Ronald Fedkiw Stanford University Industrial Light + Magic

1. Ronald Fedkiw Stanford University Industrial Light + Magic

2. What is plausible simulation? • There is no safe place to start... • One cannot even assume that the correct answer is plausible • suppose a character threw a ball forward at the ground and it bounced back into their hand like a boomerang • this is a backward scattering of the ball, and does have some small probability of occurring • unfortunately, it is not very likely and the audience will be hard pressed to accept it • in fact, one would expect the characters to be rather surprised at the outcome • that is, character emotion is needed to aid plausibility in this case

3. What is plausible simulation? • If 10,000 balls are thrown forward at the ground, and 1 bounces back that might be ok • the collision result of the ball with the ground obeys a “BRDF-like” probability for the scattering direction • one might believe that 1 out of 10,000 balls backscatters • in fact, one might expect some backscatter • in this instance the audience witnesses a reasonable probability distribution first hand (lots of balls) • but when only one ball is thrown, the audience expects the most likely solution, and insists that the characters expect this too • however, this can appear too sterile (e.g. mirror reflection)

4. What is plausible simulation? • When rendering, many rays are cast per pixel, and both sampling and averaging occurs until a single solution is converged upon • rendering algorithms tend to pick a solution near the average value • in dynamics it is not always sensical to compute and display an average solution • can only depict a single solution (unless we have more balls) • so it never makes sense to show something improbable unless the characters will act surprised by the event • plausible simulation seems somehow connected with story telling, imagination, etc.

5. What is plausible simulation? • Many times, one cannot compute the correct solution anyway • for turbulence we have no idea what the correct equations even are • engineers use turbulence models with the goal of reproducing some aspect of the physics • they tune parameters to get what they want • i.e. they work to match experiments for lift, drag, etc. • however, they could not care less about the visual norm • these models are mostly useless for visually plausible simulation

6. What is plausible simulation? • Many times, one cannot compute the correct solution anyway • for collisions the result depends on the microstructure of the material • the microstructure occurs on a microscopic scale that is difficult to measure and even more difficult to model • since we cannot see the microstructure, we cannot expect the audience to believe all high probability microstructure events • the audience will instead assume a microstructure and we need to be faithful to that

7. What is plausible simulation? • Many times, one cannot compute the correct solution anyway • for collisions the result depends on the approach velocity, angular velocity, internal response, etc. • these are all difficult to measure and model exactly • the flight conditions of a thrown ball depend on the texture of the skin on the fingers, how it was released, the firing of muscles in the hand, fingers and arm, etc. • quite a lot is unknown, unmodeled, etc.

8. What is plausible simulation? • Summary • the correct physics may seem implausible • the correct physics may be impossible to compute • often, engineers only care about macroscopic integral quantities like lift and drag, not high frequency visual features • characters probably need to react a certain way in order to make the correct physics seem plausible

9. What is plausible simulation? • I have no idea • But, let me tell you how to take advantage of it...

10. Mathematical Equations • Physicists, chemists, engineers, etc. are all interested in mathematical descriptions of the world around us • these are models! • they are intended to capture some aspect of the problem • they are not intended to be exact, true, or delivered from your favorite deity • that does not mean that they are not useful • e.g., even though electrons do not orbit atoms like planets, that model does account for certain phenomena, and can be quite useful

11. Numerical Algorithms • Since the equations are not necessarily correct, there is no reason to force the numerical algorithms to exactly mimic the equations • consistency is over-rated • if your algorithm does not solve the given equations, it may solve some other set of equations that are interesting in their own right • accuracy is over-rated • there is no reason to strive for 8 digits of accuracy when solving a set of equations that is inconsistent with the physics in the second or third decimal pace

12. Examples • The Navier-Stokes equations do not allow 2 water drops to merge into a single drop • The Navier-Stokes equations do not allow a falling drop to actually hit the ground • the numerous models for frictional contact and collision do not accurately account for microstructure • For large deformations, accurate stress strain relationships are unknown (a nonlinear Green strain coupled to linear elasticity is useless) • bending of shells and cloth, plasticity, fracture, damping, etc...

13. What do numerical analysts do? • They sit in their office waiting for some scientist or engineer to walk in with a set of equations and say “simulate this” • a numerical analyst does not know the underlying physics • thus, the best they can do is to faithfully and consistently solve the equations they are given • if they’re algorithm is not consistent with the equations, they are essentially constructing a new model • this is less than ideal since they do not know the physics of the problem at hand • once they have consistency, they strive for accuracy in order to get a more efficient algorithm

14. We are the “scientists” • We understand the underlying problem and we know what we want... plausible simulation • I don’t know what it is, but I know that I want it • it makes lot’s of $$ at the box office • it makes graduate students work • it makes our friends think we are cool

15. We are the “scientists” • We can write down a set of equations • We understand the visual norm • We can consult numerical textbooks to solve the equations • ode and pde solvers • linear algebra routines • optimization • solid and fluid mechanics • statistics • We can consult with the numerical analysts directly

16. Caution • Be careful borrowing equations and numerical algorithms from other fields • most likely, the equations were not written down to respect the visual norm • most likely, the numerical methods were not concerned with the visual norm • simple import of existing technology has little value • instead, novel contributions are needed to get equations and algorithms that respect and flourish under the visual norm

17. One way to exploit plausible simulation • There are many examples of visually rich phenomena that are nearly impossible or completely impossible to simulate • We can choose or derive a set of equations and a numerical method that is faithful to our needs, the visual norm • This is no worse than a turbulence model that gets the correct lift and drag but is inaccurate on the small scale eddies • This is no worse than any number of friction models

18. One way to exploit plausible simulation • If we choose the equations and the numerical method wisely, impossible problems may become practical or even trivial • e.g. it would be quite costly to simulate all the waves in the ocean with the nonlinear three dimensional Navier-Stokes equations • instead the equations can be both linearized and reduced to two dimensions based on a height field • this made the Titanic sink, a perfect storm, and 300 million dollars for Pixar • that’s plausible enough for me

19. One Way Wave Equation central upwind exact solution initial data central upwind

20. Higher Order Approximation - add a new term to cancel out the leading order error terms central upwind

21. Thoughts... • in traditional CFD, the results of numerical calculations are only • meaningful when the computed solution is well-resolved • well-resolved computations are within the convergent asymptotic • regime where the numerical errors are proportional to the mesh • spacing • the only sensical terms are those that accelerate convergence in the asymptotic regime, i.e. high order methods that cancel error • What happens in very complex flow fields where one cannot possibly use enough grid points to resolve all the important features? In general, one can claim very little about under resolved calculations on relatively coarse grids.

22. high order method asymptoticregime new method? truesolution

23. Vorticity Confinement – coarse grid fix - vorticity needs help to overcome coarse grid dissipation - locate the vorticity with • calculate the magnitude and direction of the force that the vorticity • should exert • scale the force so that it vanishes for consistency, but still gives a good answer on a coarse grid

24. Smoke

25. Simulation of Large Scale Phenomena • The computational cost of 3D simulations can be overwhelming • use 2D fluid simulations to obtain the desired detail and synthesize rich motion • “define” a 3D velocity field from 2D fluid simulations and interpolation • advect particles with the virtual 3D velocity field and a spatially tiled Komolgorov spectrum

26. Velocity Field Construction 2D Simulation 2D Simulation Interpolation 3D Velocity Field 2D Simulation Kolmogorov Spectrum

27. Velocity Field Construction

28. Velocity Field Construction

29. Simulation of Large Scale Phenomena

30. NIH Visible Human

31. NIH Visible Human