MODELING AND NUMERICAL SIMULATION OF A PIANO - Inria

MODELING AND NUMERICAL SIMULATION OF A PIANO Juliette Chabassier UMR POEMS 2706 (INRIA / ENSTA / CNRS) UME (ENSTA ParisTech) Antoine Chaigne – Patrick Joly

Motivations   OBJECTIVE (PhD) : •  Numerical simulation of a grand piano based on physical models •  Understand phantoms & precursor   INTO CONSIDERATION : •  Hammer •  String : transversal + longitudinal vibrations •  Soundboard •  Sound radiation •  Choirs + duplex scales   CHALLENGES : •  Multi dimensional problem •  Coupling : hammer / strings / bridge / soundboard / air •  Guaranteeing numerical… •  stability  Energy method •  accuracy  Numerical dispersion •  efficiency  C++ , openMP , MPI Conklin : Piano strings and « phantom » partials (JASA 1997) 1

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Stiff nonlinear string

Morse & Ingard, Theoretical Acoustics (1968) 2

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Space and time discretization   SPACE : High order finite elements.   TIME : Difficulty = preserve a discrete (and positive) energy  stability  Non classical scheme for nonlinear systems under the form

 Consistent dissipative scheme for fluid damping

Chabassier & Joly : Energy Preserving Schemes for Nonlinear Hamiltonian Systems of Wave Equations. Application to the Vibrating Piano String (in CMAME vol 199, 2010) 3

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Interaction with a nonlinear hammer

Stulov : Dynamic behavior and mechanical features of wool felt (Acta Mechanica 2004) Rhaouti : Timpani simulation (PhD 1999)

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Plate equation : Reissner Mindlin model

WHY REISSNER MINDLIN OVER KIRCHHOFF LOVE ?   preserves an energy   more relevant physically for thick plates (KL is poor above 2kHz)   better theoretical properties (limit on propagation speed)   easier numerical discretization (high order FEM in space)

pianotreasure.com 5

Duruflé : C++ code MONTJOIE

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Time scheme Derveaux : Numerical modelization of acoustic guitar (PhD 2002)

LEAP FROG

UNCOUPLED SCALAR ODEs :   Analytic resolution   No numerical dispersion in time   Frequency dependant damping 6

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Piano soundboard : Steinway D

Space discretization

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Piano soundboard modal shapes

32.4 Hz

51.0 Hz

77.2 Hz

101.1 Hz

128.7 Hz

145.1 Hz

166.4 Hz

178.4 Hz

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208.9 Hz

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Bridge model STRING vibrations  SOUNDBOARD vibrations   Reciprocal coupling condition  energy preserving   slight ANGLE at the bridge  precursor

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Sound radiation : vibroacoutics SOUNDBOARD vibrations  AIR vibrations   Reciprocal coupling condition  energy preserving   Continuity of mechanical and acoustical velocities   Numerical unknown : primitive of the pressure   Space discretisation : high order FEM  Time discretisation : Leap-Frog scheme (stability condition)   Absorbing conditions

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Numerical resolution of the coupled problem

  Use of Lagrange Multipliers and Schur Complement methods: •  NONLINEAR resolution of the sub-system Hammer / Strings / LM (Newton method) •  ANALYTIC resolution of the sub-system Soundboard •  NUMERICAL resolution of the sub-system Air   High performance computing : •  Strings : OpenMP + multithreaded factorization •  Air : Distributed parallel computing with MPI   C++ : modularity (object-oriented programming) 11

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Numerical simulation : C2  

               Reality                                                                      Simula,on   Wrapped string Choir : triplet Coupled hammer

Homogenized string Only one string Measured hammer force as input

1,500 dof on the string 1,928,280 dof in the volume 24,641 dof on the plate 500 modes (up to 5,000 Hz) Time step = 1e-6 sec Movie : 24 first millisec

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Numerical simulation : C2

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Numerical simulation : C2  

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Linear / Nonlinear string : precursor at the bridge WITHOUT   LONGITUDINAL   WAVES  

WITH   LONGITUDINAL   WAVES  

MEASURED  

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Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Precursor in pressure signal

WITHOUT   LONGITUDINAL   WAVES  

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WITH   LONGITUDINAL   WAVES  

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano  

Conclusions & Prospects   COMPLETE PIANO MODEL : •  Nonlinear hammer •  Stiff nonlinear string (transversal + longitudinal) • Thick soundboard •  Sound radiation   Angle at the bridge + nonlinear string : explains the precursor   FUTURE PHYSICAL SIMULATIONS : •  Spectra and phantoms •  Double decay on choirs • Vibration of duplex scales   IN A FUTURE MODEL : •  Hammer (shank ?) •  Bridge (beam-plate coupling ?) •  Soundboard : string bearing and crown •  Strings : -  Non planar motion - Torsion - Viscoelastic damping •  Structure shock 17

Julie&e  Chabassier  –  ASA  mee2ng  –  Sea&le  2011      Numerical  Simula2on  of  a  Piano