– Temperature changes

• Water continuity

– Evaporation, condensation

Water vapor mixing ratio: qv

Liquid water mixing ratio: qc

Velocity:

Pressure: p u (u,v,w)

Potential temperature: (see dissertation)

ThermodynamicsThermodynamics

• Temperature affected by– Heat sources– Advection– Latent heat released / absorbed during condensation /

evaporation

temperature = advection + latent heat release

+ temperature input

Water ContinuityWater Continuity

• Water is conserved– Condensation balances Evaporation– H2O vapor = - H2O liquid– = Phase changes + advection

• Liquid water is what is rendered– Convert water field to a “density” texture

Cloud DynamicsCloud Dynamics

Simulation AlgorithmSimulation Algorithm

• Advect quantities q, qv, qc and u

– 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]

Simulation AlgorithmSimulation Algorithm

• Most steps are simple– Most use one fragment program, one pass– Programs come directly from equations

• Tricky parts:– Staggered grid discretization

• See dissertation (harris, 2003)– Stable Fluids projection step

• Vectorized solver (see earlier talks)– Boundary conditions (Static Branch Resolution – earlier talks)– 3D Simulation

Flat 3D TexturesFlat 3D Textures

Flat 3D TexturesFlat 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

ConclusionConclusion

• GPUs are a capable, efficient, and flexible platform for physically-based visual simulation

• The possibilities are endless!

AcknowledgementsAcknowledgements

• Fellow UNC Students– Bill Baxter, Greg Coombe, Thorsten Scheuermann

• Anselmo Lastra and other UNC Faculty

• Sponsors:– NVIDIA Corporation– US National Institutes of Health– US Office of Naval Research – US Department of Energy ASCI program– US National Science Foundation

Selected ReferencesSelected References

• Chorin, A.J., Marsden, J.E. A Mathematical Introduction to Fluid Mechanics. 3rd ed. Springer. New York, 1993

• Fedkiw, R., Stam, J. and Jensen, H.W. Visual Simulation of Smoke. In Proceedings of SIGGRAPH 2001, ACM Press / ACM SIGGRAPH. 2001.

• Harris, M., Coombe, G., Scheuermann, T., and Lastra, A. Physically-Based Visual Simulation on Graphics Hardware.. Proc. 2002 SIGGRAPH / Eurographics Workshop on Graphics Hardware 2002.

• Harris, Baxter, Scheuermann, Lastra. Simulation of Cloud Dynamics on Graphics Hardware. Proc. Graphics Hardware 2003.

• Harris, M. Real-Time Cloud Simulation and Rendering. Ph.D. Dissertation. http://www.markmark.net/dissertation

• Kaneko, K. (ed.), Theory and applications of coupled map lattices. Wiley, 1993.

• Nishimori, H. and Ouchi, N. Formation of Ripple Patterns and Dunes by Wind-Blown Sand. Physical Review Letters, 71 1. 197-200. 1993.

• Pearson, J.E. Complex Patterns in a Simple System. Science, 261. 189-192. 1993.

• Stam, J. Stable Fluids. In Proceedings of SIGGRAPH 1999, ACM Press / ACM SIGGRAPH, 121-128. 1999.

• Turk, G. Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion. In Proceedings of SIGGRAPH 1991, ACM Press / ACM SIGGRAPH, 289-298. 1991.

• Witkin, A. and Kass, M. Reaction-Diffusion Textures. In Proceedings of SIGGRAPH 1991, ACM Press / ACM SIGGRAPH, 299-308. 1991.

More ReferencesMore References

• Gomez, M. Interactive Simulation of Water Surfaces. in Game Programming Gems. Charles River Media, 2000. p 187.

• Lengyel, E. Mathematics for 3D Game Programming & Computer Graphics. Charles River Media, 2002. Chapter 12, p 327.

• James, G. Operations for Hardware-Accelerated Procedural Texture Animation. in Game Programming Gems II. Charles River Media, 2001. p 497.

• Strzodka, R. Virtual 16 Bit Precise Operations on RGBA8 Textures. Proceedings VMV 2002, 2002

• Strzodka, R., Rumpf, M. Using Graphics Cards for Quantized FEM Computations. In Proceedings VIIP 2001, 2001.

• Yanagita, T. Phenomenology of boiling: A coupled map lattice model. Chaos, 2 3. 343-350. 1992.

• Yanagita, T. and Kaneko, K. Coupled map lattice model for convection. Physics Letters A, 175. 415-420. 1993.

• Yanagita, T. and Kaneko, K. Modeling and Characterization of Cloud Dynamics. Physical Review Letters, 78 22. 4297-4300. 1997

Extra SlidesExtra Slides

3D simulation3D simulation

• Lattice is now 3D– 3D texture or stack of 2D textures– Or tile slices onto a large 2D texture

• Neighbor sampling is 3D– Texels in nearby slices– If it’s a 3D texture, just perturb the r texture coordinate to get

slice neighbors– Otherwise bind neighbor slice textures

Fluid Simulation DetailsFluid Simulation Details

Divergence-Free?Divergence-Free?

• In any element of fluid, inward velocity is balanced by outward velocity– No sources or sinks

• Ensures mass / momentum conservation

AlgorithmAlgorithm

• Break it down [Stam 2000]:

– Add forces:

– Advect:

– Diffuse:

– Solve for pressure:

– Subtract pressuregradient:

w1u(x,t) f (x,t)t

2 p w

3

u(x,t t) w3 p

I t2 w3

w2

w2w

1(x w

1t)

AlgorithmAlgorithm

• Break it down [Stam 2000]:

– Add forces: w1u(x,t) f (x,t)t

Add ForcesAdd Forces

• Scale force by time step, add to velocity

• In demo, we just “splat”– Color of splat encodes direction and strength of force

(determined by mouse motion)– Simple fragment program adds gaussian splat to velocity

texture

w1u(x,t) f (x,t)t

AlgorithmAlgorithm

• Break it down [Stam 2000]:

– Add forces:

– Advect:

w1u(x,t) f (x,t)t

w2w

1(x w

1t)

AdvectionAdvection

• Advection: quantities in a fluid are carried along by its velocity

• Want velocity at position x at new time t + t

• Follow velocity field back in time from x : (x - w1t)

– Like tracing particles!– Simple in a fragment program

u(x, t+t)

u(x’, t)

Path of fluid

Trace back in time

w2(x) w

1(x w

1t)

AlgorithmAlgorithm

• Break it down [Stam 2000]:

– Add forces:

– Advect:

– Diffuse:

w1u(x,t) f (x,t)t

I t2 w3

w2

w2w

1(x w

1t)

Viscous DiffusionViscous Diffusion

• Viscous fluid exerts drag on itself– Causes diffusion of velocity

• Implicit, discrete form of– Explicit form is unstable

• Solution is just like the next step– Simpler to explain the next step…

I t2 w3

w2

w2

t2w

2

AlgorithmAlgorithm

• Break it down [Stam 2000]:

– Add forces:

– Advect:

– Diffuse:

– Solve for pressure:

w1u(x,t) f (x,t)t

2 p w

3

I t2 w3

w2

w2w

1(x w

1t)

Poisson-Pressure SolutionPoisson-Pressure Solution

• Poisson Equation– Discretize, solve using iterative solver (relaxation)– Jacobi, Gauss-Seidel, Multigrid, etc.

• Jacobi easy on GPU, the rest are trickier• Demo uses Jacobi iteration (15-20 iters.)

– Boils down to repeated evaluation of:

2 p w

3

pi, jn1

1

4p

i1, jn p

i 1, jn p

i, j1n p

i, j 1n 2(w

3) ,

w3

1

2(u3

i1, j u3

i 1, j v3

i, j1 v3

i, j 1) = grid spacing

u, v = components of w3

i, j = grid coordinatesn = solution iteration

AlgorithmAlgorithm

• Break it down [Stam 2000]:

– Add forces:

– Advect:

– Diffuse:

– Solve for pressure:

– Subtract pressure gradient:

w1u(x,t) f (x,t)t

2 p w

3

u(x,t t) w3 p

I t2 w3

w2

w2w

1(x w

1t)

Subtract Pressure GradientSubtract Pressure Gradient

• Last computation of the time step– u is now a divergence-free velocity field

• Very simple fragment program

u(x,t t) w3 p

ui, j

u3i, j

1

2 p

i1, j p

i 1, j ,v

i, jv3

i, j

1

2 p

i, j1 p

i, j 1 = grid spacingi, j = grid coordinatesu, v = components of uu3,v3= components of w3

p = pressure

Fluid Simulator Demo (more)Fluid Simulator Demo (more)

Boundary ConditionsBoundary Conditions

• Various types of BCs

– No-slip, free-slip, outflow, inflow, periodic, etc.

• Demo uses “no-slip” velocity BCs

– V = 0 at boundaries.

• Pressure: pure Neumann BCs

– Have to set them each iteration of solver

– p(boundary) = p(nearest non-boundary neighbor)

p

n0

Cloud DetailsCloud Details

Fluid DynamicsFluid Dynamics

• Incompressible Euler equations

• Same as previous fluid simulation– Viscosity = 0

• Buoyancy term– Temperature, pressure, water changes induce motion

u

t (u)u

1

p B ök f

u 0

Projection StepProjection Step

• Enforces divergence-free velocity– Iterative solution of Poisson-pressure eq.

• Poisson solver options – Conjugate Gradient– Multigrid – Fast Fourier Transform– Jacobi, Red-Black Gauss Seidel

Vectorized SolversVectorized Solvers

• Pressure is scalar, texels are RGBA – Pack 4 pressure values in each texel– Process 4x fewer fragments each iteration

Simulation AmortizationSimulation Amortization

• Simulation takes ~200 ms per step– 64x64x64 volume, GeForce FX 5900 Ultra

• But simulation time step represents 3s of actual time– On the order of visual change in real clouds– So, faster than real time, but low frame rate

Simulation AmortizationSimulation Amortization

• Non-interactive application:– Simulate as fast as possible– Frame rate suffers

20ms

Simulation AmortizationSimulation Amortization

• Interactive frame rate!– Simulation still proceeds pretty fast

10 20ms