Dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.

Juliette Chabassier 1 Sébastien Imperiale 2
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : We study the implicit time discretization of Tim- oshenko prestressed beams. This model features two types of waves: flexural and shear waves, that propa- gate with very different velocities. We present a novel implicit time discretization adapted to the physical phenomena occuring at the continuous level. Af- ter analyzing the continuous system and the two branches of eigenfrequencies associated with the standing modes, the classical θ-scheme is studied. A dispersion analysis recalls that θ = 1/12 re- duces the numerical dispersion, but yields a severely constrained stability condition for our application. Therefore we propose a new θ-like scheme based on two parameters adapted to each wave velocity, which reduces the numerical dispersion while relaxing this stability condition. Numerical experiments success- fully illustrate the theoretical results on the specific cas of a realistic piano string. This motivates the extension of the proposed approach for more chal- lenging physics.
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Juliette Chabassier, Sébastien Imperiale. Dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.. WAVES 13 : 11th International Conference on Mathematical and Numerical Aspects of Waves, Jun 2013, tunis, Tunisia. ⟨hal-00873632⟩

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