Download Simulations of Nonlinear Plate Dynamics: An Accurate and Efficient Modal Algorithm
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.
Download A Numerical Scheme for Various Nonlinear Forces, Including Collisions, Which Does Not Require an Iterative Root Finder
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.
Download Non-Iterative Solvers For Nonlinear Problems: The Case of Collisions
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.
Download Large-scale Real-time Modular Physical Modeling Sound Synthesis
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.
Download Non-Iterative Schemes for the Simulation of Nonlinear Audio Circuits
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.
Download Applications of Port Hamiltonian Methods to Non-Iterative Stable Simulations of the Korg35 and Moog 4-Pole Vcf
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.
Download Dynamic Grids for Finite-Difference Schemes in Musical Instrument Simulations
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.
Download A Physical Model of the Trombone Using Dynamic Grids for Finite-Difference Schemes
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.