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.

翻译：NVIDIA研究人员开创了一种显式方法——位置动力学（PBD），用于模拟接触力系统，在计算机图形学和动画领域得到广泛应用。尽管该方法能以惊人的数值稳定性生成视觉逼真的实时模拟，但其科学有效性因缺乏严格分析而受到质疑。本文针对一阶动力学中的PBD应用，提出了一种全新的数学收敛性分析。通过利用新推导的一致近正则集投影界，我们的证明拓展了经典的紧致性论证。本研究为PBD在各类科学与工程领域的可靠应用铺平了道路，包括体积排除粒子模拟、数学生物学中的个体基模型以及不等式约束梯度流模型。