Partial Differential Equations for Computer Animation

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1 Aknowledgemen Paral Dfferenal Eqaons for Comper Anmaon A maor par of s slde se and all aompanyng demos are oresy of Maas Müller ETH Zr NovodeX AG. Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Movaon Inrodory Eample Mos dynam effes and pysal proesses an be desrbed by paral dfferenal eqaons PDEs We gve a sor overvew of a large resear feld n maemas and pyss appled o e felds of anmaon and omper graps Goals Yo know wa a PDE s Yo know PDEs for spef effes Yo know ow o solve ese PDEs nmerally Yo an se yor knowledge o mplemen yor own effes and anmaons Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Smlae emperare evolon One spae dmenson me T T0) Gven e emperare dsrbon T0) a me 0 and wnd speed. Fnd PDE for e emperare evolon T) a me! Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory

2 Adveon How does e emperare ange n one me sep? T T T T) T 0 T) PDE for T) s -d adveon ranspor) eqaon. Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory T T or T T Analyal Solon Any T) of e form T ) f ) solves T T Te pysal solon also needs o sasfy e nal ondon T 0) T0 ) Ts e solon s T ) T0 ) T s obaned by sfng T 0 rog a dsane wo any ange of sape of T 0. Only smple PDEs an be solved analyally. Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Nmeral Solon Fne Dfferenes Olne Sample T): T T ).. n) 0..) Dsreze e dervaves e.g. frs-order bakward seme) T T T T T T Solvng for T yelds e epl pdae rle T T T T Before sarng e smlaon nalze T 0 Possble bondary ondons are perod bondares T 0 T n T n T Maemaal Bakgrond Wa s a PDE? Types of PDEs Bondary Condons Solon Tenqes Analyal Meods Nmeral Meods Eamples Adveon Dffson Te Wave Eqaon Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory

3 Maemaal Defnon of a PDE PDEs n Pyss Gven a fnon of wo or more ndependen varables e.g. yz) A Paral Dfferenal Eqaon PDE) s an eqaon a deermnes e beavor of n erms of paral dervaves of e.g. y self and e ndependen varables e.g. yz Eample: Noaon: y..) y y..) y Independen varables are Sa Problem: D) y D) yz 3D) Dynam Problem: D) y D) yz 3D) Unknown fnon an be..) a salar e.g. emperare T densy ρ..)..) v..) w..) T a veor e.g. dsplaemen veloy v Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory PDE Classfaon PDE Classfaon Order Order of PDE Order of ges paral dervave Lnear and s paral dervaves only or lnearly oeffens may be fnons of ndependen varables nd order lnear PDE of ndependen varables: f f f f Non-lnear eample y 5 y y f f f Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory yy 7 Classfaon of nd order lnear PDEs A B C F y ) y yy y Hyperbol B -AC > 0 Parabol B -AC 0 Ellp B -AC < 0 Geomer movaon Dfferen maemaal and pysal beavor Deermnes ype of bondary ondons needed For more an ndependen varables se generalzed reron see ebook) Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory 3

4 PDE Classfaon Bondary Condons Hyperbol me-dependen proesses no evolvng o a seady sae reversble propagae beavor ndmnsed e. g. wave moon Parabol me-dependen proesses evolvng o a seady sae rreversble dsspave e. g. ea dffson Ellp me-ndependen already n seady sae Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Generally ere are many w solve e PDE In pysal applaons only one solon s epeed Typally s only defned n a regon D Te pysal solon s reqred o sasfy eran ondons on e bondary δd of D ) f ) 0) o ) Inal ondon 0 D Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Vale presrbed Drle ondon ) f ) 0 0 Dervave presrbed Nemann ondon Olne Analyal Solon Tenqes Maemaal Bakgrond Wa s a PDE? Types of PDEs Bondary Condons Solon Tenqes Analyal Meods Nmeral Meods Eamples Adveon Dffson Te Wave Eqaon Analyal solons only es for small and smple problems lnear eqaons smple geomery nal and bondary ondons Eamples for analyal solon enqes see ebooks): Meod of separaon of varables Teory of araerss Green fnon meod Laplae ransform Real-world problems are ypally solved nmerally Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory

5 Nmeral Solon - Overvew Dsrezaon Governng Eqaons IC / BC Connos Solon Dsrezaon Fne Dfferene Fne Volme Fne Elemen Speral Bondary Elemen Sysem of Algebra Eqaons Dsree Nodal Vales Eqaon Mar) Solver Epl solon CG Appro. Solon Tme dervaves Fne-dfferene meods Spaal dervaves Fne-dfferene meod FDM Fne-elemen meod FEM Fne-volme meod FVM Speral meods Bondary elemen meods Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Te Fne Dfferene Meod Te Fne Dfferene Meod Smples meod o ndersand and mplemen Sample fnon on a reglar grd e.g. ).. n).. m) 0...) y w grd spang and me sep Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Appromaon of e emporal dervave O ) Appromaon of spaal dervaves ree possbles) O ) forward seme O ) bakward seme O ) enral seme If nformaon moves from lef o rg as n - ) bakward s pwnd pwnd s bes oe) Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory 5

6 6 Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Te Fne Dfferene Meod Appromaon of ger dervaves ) O Hger-order appromaons ) 8 8 O ) O Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Te Fne Dfferene Meod PDE beomes sysem of algebra eqaons for e grd vales Lnear PDE beomes sysem of lnear eqaons for e grd vales Smple sably rle: Informaon ms no ravel more an one grd ell n one me sep / < < Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Epl Semes Lnear Salar Adveon pwnd >0) 0 ) downwnd >0) 0 ) enered Forward me enered spae FTCS) 0 ) Leap-frog 0 ) Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Epl Semes Lnear Salar Adveon La-Wendroff ) ) Beam-Warmng >0) La-Fredr CFL nmber Coran-Fredrs-Lew mporan n sably analyss ) ) 3 ) ) σ

7 Olne Adveon n D and 3D Maemaal Bakgrond Wa s a PDE? Types of PDEs Bondary Condons Solon Tenqes Analyal Meods Nmeral Meods Eamples Adveon Dffson Te Wave Eqaon For D adveon we ad T T In D and 3D speed s a veor w dreon n and leng We ave e spaal dervave of T n dreon n T T / n T n D : T n Ty / y Ts we ave T n T) Sne n e general adveon PDE reads T T Nabla operaor Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Fne Dfferenes Solon Dffson n D Te pdae rle n D T T T T T T T y Make e wnd veloy a fnon of e loaon T T s adveed by a general wnd feld Use veloy array replae by and y by y PDE sll frs order and lnear Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory A d mρ)ad Te densy ρ) desrbes e onenraon of a sbsane n a be w ross seon A. Condon law: Te mass flow rog an area A fl) s proporonal o e graden of e densy normal o A: dm da dρ ) Ad k ρ da ρ ρ d ρ k d d kρ k ρ Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory A ρ kρ Condvy 7

8 Dffson n 3D Fne Dfferenes Solon Te dffson PDE n 3D ρ k ρ ρ ρ ) k ρ k ρ yy zz Laplae operaor Inon for seond spaal dervaves Consan onenraon graden ρ)ab as no effe Te pdae rle n D ρ ρ ρ ρ ρ ρ ρ k Inon: If e average onenraon of e negbor ells s larger an e onenraon of e ell s onenraon nreases and ve versa. Fl b no ange Fl and posve ange Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Te D Wave Eqaon Analyal Solon f A f Fnon ) s dsplaemen of e srng normal o -as Assmng small dsplaemens and onsan sress σ Fore ang normal o ross seon A s f σa Componen n -dreon Vbrang srng d f σ A Newon s nd law for an nfnesmal segmen ρ σ ρad) σa σa d Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory For e srng we ave ρ σ Te sandard form s w e analyal solon a f ) b f ) were so s e speed w w waves ravel Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory s? 8

9 D Wave Eqaon Fne Dfferenes Solon Te wave eqaon n D ) yy Waves propagae w veloy For e vbrang srng we ave σ/ρ Te pdae rle n D v v v Very smple o mplemen Yelds ool waer srfae anmaon Bondary ondons assmng.. n) Perod: 0 n n Mrror: Analog for 0 n n Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Demo Lerare Combnaon of wave eqaon dffson adveon D) d w a wave eqaon dffson adveon Alan Jeffrey Appled Paral Dfferenal Eqaons An Inrodon Aadem Press Amserdam ISBN Jon D. Anderson Compaonal Fld Dynams Te Bass w Applaons MGraw-Hll In. New York ISBN Epl Leap-frog negraon v v w v ) d a Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory Unversy of Frebrg - Inse of Comper Sene - Comper Graps Laboraory 9