Physical simulation for animation

Transcription

1 Physical simulation fo animation Case study: The jello cube The Jello Cube Mass-Sping System Collision Detection Integatos Septembe

2 Announcements Pogamming assignment 3 is out. It is due Tuesday, Novembe 5th midnight. Midtem exam: Next week on Thusday in class 2

3 The jello cube Undefomed cube Defomed cube The jello cube is elastic, Can be bent, stetched, squeezed,, Without extenal foces, it eventually estoes to the oiginal shape. 3

4 Physical simulations Model natue by using the laws of physics Often, the only way to achieve ealism Altenative: ty vaious non-scientific ticks to achieve ealistic effects Math becomes too complicated vey quickly Isn t it incedible that natue can compute eveything (you, me, and the whole univese) on the fly, it is the fastest compute eve. Impotant issues: simulation accuacy and stability 4

5 Simulation o eal? 5

6 Mass-Sping System Seveal mass points Connected to each othe by spings Spings expand and stetch, exeting foce on the mass points Vey often used to simulate cloth Examples: A 2-paticle sping system Anothe 2-paticle example Cloth animation example 6

7 Newton s Laws Newton s 2nd law: F = ma Tells you how to compute acceleation, given the foce and mass Newton s 3d law: If object A exets a foce F on object B, then object B is at the same time exeting foce -F on A. F F 7

8 Single sping Obeys the Hook s law: F = k (x - x 0 ) x 0 = est length k = sping elasticity (aka stiffness) Fo x<x 0, sping wants to extend Fo x>x 0, sping wants to contact 8

9 Hook s law in 3D Assume A and B two mass points connected with a sping. Let L be the vecto pointing fom B to A Let R be the sping est length Then, the elastic foce exeted on A is: L F = k ( L R) Hook L 9

10 Damping Spings ae not completely elastic They absob some of the enegy and tend to decease the velocity of the mass points attached to them Damping foce depends on the velocity: F = k d v k d = damping coefficient k d diffeent than k Hook!! 10

11 Damping in 3D Assume A and B two mass points connected with a sping. Let L be the vecto pointing fom B to A Then, the damping foce exeted on A is: F = k d ( v A v B L ) L L L Hee v A and v B ae velocities of points A and B Damping foce always OPPOSES the motion 11

12 A netwok of spings Evey mass point connected to some othe points by spings Spings exet foces on mass points Hook s foce Damping foce Othe foces Extenal foce field» Gavity» Electical o magnetic foce field Collision foce 12

13 How to oganize the netwok (fo jello cube) To obtain stability, must oganize the netwok of spings in some cleve way Jello cube is a 8x8x8 mass point netwok 512 discete points Must somehow connect them with spings Basic netwok Stable netwok Netwok out of contol 13

14 Solution: Stuctual, Shea and Bend Spings Thee will be thee types of spings: Stuctual Shea Bend Each has its own function 14

15 Stuctual spings Connect evey node to its 6 diect neighbous Node (i,j,k) connected to (i+1,j,k), (i-1,j,k), (i,j-1,k), (i,j+1,k), (i,j,k-1), (i,j,k+1) (fo suface nodes, some of these neighbos might not exists) Stuctual spings establish the basic stuctue of the jello cube The pictue shows stuctual spings fo the jello cube. Only spings connecting two suface vetices ae shown. 15

16 Shea spings Disallow excessive sheaing Pevent the cube fom distoting Evey node (i,j,k) connected to its diagonal neighbos Stuctual spings = white Shea spings = ed A 3D cube (if you can t see it immediately, keep tying) Shea sping (ed) esists stetching and thus pevents sheaing 16

17 Bend spings Pevent the cube fom folding ove Evey node connected to its second neighbo in evey diection (6 connections pe node, unless suface node) white=stuctual spings yellow=bend spings (shown fo a single node only) Bend sping (yellow) esists contacting and thus pevents bending 17

18 Extenal foce field If thee is an extenal foce field, add that foce to the sum of all the foces on a mass point F = F + F + total Hook damping F foce field Thee is one such equation fo evey mass point and fo evey moment in time 18

19 Collision detection The movement of the jello cube is limited to a bounding box Collision detection easy: Check all the vetices if any of them is outside the box Inclined plane: Equation: F( x, y, z) = ax + by + cz + d = Initially, all points on the same side of the plane F(x,y,z)>0 on one side of the plane and F(x,y,z)<0 on the othe Can check all the vetices fo this condition 0 19

20 Collision esponse When collision happens, must pefom some action to pevent the object penetating even deepe Object should bounce away fom the colliding object Some enegy is usually lost duing the collision Seveal ways to handle collision esponse We will use the penalty method 20

21 The penalty method When collision happens, put an atificial collision sping at the point of collision, which will push the object backwads and away fom the colliding object Collision spings have elasticity and damping, just like odinay spings Bounday of colliding object F v Collision sping 21

22 Integatos Netwok of mass points and spings Hook s law, damping law and Newton s 2nd law give acceleation of evey mass point at any given time F=ma Hook s law and damping povide F m is point mass The value fo a follows fom F=ma Now, we know acceleation at any given time fo any point Want to compute the actual motion 22

23 Integatos (contd.) The equations of motion: 2 d x 2 dt dx dt = = v dv dt = a( t) = 1 m ( F Hook + F damping + F foce field ) x = point position, v = point velocity, a = point acceleation They descibe the movement of any single mass point F hook =sum of all Hook foces on a mass point F damping = sum of all damping foces on a mass point 23

24 Integatos (contd.) When we put these equations togethe fo all the mass points, we obtain a system of odinay diffeential equations. In geneal, impossible to solve analytically Must solve numeically Methods to solve such systems numeically ae called integatos Most widely used: Eule Runge-Kutta 2nd ode (aka the midpoint method) (RK2) Runge-Kutta 4th ode (RK4) 24

25 Integato design issues Numeical stability If time step too big, method explodes t = is a good stating choice fo the assignment Eule much moe unstable than RK2 o RK4» Requies smalle time-step, but is simple and hence fast Eule aely used in pactice Numeical accuacy Smalle time steps means moe stability and accuacy But also means moe computation Computational cost Tadeoff: accuacy vs computation time 25

26 Integatos (contd.) RK4 is often the method of choice RK4 vey popula fo engineeing applications The time step should be invesely popotional to the squae oot of the elasticity k [Couant condition] Fo the assignment, we povide the integato outines (Eule, RK4) void Eule(stuct wold * jello); void RK4(stuct wold * jello); Calls to thee outines make the simulation pogess one time-step futhe. State of the simulation stoed in jello and automatically updated 26

27 Tips Use double pecision fo all calculations (double) Do not ovestetch the z-buffe It has finite pecision Ok: glupespective(90.0,1.0,0.01,1000.0); Bad: glupespective(90.0,1.0,0.0001, ); Choosing the ight elasticity and damping paametes is an at Tial and eo Fo a stat, can set the odinay and collision paametes the same Read the webpage fo updates and check the bulletin boad 27