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## Introduction to Non-Rigid Body Dynamics

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### Elastically Deformable Models, by Terzopoulos, Platt, Barr, and Fleischer, Proc. ... Curves & surfaces are represented by a set of control points ... – PowerPoint PPT presentation

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Title: 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.

8
Free-Form Deformation (FFD)
• Linear Combination of Node Positions

9
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)

10
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

11
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

13
An Example
• System Demonstration
• inTouch Video

14
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.

15
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.

16
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.

17
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.
• ?

18
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.

19
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

20
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

21
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.

22
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.

23
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.

24
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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