Juliette Chabassier | Time discretisation for nonlinear wave equations

In this page are gathered several integration strategies guaranteeing energy identity at the discrete level, in order to solve weakly nonlinear partial differential equations involved in piano modeling. Two contributions are adapted to wave propagation in the piano string under large deformations, leading to a model with nonlinear potential energy. Moreover a contribution is tailored to wave propagation in the rotating hammer neck, leading to a model with additional non-linear kinetic energy.

References

JSV

Energy based simulation of a Timoshenko beam in non-forced rotation. Application to the flexible piano hammer shank.

Chabassier, Juliette, and Duruflé, Marc

Journal of Sound and Vibration Apr 2014

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A nonlinear model for a vibrating Timoshenko beam in non-forced unknown rotation is derived from the virtual work principle applied to a system of beam with mass at the end. The system represents a piano hammer shank coupled to a hammer head. An energy-based numerical scheme is then provided, obtained by non-classical approaches. A major difficulty for time discretization comes from the nonlinear behavior of the kinetic energy of the system. This new numerical scheme is then coupled to a global energy-preserving numerical solution for the whole piano. The obtained numerical simulations show that the pianistic touch clearly influences the spectrum of the piano sound of equally loud isolated notes. These differences do not come from a possible shock excitation on the structure, or from a changing impact point, or a “longitudinal rubbing motion” on the string, since neither of these features is modeled in our study.
preprint

Numerical analysis of quadratized schemes. Application to the simulation of the nonlinear piano string

Castera, Guillaume, and Chabassier, Juliette

Research Report Aug 2023

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In this report, we present energy quadratization techniques for a Hamiltonian system of nonlinear wave equations formulated at order 2 in time. On a generic system, we present the so-called Invariant Energy Quadratization (IEQ) and Scalar Auxiliary Variable (SAV) methods as well as their energy conservation properties and discretization strategies in space and time. Unlike the iterative techniques commonly used for nonlinear systems to guarantee certain invariances, these two methods lead to algorithms whose complexity is known in advance and rely on the simple inversion of a linear system at each time step. In spite of an unconditional stability and an attractive complexity, the literature mentions problematic application cases with an uncontrolled accuracy. The numerical properties (stability, consistency and uniform convergence in time with respect to the CFL) of schemes obtained by hybridization between theta scheme and quadratization are studied for two classes of nonlinear terms: a nonlinearity concerning the solution field, and a nonlinearity concerning its gradient. These results are then applied to a geometrically exact nonlinear piano string for which numerical results are presented. The influence of the discretization parameters is studied and related to the theoretical results. The choices for the best accuracy or computational cost are illustrated. Some parameters can induce the space-time non-convergence of the schemes for a nonlinearity on the gradient, as it is the case for the piano string.
CMAME

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

Chabassier, Juliette, and Joly, Patrick

Computer Methods in Applied Mechanics and Engineering Aug 2010

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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.