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Introduction to Non-Rigid Body Dynamics


Introduction to Non-Rigid Body Dynamics A Survey of Deformable Modeling in Computer Graphics, by Gibson & Mirtich, MERL Tech Report 97-19 Elastically Deformable ... – PowerPoint PPT presentation

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Title: Introduction to Non-Rigid Body Dynamics

Introduction to Non-Rigid Body Dynamics
  • A Survey of Deformable Modeling in Computer
    Graphics, by Gibson Mirtich, MERL Tech Report
  • Elastically Deformable Models, by Terzopoulos,
    Platt, Barr, and Fleischer, Proc. of ACM SIGGRAPH
  • others on the reading list

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.

  • Modify Geometry
  • Space Transformation

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

Non-Physically-Based Models
  • Splines Patches
  • Free-Form Deformation
  • Subdivision Surfaces

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

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.

Free-Form Deformation (FFD)
  • Linear Combination of Node Positions

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
  • Lattices with different sizes, resolutions and
    geometries (Coquillart, SIGGRAPH90)
  • Direct manipulation of curves surfaces with
    minimum least-square energy (Hsu et al,
  • Lattices with arbitrary topology using a
    subdivision scheme (M J, SIGGRAPH96)

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

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

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

An Example
  • System Demonstration
  • inTouch Video

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.

  • 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.

  • 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
  • Other strain measures are
  • e (L2 - L02 )/2L2 ? (L2 - L02 )/2L02
  • NOTE There are other strain measures.

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.
  • ?

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
  • The total potential energy of a deformable system
  • ? ? - 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.

  • 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
  • Boundary Element Methods
  • use nodes sampled on the object surface only
  • limited to linear DEs, not suitable for
    nonlinear elastic bodies

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

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.

General FEM
  • The system is discretized by representing the
    desired function within each element as a finite
    sum of element-specific interpolation, or shape,
  • 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
  • Solving the equilibrium equation becomes a matter
    of deterimining the finite set of node values ?i
    that minimize the total potential energy in the

Basic Steps of Solving FEM
  1. Derive an equilibrium equation from the potential
    energy equation in terms of material
  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
  5. Use the node displacements and the interpolation
    functions of a particular element to calculate
    displacements (or other quantities) for points
    within the element.

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