The kinematic techniques still used in most commercial animation systems leave to the animator the task of estimating the object motions according to the laws of physics. Physically-based simulation offers a promising alternative.
The goal of this project is to develop techniques and software for simulating the dynamics of viscoelastic bodies subject to equality and inequality constraints.
We have developed an animation system that simulates the dynamic behavior of elastic bodies, according to the laws of Newtonian mechanics. Each body is modeled by a collection of tetrahedral elements, glued together by their faces. Each element is allowed to deform only by affine transformations, so that its shape remains tetrahedral along the simulation.
Here are two animations we produced with our system:
The equations of motion of such bodies can be derived from general formulas for the elastic and kinetic energy, the viscous power loss, and mechanical constraints, in terms of generalized coordinates. In our papers we have explained the general method for deriving such equations from Lagrange's formula, if one knows formulas for the kinetic and elastic energy of the body, and the rate of energy loss due to internal viscosity, in terms of those generalized coordinates.
The Lagrangian approach allows us to model general mechanical constraints on the positions of the bodies, expressed as algebraic equalities on their coordinates. These constraints can be used to control the animation: to keep a body fixed, to force it to follow a predefined trajectory, to establish a mechanical linkage between two bodies, and so on.
The forces arising in reaction to deformation of elastic materials have often been modeled by some variant of Hooke's law, which says that the forces are proportional to the amount of deformation. One undesirable feature of these mdoels is that they allow the body to be compressed to zero or negative volume by a finite force.
One of our contributions is a convenient two-parameter non-linear model for the elastic forces, that agrees with Hooke's law for small deformations, but does not allow the material to be compressed to zero or negative volume. In particular, we derive the equations of motion for elastic bodies modeled by tetrahedral finite elements with affine deformations.
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We also describe a convenient two-parameter mathematical model for the elastic properties of simulated isotropic materials. The model is non-linear in the deformation measures, and was designed to allow large deformations without allowing the bodies to be compressed to zero or negative volume. This model, nevertheless, reduces to Hooke's linear model for small deformations. The two parameters are then identified with the two elastic moduli that define the material's resistance to static compression and shearing.
A similar two-parameter model is given for the viscous forces that resist changes in the material's deformation. This model too reduces to the standard Newton model for slow deformations, its two parameters then defining the material's resistance to dynamic compression and shearing.
The independent control of these four physical parameters allows for the realistic simulation of a wide range of materials, such as solid rubber, plastic foam, putty, protoplasm --- and, of course, Jello®.
We have also developed an efficient and accurate collision detection procedure for our models, by combining Hermite interpolation of the non-penetration constraints with Lin and Manocha's bounding box tests.
An essential feature of a practical dynamic animation system is the ability to automatically detect and handle collisions between the simulated bodies (or different parts of the same body). We show how the timing of such events can be accurately computed, at relatively low cost, by using Hermite interpolation on the formulas that define the non-penetration constraints. Finally, we show how to drastically reduce the number of such tests by a bounding-box technique due to Lin and Manocha (1994).
Last edited on 2003-06-10 00:51:02 by stolfi