Physically Based Modeling

Suppose we wanted to implement a particle simulator with a “floor”: a solid horizontal plane which particles could bounce off of, or lie on. Assuming all the particles started out above the floor, they’d all have to stay there for the rest of the simulation. How could we go about implementing this? Clearly, there are two problems. We need to detect when particles are near the floor, and when they are, we need to do something to stop them from passing through the floor. One obvious thing to do is to impose a vertical spring force of−ky(t)whenever the y-coordinate y(t) of the particle is less than zero. (We’ll suppose the floor is simply the line y = 0.) If we make k large enough, than we can prevent the particle from moving too far beneath the floor. The problem is, that by making k large, we set ourselves up for stiff ODE’s, as mentioned in earlier notes. Also, no matter what value of k you pick, I can always choose a heavy enough particle (or a particle moving downwards with a high enough velocity) so that the upwards −ky(t) spring-force isn’t strong enough to stop the particle from moving substantially below the line y = 0. Of course, we could modify the spring force to completely stop the particle from moving below the floor, no matter what we throw at it. Suppose the spring force is 1 ky(t)2 e 1 ky(t) . Then as y(t) decreases towards zero, the spring force becomes strong quickly; in fact, it is impossible for the particle to ever attain y(t) = 0 given this spring force. Note that in this case we have “action at a distance”: the spring force acts even before y(t) becomes negative. By making k large though, we can make the force fall off for positive values of y(t) quickly. Of course, this is a pretty stiff force function; −ky(t) is a walk in the park in comparison...