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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin Deformation Modify Geometry Space Transformation (x,y,z)  (x,y,z)

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

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

6. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill 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.

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

9. UNC Chapel Hill M. C. Lin Generalized FFD f i : U i  R 3 where { U i } is the set of 3D cells defined by the grid and f i mappings 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)

10. UNC Chapel Hill 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

11. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin An Example System Demonstration : inTouch Video

14. UNC Chapel Hill M. C. Lin Axioms of Continuum Mechanics 1. A material continuum remains continuum under the action of forces. 2. Stress and strain can be defined everywhere in the body. 3. 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. UNC Chapel Hill M. C. Lin Stress Stress Vector T v = 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. x y  xx  yy y x  xy  yx

16. UNC Chapel Hill M. C. Lin Strain Consider a string of an initial length L 0. It is stretched to a length L. The ratio = L / L 0 is called the stretch ratio. The ratios ( L - L 0 )/ L 0 or ( L - L 0 )/ L are strain measures. Other strain measures are e = ( L 2 - L 0 2 )/ 2L 2  = ( L 2 - L 0 2 )/ 2L 0 2 NOTE: There are other strain measures.

17. UNC Chapel Hill 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. 

18. UNC Chapel Hill 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, 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. UNC Chapel Hill 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

20. UNC Chapel Hill M. C. Lin Mass-Spring Models: Review There are N particles in the system and X represents a 3 N x 1 position vector: M (d 2 X/dt 2 ) + 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 3 N - dimensional force vector. The system is evolved by solving: dV/dt = M –1 ( - CV - KX + F) dX/dt = V

21. UNC Chapel Hill 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.

22. UNC Chapel Hill 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)   h i (x,y,z)  i where the h i 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  i that minimize the total potential energy in the body.

23. UNC Chapel Hill M. C. Lin 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 element’s 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. UNC Chapel Hill 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