Flexible Objects

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Flexible Objects
Abdennour El Rhalibi
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Flexible Objects Elastic and inelastic behavior,
viscoelasticity, plasticity, fracture
Elastically Deformable Models Terzopoulos et
al SIGGRAPH 87
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Modeling Inelastic Deformation Viscoelasticity,
Plasticity, Fracture Terzopoulos and
Fleiseher SIGGRAPH 88
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Graphical Modeling and Animation of Brittle
Fracture OBrien and Hodgins SIGGRAPH 99
Simulation of Object and Human Skin Deformations
in a Grasping Task Gourred et al SIGGRAPH 89
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Graphical Modeling and Animation of Ductile
Fracture OBrien et al SIGGRAPH 02
http//www.cs.berkeley.edu/job/Projects/Fracture/
fracture.html
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Spring-Mass Systems

• Model objects as systems of springs and masses
• The springs exert forces, and you control them by
  changing their rest length
• A reasonable, but simple, physical model for
  muscles
• Advantage Good looking motion when it works
• Disadvantage Expensive and hard to control

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Flexible Objects ? SPRING-MASS SYSTEMS The
simplest, most common approach
Straightforward strategy
Point Mass
Spring (rest length edge length)
External Forces (collisions, gravity, wind, )
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Spring mass fish
Due to Xiaoyuan Tu, http//www.dgp.toronto.edu/peo
ple/tu
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Spring mass fish
http//www.dgp.toronto.edu/tu/animations.html
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Strings

• A whole line of points attached together with
  springs
• Simple to model, great for making realistic
  straps of bullets for chain guns, tails on
  animals, bungie ropes.
• The springs have a normal length of, say, one
  unit.
• If the adjacent points move further than one unit
  of length apart, they experience a force towards
  each other proportional to the extension of the
  spring that connects them.
• Likewise, if they move closer than one unit
  apart, they experience a force pushing them
  apart.

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Strings

• Two ways to model the force on the points
• With mass ? If you are creating animations
• Without mass ? If you are just trying to find the
  optimum shape of a string hanging over a certain
  object

Forces between Two Springs
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Strings without Mass Forces affect the position
of the point
i
Normal length
Small amount (0.01 or so) makes the string move
slowly
gravity
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Strings with Mass Forces affect the velocity of
the point
i

• If you make a string like this, you will notice
  that it is extremely flexible.
• To make it stiffer, you can compare each point
  with its 4 or even 6 closest neighbors, instead
  of 2.

Damping (between about 0.95 and 0.99), is the
energy loss from the string. If you set it to 1,
then the string will never stop swinging around,
and setting it to more than 1 will make the
string increase its swing by itself and
eventually fly off the screen.
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Cloths

• Simply a whole load of interwoven strings!
• We need to add an extra dimension to our string
  routine.
• Imagine a cloth to be a sheet of points all
  connected together by springs.
• If two points get pulled further apart, then they
  experience a force pulling them together and vice
  versa.
• This very simple model of a cloth is reasonably
  accurate!

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Cloth Behavior
If you compare each point with its 4 nearest
neighbors ? a fisherman's net.
If you compare each point with its 8 nearest
neighbors ? a very flexible cloth
If you compare each point with its 24 nearest
neighbors ? a more realistic, stiffer cloth,
though it's much slower to compute
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Massless Cloths

• Every point on the cloth moves at a rate
  proportional to the sum of the forces acting on
  it from the neighboring points.
• Create a 2-dimensional array of co-ordinates to
  hold the x, y and z positions of the cloth in
  space.
• Initialize the values of cloth(p,q) to (p,q,0).
• We will need two of these arrays. One to hold the
  current state of the cloth, and the other to hold
  the new cloth that is being calculated.
• When we have finished calculating the cloth, copy
  all the values from our second array back to the
  first.

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cloth1 (0 to 31, 0 to 31) cloth2 (0 to 31, 0 to
31) Variables VECTOR MovementVector VECTOR
SpringVector VECTOR ForceVector VECTOR Gravity
(initialised to (0, 0, g) where g is gravity,
0.02 is a good number) REAL Length REAL
ForceScaler REAL NormalLength
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For every point (p,q) on the cloth
MovementVector Gravity For each of the
24 neighboring points SpringVector
(position in space of neighbour) - (position in
space of point (p,q)) Length length
of SpringVector NormalLength The
length SpringVector would be if the cloth were
unstretched ForceScaler (Length -
NormalLength) / NormalLength
SpringVector SpringVector (1/Length)
ForceVector SpringVector ForceScaler
ForceVector ForceVector SmallAmount
add ForceVector to MovementVector
end of loop Add MovementVector to
cloth1(p,q) and store it in cloth2(p,q)
make sure this point does not move inside an
object end of loop Copy all the values in
cloth2 to cloth1 keep doing all this forever
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Cloth Interacting with Objects

• We will need some objects for the cloth to
  interact with.
• The simplest is a floor.
• Check each point on the cloth to see if it is
  below the floor, and if it is, then move it to
  the surface.
• It is quite easy to make a sphere for the cloth
  to fall over!
• Check each point to see if it is inside the
  sphere.
• If it is, then move it to the nearest point on
  the surface of the sphere.

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Cloth with Sphere
REAL Distance Distance distance from the
point(p,q) to the center of the sphere if
Distance lt (radius of sphere) then
ForceVector (position of point in space) -
(center of sphere) ForceVector
Forcevector / Distance radius point(p,q)
(center of sphere) ForceVector end if
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Adding Wind

• Adding wind to the cloth allows us to simulate
  the fluttering of flags and other clothwind kind
  of situations.
• This model is not totally accurate.
• The wind affects the cloth, but the cloth does
  not affect the wind, to do this would require a
  vast amount of fluid dynamic calculation.
• However, it produces reasonable looking
  fluttering effects.
• For this we will need to be modeling cloth with
  mass.

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Adding Wind

• First the cloth must be broken down into
  triangles.
• This is easy to do, since the cloth is already
  described as an array of points.
• The effect of the wind on the cloth is calculated
  on each of these triangles individually.
• At each point of the cloth, the sum of the effect
  of the wind on the surrounding triangles is
  calculated.

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Adding Wind

• The force acting on a triangle due to air
  molecules bouncing off it will always be in the
  direction of the normal vector of that triangle.
• The normal vector for each triangle will
  obviously have to be calculated every frame
  because it will be constantly changing.

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Adding Wind

• The force will be proportional to
• the surface area of the triangle,
• the angle at which the wind hits the triangle,
• and the speed of the wind.
• When we use the Cross Product to calculate the
  normal vector of the triangle, the length of that
  vector is proportional to the area of the
  triangle, which makes things a little simpler.

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VECTOR force VECTOR normal VECTOR wind set
force vector to (0,0,0) on all points on
cloth loop through all triangles force
unitvector(normal) dotproduct(normal, wind)
add force to all points making up this triangle
end of loop loop through all points on
cloth add gravity to force add force
to velocity end of loop -- rest of cloth
routine --