This paper presents simulations of nonlinear plate vibrations in relation to sound synthesis of gongs and cymbals. The von Kármán equations are shown and then solved in terms of the modes of the associated linear system. The modal equations obtained constitute a system of nonlinearly coupled Ordinary Differential Equations which are completely general as long as the modes of the system are known. A simple second-order time-stepping integration scheme yields an explicit resolution algorithm with a natural parallel structure. Examples are provided and the results discussed.

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Nonlinear forces are ubiquitous in physical systems, and of prominent importance in musical acoustics. Though many models exist to describe such forces, in most cases the associated numerical schemes rely on iterative root finding methods, such as NewtonRaphson or gradient descent, which are computationally expensive and which therefore could represent a computational bottleneck. In this paper, a model for a large class of nonlinear forces is presented, and a novel family of energy-conserving finite difference schemes given. The schemes only require the evaluation of the roots of a quadratic function. A few applications in the lumped case are shown, and the robustness and accuracy of the scheme tested.

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Nonlinearity is a key feature in musical instruments and electronic circuits alike, and thus in simulation, for the purposes of physics-based modeling and virtual analog emulation, the numerical solution of nonlinear differential equations is unavoidable. Ensuring numerical stability is thus a major consideration. In general, one may construct implicit schemes using well-known discretisation methods such as the trapezoid rule, requiring computationally-costly iterative solvers at each time step. Here, a novel family of provably numerically stable time-stepping schemes is presented, avoiding the need for iterative solvers, and thus of greatly reduced computational cost. An application to the case of the collision interaction in musical instrument modeling is detailed.

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Due to recent increases in computational power, physical modeling synthesis is now possible in real time even for relatively complex models. We present here a modular physical modeling instrument design, intended as a construction framework for string- and bar- based instruments, alongside a mechanical network allowing for arbitrary nonlinear interconnection. When multiple nonlinearities are present in a feedback setting, there are two major concerns. One is ensuring numerical stability, which can be approached using an energy-based framework. The other is coping with the computational cost associated with nonlinear solvers—standard iterative methods, such as Newton-Raphson, quickly become a computational bottleneck. Here, such iterative methods are sidestepped using an alternative energy conserving method, allowing for great reduction in computational expense or, alternatively, to real-time performance for very large-scale nonlinear physical modeling synthesis. Simulation and benchmarking results are presented.

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In this work, a number of numerical schemes are presented in the context of virtual-analog simulation. The schemes are linearlyimplicit in character, and hence directly solvable without iterative methods. Schemes of increasing order of accuracy are constructed, and convergence and stability conditions are proven formally. The schemes are able to handle stiff problems very efficiently, because of their fast update, and can be run at higher sample rates to reduce aliasing. The cases of the diode clipper and ring modulator are investigated in detail, including several numerical examples.

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This paper presents an application of the port Hamiltonian formalism to the nonlinear simulation of the OTA-based Korg35 filter circuit and the Moog 4-pole ladder filter circuit. Lyapunov analysis is used with their state-space representations to guarantee zero-input stability over the range of parameters consistent with the actual circuits. A zero-input stable non-iterative discrete-time scheme based on a discrete gradient and a change of state variables is shown along with numerical simulations. Simulations show behavior consistent with the actual operation of the circuits, e.g., self-oscillation, and are found to be stable and have lower computational cost compared to iterative methods.

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For physical modelling sound synthesis, many techniques are available; time-stepping methods (e.g., finite-difference time-domain (FDTD) methods) have an advantage of flexibility and generality in terms of the type of systems they can model. These methods do, however, lack the capability of easily handling smooth parameter changes while retaining optimal simulation quality and stability, something other techniques are better suited for. In this paper, we propose an efficient method to smoothly add and remove grid points from a FDTD simulation under sub-audio rate parameter variations. This allows for dynamic parameter changes in physical models of musical instruments. An instrument such as the trombone can now be modelled using FDTD methods, as well as physically impossible instruments where parameters such as e.g. material density or its geometry can be made time-varying. Results show that the method does not produce (visible) artifacts and stability analysis is ongoing.

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In this paper, a complete simulation of a trombone using finitedifference time-domain (FDTD) methods is proposed. In particular, we propose the use of a novel method to dynamically vary the number of grid points associated to the FDTD method, to simulate the fact that the physical dimension of the trombone’s resonator dynamically varies over time. We describe the different elements of the model and present the results of a real-time simulation.

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The motion of a bowed string is a typical nonlinear phenomenon resulting from a friction force via interaction with the bow. The system can be described using suitable differential equations. Implicit numerical discretisation methods are known to yield energy consistent algorithms, essential to ensure stability of the timestepping schemes. However, reliance on iterative nonlinear root finders carries significant implementation issues. This paper explores a method recently developed which allows nonlinear systems of ordinary differential equations to be solved non-iteratively. Case studies of a mass-spring system and an ideal string coupled with a bow are investigated. Finally, a stiff string with loss is also considered. Combining semi-discretisation and a modal approach results in an algorithm yielding faster than real-time simulation of typical musical strings.

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Modal methods are a long-established approach to physical modeling sound synthesis. Projecting the equation of motion of a linear, time-invariant system onto a basis of eigenfunctions yields a set of independent forced, lossy oscillators, which may be simulated efficiently and accurately by means of standard time-stepping methods. Extensions of modal techniques to nonlinear problems are possible, though often requiring the solution of densely coupled nonlinear time-dependent equations. Here, an application of recent results in numerical simulation design is employed, in which the nonlinear energy is first quadratised via a convenient auxiliary variable. The resulting equations may be updated in time explicitly, thus avoiding the need for expensive iterative solvers, dense linear system solutions, or matrix inversions. The case of a network of interconnected distributed elements is detailed, along with a real-time implementation as an audio plugin.

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