Transcript Elastically Deformable Models

Elastically Deformable Models
Demetri Terzopoulos
John Platt
Kurt Fleischer
1987
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Outline
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Dynamics of Deformable Models
Energies of Deformation
Applied Forces
Implementation of Deformable Models
Simulation Examples
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Dynamics of Deformable Models
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Lagrange’s form:
Inertial force
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external force
damping force elastic force
a: a point in a body
r(a,t): position of a at time t
μ(a): mass density of the body at a
γ(a): damping density of the body at a
ε(r): a function, potential energy of deformation
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Energies of Deformation
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Develop potential energies of deformation ε(r)
associated with the elastically deformable
models.
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Analysis of Deformation
Energies for Curves, Surfaces, and Solids
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Analysis of Deformation
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Distance between two point in Euclidean 3-space:
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Metric tensor G:
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Curvature tensor B:
n: unit surface normal
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For space curve:
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Arc length: s(r(a)) Curvature: κ(r(a)) Torsion: τ(r)
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Energies for Curves, Surfaces, and Solids
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Curve:
resistance: α-streching, β-bending , γ-twisting
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Surface:
Matrix norm
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Solid:
!! For rigid motion, ε(r) = 0.
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Applied Forces
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Gravitational force:
g: gravitational field
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Spring force:
k: spring constant
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Force on the surface of body:
c: strength of the fluid force n(a): unit normal on the surface
v(a,t): velocity of the surface relative to some constant stream velocity
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Net external force:
f(r,t) = fgravity + fspring + fviscous + fcollision
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Implementation of Deformable Models
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A Simplified Elastic Force
Discretization
Numerical Integration Through Time
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A Simplified Elastic Force
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Simplified deformation energy for a surface:
first
variational
derivative
 ij , ij
δεr

δr
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are weighting function.
δε(r)

δr
 ij : tension
 ij : rigidity
 ij,
 ij : resistance
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Discretization
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1/2
continuous → discrete
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Forward first differnece oprators:
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Backward first differnece oprators:
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Forward and backward cross difference operators:
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Central second difference operators:
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Discretization
2/2
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 elastic force:
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discrete form equations (1):
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Numerical Integration Through Time
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t = 0 to t = T is subdivided into equal time steps △t
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Simulation Examples
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Two different static behaviors of an elastic surface.
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a: simulates a thin plate. (  ij= 0,  ij = positive constant)
b: simulates a membrane. ( ij > 0,  ij = 0)
 ij
(a)
(b)
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Simulation Examples
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A ball resting on a supporting elastic solid.
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The solid has a metric tensor.
The internal elastic force interacts with the collision force to deform the
solid.
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Simulation Examples
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A shrink wrap effect.
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a: a model of a rigid jack.
b: a spherical membrane is stretched to surround the jack.
(a)
(b)
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Simulation Examples
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Simulation of a flag waving in the wind.
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Simulation Examples
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Simulation of a carpet falling onto two rigid bodies in a
gravitational field.
 Modeled as a membrane. (  ij= 0,  ij = positive constant)
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The carpets slides off the bodies due to the interaction between gravity
and repulsive collision force.
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