[hal-00873632, v1] Dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string

We study the implicit time discretization of Timoshenko prestressed beams. This model features two types of waves: flexural and shear waves, that propagate with very different velocities. We present a novel implicit time discretization adapted to the physical phenomena occuring at the continuous level. After 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 reduces 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 successfully illustrate the theoretical results on the specific cas of a realistic piano string. This motivates the extension of the proposed approach for more challenging physics. Introduction Piano strings can be modeled as simply supported Timoshenko prestressed beams. This model introduced in [2] accounts for inharmonicity of the transversal displacement, via a coupling with a shear angle resulting in the propagation of flexural and shear waves with very different speeds. Our concern in this work is to develop a new implicit time discretization, which will be associated with finite element methods in space, in order to reduce the numerical dispersion of flexural waves while allowing the use of a large time step in spite of the high shear velocity (compared to the maximal time step allowed with the explicit leap-frog scheme). 1 Continuous system The prestressed Timoshenko model considers two unknowns (u, φ) which stand respectively for the transversal displacement and the shear angle of the cross section of the the string. We assume that the physical parameters (see [1] for definition) are positive and that ES > T0 (which is true in practice for piano strings). We consider “simply supported” boundary conditions (zero displacement and zero torque). It reads: Find (u, φ) such that ∀x ∈]0, L[, ∀ t > 0,  ρS ∂2u ∂t2 − T0 ∂2u ∂x2 + SGκ ∂ ∂x ( φ− ∂u ∂x ) = σ, ρI ∂2φ ∂t2 − EI ∂ 2φ ∂x2 + SGκ ( φ− ∂u ∂x ) = 0, (1) with boundary conditions ∂xu(x = 0, t) = 0, ∂xu(x = L, t) = 0, ∂xφ(x = 0, t) = 0, ∂xφ(x = L, t) = 0, (2) where σ stands for a source term. Standard energy techniques for systems of wave equations can be used to show a priori estimates on this system thanks to the following energy identity: