Context. Local helioseismology has so far relied on semi-analytical methods to compute the spatial sensitivity of wave travel times to perturbations in the solar interior. These methods are cumbersome and lack flexibility. Aims. Here we propose a convenient framework for numerically solving the forward problem of time-distance helioseismology in the frequency domain. The fundamental quantity to be computed is the cross-covariance of the seismic wavefield. Methods. We choose sources of wave excitation that enable us to relate the cross-covariance of the oscillations to the Green's function in a straightforward manner. We illustrate the method by considering the 3D acoustic wave equation in an axisymmetric reference solar model, ignoring the effects of gravity on the waves. The symmetry of the background model around the rotation axis implies that the Green's function can be written as a sum of longitudinal Fourier modes, leading to a set of independent 2D problems. We use a high-order finite-element method to solve the 2D wave equation in frequency space. The computation is 'embarrassingly parallel', with each frequency and each azimuthal order solved independently on a computer cluster. Results. We compute travel-time sensitivity kernels in spherical geometry for flows, sound speed, and density perturbations under the first Born approximation. Convergence tests show that travel times can be computed with a numerical precision better than one millisecond, as required by the most precise travel-time measurements. Conclusions. The method presented here is computationally efficient and will be used to interpret travel-time measurements in order to infer, e.g., the large-scale meridional flow in the solar convection zone. It allows the implementation of (full-waveform) iterative inversions, whereby the axisymmetric background model is updated at each iteration.
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 . . . ). 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.
A time-domain global modeling of a grand piano is presented. The string model includes internal losses, stiffness, and geometrical nonlinearity. The hammer-string interaction is governed by a nonlinear dissipative compression force. The soundboard is modeled as a dissipative bidimensional orthotropic Reissner-Mindlin plate where the presence of ribs and bridges is treated as local heterogeneities. The coupling between strings and soundboard at the bridge allows the transmission of both transverse and longitudinal waves to the soundboard. The soundboard is coupled to the acoustic field, whereas all other parts of the structure are supposed to be perfectly rigid. The acoustic field is bounded artificially using perfectly matched layers. The discrete form of the equations is based on original energy preserving schemes. Artificial decoupling is achieved, through the use of Schur complements and Lagrange multipliers, so that each variable of the problem can be updated separately at each time step. The capability of the model is highlighted by series of simulations in the low, medium, and high register, and through comparisons with waveforms recorded on a Steinway D piano. Its ability to account for phantom partials and precursors, consecutive to string nonlinearity and inharmonicity, is particularly emphasized.
Abstract. This work offers some contributions to the numerical study of acoustic waves propagating in the Sun and its atmosphere. The main goal is to provide boundary conditions for outgoing waves in the solar atmosphere where it is assumed that the sound speed is constant and the density decays exponentially with radius. Outgoing waves are governed by a Dirichlet-to-Neumann map which is obtained from the factorization of the Helmholtz equation expressed in spherical coordinates. For the purpose of extending the outgoing wave equation to axisymmetric or 3D cases, different approximations are implemented by using the frequency and/or the angle of incidence as parameters of interest. This results in boundary conditions called Atmospheric Radiation Boundary Conditions (ARBC) which are tested in ideal and realistic configurations. These ARBCs deliver accurate results and reduce the computational burden by a factor of two in helioseismology applications.Résumé. Ce travail apporte quelques contributions à l'étude numérique des ondes acoustiques se propageant dans le Soleil et son atmosphère. Il se base sur la caractérisa-tion des ondes sortantes dans l'atmosphère représentée par une vitesse constante et une densité décroissant exponentiellement. Les ondes sortantes sont régies par un opérateur Dirichlet-to-Neumann qui est obtenu par la factorisation de l'équation de Helmholtz formulée dans les coordonnées sphériques. Afin d'étendre l'équation des ondes sortantes à des géométries axisymétriques ou 3D, différentes approximations sont menées en utilisant la fréquence et/ou l'angle d'incidence comme paramètres d'intérêt. Ceci mène à des conditions de frontière que nous appelons Conditions de Radiation Atmosphériques (ARBC) et qui sont testées en configuration idéalisées et réalistes. Ces conditions ARBC offrent des résultats précis et réduisent le coût de calcul d'un facteur deux pour le cas du Soleil.1991 Mathematics Subject Classification. 00A71, 35L05, 85A20, 33C55, 65M60.December 1, 2017.
The temporal covariance between seismic waves measured at two locations on the solar surface is the fundamental observable in time-distance helioseismology. Above the acoustic cut-off frequency (∼5.3 mHz), waves are not trapped in the solar interior and the covariance function can be used to probe the upper atmosphere. We wish to implement appropriate radiative boundary conditions for computing the propagation of high-frequency waves in the solar atmosphere. We consider the radiative boundary conditions recently developed by Barucq et al. (2017) for atmospheres in which sound-speed is constant and density decreases exponentially with radius. We compute the cross-covariance function using a finite element method in spherical geometry and in the frequency domain. The ratio between first-and second-skip amplitudes in the time-distance diagram is used as a diagnostic to compare boundary conditions and to compare with observations. We find that a boundary condition applied 500 km above the photosphere and derived under the approximation of small angles of incidence accurately reproduces the 'infinite atmosphere' solution for high-frequency waves. When the radiative boundary condition is applied 2 Mm above the photosphere, we find that the choice of atmospheric model affects the time-distance diagram. In particular, the time-distance diagram exhibits double-ridge structure when using a VAL atmospheric model.
International audienceA new class of high order, implicit, three time step schemes for semi-discretized wave equations is introduced and studied. These schemes are constructed using the modified equation approach, generalizing the $\theta$-scheme. Their stability properties are investigated via an energy analysis, which enables us to design super convergent schemes and also optimal stable schemes in terms of consistency errors. Specific numerical algorithms for the fully discrete problem are tested and discussed, showing the efficiency of our approach compared to second order $\theta$-schemes.Nous introduisons et étudions une nouvelle classe de schémas d'ordre élevé, implicites et à trois pas de temps pour les équations d'ondes semi-discrètes. Ces schémas sont construits sur le principe de l'équation modifiée et généralisent le theta-schéma. Nous étudions leurs propriétés de stabilité via des techniques d'énergie, ce qui nous permet de concevoir des schémas super convergents ainsi que des schémas optimaux en terme d'erreur de consistance. Des algorithmes numériques de résolution pour le problème totalement discrétisé sont testés et critiqués, montrant la supériorité de notre approche comparée aux theta shémas classiques du second ordre
Abstract. The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependent damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties.Mathematics Subject Classification. 00A71, 00A65, 65P05, 65N25, 35Q72, 35L05.
International audienceTime domain simulation of realistic highly heterogeneous media or strongly refined geometries can be a computational challenge when using explicit schemes because they impose a time step restriction that can be extremely penalizing. In this work, we present fourth order locally implicit schemes. The domain of interest is decomposed into several regions where different (explicit or implicit) fourth order time discretization are used. Whilst implicit schemes tolerate the use of larger time steps, they can induce greater numerical dispersion. Fourth order accuracy reduces this lack of precision, and makes this family of schemes attractive compared to other approaches as local time stepping
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