Fluid Simulation Overview. Presentation Date: Sep 15 th, 2009 Chrissie C. Cui

Transcription

1 Fld Smlaton Overvew Presentaton Date: Sep 5 th, 2009 Chrsse C. C

2 Introdcton Fld Characterstcs Otlne Naver-Stokes Eqaton Eleran vs. Lagragan approach Fld Smlaton Stages Sold Fld Coplng Read-tme flds

3 Introdcton Applcaton Compter Games Scentfc Comptaton Specal Move Effects Medcal Smlaton (Blood Flow) What can we acheved so far? Smoke Granlar flow (sand) Newtonan fld (Water, ocean) Non-Newtonan fld (Blood, Honey) Mcroscopc phenomena

4 Characterstcs -Basc Fld Propertes Vector Feld Velocty (v) Pressre (p) Addtonal Force (f) Srface Tenson (t) Scalar Feld Densty (ρ) Vscosty (ѵ)

5 Characterstcs Fld Types Incompressble vs. Compressble Flds Incompressble Flds: Flds doesn t change volme very mch Compressble Flds: Flds change ther volme sgnfcantly Vscos vs. Invscd Flds(deal): Vscos Flds: Flds tend to resst a certan degree of deformaton Invscd Flds: Flds don t have resstance to the shear stress Trblent vs. Lamnar (streamlne) Flow: Trblent Flow: Flow that appears to have chaotc and random changes Lamnar Flow: Flow that has smooth behavor Newtonan vs. Non-Newtonan Flds: Flds contne to flow, regardless of the force actng on t Flds that have non-constant vscosty

6 Naver-Stokes Eqaton Momentm Eqaton t = k 2 ( ) p f Change n velocty Dffson/V scosty Incompressblty =0 Advecton Pressre Body Forces : the velocty feld k: knematc vscosty Common technqes for solvng Naver Stoke s eqaton: Eleran approach (grd-based) Lagrangan approach (partcle-based) Spectral method Lattce Boltzmann method

7 EleranApproach Dscretze the doman sng fnte dfferences Defne scalar & vector felds on the grd Use the operator splttng technqe to solve each term separately Evalaton: Dervatve approxmaton Adaptve tme step/solver Memory sage & speed Grd artfact/resolton lmtaton

8 LagranganApproach Treat the fld as dscrete partcles Apply nteracton forces (.e. pressre/vscosty) accordng to certan pre-defned smoothng kernels Evalatons: Mass / Momentm conservaton More nttve Fast, no lnear system solvng Connectvty nformaton/srface reconstrcton

9 Fld Smlaton Stages Man Loop

10 Fld Smlaton Stages Advecton I Sometmes called Convecton or Transport Defne how a qantty moves wth the nderlyng velocty feld Ths term ensres the conservaton of momentm Advecton eqaton: t Approaches: = ( ) Forward Eler (nstable) Sem-Lagraganadvecton (stable for large tme steps, bt sffers from the dsspaton sse)

11 Fld Smlaton Stages Advecton II

12 Fld Smlaton Stages Dffson I Defne how a qantty n a cell nter-changes wth ts neghbors Dffson = Blrrng The vscos fld can be acheved by applyng dffson to the velocty feld Low Vscosty Hgh Vscosty Fgres from [Carlson, Mcha, Trk] Meltng and Flowng, SCA 02

13 Fld Smlaton Stages Dffson II Dffson eqaton: Approaches: Explct formlaton k t 2 = -4 Explct formlaton Implct formlaton (for hgh vscosty) ) 4 (,,,,, j j j j j t t k = ) 4 (,,,,,,, j n j n j n j n j n j n j n t k = Unknowns

14 Fld Smlaton Stages Dffson III

15 Fld Smlaton Stages Pressre Solver I It s sometmes called Pressre Projecton What does the pressre do? Keep the fld at constant volme (ncompressble, conservaton of mass). Make sre the velocty feld stays dvergence-free flx = 0 all _ faces Incompressble Compressble

16 Fld Smlaton Stages Pressre Solver II Eqaton to solve: n = n p ρ How to solve for pressre: s.t. n = 0 () Unknowns Takng dvergence of both sdes of (), we wll have 2 p= ρ n (Posson Eqaton) Bld a system of eqatons and solve Ap= d sng an teratve method sch as Conjgate Gradent Update the velocty feld from the pressre gradent

17 Fld Smlaton Stages Pressre Solver III What abot the pressre on bondary nodes? Free srface: The fld can evolve freely (p = 0) Sold wall: The fld can tpenetrate the wall bt can flow freely n tangental drectons (Nemann BC) bondary n = sold n

18 Sold Fld Coplng One-way coplng: Sold-Fld nteracton: The fld has no nflence on the sold Fld-Sold nteracton: The sold has no nflence on the fld Two-way coplng: Manplate the bondary condtons Fnte Element technqes: ALE & DLM Rgd Fld: Treat the sold as flds and enforce the rgdty constrant

19 Prncples: Cheap to compte Low memory consmpton Stablty Plasblty Interactvty Common technqes: Real-tme Flds H(x,y) Procedral water: Spermpose sne waves of a varety of ampltdes and drectons. Heght feld approxmatons: If the srface s the only nterest, t can be represented sng a 2D heght feld and anmated by 2d wave eqatons wth nteracton forces Partcle systems: Ths approach s good at smlatng a small amont of water sch as a pddle, a bbble or splashng flds f (x,x j ) = k x x j m k 2 x x j n x x j x x j

20 Agenda Sep 5 th - Partcle-Based Fld Smlaton for Interactve Applcatons Sep 5 th - Hardware-Aware Analyss and Optmzaton of Stable Flds Sep 22 nd Drect Forcng for LagranganRgd-Fld Coplng Sep 22 nd Drectable, Hgh-Resolton Smlaton of Fre on the GPU