This article is the second of a series of two papers devoted to the numerical simulation of the piano. It concerns the numerical aspects of the work, the implementation of a piano code and the presentation of corresponding simulations. The main difficulty is time discretization and stability is achieved via energy methods. Numerical illustrations are provided for a realistic piano and compared to experimental recordings.
DOI : 10.1051/m2an/2015007
Mots-clés : Piano model, energy preserving schemes, numerical methods
@article{M2AN_2016__50_1_93_0,
author = {Chabassier, Juliette and Durufl\'e, Marc and Joly, Patrick},
title = {Time domain simulation of a piano. {Part} 2: numerical aspects},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {93--133},
publisher = {EDP-Sciences},
volume = {50},
number = {1},
year = {2016},
doi = {10.1051/m2an/2015007},
zbl = {1381.00017},
mrnumber = {3460103},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015007/}
}
TY - JOUR AU - Chabassier, Juliette AU - Duruflé, Marc AU - Joly, Patrick TI - Time domain simulation of a piano. Part 2: numerical aspects JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 93 EP - 133 VL - 50 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015007/ DO - 10.1051/m2an/2015007 LA - en ID - M2AN_2016__50_1_93_0 ER -
%0 Journal Article %A Chabassier, Juliette %A Duruflé, Marc %A Joly, Patrick %T Time domain simulation of a piano. Part 2: numerical aspects %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 93-133 %V 50 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015007/ %R 10.1051/m2an/2015007 %G en %F M2AN_2016__50_1_93_0
Chabassier, Juliette; Duruflé, Marc; Joly, Patrick. Time domain simulation of a piano. Part 2: numerical aspects. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 93-133. doi : 10.1051/m2an/2015007. http://archive.numdam.org/articles/10.1051/m2an/2015007/
, , R. Kronland-Martinet and J. O. Smith III, The simulation of piano string vibration: from physical models to finite difference schemes and digital waveguides. J. Acoust. Soc. Am. 114 (2003) 1095–1107. | DOI
D.P. Bertsekas, Nonlinear Programming, 2nd edn.. Athena Scientific (1999) 1–780. | MR | Zbl
, and , Piano soundboard: structural behavior, numerical and experimental study in the modal range. Appl. Acoust. 64 (2003) 1113–1136. | DOI
, Conservative numerical methods for nonlinear strings. J. Acoust. Soc. Am. 118 (2005) 3316–3327. | DOI
, Model for piano hammers: Experimental determination and digital simulation. J. Acoust. Soc. Am. 83 (1988) 746–754. | DOI
B. Bank and L. Sujbert, A piano model including longitudinal string vibrations. Proc. of the Digital Audio Effects Conference (2004) 89–94.
J. Chabassier, Modélisation et simulation numérique d’un piano par modèles physiques. Ph.D. thesis, École Polytechnique (2012).
J. Chabassier and M. Duruflé, Physical parameters for piano modeling. Technical Report RT-0425, INRIA (2012).
and , Energy based simulation of a Timoshenko beam in non-forced rotation. Influence of the piano hammer shank flexibility on the sound. J. Sound Vibr. 333 (2014) 7198–7215. | DOI
and , Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string. Wave Motion 50 (2012) 456–480. | DOI | MR | Zbl
and , Introduction and study of fourth order theta schemes for linear wave equations. J. Comput. Appl. Math. 245 (2013) 194–212. | DOI | MR | Zbl
and , Energy preserving schemes for nonlinear hamiltonian systems of wave equations. application to the vibrating piano string. Comput. Meth. Appl. Mech. Eng. 199 (2010) 2779–2795. | DOI | MR | Zbl
, and , Time domain simulation of a piano. Part I: Model description. ESAIM: M2AN 48 (2014) 1241–1278. | DOI | Numdam | MR | Zbl
and , Numerical simulation of piano strings I. A physical model for a struck string using finite-difference methods. J. Acoust. Soc. Am. 95 (1994) 1112–1118. | DOI
and , Numerical simulations of piano strings II. Comparisons with measurements and systematic exploration of some hammer string parameters. J. Acoust. Soc. Am. 95 (1994) 1631–1640,. | DOI
and , Mixed higher order spectral finite elements for reissner–mindlin equations. SIAM J. Sci. Comput. 29 (2007) 986–105. | DOI | MR | Zbl
and , Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66 (2001) 294–307. | DOI
, Piano strings and “phantom” partials. J. Acoust. Soc. Am. 102 (1997) 659. | DOI
R. Dautray, J.L. Lions, C. Bardos, M. Cessenat, P. Lascaux, A. Kavenoky, B. Mercier, O. Pironneau, B. Scheurer and R. Sentis, Mathematical analysis and numerical methods for science and technology. In vol. 6. Springer (2000). | Zbl
, , and , Time-domain simulation of a guitar: Model and method. J. Acoust. Soc. Am. 114 (2003) 3368–3383. | DOI
and . Energy conserving explicit local time-stepping for second-order wave equations. Siam J. Sci. Comput. 31 (2009) 1985–2014. | DOI | MR | Zbl
, and , Vibroacoustics of the piano soundboard: (Non)linearity and modal properties in the low- and mid-frequency ranges. J. Sound Vibr. 332 (2013) 1288–1305. | DOI
and , Mixed finite elements with mass-lumping for the transient wave equation. J. Comput. Acoustics 8 (2000) 171–188. | DOI | MR | Zbl
and , Conservative space-time mesh refinement methods for the fdtd solution of maxwell’s equations. J. Comput. Phys. 211 (2006) 9–35. | DOI | MR | Zbl
, Simple model of a piano soundboard. J. Acoust. Soc. Am. 102 (1997) 1159–1168. | DOI
and , Physical modeling of the piano. EURASIP J. Appl. Signal Process. 2004 (2004) 926–933.
, Shear coefficients for timoshenko beam theory. J. Appl. Mech. 68 (2000) 87–92. | DOI | Zbl
, and , Dynamic modeling and experimental testing of a piano action mechanism with a flexible hammer shank. J. Comput. Nonlin. Dyn. 3 (2008) 1–10.
P. Joly, Variational methods for time-dependent wave propagation problems. In vol. 31: Topics in Computational Wave Propagation. Springer, Berlin (2003) 201–264. | MR | Zbl
and , Vibrations of a piano soundboard: Modal analysis and finite element analysis. J. Acoust. Soc. Am. Suppl. 1 81 (1987) S61. | DOI
, and , Numerical simulation of a piano soundboard under downbearing. J. Acoust. Soc. Am. 123 (2008) 2401–2406. | DOI
L. Ortiz-Berenguer and F. Casajus-Quiros, Modeling of piano sounds using fem simulation of soundboard vibration. Proc. of Acoustics 08 Paris (2008) 6252–6256.
and , Discrete gradient methods for solving odes numerically while preserving a first integral. J. Phys. A 29 (1996) 341–349. | DOI | MR | Zbl
, and , Time-domain modeling and numerical simulation of a kettledrum. J. Acoust. Soc. Am. 105 (1999) 3545–3562. | DOI
and , Numerical solution of a nonlinear klein-gordon equation. J. Comput. Phys. 28 (1978) 271–278. | DOI | MR | Zbl
, Coupled piano strings. J. Acoust. Soc. Am. 62 (1977) 1474–1484. | DOI
Cité par Sources :
