Non-iterative numerical simulation techniques for nonlinear string vibration in musical acoustics

Physics-based sound synthesis of musical instruments has seen growing interest in recent years, as it allows for reproducing realistic and natural sounds while offering great flexibility and minimal storage requirements. This research falls within the scope of the NEMUS project, which is dedicated to the digital reproduction of the sound of ancient stringed instruments using physical modelling techniques. Specifically, this work focuses on the numerical simulation of nonlinear string vibration. Nonlinearities are a critical factor in accurately replicating the sound of real-world instruments. Much of the recent literature has employed energy-based methods to ensure algorithmic stability when nonlinear behaviour is present, often resulting in fully implicit schemes requiring iterative root-finding methods. While effective, these schemes are computationally expensive and introduce additional complexities. Recent developments in numerical analysis have, in some cases, enabled real-time simulation of strongly nonlinear systems using non-iterative algorithms. However, several challenges remain unresolved. This thesis aims to advance the use of finite-difference time-domain and modal methods to address nonlinearities in string vibration, which capture salient perceptual features. The emphasis is on the efficiency of the algorithms, while also developing a framework for the sound synthesis of nonlinear strings. The work begins with a comprehensive review of string models and simulation techniques, covering both historical and modern approaches. Linear models are then used as a starting point, allowing for the introduction of impedance-type boundary conditions. The research then investigates typical nonlinear effects in string vibration, such as geometric nonlinearities, collisions, and friction, using newly developed non-iterative approaches, including quadratisation-based methods for conservative forces. These techniques significantly reduce computation times, making real-time simulation feasible for most systems. However, the quality of the simulations is still highly dependent on tailored discretisation choices. The thesis concludes with two case studies that apply these methods to physical models of musical instruments.