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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 ……

- 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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(x,y,z)

(x,y,z)

- Modify Geometry
- Space Transformation

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

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- Splines & Patches
- Free-Form Deformation
- Subdivision Surfaces

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- 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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- 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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- Linear Combination of Node Positions

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- 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 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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- 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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- 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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System Demonstration:

inTouch Video

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- 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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y

y

xx

xy

x

x

yy

yx

- Stress Vector Tv =dF/dS (roughly) wherevis the normal direction of the area dS.
- Normal stress, sayxxacts on a cross section normal to the x-axis and in the direction of the x-axis. Similarly foryy .
- Shear stressxyis a force per unit area acting in a plane cross section to the x-axis in the direction of y-axis. Similarly foryx.

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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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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