Simulation - Spheres in Collision

This scene models a set of particles representing spheres falling under gravity. Each particle is associated to the following parameters: position \(p\), speed \(v\), forces \(f\), as well as radius and color. The position and speed of each particle is updated through time using a discrete integration in the file simulation/simulation.cpp

• - \(v^{k+1} = v^{k} + \Delta t\;f^{k}\)

• - \(p^{k+1} = p^{k} + \Delta t\;v^{k+1}\)

In the current state of the program, only the gravity is taken into account. But the collision with the cube boundaries and between the sphere need to be implemented.

Collisions

Add the collision detection and response between the spheres and boundary faces of the cube using impulse based response on the sphere velocities. The collision response can be implemented in two steps:

• - First applying impulse response to generate the bouncing (with the cube, and other particles)
• - Second canceling the velocity components that still generate a penetration (with the cube, and other particles)

Once the velocity is computed, then the sphere positions are finally updated Note: You may perform several iterations of the numerical integration at each frame in order to speed up the simulation, while keeping a constant (or smaller) time step.

Possible extension

• Allow the user to move the box interactively. (In this case, the particles that goes out of the cube will need to get their position projected back inside the cube.)

• Speed-up the collision detection in building a spatial acceleration structure such as a regular grid storing neighboring particles.