“Optimization-based animation” by Milenkovic and Schmidl

Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex three-dimensional bodies settling into stacks and other “crowded” arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta and accelerations strictly at frame times. Contact points are synchronized at the end of each frame. The extremely small integration steps inherent to traditional simulation techniques are avoided. Non-convex bodies are simulated as unions of convex bodies. Links and joints are simulated successfully with bi-directional constraints. A hybrid of OBA and retroactive detection (RD) has been implemented as well. A review of existing work finds no other packages that can simulate similarly complex scenes in a practical amount of time.

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