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

3. (x,y,z) (x,y,z) Deformation • Modify Geometry • Space Transformation M. C. Lin

4. Applications • Shape editing • Cloth modeling • Character animation • Image analysis • Surgical simulation M. C. Lin

5. Non-Physically-Based Models • Splines & Patches • Free-Form Deformation • Subdivision Surfaces M. C. Lin

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 M. C. Lin

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. • 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

8. Free-Form Deformation (FFD) • Linear Combination of Node Positions M. C. Lin

9. 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

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 M. C. Lin

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 M. C. Lin

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 M. C. Lin

13. An Example System Demonstration: inTouch Video M. C. Lin

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. M. C. Lin

15. 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

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 )/Lare strain measures. • Other strain measures are e =(L2 - L02 )/2L2  =(L2 - L02 )/2L02 NOTE: There are other strain measures. M. C. Lin

17. 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

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, 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

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 DE’s, not suitable for nonlinear elastic bodies M. C. Lin

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 = M–1 ( - CV - KX + F) dX/dt = V M. C. Lin

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. M. C. Lin

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 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

23. 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

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 M. C. Lin