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Research Papers: Design Automation

Brass Instruments Design Using Physics-Based Sound Simulation Models and Surrogate-Assisted Derivative-Free Optimization

[+] Author and Article Information
Robin Tournemenne

Institut de Recherche en Communications
et Cybernétique de Nantes,
UMR CNRS 6597,
École Centrale Nantes,
1 rue de la Noë,
Nantes 44300, France
e-mail: robin.tournemenne@irccyn.ec-nantes.fr

Jean-François Petiot

Institut de Recherche en Communications et
Cybernétique de Nantes,
UMR CNRS 6597,
École Centrale Nantes,
1 rue de la Noë,
Nantes 44300, France
e-mail: jean-francois.petiot@irccyn.ec-nantes.fr

Bastien Talgorn

Department of Mechanical Engineering,
GERAD and McGill University,
Montréal, QC H3T1J4, Canada
e-mail: bastien.talgorn@mail.mcgill.ca

Michael Kokkolaras

Department of Mechanical Engineering,
GERAD and McGill University,
Montréal, QC H3T1J4, Canada
e-mail: michael.kokkolaras@mcgill.ca

Joël Gilbert

Laboratoire d'Acoustique
de l'Université du Maine,
UMR CNRS 6613,
Université du Maine,
Le Mans 72085, France
e-mail: joel.gilbert@univ-lemans.fr

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 4, 2016; final manuscript received December 13, 2016; published online January 31, 2017. Assoc. Editor: Gary Wang.

J. Mech. Des 139(4), 041401 (Jan 31, 2017) (9 pages) Paper No: MD-16-1482; doi: 10.1115/1.4035503 History: Received July 04, 2016; Revised December 13, 2016

This paper presents a method for design optimization of brass wind instruments. The shape of a trumpet's bore is optimized to improve intonation using a physics-based sound simulation model. This physics-based model consists of an acoustic model of the resonator, a mechanical model of the excitator, and a model of the coupling between the excitator and the resonator. The harmonic balance technique allows the computation of sounds in a permanent regime, representative of the shape of the resonator according to control parameters of the virtual musician. An optimization problem is formulated in which the objective function to be minimized is the overall quality of the intonation of the different notes played by the instrument. The design variables are the physical dimensions of the resonator. Given the computationally expensive function evaluation and the unavailability of gradients, a surrogate-assisted optimization framework is implemented using the mesh adaptive direct search algorithm (MADS). Surrogate models are used both to obtain promising candidates in the search step of MADS and to rank-order additional candidates generated by the poll step of MADS. The physics-based model is then used to determine the next design iterate. Two examples (with two and five design optimization variables) demonstrate the approach. Results show that significant improvement of intonation can be achieved at reasonable computational cost. Finally, the perspectives of this approach for computer-aided instrument design are evoked, considering optimization algorithm improvements and problem formulation modifications using for instance different design variables, multiple objectives and constraints or objective functions based on the instrument's timbre.

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Figures

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Fig. 1

Typical input impedance Z of a Bb trumpet (magnitude), showing the resonances 2, 3, 4, 5 of the instrument

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Fig. 2

Definition of the main parts of the trumpet: the mouthpiece (in light gray), the leadpipe (in black), and the flaring bell

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Fig. 3

Bore geometry of a trumpet, described as a series of conical and cylindrical segments

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Fig. 4

Musical notation of the notes Bb3, F4, Bb4, and D5 of the Bb trumpet that are simulated in this study

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Fig. 5

Flowchart of the optimization process

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Fig. 6

MADS poll and mesh sizes for a two-dimensional problem

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Fig. 7

Representation of the leadpipe inner radius along the instrument axis; the black dotted line to the initial geometry (measured on the Yamaha trumpet); each other line corresponds to the best design found by one of the four methods over 20 runs

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Fig. 8

Exhaustive computation of the objective function for the two-dimensional design example; the black cross denotes the initial geometry; the other dots denote the best solutions of the four employed strategies

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Fig. 9

Evolution of the objective in the 2D problem

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Fig. 10

Details of the intonation improvements obtained using the OECV method for the 2D case: the dark gray columns correspond to the initial geometry while the light gray columns correspond to the optimum; the two left columns are the objective function mean value while the six other columns represent the detailed mean absolute value of the ETDs; the black bars on each column correspond from bottom to top to the first quartile the median and the third quartile of distributions of the 20 runs

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Fig. 11

Evolution of the objective in the 5D problem

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Fig. 12

Details of the intonation improvements obtained using the OECV method for the 5D case: the dark gray columns correspond to the initial geometry while the light gray columns correspond to the optimum; the two left columns are the objective function mean value while the six other columns represent the detailed mean absolute value of the ETDs; the black bars on each column correspond from bottom to top to the first quartile the median and the third quartile of distributions of the 20 runs

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Fig. 13

Model selection map Si,k for PRESS and OECV strategies on the 5D problem; the upper (respectively, lower) map indicates which models were selected with the PRESS (respectively, OECV) error metric; the gray intensity indicates how often the model k was selected during blackbox evaluation i over the 20 runs; darker gray indicates a model selected more frequently (a) selection with PRESS metric, (b) selection with OECV metric, and (c) legend: never selected always selected

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