Kinematic Match: a smooth-contact framework for deformable impacts
A geometric constraint that makes collisions stable, accurate, and optimization-ready
Turning impacts into equations that behave
Contact mechanics is a discontinuity problem: two surfaces that were separate become instantaneously constrained at collision. Most numerical contact models handle this by inserting stiff penalty forces or toggling hard constraints between “touching” and “not touching.” Both approaches make solver stiffness mesh-dependent, suppress correct energy transfer at the interface, and block gradient-based optimization.
The Kinematic Match (KM) framework replaces the contact condition entirely. Instead of penalizing interpenetration, KM imposes a single geometric requirement: the angle of incidence between the two surfaces must evolve continuously through impact. In discrete form, this couples curvature and normal vectors across the interface at each time step, producing a contact manifold that is continuously differentiable, requires no switching logic, and introduces no tuning constants. The method is compatible with finite-difference, finite-element, and interface-capturing schemes.
In Proceedings of the Royal Society A ((Agüero et al., 2022)), we applied KM to a rigid sphere striking an elastic membrane, matching experimental deformation profiles and energy-transfer rates. In Journal of Fluid Mechanics ((Gabbard et al., 2025)), we extended the framework to droplets rebounding on fluid baths, a regime sensitive to the contact model’s treatment of capillary forces and coalescence, reproducing observations that conventional CFD misses.
Because the contact manifold remains differentiable, the same formulation integrates directly with adjoint-based inference and gradient-based design. We are extending it to multi-material and biological interfaces where surfaces can merge or tear.