Time domain simulation of a piano. Part 2: numerical aspects
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 1, pp. 93-133.

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.

Received:
DOI: 10.1051/m2an/2015007
Classification: 00A71, 00A65, 65P05, 65N25, 35Q72, 35L05
Mots-clés : Piano model, energy preserving schemes, numerical methods
Chabassier, Juliette 1; Duruflé, Marc 1; Joly, Patrick 2

1 Magique 3D team, Inria Sud Ouest, 200 avenue de la vieille tour, 33 405 Talence cedex, France
2 POems team, Inria Saclay, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau cedex, France
@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://www.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://www.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://www.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 , Volume 50 (2016) no. 1, pp. 93-133. doi : 10.1051/m2an/2015007. http://www.numdam.org/articles/10.1051/m2an/2015007/

J. Bensa, S. Bilbao, 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

J. Berthaut, M.N. Ichchou and L. Jezequel, Piano soundboard: structural behavior, numerical and experimental study in the modal range. Appl. Acoust. 64 (2003) 1113–1136. | DOI

S. Bilbao, Conservative numerical methods for nonlinear strings. J. Acoust. Soc. Am. 118 (2005) 3316–3327. | DOI

X. Boutillon, 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).

J. Chabassier and M. Duruflé, 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

J. Chabassier and S. Imperiale, 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

J. Chabassier and S. Imperiale, Introduction and study of fourth order theta schemes for linear wave equations. J. Comput. Appl. Math. 245 (2013) 194–212. | DOI | MR | Zbl

J. Chabassier and P. Joly, 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

J. Chabassier, A. Chaigne and P. Joly, Time domain simulation of a piano. Part I: Model description. ESAIM: M2AN 48 (2014) 1241–1278. | DOI | Numdam | MR | Zbl

A. Chaigne and A. Askenfelt, 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

A. Chaigne and A. Askenfelt, 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

G. Cohen and P. Grob, Mixed higher order spectral finite elements for reissner–mindlin equations. SIAM J. Sci. Comput. 29 (2007) 986–105. | DOI | MR | Zbl

F. Collino and C. Tsogka, Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66 (2001) 294–307. | DOI

H.A. Conklin, 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

G. Derveaux, A. Chaigne, P. Joly and E. Bécache, Time-domain simulation of a guitar: Model and method. J. Acoust. Soc. Am. 114 (2003) 3368–3383. | DOI

J. Diaz and M. Grote. Energy conserving explicit local time-stepping for second-order wave equations. Siam J. Sci. Comput. 31 (2009) 1985–2014. | DOI | MR | Zbl

K. Ege, X. Boutillon and M. Rébillat, 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

S. Fauqueux and G. Cohen, Mixed finite elements with mass-lumping for the transient wave equation. J. Comput. Acoustics 8 (2000) 171–188. | DOI | MR | Zbl

T. Fouquet F. Collino and P. Joly, Conservative space-time mesh refinement methods for the fdtd solution of maxwell’s equations. J. Comput. Phys. 211 (2006) 9–35. | DOI | MR | Zbl

N. Giordano, Simple model of a piano soundboard. J. Acoust. Soc. Am. 102 (1997) 1159–1168. | DOI

N. Giordano and M. Jiang, Physical modeling of the piano. EURASIP J. Appl. Signal Process. 2004 (2004) 926–933.

J.R. Hutchinson, Shear coefficients for timoshenko beam theory. J. Appl. Mech. 68 (2000) 87–92. | DOI | Zbl

A. Izadbakhsh, J. Mcphee and S. Birkett, 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

J. Kindel and I. Wang, Vibrations of a piano soundboard: Modal analysis and finite element analysis. J. Acoust. Soc. Am. Suppl. 1 81 (1987) S61. | DOI

A. Mamou-Mani, J. Frelat and C. Besnainou, 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.

G.R.W. Quispel and G.S Turner, Discrete gradient methods for solving odes numerically while preserving a first integral. J. Phys. A 29 (1996) 341–349. | DOI | MR | Zbl

L. Rhaouti, A. Chaigne and P. Joly, Time-domain modeling and numerical simulation of a kettledrum. J. Acoust. Soc. Am. 105 (1999) 3545–3562. | DOI

W. Strauss and L. Vazquez, Numerical solution of a nonlinear klein-gordon equation. J. Comput. Phys. 28 (1978) 271–278. | DOI | MR | Zbl

G. Weinreich, Coupled piano strings. J. Acoust. Soc. Am. 62 (1977) 1474–1484. | DOI

Cited by Sources: