Introduction to Physically-Based Modeling of Deformable Bodies

Introduction to Physically-Based Modeling of Deformable Bodies Ming C. Lin

Outline • Motivation, Background • Non-physically based (geometric) models • Physically based models • Continuum mechanics, strain, stress • Lagrangian mesh based methods • Method of Finite Differences (1) • Finite Element Method (2) • Boundary Element Method (3) • Mass-spring System (4) • Others

Motivation • Quest for visually, physically plausible animation of deformable objects and fluids • > beautiful dynamic scenes • Many proposed methods • No one model is suited best for any given application • Take the application’s scenario into account • Model representation • Range of physical parameters • Topological changes • Real-time or interactive • etc

Background • Interdisciplinary field • Newtonian dynamics, continuum mechanics, numerical computation, differential geometry, vector calculus, approximation theory • Applications • Medicine: surgical simulation • Entertainment: digital art • Movies and video games • Character animation, hair and cloth modeling • Physical simulation • Elastic, plastic, melting objects, fluids, gases, interaction between different media, tear and fracture

Eye Candy • Videos • Material fracture • O'Brien, J. F., Hodgins, J. K., "Animating Fracture." Communications of the ACM, July 2000, Vol. 43, No. 7, pp. 68-75. • http://www.cs.berkeley.edu/b-cam/Papers/obrien-2000-AF/ • Interplay between rigid bodies and fluids • "Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid" Mark Carlson, Peter J. Mucha and Greg Turk. To appear in ACM SIGGRAPH 2004. • http://www.cc.gatech.edu/~carlson/ • Cloth simulation • “Fast Collision Detection for Deformable Models using Representative-Triangles” Sean Curtis, 2007.

Basic Definition • Deformable object • Defined by • Undeformed shape (equilibrium configuration/rest shape/initial shape) • Set of material parameters that define how it deforms under applied forces • Deformation • A mapping of the positions of every particle in the original object to those in the deformed body

(x,y,z) (x,y,z) Deformation • Modify Geometry • Space Transformation

Non-Physically-Based Methods • Geometrically based • Quick overview • Splines & Patches (1) • Free-Form Deformation (2) • Subdivision Surfaces (3)

Splines & Patches (1) • 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

Free-Form Deformation (2) • FFD (space deformation) change the shape of an object by deforming the space (lattice) in which the object lies within. • Versatile: deformation is defined independent of the representation of geometry • 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.

Free-Form Deformation (2) • Linear Combination of Node Positions

Thomas W. Sederberg, Scott R. Parry, Free-Form Deformation of Solid Geometric Models, Computer Graphics (Proceedings of SIGGRAPH 86). 20(4), pp. 151-160, 1986.

Generalized FFD (2) • 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)

Subdivision Surfaces (3) • 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

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

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

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 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 temperature, electric charge, ion transport, etc.

Revisiting Deformation • Deformable object • Continuous connected subset M of R3 where m in M are coordinates of points in the object • Process of physically based deformation • Force applied to object -> it deforms • Point at location m moves to new location x(m,t) at time t • Deformation of M specified by displacement vector field u(m) = x(m) – m • Elastic strain • Measured in terms of spatial variations of displacement field u = u(m) = (u,v,w)T

Strain • Consider a 1D example: a string of an initial length L0 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 • Popular choices in computer graphics: • G is Green’s nonlinear strain tensor; C is its linearization, Cauchey’s linear strain tensor; del of u is the gradient of the displacement field

y y xx xy x x yy yx Stress • Internal stress tensor  for each material point m is based on strain  at that point • measured as a force per unit area • 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

 Hooke’s Law • Use Hooke’s linear material law • For an infinitesimal strain in uniaxial stretching, a relation like  = 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.

Continuum Model • The full continuum model of a deformable object considers the equilibrium of a general buoy 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.

Time Integration • To simulate a dynamic deformable object, need to knowx(m,t) of all points in M • Givenx(m,t), we can display current state of the object at fixed time steps ∆t in the simulation • x(t0) -> x(∆t) -> x(2∆t) … = animated deformable object • Calculate x(t) implicitly from the differential equations representing by Newton’s 2nd law of motion • Integrating this system of equations will give the unknown vector field x and thus allow us to display the deformable object on the screen

Physically-Based Methods • Different numerical methods and models can be used for solving the problem of physical deformation • Method of Finite Differences • Finite Element Method • Boundary Element Method • Mass-spring System

Physically-Based Method 1 • Method of Finite Differences • Finite Element Method • Boundary Element Method • Mass-spring System

Method of Finite Differences • One of the earliest proposed methods • Terzopolous D., Platt J., Fleischer K.: Elastically Deformable Models (SIGGRAPH ‘87) • Extended to plasticity and fracture • Terzopolous D., Fleischer K.: Modeling inelastic deformation: viscoelasticity, plasticity, fracture (SIGGRAPH ‘88) • Advantage • Easier to implement that FEM • Disadvantage • Requires a regular mesh grid (constrains choices of geometric representations) • Difficult to approximate an arbitrary object with a regular mesh • No local adaptation

Method of Finite Differences(continued) • Algorithm • Potential energy stored in elastically deformable body dictates its dynamics • Deformation energy = weighted matrix norm of the difference between the metric tensors of the deformed and original shape, integrated over the entire continuum • Continuous directional derivative of the elastic force is discretized using the Method of Finite Differences

Finite Element Methods • One of the most popular methods to solve PDE’s on irregular grids • General, versatile and accurate • But, computationally expensive and mathematically sophisticated • Explicit FEM: simpler to understand and implement • O’Brien J. F., Hodgins J. K.: Graphical modeling and animation of brittle fracture (SIGGRAPH ‘99) • DEBUNNE G., DESBRUN M., CANI M.-P., BARR A.: Dynamic real-time deformations using space & time adaptive sampling (SIGGRAPH ‘01) • M. Müller, J. Dorsey, L. McMillan, R. Jagnow, and B. Cutler, Stable Real-Time Deformations (SIGGRAPH ’02)

Finite Element Methods (continued) • 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 a finite number of disjoint elements and we 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. • PDE governing dynamic elastic materials • p is density of material • f is externally applied forces (gravity, collision)

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 determining the finite set of node values ithat minimize the total potential energy in the body.

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.

Boundary Element Method • Alternative to FEM • Good introduction: Hunter P.: FEM/BEM Notes. University of Oakland, New Zealand, 2005. • All computations done on the surface nodes of the elastic body instead of on its volume (interior) • Substantial speedup: 3D problem essentially reduced to 2D • Real-time deformation achieved • D. James and D. Pai, Art Defo - Accurate Real Time Deformable Objects (SIGGRAPH ‘99) • Limited to objects with linear elastic bodies, i.e. homogeneous interior • Topological changes (fractures) more difficult to handle

Boundary Element Method • Algorithm • Integral form of the equation of motion is transformed into a surface integral by applying the Green-Gauss theorem • More work • James D. L., Pai D. K.: Real time simulation of multizone elastokinematic models (Intl. Conference on Robotics and Automation ‘02) • JAMES D. L., PAI D. K.: Multiresolution Green’s function methods for interactive simulation of large-scale elastostatic objects. ACM Transactions on Graphics 22, 1 (2003)

Mass-spring System • Simplest and most intuitive model • Discrete point masses (~particle system) connected together by a network of massless springs • Has been used to model skin, fat, muscle, cloth, snakes

Mass-Spring System • Algorithm • 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

Mass-Spring System (continued) • Advantages • Computationally efficient and handles large deformations with ease • Disadvantages • Not convergent; instead dependent on the mesh resolution and topology • Not necessarily accurate: not built upon elasticity theory like FEM and FDM • Spring constants chosen arbitrarily • Difficult to model continuum properties • Do not generalize easily to triangular or tetrahedral meshes • Works • D. Baraff, A. P. Witkin, Large Steps in Cloth Simulation, (SIGGRAPH ’98)

Other Physically-Based Methods • Mesh-based • Finite Volume Method • Mesh Free • Point-based Animations • Shape Matching • Modal Analysis • Eulerian and Semi-Lagrangian

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

References [1] A. Nealen, M. Müller, R. Keiser, E. Boxerman, M. Carlson, Physically Based Deformable Models in Computer Graphics, Computer Graphics Forum, Vol. 25, issue 4, pages 809-836, previously appeared as EG STAR 2005. [2] Dr. Ming Lin’s Slides “Introduction to Non-Rigid Body Dynamics”, COMP 768: Physically Based Modeling, Simulation and Animation, University of North Carolina at Chapel Hill.

Introduction to Physically-Based Modeling of Deformable Bodies