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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 ……. Basic Definition.

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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 Models, by Terzopoulos, Platt, Barr, and Fleischer, Proc. of ACM SIGGRAPH 1987

…… others on the reading list ……

basic definition
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)

wherep represents the original position and (t, p) represents the position at time t.

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  • Modify Geometry
  • Space Transformation

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

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non physically based models
Non-Physically-Based Models
  • Splines & Patches
  • Free-Form Deformation
  • Subdivision Surfaces

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splines patches
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

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free form deformation ffd
Free-Form Deformation (FFD)
  • FFD (space deformation) change the shape of an object by deforming the space (lattice) in which the object lies within.
  • Barr’s space warp defines deformation in terms of geometric mapping (SIGGRAPH’84)
  • 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 ffd8
Free-Form Deformation (FFD)
  • Linear Combination of Node Positions

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generalized ffd
Generalized FFD
  • fi: Ui R3 where {Ui} is the set of 3D cells defined by the grid and fimappings define how different object representations are affected by deformation
  • Lattices with different sizes, resolutions and geometries (Coquillart, SIGGRAPH’90)
  • Direct manipulation of curves & surfaces with minimum least-square energy (Hsu et al, SIGGRAPH’90)
  • Lattices with arbitrary topology using a subdivision scheme (M & J, SIGGRAPH’96)

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subdivision surfaces
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
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

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surface rules
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
An Example

System Demonstration:

inTouch Video

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axioms of continuum mechanics
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 Vector Tv =dF/dS (roughly) wherevis the normal direction of the area dS.
  • Normal stress, sayxxacts on a cross section normal to the x-axis and in the direction of the x-axis. Similarly foryy .
  • Shear stressxyis 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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  • 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 )/Lare strain measures.
  • Other strain measures are

e =(L2 - L02 )/2L2  =(L2 - L02 )/2L02

NOTE: There are other strain measures.

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hooke s law
Hooke’s Law
  • For an infinitesimal strain in uniaxial stretching, a relation like

 = E e

where E is a constant called Young’s 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
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, andWis 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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  • 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 DE’s, not suitable for nonlinear elastic bodies

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mass spring models review
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 = M–1 ( - CV - KX + F)

dX/dt = V

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intro to finite element methods
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
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 element’s node points.

  • Solving the equilibrium equation becomes a matter of deterimining the finite set of node values ithat minimize the total potential energy in the body.

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basic steps of solving fem
Basic Steps of Solving FEM
  • Derive an equilibrium equation from the potential energy equation in terms of material displacement.
  • Select the appropriate finite elements and corresponding interpolation functions. Subdivide the object into elements.
  • For each element, reexpress the components of the equilibrium equation in terms of interpolation functions and the element’s node displacements.
  • 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.
  • 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
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

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