g(x,t) Particle Systems Reading Particle in a flow field What are particle systems? Brian Curless CSE 457 Spring 2015

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1 Readig article Systes Bria Curless CSE 457 Sprig 05 Required: Witki, article Syste Dyaics, SIGGRAH 0 course otes o hysically Based Modelig. (olie hadout) Witki ad Bara, Dieretial Equatio Basics, SIGGRAH 0 course otes o hysically Based Modelig. (olie hadout) Optioal Hockey ad Eastwood. Coputer siulatio usig particles. Ada Hilger, ew York, 988. Gai Miller. The otio dyaics o sakes ad wors. Coputer Graphics :69-78, 988. What are particle systes? A particle syste is a collectio o poit asses that obeys soe physical laws (e.g, graity, heat coectio, sprig behaiors, ). article systes ca be used to siulate all sorts o physical pheoea: article i a low ield We begi with a sigle particle with: ositio, y Velocity, d d / dt dt dy / dt Suppose the elocity is actually dictated by a driig uctio, a ector low ield, g: g(, t) y g(,t) I a particle starts at soe poit i that low ield, how should it oe? 3 4

2 Di eqs ad itegral cures Euler s ethod The equatio g(, t) is actually a irst order dieretial equatio. We ca sole or through tie by startig at a iitial poit ad steppig alog the ector ield: Oe siple approach is to choose a tie step, t, ad take liear steps alog the low: ( tt) ( t) ( t) t t () t t () t () t tg( (),) t t Writig as a tie iteratio: i i i i i t g g g t with (, i t) Start Here This approach is called Euler s ethod ad looks like: This is called a iitial alue proble ad the solutio is called a itegral cure. 5 roperties: Siplest uerical ethod Bigger steps, bigger errors. Error ~ O(t ). eed to take pretty sall steps, so ot ery eiciet. Better (ore coplicated) ethods eist, e.g., adaptie tiesteps, Ruge-Kutta, ad iplicit itegratio. 6 article i a orce ield Secod order equatios ow cosider a particle i a orce ield. This equatio: I this case, the particle has: Mass, d d Acceleratio, a dt dt The particle obeys ewto s law: a (,, t) is a secod order dieretial equatio. Our solutio ethod, though, worked o irst order dieretial equatios. So, gie a orce, we ca sole or the acceleratio: The orce ield ca i geeral deped o the positio ad elocity o the particle as well as tie. We ca rewrite the secod order equatio as: (,, t) or (,, t) Thus, with soe rearrageet, we ed up with: (,, t) where we substitute i ad its deriatie to get a pair o coupled irst order equatios. 7 8

3 hase space Dieretial equatio soler / Cocateate ad to ake a 6-ector: positio i phase space. Takig the tie deriatie: aother 6-ector. A ailla st -order dieretial equatio. Startig with: / Applyig Euler s ethod: ( tt) ( t) t ( t) ( tt) ( t) t ( t) Ad akig substitutios: ( tt) () t t() t ( ( t), ( t), t) ( tt) ( t) t Writig this as a iteratio, we hae: t i i i i i i t with i i, i, t Agai, perors poorly or large t. 9 0 article structure Sigle particle soler iterace How do we represet a particle? ositio i phase space positio elocity orce accuulator ass getdi getstate setstate derieal 6 /

4 article systes article syste soler iterace I geeral, we hae a particle syste cosistig o particles to be aaged oer tie: For particles, the soler iterace ow looks like: particles tie particles tie get/setstate getdi derieal article syste di. eq. soler Forces We ca sole the eolutio o a particle syste agai usig the Euler ethod: i i i i i i / t i i i i i i / Each particle ca eperiece a orce which seds it o its erry way. Where do these orces coe ro? Soe eaples: Costat (graity) ositio/tie depedet (orce ields) Velocity-depedet (drag) -ary (sprigs) How do we copute the et orce o a particle? 5 6

5 article systes with orces Graity ad iscous drag Force objects are black boes that poit to the particles they iluece ad add i their cotributios. We ca ow isualize the particle syste with orce objects: particles tie orces The orce due to graity is siply: gra G p-> += p-> * F->G Ote, we wat to slow thigs dow with iscous drag: F F F drag kdrag p-> -= F->k * p-> 7 8 Daped sprig derieal A sprig is a siple eaples o a -ary orce. Recall the equatio or the orce due to a D sprig: With dapig: k ( sprig r ) [ k ( r) k ] sprig I D or 3D, we get: dap r p p k ( ) ˆ ˆ sprig r kdap ote: sti sprig systes ca be ery ustable uder Euler itegratio. Siple solutios iclude heay dapig (ay ot look good), tiy tie steps (slow), or better itegratio (Ruge-Kutta is straightorward). ˆ r = rest legth 9. Clear orces Loop oer particles, zero orce accuulators. Calculate orces Su all orces ito accuulators 3. Retur deriaties Loop oer particles, retur ad / 00 0 Apply orces to particles 3 Clear orce accuulators F Retur deriaties to soler F F 3 F 0

6 Boucig o the walls Hadlig collisios is a useul add-o or a particle siulator. Collisio Detectio How do you decide whe you e ade eact cotact with the plae? For ow, we ll just cosider siple poit-plae collisios. A plae is ully speciied by ay poit o the plae ad its oral. oral ad tagetial elocity Collisio Respose To copute the collisio respose, we eed to cosider the oral ad tagetial copoets o a particle s elocity. T k resitutio beore ater T T T ( ) The respose to collisio is the to iediately replace the curret elocity with a ew elocity: k T restitutio The particle will the oe accordig to this elocity i the et tiestep. 3 4

7 Collisio without cotact I geeral, we do t saple oets i tie whe particles are i eact cotact with the surace. There are a ariety o ways to deal with this proble. The ost epesie is backtrackig: deterie i a collisio ust hae occurred, ad the roll back the siulatio to the oet o cotact. A siple alteratie is to deterie i a collisio ust hae occurred i the past, ad the preted that you re curretly i eact cotact. Very siple collisio respose How do you decide whe you e had a collisio durig a tiestep? A proble with this approach is that particles will disappear uder the surace. We ca reduce this proble by essetially osettig the surace: 3 3 Also, the respose ay ot be eough to brig a particle to the other side o a wall I that case, detectio should iclude a elocity check: 5 6 More coplicated collisio respose Aother solutio is to odiy the update schee to: detect the uture tie ad poit o collisio relect the particle withi the tie-step article-sphere collisio Suppose a particle collides with a sphere : How would we detect this collisio? What oral should we use or collisio respose? 7 8

8 article rae o reerece The OpeGL geoetry pipelie Let s say we had our robot ar eaple ad we wated to lauch particles ro its tip. How would we go about startig the particles ro the right place? First, we hae to look at the coordiate systes i the OpeGL pipelie 9 30 rojectio ad odeliew atrices Robot ar code, reisited Ay piece o geoetry will get trasored by a sequece o atrices beore drawig: p = M project M iew M odel p The irst atri is OpeGL s GL_ROJECTIO atri. The secod two atrices, take as a product, are aitaied o OpeGL s GL_MODELVIEW stack: M odeliew = M iew M odel Recall that the code or the robot ar looked soethig like: glrotate( theta, 0.0,.0, 0.0 ); base(h); gltraslate( 0.0, h, 0.0 ); glrotate( phi, 0.0, 0.0,.0 ); upper_ar(h); gltraslate( 0.0, h, 0.0 ); glrotate( psi, 0.0, 0.0,.0 ); lower_ar(h3); All o the GL calls here odiy the odeliew atri. ote that ee beore these calls are ade, the odeliew atri has bee odiied by the iewig trasoratio, M iew. 3 3

9 Coputig the particle lauch poit To id the world coordiate positio o the ed o the robot ar, you eed to ollow a series o steps:. Figure out what M iew is beore drawig your odel.. Draw your odel ad add oe ore trasoratio to the tip o the robot ar. 3. Copute Mat4 Miew = glgetmodelviewmatri(); gltraslate( 0.0, h3, 0.0 ); M - odel MiewModeliew Suary What you should take away ro this lecture: The eaigs o all the boldaced ters Euler ethod or solig dieretial equatios Cobiig particles ito a particle syste hysics o a particle syste Various orces actig o a particle Siple collisio detectio with a plae ad a sphere How to hook your particle syste ito the coordiate rae o your odel Mat4 particlexor = getworldxor(miew); 4. Trasor a poit at the origi by the resultig atri. Vec3 particleorigi = particlexor * Vec3(0,0,0); ow you re ready to lauch a particle ro that last coputed poit! 33 34