An Energy Conserving Finite Difference Scheme for the Simulation of Collisions in Snare Drums

In this paper, a physics-based model for a snare drum will be dis-cussed, along with its finite difference simulation. The interac-tions between a mallet and the membrane and between the snares and the membrane will be described as perfectly elastic collisions. A novel numerical scheme for the implementation of collisions will be presented, which allows a complete energy analysis for the whole system. Viscothermal losses will be added to the equation for the 3D wave propagation. Results from simulations and sound examples will be presented. 1

Percussion instrument modelling In 3D: sound synthesis through time domain numerical simulation

This work is concerned with the numerical simulation of percussion instruments based on physical principles. Three novel modular environments for sound synthesis are presented: a system composed of various plates vibrating under nonlinear conditions, a model for a nonlinear double membrane drum and a snare drum. All are embedded in a 3D acoustic environment. The approach adopted is based on the finite difference method, and extends recent results in the field. Starting from simple models, the modular instruments can be created by combining different components in order to obtain virtual environments with increasing complexity. The resulting numerical codes can be used by composers and musicians to create music by specifying the parameters and a score for the systems. Stability is a major concern in numerical simulation. In this work, energy techniques are employed in order to guarantee the stability of the numerical schemes for the virtual instruments, by imposing suitable coupling conditions between the various components of the system. Before presenting the virtual instruments, the various components are individually analysed. Plates are the main elements of the multiple plate system, and they represent the first approximation to the simulation of gongs and cymbals. Similarly to plates, membranes are important in the simulation of drums. Linear and nonlinear plate/membrane vibration is thus the starting point of this work. An important aspect of percussion instruments is the modelling of collisions. A novel approach based on penalty methods is adopted here to describe lumped collisions with a mallet and distributed collisions with a string in the case of a membrane. Another point discussed in the present work is the coupling between 2D structures like plates and membranes with the 3D acoustic field, in order to obtain an integrated system. It is demonstrated how the air coupling can be implemented when nonlinearities and collisions are present. Finally, some attention is devoted to the experimental validation of the numerical simulation in the case of tom tom drums. Preliminary results comparing different types of nonlinear models for membrane vibration are presented

SIMULATION OF THE SNARE-MEMBRANE COLLISION IN MODAL FORM USING THE SCALAR AUXILIARY VARIABLE (SAV) METHOD

Collisions play an essential role in the sound production of many musical instruments, such as in the snare drum. Here, collisions occur between the stick and the batter head and between the snares and the bottom head. The latter involve interactions between fully distributed objects and are the subject of this work. From a simulation standpoint, simple explicit or semi-implicit schemes are prone to unstable numerical behaviour and an appropriate energy-conserving framework is required for stable simulation designs. Usually, this is accomplished via fully-implicit designs that are known to conserve energy but that require iterative solvers such as Newton-Raphson. Other than representing a computational bottleneck, iterative schemes present a variable operational cost per timestep and, furthermore, are serial in nature. This work will explore the possibility of simulating the snare-membrane collision using explicit designs obtained via a quadratisation of the nonlinear potential energy. A modal function basis will be employed for the spatial discretisation, allowing for fine-tuning damping ratios and natural frequencies

Physical modelling of the bowed string and applications to sound synthesis

This work outlines the design and implementation of an algorithm to simulate two-polarisation bowed string motion, for the purpose of realistic sound synthesis. The algorithm is based on a physical model of a linear string, coupled with a bow, stopping fi ngers, and a rigid, distributed fingerboard. In one polarisation, the normal interaction forces are based on a nonlinear impact model. In the other polarisation, the tangential forces between the string and the bow, fingers, and fingerboard are based on a force-velocity friction curve model, also nonlinear. The linear string model includes accurate time-domain reproduction of frequency-dependent decay times. The equations of motion for the full system are discretised with an energy-balanced finite difference scheme, and integrated in the discrete time domain. Control parameters are dynamically updated, allowing for the simulation of a wide range of bowed string gestures. The playability range of the proposed algorithm is explored, and example synthesised gestures are demonstrated

Finite difference and finite volume methods for wave-based modelling of room acoustics

Wave-based models of sound propagation can be used to predict and synthesize sounds as they would be heard naturally in room acoustic environments. The numerical simulation of such models with traditional time-stepping grid-based methods can be an expensive process, due to the sheer size of listening environments (e.g., auditoriums and concert halls) and due to the temporal resolution required by audio rates that resolve frequencies up to the limit of human hearing. Finite difference methods comprise a simple starting point for such simulations, but they are known to suffer from approximation errors that may necessitate expensive grid refinements in order to achieve sufficient levels of accuracy. As such, a significant amount of research has gone into designing finite difference methods that are highly accurate while remaining computationally efficient. The problem of designing and using accurate finite difference schemes is compounded by the fact that room acoustics models require complex boundary conditions to model frequency-dependent wall impedances over non-trivial geometries. The implementation of such boundary conditions in a numerically stable manner has been a challenge for some time. Stable boundary conditions for finite difference room acoustics simulations have been formulated in the past, but generally they have only been useful in modelling trivial geometries (e.g., idealised shoebox halls). Finite volume methods have recently been shown to be a viable solution to the problem of complex boundary conditions over non-trivial geometries, and they also allow for the use of energy methods for numerical stability analyses. Finite volume methods lend themselves naturally to fully unstructured grids and they can simplify to the types of grids typically used in finite difference methods. This allows for room acoustics simulation models that balance the simplicity of finite difference methods for wave propagation in air with the detail of finite volume methods for the modelling of complex boundaries. This thesis is an exploration of these two distinct, yet related, approaches to wave-based room acoustic simulations. The overarching theme in this investigation is the balance between accuracy, computational efficiency, and numerical stability. Higher-order and optimised schemes in two and three spatial dimensions are derived and compared, towards the goal of finding accurate and efficient finite difference schemes. Numerical stability is analysed using frequency-domain analyses, as well as energy techniques whenever possible, allowing for stable and frequency-dependent boundary conditions appropriate for room acoustics modelling. Along the way, the use of non-Cartesian grids is investigated, geometric relationships between certain finite difference and finite volume schemes are explored, and some problems associated to staircasing effects at boundaries are considered. Also, models of sound absorption in air are incorporated into these numerical schemes, using physical parameters that are appropriate for room acoustic scenarios

Proceedings of the 19th Sound and Music Computing Conference

Proceedings of the 19th Sound and Music Computing Conference - June 5-12, 2022 - Saint-Étienne (France). https://smc22.grame.f