Reading. Particle Systems. What are particle systems? Overview

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1 Readig Particle Systems Required: Witki, Particle System Dyamics, SIGGRAPH 97 course otes o Physically Based Modelig. Optioal Witki ad Baraff, Differetial Equatio Basics, SIGGRAPH 97 course otes o Physically Based Modelig. Hockew ad Eastwood. Computer simulatio usig particles. Adam Hilger, New York, 988. Gai Miller. The motio dyamics of sakes ad worms. Computer Graphics :69-78, 988. What are particle systems? A particle system is a collectio of poit masses that obeys some physical laws (e.g, graity or sprig behaiors). Particle systems ca be used to simulate all sorts of physical pheomea: Smoke Sow Fireworks Hair Cloth Sakes Fish. Oe lousy particle. Particle systems 3. Forces: graity, sprigs 4. Implemetatio Oeriew 3 4

2 Particle i a flow field We begi with a sigle particle with: Positio, Velocity, = y d d / dt & = = dt dy / dt Suppose the elocity is dictated by some driig fuctio g: & = g(, t) y g(,t) Vector fields At ay momet i time, the fuctio g defies a ector field oer : How does our particle moe through the ector field? 5 6 Diff eqs ad itegral cures The equatio & = g(, t) is actually a first order differetial equatio. We ca sole for through time by startig at a iitial poit ad steppig alog the ector field: Start Here This is called a itial alue problem ad the solutio is called a itegral cure. Euler s method Oe simple approach is to choose a time step, t, ad take liear steps alog the flow: ( t+ t) = ( t) + t & ( t) = () t + t g(,) t This approach is called Euler s method ad looks like: Properties: Simplest umerical method Bigger steps, bigger errors Need to take pretty small steps, so ot ery efficiet. Better (more complicated) methods eist, e.g., Ruge-Kutta. 7 8

3 Particle i a force field Secod order equatios Now cosider a particle i a force field f. I this case, the particle has: Mass, m Acceleratio, d d a && = = dt dt The particle obeys Newto s law: f = ma= m&& The force field f ca i geeral deped o the positio ad elocity of the particle as well as time. Thus, with some rearragemet, we ed up with: (,, t) = f & && m 9 This equatio: is a secod order differetial equatio. Our solutio method, though, worked o first order differetial equatios. We ca rewrite this as: (,, t) && = f m & = f (,, t) & = m where we hae added a ew ariable to get a pair of coupled first order equatios. 0 Phase space Particle structure & & & = / m & f Cocateate ad to make a 6-ector: positio i phase space. Takig the time deriatie: aother 6-ector. A ailla st -order differetial equatio. f m positio elocity force accumulator mass Positio i phase space

4 Soler iterface Particle systems f m getdim getstate setstate derieal [ 6] / m f particles time 3 3 L f f f3 f m m 3 m m 3 4 Soler iterface Forces particles time get/setstate getdim Costat (graity) Positio/time depedet (force fields) Velocity-depedet (drag) N-ary (sprigs) derieal 6 L f f f L m m m 5 6

5 Graity Viscous drag Force law: f gra = mg p->f += p->m * F->G Force law: f drag = kdrag p->f -= F->k * p-> 7 8 Damped sprig Particle systems with forces Force law: f = ks( r) + k f = f d r = rest legth = = particles time forces L f f f m m m f F F F F 9 0

6 derieal loop derieal Loop. Clear forces Loop oer particles, zero force accumulators. Calculate forces Sum all forces ito accumulators 3. Gather Loop oer particles, copyig ad f/m ito destiatio array L f f f m m m Clear force accumulators F F F Apply forces to particles F L f f f m m m 3 Retur [,f/m, ] to soler Soler iterface Differetial equatio soler particles time get/setstate derieal getdim 6 L f f f L m m m & = / m & f Euler method: i+ i i i+ i i f / m M = M + t M i+ i i i+ i i f / m 3 4

7 Boucig off the walls Add-o for a particle simulator For ow, just simple poit-plae collisios Normal ad tagetial compoets V N P N V V VT VN = ( N V) N V = V V T N 5 6 Collisio Detectio Collisio Respose ( X P) N<ε Withi ε of the wall N V < 0 Headig i V N k r V N V X P N V before VT V T V after V = VT krvn 7 8

8 Cloth Simulatio Artificial Fish Cloth forces: Blue (short horizotal & ertical) = stretch sprigs Gree (diagoal) = shear sprigs Red (log horizotal & ertical) = bed sprigs 9 30 Summary What you should take away from this lecture: The meaigs of all the boldfaced terms Euler method for solig differetial equatios Combiig particles ito a particle system Physics of a particle system Various forces actig o a particle Simple collisio detectio 3

Reading. Required: Witkin, Particle System Dynamics, SIGGRAPH 97 course notes on Physically Based Modeling.

Reading. Required: Witkin, Particle System Dynamics, SIGGRAPH 97 course notes on Physically Based Modeling. Particle Systems Readig Required: Witki, Particle System Dyamics, SIGGRAPH 97 course otes o Physically Based Modelig. Optioal Witki ad Baraff, Differetial Equatio Basics, SIGGRAPH 97 course otes o Physically