International audienceThe linear wave equation does not describe the com- plexity of the piano strings vibration enough for physics based sound synthesis. The nonlinear cou- pling between transversal and longitudinal modes has to be taken into account, as does the "geometrically exact" model. This system of equations can be clas- sified among a general energy preserving class of sys- tems. We present an implicit, centered, second order accurate, numerical scheme that preserves a discrete energy, leading to unconditional stability of the nu- merical scheme. The complete model takes into ac- count the bridge coupling the strings, and the ham- mer non linear attack on the strings

Chabassier, Juliette

Joly, Patrick

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HAL Id: hal-00688959https://hal.archives-ouvertes.fr/hal-00688959Submitted on 19 Apr 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Energy preserving scheme for non linear systems of waveequations. Application to piano strings.Juliette Chabassier, Patrick JolyTo cite this version:Juliette Chabassier, Patrick Joly. Energy preserving scheme for non linear systems of wave equations.Application to piano strings.. WAVES 2009 : The 9th International Conference on Mathematical andNumerical Aspects of Wawes Propagation, Jun 2009, Pau, France. pp.00. ￿hal-00688959￿Energy preserving scheme for non linear systems of wave equations.Application to piano strings.J. Chabassier1,∗, P. Joly11 Projet POems, INRIA Rocquencourt, Rocquencourt, France∗Email: juliette.chabassier@inria.frAbstractThe linear wave equation does not describe the com-plexity of the piano strings vibration enough forphysics based sound synthesis. The nonlinear cou-pling between transversal and longitudinal modes hasto be taken into account, as does the “geometricallyexact” model. This system of equations can be clas-sified among a general energy preserving class of sys-tems. We present an implicit, centered, second orderaccurate, numerical scheme that preserves a discreteenergy, leading to unconditional stability of the nu-merical scheme. The complete model takes into ac-count the bridge coupling the strings, and the ham-mer non linear attack on the strings.1 Non linear string vibrationsObservations on piano strings spectra have shownthat the linear wave equation is not enough to de-scribe the physical phenomenon of piano strings vi-bration. Unusual frequency series have been noticed,which Conklin [2] called “phantom partials”. UsingFigure 1: Displacement of one string point :transverse motion u and longitudinal motion v.a non linear geometric description of the transver-sal and longitudinal motions of the string leads tothe “geometrically exact” model (see Morse & In-gard [5]). We use adimensionned variables, let E bethe Young’s modulus, A the section and T0 the resttension of the string, and define α = EA−T0EA.The system can be written:∂2u∂t2−∂∂x∂u∂x− α∂u∂x√(∂u∂x)2+(1 + ∂v∂x)2 = 0∂2v∂t2−∂∂x∂v∂x− α1 + ∂v∂x√(∂u∂x)2+(1 + ∂v∂x)2 = 0This system is a particular case of:Find u : Ω × R+ → RN∂2u∂t2−∂∂x[∇H(∂u∂x)]= 0(1)where H : RN → R. String model corresponds toN = 2, u = (u, v) andH(u, v) =12u2 +12v2 − α[√u2 + (1 + v)2 − (1 + v)] (2)This quasilinear system is hyperbolic when H is con-vex, which is the case with (2) for small values of(u, v). The theory for these equations is now wellknown, see Li TaTsien [4]. An important result isthe energy conservation for this general type of sys-tems:Theorem 1. Let u be a sufficiently smooth solutionof (1). Then it satisfies :ddt∫Ω{12∣∣∣∂u∂t∣∣∣2+ H(∂u∂x)}dx = 0 (3)Energy preservation can lead to H1 stability on thesolution, under a coercivity condition on H.2 Energy preserving schemeA classic way of building stable numerical schemes forPDEs is to preserve, on a discrete level, a continuousconserved quantity, like an energy for instance. If thisis relatively easy and well-known for scalar equations(N = 1, see Furihata [3]), this is less obvious forsystems for which existing solutions that we foundin the litterature (see for instance Bilbao [1]) seemto be limited to specific hamiltonians. In the generalcase, we developped an implicit, energy preservingscheme which is also centered in time. The schemecan be written for any value of N and any function H.It is second order accurate, unconditionnally stableand its algebraic complexity grows as 2N−1. Thescheme is based on a variational approach which canlead to finite elements for space variable. The maindifficulty remains in the time discretisation, which webased on finite differences. In the following, Vh is afinite dimensional subspace of H10 , uk,h denotes thekth coordinate of the discrete solution uh (k ∈ [1,N ]).For simplicity, we will write abusively H(uk, ul 6=k) forH(u). Next, we define the directional finite differenceδkH(uk,1, uk,2;ul 6=k) :=H(uk,1, ul 6=k) − H(uk,2, ul 6=k)uk,1 − uk,2This will be used in the discretisation of the kth equa-tion of (1) with uk,1 = un+1k,h and uk,2 = un−1k,h . Thedifficult point is to decide a which times the N − 1other quantities ul 6=k are taken. The choice is madein order to preserve a discrete energy.Let us introduceΣk ={σ : Jk → {−1,+1}, Jk = {1, · · · , N} \ {k}}and for σ ∈ Σk, defineθ(σ) = µ(σ)! (N − 1 − µ(σ))! / N !where µ(σ) = #{l ∈ Jk, σ(l) = +1}.The scheme can be written, for any vh ∈ Vh:∮ L0un+1k,h − 2unk,h + un−1k,h∆t2vh +∑σ∈Σk∮ L0θσ δkH(∂xun+1k,h , ∂xun−1k,h ; ∂xun+σ(l)l 6=k,h ) ∂xvh = 0where∮refers to any chosen numerical integration.The main theoretical property of this scheme is:Theorem 2. Our scheme preserves the energy:En+12 =N∑k=1[ 12∮ L0[un+1k,h − unk,h∆t]2]+∮ L0H(∂xun+1h ) + H(∂xunh)23 Coupled strings and hammerPiano strings are mostly grouped by three in order toproduced one note. The coupling point is the bridgeat which there is actually a displacement, that is thesame for all the strings. For now, we modelled thismotion by a simple damped oscillating movement,expecting a better modelisation in the future. Thehammer has also been taken into account, coupledwith strings by a damped non linear law with hys-teresis. Figure (2) shows the conservation of totalenergy (full black line) without any damping, andthe evolution of internal energies (hammer, string,bridge) according to the time steps.Figure 2: Energy values according to time steps.References[1] S. Bilbao. Conservative numerical methods fornonlinear strings. Acoustical Society of AmericaJournal, 118:3316–3327, November 2005.[2] H. A. Conklin. Piano strings and “phantom”partials. Acoustical Society of America Journal,102:659, July 1997.[3] D. Furihata. Finite-difference schemes for non-linear wave equation that inherit energy conser-vation property. Journal of Computational andApplied Mathematics, 134(1-2):37–57, 2001.[4] T.-T. Li. Global classical solutions for quasilinearhyperbolic systems. Masson Chichester, 1994.[5] P.M. Morse and K.U. Ingard. Theoretical Acous-tics. McGraw-Hill, 1968.

Energy preserving scheme for non linear systems of wave equations. Application to piano strings.

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Energy preserving scheme for non linear systems of wave equations. Application to piano strings.

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