More PBM

Ruth Aylett

Overview

!  Collisions !  Deformable objects

– Hookes law – Cloth – Flesh

!  Fluids !  Plants !  Physics engines

Collision response

!  When objects collide they apply forces on each other

!  Collisions can be: – elastic, where no kinetic energy is lost

during the collision –  inelastic, where kinetic energy is lost

Collision response

!  Consider a collision between 2 balls !  An impulse force, P, acts for a very

short time m1 ( v1

after - v1before ) = -f

m2 ( v2after - v2

before ) = f

!  Therefore m1v1

after + m2v2after = m1v1

before + m2v2

before

f

f

v1

v2

m1

m2

Collision response

!  We can define the velocity at which the 2 balls approach each other as:

vrelbefore = v1

before - v2before

!  and the speed at which they separate as: vrel

after = v1after - v2

after

!  Inelastic collisions can be simulated with a coefficient of restitution, e (between 0 and 1)

vrelafter = -e vrel

before

Collision response

!  We can calculate the impulse force, f, for a perfectly elastic collision

f = 2 m1 m2 * ((v1 x v2).n) * n

m1+m2

!  where n is the normal at the point of collision (equal and opposite for each ball)

!  for 2 spheres n is the normalised vector between the centres of the 2 spheres

Collision response

!  We can implement a simulation of collisions in the following way: –  test for collision (check distance between the

centres of the balls) –  if a collision occurs, calculate the impulse

force, P – add that force to the total force for the object

and continue the standard simulation loop

Deformable objects

!  Deformable or soft objects can also be simulated using physical laws –  jelly – cloth –  flesh

Deformable objects

!  Modelling Approach – Spring and dashpot elements – Hooke’s law – Replace polygon edges with springs – Extend to allow cutting and fracture

Hooke’s Law !  Theory

– Consider a spring with rest length, L

– Let the extension of the spring be !L

– Let the coefficient of restitution be k

L+!L L

Hooke’s Law

!  The force generated in the spring is given by:

f = -k !L L

–  The same expression works for spring compression, except !L is negative

–  The dashpot element is included in the system to add some drag, reducing total energy

Modelling cloth

!  To create a model of a deformable object we create a mesh with each edge made from a spring & dashpot

Modelling cloth

!  When a node on the mesh is moved, then all the connected spring elements will attempt to pull or push the node back

p1 p3

p4

p2

p

Modelling cloth

!  The new lengths of the springs will be:

L1 = |p - p1| L2 = |p - p2| L3 = |p - p3| L4 = |p - p4|

Modelling cloth

!  So the force on the node is: f1 + f2 + f3 + f4

where fi = k * (Li - L) * ( pi - p ) L

–  note that each fi points in the direction of the associated spring

–  once the forces have been added, the acceleration can be found ( f / m = a)

normalised

Modelling cloth

!  We can then use our standard simulation loop to calculate new velocities and positions for the node

!  When we have a mesh with many nodes we calculate each step on all nodes simultaneously

Problems

!  Retaining the original structure –  When these routines are applied to 3D objects,

they can easily lose their original shape

!  Solution –  Record the original locations of the object’s nodes –  Connect extra imaginary springs between the

original and current location of each node –  Use a smaller value of k for these imaginary

springs

Problems

!  The cloth is too floppy – 4 springs for each element of the mesh

isn’t enough to give rigidity

!  Solution – Add extra springs

Modelling flesh

!  Case with no cutting / fracture – same as for cloth, but create a 3D mesh

with cubic elements

Modelling flesh

!  Case with cutting and fracture – More complex, need to introduce a method

to cut the forces between nodes –  Introduce the concept of plates

• Each plate is a separate element • Each plate is connected to its neighbours by

filaments • Each filament is non elastic - it merely conducts

forces between neighbouring plate nodes

Plates

filament

plate

Controlling movement !  When no cut has taken place, the filaments

act to: –  keep the nodes of the plates together –  transmit forces from one node to another

!  Process is to calculate the forces on all nodes within each plate

!  If a node n1 from one plate and a node n2 from another plate are connected then apply the forces from n1 to n2 and vice versa

Cutting

!  To cut the flesh, merely delete the connections for appropriate filaments

!  To get the flesh to peel back when cutting / tearing, stretch the flesh before the simulation starts

Fracture

!  To simulate fracture (ripping, tearing), set a maximum force that the filament can conduct

!  If the total force conducted exceeds the fracture force then cut the filament automatically

!  By setting slightly different fracture forces for each filament, we get irregular ripping effects

Problems

!  This method can be unstable !  Physical values have to be chosen

carefully !  Structure is very important

Fluid simulation

!  Three methods used for this: – Lattice Boltzmann Method (LBM); – Navier-Stokes (NS) solvers – Smoothed Particle Hydrodynamics (SPH)

!  All require substantial amounts of computation

!  Simplified simulation also used: O’Brien & Hodgins

Computational Fluid Dynamics

!  Start with: – an arbitrary distribution of fluid – submerged or semi-submerged obstacles

!  Discretise the fluid volume into a 3D array of box shaped elements

CFD

!  Velocity and pressure are: – defined throughout the fluid – updated throughout the simulation, using a

set of finite difference expressions – used to drive a height field equation to

create a surface for liquids

Boundary conditions

!  Boundary conditions can be used to –  represent the edges of the mesh

• open, allowing fluid to flow in and out •  closed

–  represent the surface of a liquid –  represent stationary obstacles

•  sea bed •  rocks

CFD

!  A variety of effects occur naturally from this method –  waves

•  reflection •  refraction •  diffraction

–  rotational effects •  eddies •  vorticity •  splashing

CFD

!  We can visually represent the fluid in 2 ways – as a heightfield to represent the surface of

a liquid – as a collection of massless particles moving

through the fluid

Simple fluids

!  Navier Stokes provides an accurate, but computationally intensive solution

!  O’Brien and Hodgins provide an alternative solution which is – simpler –  less accurate –  less detailed

Simplified representation

!  This method starts by considering the fluid body as a 2D grid rather than a 3D grid

!  Each element represents a column of fluid

Connecting elements

!  Each 2D element is connected to its neighbours (or the outside world) by 8 ‘pipes’

!  The pipes can allow –  the fluid to flow freely through

from one element to another –  an element to have fluid

injected or extracted

Fluid flow in pipes

!  The equations to determine the fluid flow in the pipes are based on laws for hydrostatic pressure

!  The static pressure, H, in column i,j is Hij = hij"g + p0 + Eij

!  where –  h is the column height –  g is gravity –  " is the density of the fluid –  p0 is the atmospheric pressure –  Eij is an external pressure applied by other objects #

Constraints

!  We must also apply some constraints to the system – The flows at each end of a pipe must be

equal and opposite – A column may not have a negative volume

Checking pipes

!  To ensure that no column has a negative volume – at the end of each update, the volume of

each column is tested –  if the volume is negative, scale back the

pipes removing fluid from the column –  repeat this procedure until all columns

have a non-negative volume

Setting the boundaries

!  Columns that are on the edge of the grid have pipe connections to the outside world

!  As there is no column at the other end, we must specify the flow in the pipe – barrier - set the flow to zero –  fluid source - positive flow –  fluid sink - negative flow

Fluid surface

!  We can represent the fluid with a heightfield surface

!  The surface is made of triangles / quads, joining control points

Plants

!  Apart from L-Systems (Lindenmeyer) can also model plants with voxels – A voxel is like a 3D pixel

!  Voxel space automata (Greene)

Voxel automata

!  This technique models plants with stochastic (random) growth

!  The space in which the plant grows is discretised into voxels

!  This technique is particularly good for modelling climbing / rambling plants

Growing the plant

!  The growth of the plant is affected by – occlusion (availability of light) –  intersection (colliding with objects) – proximity (stick to walls, ramble over other

objects) !  These features are recorded within the

voxels

Probabilistic growth

!  The growth of the plant is probabilistic –  test several different options for the next

bit of growth –  for each option assess how good the

conditions are in the new position – choose and apply the best option

Light levels

!  Light levels are modelled with the assessment of occlusion

!  Each voxel assesses – amount of sunlight [ 0 .. 1 ] – amount of sky light [ 0 .. 1 ]

!  The amounts of these types of light are calculated using ray casting

Light from the sky

!  Sky light –  rays are cast from a voxel to points all over

the sky –  if the ray hits an object before it gets to

the sky it is occluded [0] – otherwise it is exposed [1] –  the average occlusion of all the rays can

then be calculated

Light from the sun

!  Sunlight –  the 180 degree arc forming the trajectory

of the sun is determined –  rays are cast from the voxel to the arc –  rays are assessed for occlusion / exposure

!  The average occlusion is calculated

Growth cycle

!  Growth is affected by the amounts of light available –  The plant has acceptable max, min light levels

specified

!  Also assess all voxels for the presence of objects within each voxel (including bits of the plant) –  Allows assessment of proximity, iintersection

!  Once we know the conditions for all voxels, we can proceed with one stage of growth

Updating growth

!  Once a stage of growth has occurred the voxel space is updated

!  Identify changes in – occlusion –  intersection – proximity

!  Then another stage of growth can be started

Physics engines

!  Graphics programmers do not want to learn physics –  So specialist graphics libraries seemed a really

good idea

!  Now incorporated into most games engines –  UT, Half-life, Unity etc –  Also in Director, Flash (2D), Flex, Blender

!  Physics Processing Unit (2006) –  A physics card in spirit of a graphics card –  Physics processing now part of Nvidia and ATI

graphics cards

Physics engines available

!  Commercial – Havok

– www.havok.com

• Which took over Ipion • Used in Director, Half-life

– NVidia took over Ageia Feb 2008 • PhysX engine and PPU

Free-or-nearly physics engines

!  ODE - Open Dynamics Environment –  Rigid body dynamics, C/C++ API, mature, big community

•  http://ode.org/

!  Bullet •  http://www.bulletphysics.com/Bullet/wordpress/

!  The Physics Engine •  http://www.thephysicsengine.com/

!  Newton game dynamics •  http://www.newtondynamics.com/

!  Probably many others… !  Physics Abstraction Layer - PAL

–  Interface to variety of physics engines •  http://www.adrianboeing.com/pal/