The simulation of contact problems (such as physical objects with collisions or biological cells) usually leads to non-smooth dynamical systems, which can be treated analytically with the theory of differential inclusions (set-valued ODEs).
Numerical methods for differential inclusions often suffer from stiffness issues, which makes their application to large dimensional systems challenging since only implicit methods are applicable. However, in 2019, researchers from NVIDIA (tech company producing GPUs) found that one can fuse the implicit steps with explicit force integration steps, leading to a new explicit method which is unconditionally stable (Position-based dynamics).
Our contribution is the first mathematical convergence proof of position-based dynamics for first-order differential inclusions. We use the theory of uniform prox-regular sets (which generalises concepts from convex analysis) to show convergence of PBD. The proof is based on a classical compactness argument and new estimates for alternating projections onto uniform prox-regular sets.
This work is a collaboration with Sara Merino-Aceituno (Vienna).