Real-time simulation of the struck piano string with geometrically exact nonlinearity via a scalar quadratic energy method

This work addresses the problem of the struck piano string. This system is highly nonlinear, and a sound representation of the energy balance is therefore necessary in any time stepping routine used in simulation, in order to preserve stability. Many algorithms have been developed in previous works. Among them, some present fully-implicit discretisations, which are only approachable using iterative root finders such as Newton-Raphson. Others are linearly-implicit, but not quite suited for real-time rendering. Here, a novel approach is presented, based on the idea of energy quadratisation. It will be shown that, when the nonlinearities are consolidated into a scalar auxiliary state function, the time stepping scheme presents a fast inversion formula. A C++ implementation of the proposed scheme yields indeed compute times below real-time, for typical strings.