Shi, AlvinKim, TheodoreWang, HuaminYe, YutingVictor Zordan2023-10-162023-10-1620232577-6193https://doi.org/10.1145/3606934https://diglib.eg.org:443/handle/10.1145/3606934We analyze a wide class of penalty energies used for contact response through the lens of a reduced frame. Applying our analysis to both spring-based and barrier-based energies, we show that we can obtain closedform, analytic eigensystems that can be used to guarantee positive semidefiniteness in implicit solvers. Our approach is both faster than direct numerical methods, and more robust than approximate methods such as Gauss-Newton. Over the course of our analysis, we investigate physical interpretations for two separate notions of length. Finally, we showcase the stability of our analysis on challenging strand, cloth, and volume scenarios with large timesteps on the order of 1/40 s.CCS Concepts: Computing methodologies -> Physical simulation Collisions, Penalty Energies, Barrier Functions"Computing methodologiesPhysical simulation CollisionsPenalty EnergiesBarrier Functions"10.1145/3606934