The logical clarinet: numerical optimization of the geometry of woodwind instruments

The tone hole geometry of a clarinet is optimized numerically. The instrument is modeled as a network of one dimensional transmission line elements. For each (non-fork) fingering, we first calculate the resonance frequencies of the input impedance peaks, and compare them with the frequencies of a mathematically even chromatic scale (equal temperament). A least square algorithm is then used to minimize the differences and to derive the geometry of the instrument. Various situations are studied, with and without dedicated register hole and/or enlargement of the bore. With a dedicated register hole, the differences can remain less than 10 musical cents throughout the whole usual range of a clarinet. The positions, diameters and lengths of the chimneys vary regularly over the whole length of the instrument, in contrast with usual clarinets. Nevertheless, we recover one usual feature of instruments, namely that gradually larger tone holes occur when the distance to the reed increases. A fully chromatic prototype instrument has been built to check these calculations, and tested experimentally with an artificial blowing machine, providing good agreement with the numerical predictions

Nonlinear modes of clarinet-like musical instruments

The concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments. Using a modal expansion of the impedance of the instrument, and by projecting the equation for the acoustic pressure on the normal modes of the air column, a system of second order ordinary differential equations is obtained. The equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed. The system is treated using an amplitude-phase formulation for nonlinear modes, where the frequency and damping functions, as well as the invariant manifolds in the phase space, are unknowns to be determined. The formulation gives, without explicit integration of the underlying ordinary differential equation, access to the transient, the limit cycle, its period and stability. The process is illustrated for a model reduced to three normal modes of the air column

A gradient based optimisation algorithm for the design of brass-wind instruments

This paper presents how the shape of a brass instrument can be optimised with respect to its intonation and impedance peak magnitudes. The instrument is modelled using a one-dimensional transmission line analogy with truncated cones. The optimisation employs the Levenberg-Marquardt method, with the gradient of the objective function obtained by analytic manipulation. Through the use of an appropriate choice of design variables, the optimisation is capable of rapidly finding smooth horn profiles

Impedance boundary conditions for acoustic waves in a duct with a step discontinuity

This paper treats the use of numerically computed impedance boundary conditions for acoustic simulations. Such boundary conditions may be used to combine different methods on different parts of the computational domain. Impedance boundary conditions may be computed for each subproblem independently of each other. In order to develop insight into this approach, wave propagation in a rectangular waveguide with a step discontinuity is studied

Ill-posedness of absorbing boundary conditions applied on convex surfaces

Absorbing boundary conditions are important in many applications where partial differential equations defined on infinite domains are solved numerically. A problem that has attracted interest recently is that perfectly matched layers layers (PML) for electro-magnetic FDTD simulations applied on convex surfaces may lead to instabilities. This paper shows that these problems are not restricted to electro-magnetic calculations, but common for problems described by the classical wave equation with absorbing boundary conditions on convex surfaces. It is shown that the instabilities are independent of the numerical implementation of the absorbing boundary condition, and instead a result of unphysical assumption in the formulation of the boundary condition

Numerical Techniques for Acoustic Modelling and Design of Brass Wind Instruments

Acoustic horns are used in musical instruments and loudspeakers in order to provide an impedance match between an acoustic source and the surrounding air. The aim of this study is to develop numerical tools for the analysis and optimisation of such horns, with respect to their input impedance spectra. Important effects such as visco-thermal damping and modal conversion are shown to be localised to different parts of a typical brass instrument. This makes it possible to construct hybrid methods that apply different numerical techniques in different parts of the instrument. Narrow and slowly flaring parts are modelled using a one-dimensional transmission line analogy, and the rapidly flaring bell is modelled using a two-dimensional finite-difference method. The connection between the different regions is done by the aid of impedance boundary conditions. The use of such boundary conditions is investigated with respect to the required number of degrees of freedom. Numerical shape optimisation is employed in order to design horns with desired impedance characteristics throughout a design frequency band. A loudspeaker horn is optimised with respect to its sound power output, and a brass instrument is optimised with respect to its intonation. The horns are modelled using the finite-element method and a transmission line analogy. In order to achieve rapid convergence of the optimisation, gradient based minimisation algorithms are used. A prerequisite for success is the ability to accurately and inexpensively compute the gradient of the objective function. The gradient for the finite-element method is computed by an adjoint equation technique, whereas for the transmission line analogy, it is derived by formal differentiation of the model. In order to find smooth solutions, a smoothing technique is used, where optimisation is done with respect to the right hand side of a Poisson type equation

Continuous transportation as a material distribution topology optimization problem

The problem of moving a commodity with a given initial mass distribution to a pre-specified target mass distribution so that the total work is minimized can be traced back at least to Monge’s work from 1781. Here, we consider a version of this problem aiming to minimize a combination of road construction and transportation cost by determining, at each point, the local direction of transportation. This paper covers the modeling of the problem, highlights how it can be formulated as a material distribution topology optimization problem, and shows some results