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

TL;DR

This paper provides a rigorous convergence proof for position-based dynamics (PBD) applied to first-order inequality-constrained ODEs, enhancing its scientific validity for various simulation applications.

Contribution

The paper introduces a novel convergence analysis for PBD, employing bounds on projections onto prox-regular sets, thus establishing its reliability for scientific and engineering simulations.

Findings

Proves convergence of PBD for inequality-constrained ODEs.

Extends classical compactness arguments to PBD.

Enables reliable scientific applications of PBD.

Abstract

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.