[SOLVED] CS The Jello Cube Assignment 1, CSCI 520

The Jello Cube Assignment 1, CSCI 520
Jernej Barbic, USC
1

The jello cube
Undeformed cube Deformed cube
The jello cube is elastic,
Can be bent, stretched, squeezed, , Without external forces, it eventually
restores to the original shape. 2

Mass-Spring System
Several mass points
Connected to each other by springs
Springs expand and stretch, exerting force on the mass points
Very often used to simulate cloth
Examples:
A 2-particle spring system Another 2-particle example Cloth animation example
3

Newtons Laws
Newtons 2nd law:
Tells you how to compute acceleration, given the force and mass
Newtons 3rd law: If object A exerts a force F on object B, then object B is at the same time exerting force -F on A.
F = ma
F F
4

Single spring
Obeys the Hooks law: F = k (x x0)
x0 = rest length
k = spring elasticity
(aka stiffness)
For x x0, spring wants to contract
5

Hooks law in 3D
Assume A and B two mass points connected with a spring.
Let L be the vector pointing from B to A
Let R be the spring rest length
Then, the elastic force exerted on A is:

F=kHook(|L|R)
L |L|
6

Damping
Springs are not completely elastic
They absorb some of the energy and tend to decrease the velocity of the mass points attached to them
Damping force depends on the velocity:

F = kd v
kd = damping coefficient
kd different than kHook !!
7

Damping in 3D
Assume A and B two mass points connected with a spring.
Let L be the vector pointing from B to A
Then, the damping force exerted on A is:

(v v )L L AB
F = kd
|L| |L|
Here vA and vB are velocities of points A and B
Damping force always OPPOSES the motion
8

A network of springs
Every mass point connected to some other points by springs
Springs exert forces on mass points
Hooks force Damping force
Other forces
External force field
Gravity
Electrical or magnetic force field Collision force
9

How to organize the network (for jello cube)
To obtain stability, must organize the network of springs in some clever way
Jello cube is a 8x8x8 mass point network
512 discrete points
Must somehow connect them with springs
Basic network Stable network Network out of control
10

Solution:
Structural, Shear and Bend Springs
There will be three types of springs:
Structural Shear
Bend
Each has its own function
11

Structural springs
Connect every node to its 6 direct neighbours
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)
(for surface nodes, some of these neighbors might not exists)
Structural springs establish the basic structure of the jello cube
The picture shows structural springs for the jello cube. Only springs connecting two surface vertices are shown.
12

Shear springs
Disallow excessive shearing
Prevent the cube from distorting
Every node (i,j,k) connected to its diagonal neighbors
Structural springs = white
Shear springs = red
Shear spring (red) resists stretching and thus prevents shearing
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A 3D cube
(if you cant see it immediately, keep trying)

Bend springs
Prevent the cube from folding over
Every node connected to its second neighbor in every direction
(6 connections per node, unless surface node)
white=structural springs
yellow=bend springs (shown for a single node only)
Bend spring (yellow) resists contracting and thus prevents bending
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External force field
If there is an external force field, add that force to the sum of all the forces on a mass point

There is one such equation for every mass point and for every moment in time
F=F+F +F
total Hook damping force field
15

Collision detection
The movement of the jello cube is limited to a bounding box
Collision detection easy:
Check all the vertices if any of them is outside the box
Inclined plane:
Equation:
F(x,y,z)=ax+by+cz+d =0
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 other Can check all the vertices for this condition16 Collision responseWhen collision happens, must perform some action to prevent the object penetrating even deeperObject should bounce away from the colliding objectSome energy is usually lost during the collisionSeveral ways to handle collision responseWe will use the penalty method17 The penalty methodWhen collision happens, put an artificial collision spring at the point of collision, which will push the object backwards and awayfrom the colliding objectCollision springs have elasticity and damping, just like ordinary springs Boundary of colliding objectFv Collision spring18 Penalty forceDirection is normal to the contact surfaceMagnitude is proportional to the amount of penetrationCollision spring rest length is zero Boundary of colliding objectFCollision spring 19 IntegratorsNetwork of mass points and springsHooks law, damping law and Newtons 2nd law give acceleration of every mass pointat any given timeF = maHooks law and damping provide F m is point massThe value for a follows from F=maNow, we know acceleration at any given time for any pointWant to compute the actual motion20 Integrators (contd.)Theequationsofmotion: x=pointposition,v=pointvelocity,a=pointacceleration TheydescribethemovementofanysinglemasspointFhook=sumofallHookforcesonamasspointFdamping = sum of all damping forces on a mass point21 dx dt=v d2x dv 1= =a(t)= (F +F +F ) dt2 dt mHook damping force field Integrators (contd.)When we put these equations together for all the mass points, we obtain a system of ordinary differential equations.In general, impossible to solve analyticallyMust solve numericallyMethods to solve such systems numerically are called integratorsMost widely used: EulerRunge-Kutta 2nd order (aka the midpoint method) (RK2) Runge-Kutta 4th order (RK4)22 Integrator design issuesNumerical stabilityIf time step too big, method explodest = 0.001 is a good starting choice for the assignment Euler much more unstable than RK2 or RK4Requires smaller time-step, but is simple and hence fast Euler rarely used in practiceNumerical accuracySmaller time steps means more stability and accuracy But also means more computationComputational costTradeoff: accuracy vs computation time23 Integrators (contd.)RK4 is often the method of choiceRK4 very popular for engineering applicationsThe time step should be inversely proportional to the square root of the elasticity k [Courant condition]For the assignment, we provide the integrator routines (Euler, RK4)void Euler(struct world * jello);void RK4(struct world * jello);Calls to there routines make the simulation progress one time-step further.State of the simulation stored in jello and automatically updated24 TipsUse double precision for all calculations (double)Do not overstretch the z-bufferIt has finite precisionOk: gluPerspective(90.0,1.0,0.01,1000.0);Bad: gluPerspective(90.0,1.0,0.0001,100000.0);Choosing the right elasticity and damping parameters is an artTrial and errorFor a start, can set the ordinary and collision parameters thesameRead the webpage for updates25