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