Smooth-Rigid-Body Contact as a ReLCP: A Recursively Generated Linear Complementarity Problem

Abstract

This paper reformulates complementarity-based time-stepping for frictionless nonsmooth contact between smooth rigid bodies as a recursively generated linear complementarity problem (ReLCP), involving a sequence of LCPs of increasing dimension. Starting from a classical single-constraint shared-normal signed-distance (SNSD) LCP, the method adds unilateral constraints only when the discrete-time update predicted by the current contact set would violate nonpenetration of the underlying smooth surfaces. The resulting procedure acts directly on smooth geometry, enforces nonpenetration to a prescribed tolerance, and avoids the oversampling inherent to proxy-surface contact models such as tessellations or multi-sphere decompositions, for which improved geometric fidelity can drive rapid growth in constraint count and cost. For strictly convex bodies, we prove that an initially overlap free configuration with sufficiently small timestep sizes, imply finite termination of the adaptive augmentation, and yield a unique discrete-time velocity update. In the small timestep limit and for any fixed overlap-free discrete state with a fixed geometric overlap tolerance, we prove that the recursion terminates after the initial solve, reducing the method to the classical single-constraint SNSD LCP and retaining the usual consistency of complementarity time-stepping with the underlying differential variational inequality. Numerical tests on colliding ellipsoids, compacting ellipsoid suspensions, growing bacterial colonies, and taut chainmail networks demonstrate stable large-timestep behavior, bounded interpenetration without discretization-induced surface roughness, and substantial reductions in both active constraint counts and runtime relative to representative discrete-surface complementarity formulations.