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# Introduction to Non-Rigid Body Dynamics - PowerPoint PPT Presentation

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

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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)

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

M. C. Lin

(x,y,z)

(x,y,z)

### Deformation

• Modify Geometry

• Space Transformation

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### Applications

• Shape editing

• Cloth modeling

• Character animation

• Image analysis

• Surgical simulation

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### Non-Physically-Based Models

• Splines & Patches

• Free-Form Deformation

• Subdivision Surfaces

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### Splines & Patches

• Curves & surfaces are represented by a set of control points

• Precise specification & modification of curves & surfaces can be laborious

M. C. Lin

### 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.

M. C. Lin

### Free-Form Deformation (FFD)

• Linear Combination of Node Positions

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### 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)

M. C. Lin

### 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

• 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

M. C. Lin

### Surface Rules

• For triangular meshes

• Loop, Modified Butterfly

• Doo-Sabin, Catmull-Clark, Kobbelt

• The only other possibility for regular meshes are hexagonal but these are not very common

M. C. Lin

### An Example

System Demonstration:

inTouch Video

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### 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.

M. C. Lin

y

y

xx

xy

x

x

yy

yx

### Stress

• 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.

M. C. Lin

### 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 )/Lare strain measures.

• Other strain measures are

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

NOTE: There are other strain measures.

M. C. Lin

### 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.

M. C. Lin

### 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.

M. C. Lin

### 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 DE’s, not suitable for nonlinear elastic bodies

M. C. Lin

### 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

M. C. Lin

### 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.

M. C. Lin

### 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.

M. C. Lin

### 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.

M. C. Lin

### 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

M. C. Lin