Introduction to Non-Rigid Body Dynamics

1
Introduction to Non-Rigid Body Dynamics

• A Survey of Deformable Modeling in Computer
  Graphics, by Gibson Mirtich, MERL Tech Report
  97-19
• Elastically Deformable Models, by Terzopoulos,
  Platt, Barr, and Fleischer, Proc. of ACM SIGGRAPH
  1987
• others on the reading list

2
Basic Definition

• Deformation a mapping of the positions of every
  particle in the original object to those in the
  deformed body
• Each particle represented by a point p is moved
  by ?(?)
• p ? ? (t, p)
• where p represents the original position and
  ?(t, p) represents the position at time t.

3
Deformation

• Modify Geometry
• Space Transformation

4
Applications

• Shape editing
• Cloth modeling
• Character animation
• Image analysis
• Surgical simulation

5
Non-Physically-Based Models

• Splines Patches
• Free-Form Deformation
• Subdivision Surfaces

6
Splines Patches

• Curves surfaces are represented by a set of
  control points
• Adjust shape by moving/adding/deleting control
  points or changing weights
• Precise specification modification of curves
  surfaces can be laborious

7
Free-Form Deformation (FFD)

• FFD (space deformation) change the shape of an
  object by deforming the space (lattice) in which
  the object lies within.
• Barrs space warp defines deformation in terms of
  geometric mapping (SIGGRAPH84)
• Sederberg Parry generalized space warp by
  embedding an object in a lattice of grids.
• Manipulating the nodes of these grids (cubes)
  induces deformation of the space inside of each
  grid and thus the object itself.

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Free-Form Deformation (FFD)

• Linear Combination of Node Positions

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Generalized FFD

• fi Ui ? R3 where Ui is the set of 3D cells
  defined by the grid and fi mappings define how
  different object representations are affected by
  deformation
• Lattices with different sizes, resolutions and
  geometries (Coquillart, SIGGRAPH90)
• Direct manipulation of curves surfaces with
  minimum least-square energy (Hsu et al,
  SIGGRAPH90)
• Lattices with arbitrary topology using a
  subdivision scheme (M J, SIGGRAPH96)

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Subdivision Surfaces

• Subdivision produces a smooth curve or surface as
  the limit of a sequence of successive refinements
• We can repeat a simple operation and obtain a
  smooth result after doing it an infinite number
  of times

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Two Approaches

• Interpolating
• At each step of subdivision, the points defining
  the previous level remain undisturbed in all
  finer levels
• Can control the limit surface more intuitively
• Can simplify algorithms efficiently
• Approximating
• At each step of subdivision, all of the points
  are moved (in general)
• Can provide higher quality surfaces
• Can result in faster convergence

12
Surface Rules

• For triangular meshes
• Loop, Modified Butterfly
• For quad meshes
• Doo-Sabin, Catmull-Clark, Kobbelt
• The only other possibility for regular meshes are
  hexagonal but these are not very common

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An Example

• System Demonstration
• inTouch Video

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Axioms of Continuum Mechanics

• A material continuum remains continuum under the
  action of forces.
• Stress and strain can be defined everywhere in
  the body.
• Stress at a point is related to the strain and
  the rate of of change of strain with respect to
  time at the same point.
• Stress at any point in the body depends only on
  the deformation in the immediate neighborhood of
  that point.
• The stress-strain relationship may be considered
  separately, though it may be influenced by
  temparature, electric charge, ion transport, etc.

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Stress

• Stress Vector Tv dF/dS (roughly) where v is the
  normal direction of the area dS.
• Normal stress, say ?xx acts on a cross section
  normal to the x-axis and in the direction of the
  x-axis. Similarly for ?yy .
• Shear stress ?xy is a force per unit area acting
  in a plane cross section ? to the x-axis in the
  direction of y-axis. Similarly for ?yx.

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Strain

• Consider a string of an initial length L0. It is
  stretched to a length L.
• The ratio ? L/L0 is called the stretch ratio.
• The ratios (L - L0)/L0 or (L - L0 )/L are strain
  measures.
• Other strain measures are
• e (L2 - L02 )/2L2 ? (L2 - L02 )/2L02
• NOTE There are other strain measures.
  

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Hookes Law

• For an infinitesimal strain in uniaxial
  stretching, a relation like
• ? E e
• where E is a constant called Youngs Modulus,
  is valid within a certain range of stresses.
• For a Hookean material subjected to an
  infinitesimal shear strain is
• ? G tan ?
• where G is another constant called the shear
  modulus or modulus of rigidity.

• ?

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Continuum Model

• The full continuum model of a deformable object
  considers the equilibrium of a general boy acted
  on by external forces. The object reaches
  equilibrium when its potential energy is at a
  minimum.
• The total potential energy of a deformable system
  is
• ? ? - W
• where ? is the total strain energy of the
  deformable object, and W is the work done by
  external loads on the deformable object.
• In order to determine the shape of the object at
  equilibrium, both are expressed in terms of the
  object deformation, which is represented by a
  function of the material displacement over the
  object. The system potential reaches a minimum
  when d? w.r.t. displacement function is zero.

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Discretization

• Spring-mass models (basics covered)
• difficult to model continuum properties
• Simple fast to implement and understand
• Finite Difference Methods
• usually require regular structure of meshes
• constrain choices of geometric representations
• Finite Element Methods
• general, versatile and more accurate
• computationally expensive and mathematically
  sophisticated
• Boundary Element Methods
• use nodes sampled on the object surface only
• limited to linear DEs, not suitable for
  nonlinear elastic bodies

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Mass-Spring Models Review

• There are N particles in the system and X
  represents a 3N x 1 position vector
• M (d2X/dt2) C (dX/dt) K X F
• M, C, K are 3N x 3N mass, damping and stiffness
  matrices. M and C are diagonal and K is banded.
  F is a 3N-dimensional force vector.
• The system is evolved by solving
• dV/dt M1 ( - CV - KX F)
• dX/dt V

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Intro to Finite Element Methods

• FEM is used to find an approximation for a
  continuous function that satisfies some
  equilibrium expression due to deformation.
• In FEM, the continuum, or object, is divided into
  elements and approximate the continuous
  equilibrium equation over each element.
• The solution is subject to the constraints at the
  node points and the element boundaries, so that
  continuity between elements is achieved.

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General FEM

• The system is discretized by representing the
  desired function within each element as a finite
  sum of element-specific interpolation, or shape,
  functions.
• For example, in the case when the desired
  function is a scalar function ?(x,y,z), the value
  of ? at the point (x,y,z) is approximated by
• ?(x,y,z) ? ? hi(x,y,z) ?i
• where the hi are the interpolation functions
  for the elements containing (x,y,z), and the ?i
  are the values of ?(x,y,z) at the elements node
  points.
• Solving the equilibrium equation becomes a matter
  of deterimining the finite set of node values ?i
  that minimize the total potential energy in the
  body.

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Basic Steps of Solving FEM

1. Derive an equilibrium equation from the potential
   energy equation in terms of material
   displacement.
2. Select the appropriate finite elements and
   corresponding interpolation functions. Subdivide
   the object into elements.
3. For each element, reexpress the components of the
   equilibrium equation in terms of interpolation
   functions and the elements node displacements.
4. Combine the set of equilibrium equations for all
   the elements into a single system and solve the
   system for the node displacements for the whole
   object.
5. Use the node displacements and the interpolation
   functions of a particular element to calculate
   displacements (or other quantities) for points
   within the element.

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Open Research Issues

• Validation of physically accurate deformation
• tissue, fabrics, material properties
• Achieving realistic real-time deformation of
  complex objects
• exploiting hardware parallelism, hierarchical
  methods, dynamics simplification, etc.
• Integrating deformable modeling with interesting
  real applications
• various constraints contacts, collision
  detection