Physically Based Modeling

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Physically Based Modeling

• Let physics take over!

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Physically Based Modeling

• Account for forces in system
• Account for object interaction, e.g. friction,
  collision

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Spring-Mass-Damper

• Model jello, cloth, muscle
• Have gravity add mass to vertices
• Have stability (add stiffness to flag pole)
• Put springs on each vertex, allowing to stretch a
  finite amount
• Another use angular springs on polygon corners
  to prevent self-penetration
• Numerical integration for animation

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Governing Equations

• Hookes Law (for graphics)Fs ks(dist-len)
  where
• len rest length
• dist current length
• ks spring constant
• Fij -Fji ks(distij(t) lenij)dij
• Where dij unit vector along i-j
• t time
• Fs S Fij (sum of all edges coming out of a
  vertex)

F
V1
E12
E31
V3
V2
E23
Look at neighbors when calculating spring force!
V
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Damping Force Angular Springs

• FD -kd v(t)
• Damper force is proportional to velocity
• and acts in direction opposite to velocity
• NET FORCEF Fs Fd ks(dist-len) -kd v(t)

M2
M1

• Angular spring t (torque) ks?(t) -?(rest) -
  kd?(t)

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Object Representation

• Vertices mass
• Edges
• Spring, damping constant
• Resting length
• Vertex IDs
• Each vertex has
• Current position
• Current velocity
• Current acceleration
• Mass
• Number of edges

X, Y, Z components
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How it all comes together

• Use Newtons Law (F ma) to calculateacceleratio
  n for every vertex(big system of linear
  equations)
• Basic strategy accumulate acceleration from
  different sources (Gravity, Spring, Damper) and
  integrate 2 times to get velocity and position
• Can use Runge-Kutta, for example
• For project you can use xspringies (2D) ODE
  library or other library you can find or roll
  your own
• Make sure to cite your sources

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Details (source Paul Bourke)

• Create the particles
• Create the springs between the particles
  Initialize the particle and spring parameters
• loop in time
• Update the particle positions (solve ODEs)
  Display the results somehow

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Update Particle Positions

• Calculate force at each point
• Add Positive force Forcemass for each X,Y,Z
• Subtract Drag dragvelocity for each X,Y,Z
• Handle spring interaction
• For each spring
• For each of X,Y,Z of attached points
• Force Hookes law
• Force damping constant(?velocity of
  points)(?len of points in direction)/lenx,y,z
• Force -(?len of points)/lenx,y,z
• Add or subtract force to/from point (if the point
  is not fixed)

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Calculate derivatives for points

• We already have velocities
• dpx/dt velocityx
• Velocity derivative
• dvx/dt forcex/mass
• And solve with your favorite ODE solver
  (Runge-Kutta..)
• Update positions
• Update velocity