Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string

https://doi.org/10.1016/j.cma.2010.04.013Get rights and content

Abstract

This paper considers a general class of nonlinear systems, “nonlinear Hamiltonian systems of wave equations”. The first part of our work focuses on the mathematical study of these systems, showing central properties (energy preservation, stability, hyperbolicity, finite propagation velocity, etc.). Space discretization is made in a classical way (variational formulation) and time discretization aims at numerical stability using an energy technique. A definition of “preserving schemes” is introduced, and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is trivial. A general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear Hamiltonian systems of wave equations class. The problem of the vibration of a piano string is taken as an example. Nonlinear coupling between longitudinal and transversal modes is modeled in the “geometrically exact model”, or approximations of this model. Numerical results are presented.

Introduction

We consider in this paper a general class of nonlinear systems, namely nonlinear Hamiltonian systems of wave equations. Our main objective is the construction of energy preserving discretization schemes for such systems. The concrete problem that has motivated this work was to compute the vibrations of a piano string, with the objective to achieve the numerical simulation of a whole concert piano. The full piano model is quite complex and couples the vibrations of the string (a 1D model) with the vibrations of the soundboard (a 2D phenomenon) and with the sound radiation (a 3D phenomenon). Guaranteeing and proving the stability of a numerical method for the coupled problem is not an easy task. Having an energy approach is a very powerful approach to achieve this goal. The Hamiltonian nature of the equations governing the vibrations of the string makes it possible a priori: there is conservation of an energy for the continuous problem. Preserving such a property at the discrete level has two nice consequences: keeping after discretization an important (from both theoretical and physical points of view) property of the exact solution and getting stability results provided that the discrete property has pleasant positivity properties.

The problematic of energy preserving schemes is far from new and has already generated an intensive literature. It appears that results can easily be found in the case of scalar equations (the unknown function takes scalar values), of course in the context of ordinary differential equations ODEs (see for instance [16], [26]) but also of some partial differential equations, particularly semi-linear wave equations (see for instance [6], [11], [12], [23], [25], [31]). The case of systems has been much less investigated and it seems that most results are restricted to very particular systems: see for instance in [27], [15] or [8] in the context of systems of ODE’s and [14] (for nonlinear elasticity) or [5] (for nonlinear strings) in the context of systems of PDE’s. Finally, Refs. [3], [4], [17] investigate time FE methods for the N-body problem, nonlinear elastodynamics and are extended to more general problems and higher orders. These very interesting methods rely on the difficult seek of a good quadrature rule and reduce to the previously mentioned methods in particular cases. Our aim in this context was to find a systematic and easily computed energy preserving scheme for any system of PDE’s, while keeping a great degree of generality.

As said above, we tackle in this paper a rather general class of 1D nonlinear Hamiltonian systems of wave equations, where the unknown function takes values in RN for arbitrary N  1. The restriction to the 1D case contributes essentially to simplifying the presentation. However, most of our developments can be extended to higher space dimensions. Our article is divided into two parts. The first part (Sections 2 Nonlinear Hamiltonian systems of wave equations: general theoretical frame, 3 Finite element energy preserving numerical schemes for nonlinear Hamiltonian systems of wave equations) concerns general systems. The second part (Section 4) presents the application to the particular system which governs the vibrations of a piano string.

In Section 2, we recall the main properties of 1D nonlinear Hamiltonian systems of wave equations. We insist particularly on the most relevant (for our purpose) properties of (sufficiently) smooth solutions of such systems: energy preservation (leading to H1 stability), hyperbolicity and finite propagation velocity. Section 3, the main section of the article, is devoted to the discretization schemes. For the space discretization, we use a variational formulation and a Galerkin approximation procedure (Section 3.1). The main difficulties are encountered when looking at the time discretization using finite differences, which is the object of Section 3.2. Our desire to preserve a certain energy leads us to introduce a particular class of numerical schemes. We show that this class excludes the explicit scheme (except in the linear case, see Lemma 3.2 of Section 3.2.2) as well as partially decoupled implicit schemes (except in some very particular systems, see Section 3.2.3). Finally in Section 3.2.4, we exhibit inside our class of numerical schemes a fully implicit, second order accurate, energy preserving and unconditionally stable scheme for any nonlinear Hamiltonian system of wave equations, with any number of unknown variables. Note that the implicitness of the scheme is the price to be paid for robustness (obtained via energy conservation).

In the context of the simulation of the piano (Section 4), the implicitness of the scheme is by no means a real constraint since the time devoted to the string itself should be a small percentage of the total computational cost while the unconditional stability provides more flexibility and robustness for the coupled model. In Section 4.1 we present the nonlinear vibrating string model introduced in [28], as well as some of its approximations including the one used in [2], [5]. These models are all nonlinear Hamiltonian systems of wave equations. We give their main mathematical properties in Section 4.2. We apply the numerical scheme of Section 3.2.4 to this system (Section 4.3) and related numerical results are given in Section 4.4.

Section snippets

General formulation

This paragraph is devoted to 1D nonlinear Hamiltonian systems of wave equations. A function H:RNR, a potential energy, totally determines the system (N is the size of the system). We shall consider the following Cauchy problem (Ω is a segment of R or R itself):Findu=(u1,,uN):Ω×R+RN,tt2u-x[H(xu)]=0,xΩ,t>0,u(x,0)=u0(x),tu(x,0)=u1(x),u(x,t)=0,xΩ.

Remark 2.1

The function H is only used through its gradient, hence any H+L gives the same system of equations as H, with L a linear function on RN. Thus,

Finite element energy preserving numerical schemes for nonlinear Hamiltonian systems of wave equations

Each time one wishes to discretize in space and time an evolution problem whose solution satisfies the conservation of an energy, as in the case of the systems (1) but more generally of many mechanical models, it is a natural idea to try to construct numerical schemes that preserve rigorously a discrete energy that is equivalent of the continuous energy. As we shall see immediately in the next paragraph, in the case of (1), the use of variational techniques (such as the finite element method)

Establishment of the geometrically exact model

We are interested in the string vibration, for instance a piano string. The problem has been formulated in its nonlinear version in [28], then used and modified by several authors. The geometrically exact model uses an exact geometric description of the movement of the string: this introduces geometric nonlinearity. The most complete model takes into account the three components of the points of the string (which corresponds to N = 3 with the notation of the previous section, see Section 4.1.2)

Conclusions and perspectives

In this paper we have proposed one solution for achieving our goal : constructing energy preserving schemes for nonlinear Hamiltonian systems of wave equations. Work is currently in progress on an energy preserving discretization of a more general class of systems. The future plan for this study is to develop a mathematical and a numerical model for the full grand piano. One of the main difficulties is to consider the coupling of the strings’ movement with the vibrations of the soundboard and

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