Elastically Deformable Models Demetri Terzopoulos John Platt Kurt Fleischer 1987

Outline Dynamics of Deformable Models Energies of Deformation Applied Forces Implementation of Deformable Models Simulation Examples

Dynamics of Deformable Models Lagrange’s form: a: a point in a body r(a,t): position of a at time t μ(a): mass density of the body at a γ(a): damping density of the body at a ε(r): a function, potential energy of deformation external force Inertial force damping force elastic force

Energies of Deformation Develop potential energies of deformation ε(r) associated with the elastically deformable models. Analysis of Deformation Energies for Curves, Surfaces, and Solids

Analysis of Deformation Distance between two point in Euclidean 3-space: Metric tensor G: Curvature tensor B: For space curve: Arc length: s(r(a)) Curvature: κ(r(a)) Torsion: τ(r) n: unit surface normal

Energies for Curves, Surfaces, and Solids resistance: α-streching, β-bending , γ-twisting Matrix norm !! For rigid motion, ε(r) = 0.

Applied Forces Gravitational force: Spring force: Force on the surface of body: Net external force: f(r,t) = fgravity + fspring + fviscous + fcollision g: gravitational field k: spring constant c: strength of the fluid force n(a): unit normal on the surface v(a,t): velocity of the surface relative to some constant stream velocity

Implementation of Deformable Models A Simplified Elastic Force Discretization Numerical Integration Through Time

A Simplified Elastic Force Simplified deformation energy for a surface: 7 first variational derivative , are weighting function. : tension , : resistance : rigidity

Discretization1/2 continuous → discrete Forward first differnece oprators: Backward first differnece oprators: Forward and backward cross difference operators: Central second difference operators:

Discretization2/2 elastic force: discrete form equations (1):

Numerical Integration Through Time t = 0 to t = T is subdivided into equal time steps △t

Simulation Examples1 Two different static behaviors of an elastic surface. a: simulates a thin plate. ( = 0, = positive constant) b: simulates a membrane. ( > 0, = 0) (a) (b)

Simulation Examples2 A ball resting on a supporting elastic solid. The solid has a metric tensor. The internal elastic force interacts with the collision force to deform the solid.

Simulation Examples3 A shrink wrap effect. a: a model of a rigid jack. b: a spherical membrane is stretched to surround the jack. (a) (b)

Simulation Examples4 Simulation of a flag waving in the wind.

Simulation Examples5 Simulation of a carpet falling onto two rigid bodies in a gravitational field. Modeled as a membrane. ( = 0, = positive constant) The carpets slides off the bodies due to the interaction between gravity and repulsive collision force.