Keyframe Control of Smoke Simulations SIGGRAPH Roland Angst

Transcription

1 Keyfame Contol of Smoke Smulatons SIGGRAPH 2003

2 Keyfame Contol of Smoke Smulatons SIGGRAPH 2003 Authos: Aden eulle (Unesty of Washngton) Antone McNamaa (Unesty of Washngton) Zoan Popoc (Unesty of Washngton) Jos Stam (Alas Waefont) Pesented by: Roland Angst

3 Outlne 1. Motaton and Geneal Idea 2. Intoducton to Nae-Stokes Equatons 3. Keyfame-Contol Appoach 4. Exact Deates 5. Contol Paametes 6. Layeed Multple Shootng 7. Results 8. Poblems 9. My Own houghts

4 1. Motaton Anmaton s. Physcal based Smulaton Po Cons Complete atstc feedom Fo physcal plausble scenes t becomes quckly athe tedous Plausble scenes - computatonal esouces - lmted atstc feedom 1. Motaton and Geneal Idea

5 1.1 Physcal Based Smulaton Smulaton wth consdeng physcs Defnng an ntal state q 0 and tempoal ntegaton of physcal laws. Influence capabltes of an atst: Manpulatng the ntal state of the smulaton Leads to almost unpedctable smulaton behaou! Goal: Combne the atstc feedom of anmatons wth physcal plausblty of smulatons. 1. Motaton and Geneal Idea 1.1 Physcal Based Smulaton

6 1.2 Keyfamng tme Key Fame: A key fame ( ) s a fame n an anmated sequence of fames that was dawn o othewse constucted dectly by the use.... he compute flls n the gap ( ). hs s called tweenng. [Wkpeda] Key fame n smulaton Defnng the system state q at a fxed tme 1. Motaton and Geneal Idea 1.2 Keyfamng

7 Keyfamng fo Smoke What we hae: physcal descpton of the flud dynamcs though PDEs. What we want: physcal plausble ntepolaton o appoxmaton of the key fames Idea: Influence the dynamcs by addton of paametesed, extenal contol foces. automatc optmzaton pocess seaches fo sutable contol foce paametes to appoxmate the gen key fames. 1. Motaton and Geneal Idea 1.2 Keyfamng

8 1.3 wo Man Contbutons Optmzaton appoach: Defnton of a taget functon we hae to mnmze. Mnmzaton technque: gadent based appoach Method fo exact calculaton of the deates of the flud smulaton states. Optmzaton wth multple key fames needs a lot of computaton. New multple shootng appoach fo anmatons wth seeal key fames. 1. Motaton and Geneal Idea 1.3 wo Man Contbutons

9 2. Intoducton to Nae-Stokes Equatons Shot ntoducton o efesh of the Nae-Stokes equatons fo flud smulaton hs s not an actual pat of the pape, but t s equed fo the compehenson. Fo futhe nfomaton see pesentaton of Jos Stam: 2. Intoducton to N.-S. Equatons

10 2.1 Nae-Stokes Equatons Nae-Stokes equatons completely descbe dynamc behaou of an ncompessble flud (gas o lqud) Nae-Stokes equatons consst of a scala and a ecto alued PDE State q of a pont n a flud (gas o lqud) s descbed by: elocty feld densty feld ρ 3 DoF of + 1 DoF of ρ = 4 DoF pe pont 2. Intoducton to N.-S. Equatons 2.1 Nae-Stokes Equatons

11 Mathematcal Descpton 1. Mass conseaton n ncompessble medum: ρ + t ( ) ncompessble ρ = 0 = 0 s degence fee 2. Momentum conseaton (ow wse to undestand,.e. 3 equatons): t = ( ) + { μ Δ + f exten { p Adecton Dffuson extenal Foces pessue Gadent Feld t s a lnea combnaton of 4 tems 2. Intoducton to N.-S. Equatons 2.1 Nae-Stokes Equatons

12 2.2 Numecal Soluton State q n pont n tme : Gd of denstes and eloctes: q = ( ρ, ) Integaton of the elocty feld n tme (ex. wth Eule): t = μ Δ + f exten p Adecton Dffuson extenal Foces pessue +1 = + h ( ) { t Gadent Feld Notaton: o peent confuson wth patal tme deates the ponts n tme ae ndcated by supe- nstead of subscpts (hs s a dffeence to the notaton used n the pape ) 2. Intoducton to N.-S. Equatons 2.2 Numecal Soluton

13 Uncondtonal Stable Method Splttng computaton of +1 n fou smalle steps: 1. Add extenal foces 2. Self-adect elocty feld 3. Dffuson 4. Use emanng DoF of the densty feld ρ to ensue a degence fee elocty feld (aka. pojecton step) = 0 + ( ) f exten F Extenal foces = F F Α D P Selfadecton = AF AF + μδ Dffuson = DAF DAF p Pojecton = h + + = = h PDAF 0 + h 2. Intoducton to N.-S. Equatons 2.2 Numecal Soluton

14 Whole Smulaton Step 1. Calculate +1 by splttng t nto fou smalle steps 2. Adect the densty feld though ths newly calculated elocty feld 3. Compensate the dsspaton (nheently n uncondtonal stable methods) by a mass conseng step. Extenal foces F F Dffuson Pojecton Α D P AF DAF PDAF A ρ A ρ Selfadecton Denstyadecton Masspeseaton M MA ρ = 0 = F = AF = DAF = PDAF = APDAF = = MAPDAF 0 + h 2. Intoducton to N.-S. Equatons 2.2 Numecal Soluton

15 3. Keyfame-Contol Appoach Gen: keyfame state at tme : q * Goal: add extenal contol foces f contol (u) to Appoxmate the key fame states q * by the smulaton states q whle Mnmzng the atfcal ntoduces extenal contol foces f contol (u) 3. Keyfame-Contol Appoach

16 3.1 Optmzaton Appoach ϕ k φ s : φ k : = k d Defnton of a taget functon: φ(q 0, u) = φ s + φ k Paametese the contol foces by a paamete ecto u gadent based mnmzaton technque usable: 0 ag mn ϕ q, u ag mn ϕ + ϕ u ( ) ( ) = u Penalty tem fo added extenal contol foces f contol (u): 2 ϕ = k f s s mesteps contol Dffeence metc between keyfames q * = (ρ *, * ) and coespondng smulaton states q = (ρ, ): 2 B ρ ρ + k B mesteps wth densty ( ) ( ) * * keyfames k mesteps wth s elocty keyfames 2 3. Keyfame-Contol Appoach 3.1 Optmzaton Appoach

17 3.2 aget Functon Gadent 2 = mesteps contol s s f k ϕ contol mesteps contol s s du d f f k du d = 2 ϕ dffeentate ( ) ( ) + = 2 * 2 * d k B k B k ρ ρ ϕ ( ) ( ) + = d k du d B B k du d B B k du d * * 2 2 ρ ρ ρ ϕ dffeentate 3. Keyfame-Contol Appoach 3.2 aget Functon Gadent

18 3.3 Blung q q * 2 B ( q q ) * 2 Unblued smulaton state and keyfame Blued smulaton state and blued keyfame 3. Keyfame-Contol Appoach 3.3 Blung

19 4. Exact Deates Needed tems to compute the gadent of the taget functon φ: d f d contol u and d ρ d u and d d u 4. Exact Deates

20 4.1 hee Soluton Appoaches 1. Analytc deates of the Nae-Stokes equatons Poblem: no absolutely physcally coect numecal soluton heefoe analytc deates of Nae-Stokes equatons need not agee wth deates of numecal smulaton! 2. Fnte Dffeence Appoxmaton Poblem: Unsutable because slow and ey naccuate! 4. Exact Deates 4.1 hee Soluton Appoaches

21 3. New method: Augment the state of the smulaton q = (, ρ ) wth the needed deates: 4.1 hee Soluton Appoaches ( ),0,0,0,0,...,,...,,,,, ρ ρ ρ ρ q u u u u q = = Motated by [Popoc 2000]. 4. Exact Deates 4.1 hee Soluton Appoaches

22 4.2 New Method Remembe: Needed tems to calculate the gadent of the taget d f functon φ: contol Analytc tem deable snce contol d u foces ae dectly paametesed by u d ρ d and Idea: apply the smulaton steps not only d u d u on q = (, ρ) but also on the patal deates wth espect to eey contol foce paamete u : d q d d ρ =, d u d u d u 4. Exact Deates 4.2 New Method

23 4.3 Patal Stepped Deates Recall: One smulaton step of Nae-Stokes equatons by multple small patal steps! Cayng along the deates n tme Eey patal smulaton step has coespondng patal step fo the deates 4. Exact Deates 4.3 Patal Stepped Deates

24 Patal Extenal Foce Step = 0 = F F F F + = f exten u F u F u u F Dffeentate = u + f exten u 4. Exact Deates 4.3 Patal Stepped Deates

25 Paallel Patal Steps = 0 F = F F = Α D P AF AF = DAF DAF = PDAF PDAF A ρ = A ρ APDAF M = = MA ρ MAPDAF 0 + h Extenal foces Dffuson Pojecton Selfadecton Denstyadecton Masspeseaton u F u F u Α u AF u D u DAF u P u PDAF u ρ A u u ρ A M u u ρ MA 4. Exact Deates 4.3 Patal Stepped Deates

26 5. Contol Paametes Wnd Foces: a sngle ecto scaled by a Gaussan falloff functon u = wnd Gaussan decton cente Votex Foces: a fxed otaton matx scaled by a Gaussan falloff functon and a paamete otex cente u = 5. Contol Paametes

27 5. Contol Paametes wnd foce otex foce 5. Contol Paametes

28 6. Layeed Multple Shootng 1 st Poblem: Computng q u only fom the tmestep on when the contol foce belongng to u affected the smulaton. No need to compute q u f(u ) affects smulaton 2 nd Poblem: Local mnma of cost functon q u needed t 6. Layeed Multple Shootng

29 6.1 Idea of Multple Shootng Multple Shootng: empoally beak a complex poblem nto a set of subpoblems. Use local solutons of these subpoblems to popagate knowledge back and foth to get a global soluton. Poblem: no physcal meanngful ntepolaton to constuct a global soluton. 6. Layeed Multple Shootng 6.1 Idea of Multple Shootng

30 6.2 Layeed Multple Shootng f(u ) affects smulaton A B C D E F G No need to compute q needed u q u t Bounday ognal keyfame Intal schedule Non bounday ognal keyfame Altenate schedule Culled fom ntemedate states of the ntal segments 6. Layeed Multple Shootng 6.2 Layeed Multple Shootng

31 Paallel Pocessng 6. Layeed Multple Shootng 6.2 Layeed Multple Shootng

32 Sequental Pocessng 6. Layeed Multple Shootng 6.2 Layeed Multple Shootng

33 7. Results Keyfame 0 q * Keyfame q end * Gdsze: N. of contol foces: 20 Computaton tme: 2h on P4 2GHz 7. Results

34 7. Results Paametes: 408 Keyfames: 2 Steps: 35 me: ca. 24h 7. Results

35 8. Poblems Optmzaton pocess athe slow One sngle ealuaton of the taget functon needs a un of the whole smulaton wth augmented states!!! Local Mnma: method not fully automated (yet?) Possble Soluton: nsetng addtonal keyfame to gude the optmzaton pocess Result too contolled and not smoke-lke 8. Poblems

36 9. My Own houghts 1 st appoach to combne physcally based smulatons wth atstc ceatty (n the doman of flud smulaton) Shown esults look good But: how much fne tunng was needed to get them? Pocess s tebly slow! How does t scale fo lage gd szes and moe contol paametes? Is mnmzng a cost functon the ght way to go? 9. My Own houghts

37 Futhe Ideas Multesoluton foce famewok Othe cost functon Non-gadent based optmzaton technque 9. My Own houghts

38 End. hank you fo you attenton.