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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

- 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.

Deformation

- Modify Geometry
- Space Transformation

Applications

- 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

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)

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

- 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.

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.

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

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.

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

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,

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.

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 elements 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.

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