Introduction to non rigid body dynamics
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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

M. C. Lin

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 ffd1
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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