533D Animation Physics: Why? � Natural phenomena: passive motion � Film/TV: difficult with traditional techniques CS533D - Animation Physics • When you control every detail of the motion, it � s hard to make it look like it � s not being controlled! � Games: difficult to handle everything convincingly with prescripted motion � Computer power is increasing, audience expectations are increasing, artist power isn � t: need more automatic methods � Directly simulate the underlying physics to get realistic motion cs533d-winter-2005 1 cs533d-winter-2005 2 Web Contacting Me � www.cs.ubc.ca/~rbridson/courses/533d � Robert Bridson • X663 (new wing of CS building) � Course schedule • Drop by, or make an appointment (safer) • Slides online, but you need to take notes too! • 604-822-1993 (or just 21993) � Reading • email rbridson@cs.ubc.ca • Relevant animation papers as we go � I always like feedback! � Assignments + Final Project information • Ask questions if I go too fast… • Look for Assignment 1 � Resources cs533d-winter-2005 3 cs533d-winter-2005 4
Evaluation Topics � 4 assignments (60%) � Particle Systems • See the web for details + when they are due • the basics - time integration, forces, collisions • Mostly programming, with a little analysis (writing) � Deformable Bodies � Also a final project (40%) • e.g. cloth and flesh • Details will come later, but basically you need to � Constrained Dynamics either significantly extend an assignment or animate something else - talk to me about topics • e.g. rigid bodies • Present in final class - informal talk, show movies � Fluids � Late: without a good reason, 20% off per day • e.g. water • For final project starts after final class • For assignments starts morning after due cs533d-winter-2005 5 cs533d-winter-2005 6 Particle Systems Particle Systems � Read: Reeves, “Particle systems…”, SIGGRAPH � 83 Sims, “Particle animation and rendering using data parallel computation", SIGGRAPH '90 Miller & Pearce, “Globular dynamics…”, SIGGRAPH � 89 � Some phenomena is most naturally described as many small particles • Rain, snow, dust, sparks, gravel, … � Others are difficult to get a handle on • Fire, water, grass, … cs533d-winter-2005 7 cs533d-winter-2005 8
Particle Basics Example � Each particle has a position � Sparks from a campfire • Maybe orientation, age, colour, velocity, � Every frame (1/24 s) add 2-3 particles temperature, radius, … • Position randomly in fire • Call the state x • Initialize temperature randomly � Seeded randomly somewhere at start � Move in specified turbulent smoke flow • Maybe some created each frame • Also decrease temperature � Move (evolve state x) each frame � Render as a glowing dot (blackbody according to some formula radiation from temperature) � Eventually die when some condition met � Kill when too cold to glow visibly cs533d-winter-2005 9 cs533d-winter-2005 10 Rendering First Order Motion � We won � t talk much about rendering in this course, but most important for particles � The real strength of the idea of particle systems: how to render • Could just be coloured dots • Or could be shards of glass, or animated sprites (e.g. fire), or deforming blobs of water, or blades of grass, or birds in flight, or … cs533d-winter-2005 11 cs533d-winter-2005 12
First Order Motion Forward Euler � For each particle, have a simple 1 st order � Simplest method: x n + 1 � x n differential equation: ( ) = v x n , t n � t dx ( ) dt = v x , t Or: ( ) x n + 1 = x n + � tv x n , t n � Analytic solutions hopeless � Need to solve this numerically forward in � Can show it � s first order accurate: time from x(t=0) to • Error accumulated by a fixed time is O( � t) x(frame1), x(frame2), x(frame3), … � Thus it converges to the right answer • May be convenient to solve at some • Do we care? intermediate times between frames too cs533d-winter-2005 13 cs533d-winter-2005 14 Aside on Error Forward Euler Stability � General idea - want error to be small � Big problem with Forward Euler: • Obvious approach: make � t small it � s not very stable dx dt = � x , x (0) = 1 • But then need more time steps - expensive � Example: � Also note - O(1) error made in modeling e � t � Real solution smoothly decays to zero, • Even if numerical error was 0, still wrong! always positive • In science, need to validate against experiments • In graphics, the experiment is showing it to an � Run Forward Euler with � t=11 audience: does it look real? • x=1, -10, 100, -1000, 10000, … � So numerical error can be huge, as long as your • Instead of 1, 1.7*10 -5 , 2.8*10 -10 , … solution has the right qualitative look cs533d-winter-2005 15 cs533d-winter-2005 16
Linear Analysis The Test Equation � Get a rough, hazy, heuristic picture of the � Approximate stability of a method ) + � v � x � ( x � x � ) + � v ( ( ) � v x � , t � � t � ( t � t � ) v x , t � Note that eigenvalue � can be complex � But, assume that for real physics � Ignore all but the middle term (the one that • Things don � t blow up without bound could cause blow-up) • Thus real part of eigenvalue � is � 0 � Beware! dx dt = Ax • Nonlinear effects can cause instability • Even with linear problems, what follows assumes � Look at x parallel to eigenvector of A: constant time steps - varying (but supposedly stable) dx dt = � x the “test equation” steps can induce instability � see J. P. Wright, “Numerical instability due to varying time steps…”, JCP 1998 cs533d-winter-2005 17 cs533d-winter-2005 18 Using the Test Equation Stability Region � Can plot all the values of � � t on the � Forward Euler on test equation is x n + 1 = x n + � t � x n complex plane where F.E. is stable: � Solving gives n x 0 ( ) x n = 1 + � � t � So for stability, need 1 + � � t < 1 cs533d-winter-2005 19 cs533d-winter-2005 20
Real Eigenvalue Imaginary Eigenvalue � Say eigenvalue is real (and negative) � If eigenvalue is pure imaginary… • Corresponds to a damping motion, smoothly • Oscillatory or rotational motion coming to a halt � Cannot make � t small enough � Then need: � t < 2 � Forward Euler unconditionally unstable for � these kinds of problems! � Is this bad? � Need to look at other methods • If eigenvalue is big, could mean small time steps • But, maybe we really need to capture that time scale anyways, so no big deal cs533d-winter-2005 21 cs533d-winter-2005 22 Runge-Kutta Methods Midpoint RK2 � Second order: error is O( � t 2 ) when smooth � Also “explicit” • next x is an explicit function of previous � Larger stability region: � But evaluate v at a few locations to get a better estimate of next x � E.g. midpoint method (one of RK2) ( ) x n + 12 = x n + 1 2 � tv x n , t n ( ) x n + 1 = x n + � tv x n + 12 , t n + 12 � But still not stable on imaginary axis: no point cs533d-winter-2005 23 cs533d-winter-2005 24
Modified Euler Modified Euler (2) � Stability region for � =2/3 � (Not an official name) � Lose second-order accuracy, get stability on imaginary axis: ( ) x n + � = x n + � � tv x n , t n ( ) x n + 1 = x n + � tv x n + � , t n + � � Parameter � between 0.5 and 1 gives � Great! But twice the cost of Forward Euler trade-off between imaginary axis and real � Can you get more stability per v- axis evaluation? cs533d-winter-2005 25 cs533d-winter-2005 26 Higher Order Runge-Kutta TVD-RK3 � RK3 and up naturally include part of the � RK3 useful because it can be written as a imaginary axis combination of Forward Euler steps and averaging: can guarantee some properties even for nonlinear problems! ( ) n + 1 = x n + � tv x n , t n ˜ x ( ) n + 2 = ˜ n + 1 + � tv ˜ ˜ x x x n + 1 , t n + 1 n + 12 = 3 4 x n + 1 ˜ 4 ˜ x x n + 2 ( ) n + 32 = ˜ n + 12 + � tv ˜ ˜ x x x n + 12 , t n + 12 x n + 1 = 1 3 x n + 2 3 ˜ x n + 32 cs533d-winter-2005 27 cs533d-winter-2005 28
RK4 Selecting Time Steps � Often most bang for the buck ( ) v 1 = v x n , t n ( ) v 2 = v x n + 1 2 � tv 1 , t n + 12 ( ) v 3 = v x n + 1 2 � tv 2 , t n + 12 ( ) v 4 = v x n + � tv 3 , t n + 1 ( ) x n + 1 = x n + � t 6 v 1 + 2 6 v 2 + 2 6 v 3 + 1 1 6 v 4 cs533d-winter-2005 29 cs533d-winter-2005 30 Selecting Time Steps Time Stepping � Hack: try until it looks like it works � Sometimes can pick constant � t • One frame, or 1/8th of a frame, or … � Stability based: • Figure out a bound on magnitude of Jacobian � Often need to allow for variable � t • Changing stability limit due to changing Jacobian • Scale back by a fudge factor (e.g. 0.9, 0.5) • Difficulty in Newton converging � Try until it looks like it works… (remember all the dubious assumptions we made for linear stability • … analysis!) � But prefer to land at the exact frame time � Why is this better than just hacking around in the • So clamp � t so you can � t overshoot the frame first place? � Adaptive error based: • Usually not worth the trouble in graphics cs533d-winter-2005 31 cs533d-winter-2005 32
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