Computer Animation. Rick Parent

Transcription

1 Algorithms and Techniqes Flids

2 Sperficial models. Deep models comes p throghot graphics, bt particlarl releant here OR Directl model isible properties Water waes Wrinkles in skin and cloth Hi Hair Clods Model nderling processes that prodce the isible properties Comptational Flid Dnamics Cloth weae Phsical properties of a strand of hair Comptational Flid Dnamics

3 Sperficial Models for Water Main problem with water Changes shape Changes topolog Still waters Small amplitde waes The anatom of waes ocean waes rnning downhill

4 Simple Wae Model - Sinsoidal Distance-amplitde

5 Simple Wae Time-amplitde at a location

6 Simple Wae

7 Simple Wae

8 Sm of Sinsoidals

9 Sm of Sinsoidals Normal ector pertrbation Hihtfild Height field

10 Ocean Waes F s, t π s Ct Acos L s distance from sorce t time A maimm amplitde C propogation speed L waelength T period of wae

11 Moement of a particle In idealized wae, no transport of water Q aerage πh T πhc L π SC Q aerage orbital speed S steepness of the wae T time to complete orbit H twice the amplitde

12 Breaking waes If Qeceeds C > breaking wae If non-breaking wae, steepness is limited Obsered steepness between 0.5 and 1.0 Q aerage πh T πhc L πsc

13 Air model of waes Relates depth of water, propagation speed and waelength C g tanh κ d κ gl πd tanh κπ L L CT π κ g - grait L As depth increases, C approaches As depth decreases, C approaches gl π gd

14 Implication of depth on waes approaching beach at an angle Wae tends to straighten ot closer sections slow down

15 Wae in shallow water C an L are redced T remains the same A H remain the same or increase Q remains the same Waes break

16 See book for details of modeling ocean waes from article b Peache

17 Model for Transport ofwater h water srface b grond water elocit

18 Model for Transport ofwater t g h 0 Rlt Relates Acceleration Difference in adjacent elocities Acceleration de to grait

19 Model for Transport ofwater d t d 0 d h b Relates Temporal change in the height Spatial change in amont of water

20 Model for Transport ofwater 0 h g t 0 d t d 0 d t d t Small flid elocit t h h Slowl aring depth h gd t h U fi it diff t d l b k Use finite differences to model - see book

21 Models for Clods Basic clod tpes Phsics of clods Visal characteristics, rendering isses Earl approaches Volmetric clod modeling

22 Basic Clod Tpes Eample forces of formation Conection Conergence lifting along frontal bondaries Lifting de to montains Height water. ice composition cmls strats cirrs nimbs heap laer crl of hair rain cirr high alto mid-leel

23 Visal characteristics 3D Amorphos Trblent Comple shading Semi-transparent Self-shadowing Reflectie albedo

24 Earl Approach - Gardner Earl flight simlator research Static model for the most part Sm of oerlaping semi-transparent hollow ellipsoids Taper transparenc from edges to center See his paper from SIGGRAPH 1985

25 Other approaches Particle sstems implicit fnctions Volmetric representations

26 Dae Ebert

27 Models for Fire Procedral ld Particle sstem Other approaches

28 Models for Fire - D

29 Models for Fire - particle sstem Deried from Reees paper on particle sstems

30 Particle Sstem Fire

31 Combstion eamples Show Fedkiw s work

32 Comptational Flid Dnamics CFD Flid - a sbstance, as a liqid or gas, that is capable of flowing and dthat tchanges its shape at a stead rate when acted pon b a force. Compressible changeable densit Stead state flow motion attribtes are constant at a point Viscos resists flow; Newtonian flid has linear stresss-strain rate Vortices circlar swirls

33 General Approaches Grid-based Particle-based method Hbrid method

34 CFD eqations mass is consered momentm is consered energ is consered Usall not modeled in compter animation To sole: discretize cells discrete eqations nmericall sole

35 CFD

36 Conseration of mass D Small control olme: Δ b Δ b 1 A Δ * Δ A r Time rate of mass change in olme rate of mass entering Mass inside CV: Time rate of mass change in olme Δ Δ ΔΔ t ΔΔ t

37 Conseration of mass D Small control olme: Δ b Δ b 1 A Δ * Δ A r Time rate of mass change in olme rate of mass entering ΔΔ ΔΔ t t Amont of mass entering from left: Difference between left and right: d dt Δ Δ Δ d Δ d Δ left right

38 Conseration of mass D A r Time rate of mass change in olme rate of mass entering t 0 If incompressible t V V 0 VV 0 Diergence operator

39 Conseration of momentm Momentm in CV changes as the reslt of: Mass flowing in and ot Collisions of adjacent flid pressre Random interchange of flid at bondar

40 Conseration of momentm in D Rate of change of -momentm within CV -Momentm entering: Difference in : Difference in : mv d d p Pressre difference in : d dt

41 Conseration of momentm in D p dt d d d p dt d d d 0 0

42 C ti f t 3D Conseration of momentm 3D d direction t z w d dp t z w d dp direction t w z w w w dz dp z direction Dt DV V V V t V p & Material deriatie Dt t

43 Naier-Stokes for graphics p t V 0 V V V V& t D Dt 0 V

44 Viscosit, etc. Hooke s law: in solid, stress is proportional to strain Flid continosl deforms nder an applied shear stress Newtonian flid: stress is linearl proportional to time rate of strain h V V τ h τ μ Water, air are Newtonian; blood in non-newtonian

45 Stokes Relations Stokes Relations Etended Newtonian idea to mlti-dimensional flows z w λ μ τ μ τ τ z w λ τ μ w z z μ τ τ z λ μ τ w w z z z μ w μ τ τ z w z w zz λ μ τ z z z μ τ τ

46 Stokes Hpothesis Choose λ so that normal stresses sm to zero τ τ τ zz 0 λ 3 μ

47 C ti f t ith Conseration of momentm with iscosit w dp z τ τ τ t z z d p z d τ τ τ t z w z d dp z τ τ τ t w z w w w z dz dp z z zz τ τ τ

48 Incompressible Stead D flow Incompressible, Stead -D flow p ν 1 p ν μ ν Ki ti i it p ν Kinematic iscosit

49 D Eler Eqations no iscosit If incompressible 0 dp d d d d d dp d d d d d 0 1 dp d d d d d 1 dp d d d d d

50 D Eqations reiew t p dt d d d p dt d d d