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NVIDIA researchers have pioneered an explicit method, position-based dynamics(PBD), for simulating systems with contact forces, gaining widespread use incomputer graphics and animation. While the method yields visually compellingreal-time simulations with surprising numerical stability, its scientificvalidity has been questioned due to a lack of rigorous analysis.  In this paper, we introduce a new mathematical convergence analysisspecifically tailored for PBD applied to first-order dynamics. Utilizing newlyderived bounds for projections onto uniformly prox-regular sets, our proofextends classical compactness arguments. Our work paves the way for thereliable application of PBD in various scientific and engineering fields,including particle simulations with volume exclusion, agent-based models inmathematical biology or inequality-constrained gradient-flow models.

Convergence proof for first-order position-based dynamics: An efficient scheme for inequality constrained ODEs