Quadratic spline approximation of the contact potential for real-time simulation of lumped collisions in musical instruments

Collisions are an integral part of the sound production mechanism in a wide variety of musical instruments. In physics-based Realtime simulation of such nonlinear phenomena, challenges centred around efficient and accurate root-finding arise. Nonlinearly implicit schemes are normally ill-suited for real-time simulation as they rely on iterative solvers for root-solving. Explicit schemes overcome this issue at the cost of a slightly larger error for a given sample rate. In this paper, for the case of lumped collisions, an alternate approach is proposed by approximating the contact potential curve. The approximation is described, and is shown to lead to a non-iterative update for an energy-stable nonlinearly implicit scheme. The method is first tested on single mass-barrier collision simulations, and then employed in conjunction with a modal string model to simulate hammer-string and slide-string interaction. Results are discussed in comparison with existing approaches, and real-time feasibility is demonstrated.