Section: New Results

Mathematical modeling of multi-physics involving wave equations

Hybrid space discretization to solve elasto-acoustic coupling

Participants : Hélène Barucq, Julien Diaz, Aurélien Citrain.

Accurate wave propagation simulations require selecting numerical schemes capable of taking features of the medium into account. In case of complex topography, unstructured meshes are the most adapted and in that case, Discontinuous Galerkin Methods (DGM) have demonstrated great performance. Off-shore exploration involves propagation media which can be well represented by hybrid meshes combining unstructured meshes with structured grids that are best for representing homogeneous media like water layers. Then it has been shown that Spectral Element Methods (SEM) deliver very accurate simulations on structured grids with much lower computational costs than DGMs.

We have developed a SEM-DG numerical method for solving time-dependent elasto- acoustic wave problems. We consider the first-order coupled formulation for which we propose a SEM-DG formulation which turns out to be stable. In the 2D case, the coupling is quite straightforward due to the natural way of mixing triangles with quadrangles. 3D coupling is much more difficult and the interface between tetrahedra and hexahedra deserves a particular attention.

These results have been obtained in collaboration with Henri Calandra(TOTAL) and Christian Gout (INSA Rouen) and have been presented at the Fifth International congress on multiphysics, multiscale and optimization problems in Bilbao, the 13th World Congress on Computational Mecanics in New-York and MATHIAS conference in Paris [16], [17], [24].

Signal and noise in helioseismic holography

Participant : Hélène Barucq.

Helioseismic holography is an imaging technique used to study heterogeneities and flows in the solar interior from observations of solar oscillations at the surface. Holograms contain noise due to the stochastic nature of solar oscillations. Aims. We provide a theoretical framework for modeling signal and noise in Porter-Bojarski helioseismic holography. Methods. The wave equation may be recast into a Helmholtz-like equation, so as to connect with the acoustics literature and define the holography Green’s function in a meaningful way. Sources of wave excitation are assumed to be stationary, horizontally homogeneous, and spatially uncorrelated. Using the first Born approximation we calculate holograms in the presence of perturbations in sound-speed, density, flows, and source covariance, as well as the noise level as a function of position. This work is a direct extension of the methods used in time-distance helioseismology to model signal and noise. Results. To illustrate the theory, we compute the hologram intensity numerically for a buried sound-speed perturbation at different depths in the solar interior. The reference Green’s function is obtained for a spherically-symmetric solar model using a finite-element solver in the frequency domain. Below the pupil area on the surface, we find that the spatial resolution of the hologram intensity is very close to half the local wavelength. For a sound-speed perturbation of size comparable to the local spatial resolution, the signalto-noise ratio is approximately constant with depth. Averaging the hologram intensity over a number N of frequencies above 3 mHz increases the signal-to-noise ratio by a factor nearly equal to the square root of N. This may not be the case at lower frequencies, where large variations in the holographic signal are due to the individual contributions of the long-lived modes of oscillation. This work has been done in collaboration with Laurent Gizon, Damien Fournier, Dan Yang and Aaron C. Birch of the Max-Planck-Institut für Sonnensystemforschung at Göttingen (Germany) and published in Astronomy and Astrophysics [14]

Sensitivity kernels for time-distance helioseismology. Efficient computation for spherically symmetric solar models

Participant : Hélène Barucq.

The interpretation of helioseismic measurements, such as wave travel-time, is based on the computation of kernels that give the sensitivity of the measurements to localized changes in the solar interior. These kernels are computed using the ray or the Born approximation. The Born approximation is preferable as it takes finite-wavelength effects into account, although it can be computationally expensive. Aims.We propose a fast algorithm to compute travel-time sensitivity kernels under the assumption that the background solar medium is spherically symmetric. Methods. Kernels are typically expressed as products of Green’s functions that depend upon depth, latitude, and longitude. Here, we compute the spherical harmonic decomposition of the kernels and show that the integrals in latitude and longitude can be performed analytically. In particular, the integrals of the product of three associated Legendre polynomials can be computed. Results. The computations are fast and accurate and only require the knowledge of the Green’s function where the source is at the pole. The computation time is reduced by two orders of magnitude compared to other recent computational frameworks. Conclusions. This new method allows flexible and computationally efficient calculations of a large number of kernels, required in addressing key helioseismic problems. For example, the computation of all the kernels required for meridional flow inversion takes less than two hours on 100 cores. This work has been done in collaboration with Damien Fournier, Chris S. Hanson and Laurent Gizon of the Max-Planck-Institut für Sonnensystemforschung at Göttingen (Germany) and published in Astronomy and Astrophysics [13]

Characterization of partial derivatives with respect to material parameters in a fluid-solid interaction problem.

Participants : Izar Azpiroz Iragorri, Hélène Barucq, Ha Howard Faucher.

For a fluid-solid interaction problem with Lipschitz interface, we investigate the partial Fréchet differentiability of the solutions and the approximate far-field-pattern with respect to solid material parameters. Differentiability is shown in standard Sobolev framework, and the derivatives are characterized as solutions to inhomogeneous fluid-solid transmission problems. To validate the accuracy of the characterization, we compare analytical values with numerical ones given by Interior Penalty Discontinuous Galerkin (IPDG) in a setting with circular obstacles. Our comparisons also show that IPDG gives results with high precision and incurs almost no effect of discretization error accumulation. This work has been done in colllaboration with Rabia Djellouli (california State University at Northridge, USA). It has been published in [3].

Asymptotic modeling of multiple electromagnetic scattering problems by small obstacles

Participants : Justine Labat, Victor Péron, Sébastien Tordeux.

The detection of small conductive heterogeneities in three dimensional domains by non-destructive electromagnetic imaging is a real challenge. Basic finite element-based methods are very expensive in terms of computation time and memory burden, since they involve a huge number of degrees of freedom when the obstacles are very small compared to the testing wavelength. Using the matched asymptotic expansions method, we have developed a meshless reduced model, which consists of replacing the scatterers by equivalent point sources. This method has been numerically implemented in Matlab and its accuracy validated with analytical solutions in spherical geometries. The details of the results are given in [35] and were presented at the fifth International Congress on Multiphysics, Multiscale and Optimization Problems in Bilbao [18] and at ECCOMAS conferences in Glasgow [22]. Following the Born and Foldy-Lax models, we can extend the results for one obstacle to the multiple scattering problem, thus provide meshless methods in this case. Numerical simulations with thousands of small scatterers, up to 10000, were presented at the seminar of RWTH Aachen University [40].

Discontinuous Galerkin Trefftz type method for solving the Maxwell equations

Participant : Sébastien Tordeux.

Trefftz type methods have been developed in Magique 3D to solve Helmholtz equation. Theese methods reduce the numerical dispersion and the condition number of the linear system. This work aims in pursuing this development for electromagnetic scattering. We have adapted and tested the method for an academical 2D configuration. This work has been achieved in the context of the Master trainee of Hakon Fure in collaboration with Sébastien Pernet of ONERA Toulouse.

Comparison between Galbrun and linearized Euler models in the context of helioseismology

Participants : Hélène Barucq, Juliette Chabassier, Marc Duruflé, Nathan Rouxelin.

Helioseismology aims to probe the Sun’s internal structure thanks to surface observations and our knowledge of acoustic wave propagation. In this work we focus on modeling and simulating the propagation of waves below the surface of the Sun.

In the first part, we establish the equations for acoustic wave propagation by linearizing the Euler equations describing the fluid flow. We then compare two linearization processes based on the eulerian and lagrangian description of fluid dynamics.

In the second part, we solve those equations in time-harmonic domain using high order Discontinuous Galerkin methods. Those numerical methods seem to lack consistency and stability when applied to our problems. Specifically, we notice the presence of spurious modes in our numerical solutions.

To fully understand those results further investigations are needed. In particular, two questions seem to stand out : Is the acoustic wave propagation problem in time-harmonic domain well posed for a recirculating background flow ? Is this approach valid ? Can we really assume that the solar plasma solves the Euler equations ?

Asymptotic Models for the Electric Potential across a Highly Conductive Casing

Participant : Victor Péron.

We analyze a configuration that involves a steel-cased borehole, where the casing that covers the borehole is considered as a highly conductive thin layer. We develop an asymptotic method for deriving reduced problems capable of efficiently dealing with the numerical difficulties caused by the casing when applying traditional numerical methods. We derive several reduced models by employing two different approaches, each of them leading to different classes of models. We prove stability and convergence results for these models. The theoretical orders of convergence are supported by numerical results obtained with the finite element method. These results have been obtained with D. Pardo (UPV/EHU, BCAM, Ikerbasque) and Aralar Erdozain. It was published in Computers and Mathematics with Applications [12].

Boundary Element Method for 3D Conductive Thin Layer in Eddy Current Problems

Participant : Victor Péron.

Thin conducting sheets are used in many electric and electronic devices. Solving numerically the eddy current problems in presence of these thin conductive sheets requires a very fine mesh which leads to a large system of equations, and becoming more problematic in case of high frequencies. In this work we show the numerical pertinence of asymptotic models for 3D eddy current problems with a conductive thin layer of small thickness based on the replacement of the thin layer by its mid-surface with impedance transmission conditions that satisfy the shielding purpose, and by using an efficient discretization with the Boundary Element Method in order to reduce the computational cost. These results have been obtained in collaboration with M. Issa, R. Perrussel and J-R. Poirier (LAPLACE, CNRS/INPT/UPS, Univ. de Toulouse) and O. Chadebec (G2Elab, CNRS/INPG/UJF, Institut Polytechnique de Grenoble). This work has been accepted for publication in COMPEL - The international journal for computation and mathematics in electrical and electronic engineering. This work has been presented in the symposium IABEM 2018.

Model-based digital pianos: from physics to sound synthesis

Participant : Juliette Chabassier.

A review article has been published in IEEE Signal Processing Magazine on model-based digital pianos in collaboration with Balasz Bank [4] .

The virtual workshop : towards versatile optimal design of musical wind instruments for the makers

Participants : Juliette Chabassier, Robin Tournemenne.

Our project aims at proposing optimization solutions for wind instrument making. Our approach is based on a strong interaction with makers and players, aiming at defining interesting criteria to optimize from their point of view. After having quantified those criteria under the form of a cost function and a design parameters space, we wish to implement state-of-the-art numerical methods (finite elements, full waveform inversion, neuronal networks, diverse optimization techniques...) that are versatile (in terms of models, formulations, couplings...) in order to solve the optimization problem. More precisely, we wish to take advantage of the fact that sound waves in musical instruments satisfy the laws of acoustics in pipes (PDE), which gives us access to the full waveform inversion technique, usable in harmonic or temporal regime. The methods that we want to use are attractive because the weekly depend on the chosen criterion, and they are easily adaptable to various physical situations (multimodal decomposition in the pipe, coupling with the embouchure, ...), which can therefore be modified a posteriori. The goal is to proceed iteratively between instrument making and optimal design (the virtual workshop) in order to get close to tone quality related and playability criteria. In 2018 we have implemented a python 3 toolbox named OpenWind that includes the first simulation module. Next modules will be implemented next year. This work has been presented to the Congrès Français d'Acoustique [26].

Collaboration with Augustin Humeau, wind instrument maker

Participants : Juliette Chabassier, Robin Tournemenne.

We have initiated a strong collaboration with Augustin Humeau, bassoon maker in Dordogne, France. The goal is to develop practical tools for instrument design, in the realistic context of an artisanal workshop. Until now, an input impedance measurement setup has been developed in collaboration with Samuel Rodriguez, I2M Univ. Bordeaux. It is based on the use of five microphones and the need of one calibration. It has been specifically adapted to the small entrances of some wind instruments (bassoon, oboes). We have attended the JFIS (journées facture instrumentale et science) in November 2018, Le Mans, where the approach has been presented and demonstrated. Given the great interest showed by other instrument makers attendanting the conference, the future of this collaboration is in discussion and may integrate an Inria startup process.

Optimization of brass wind instruments based on sound simulations

Participant : Robin Tournemenne.

We exploited a new optimization method of the inner shape of brass instruments using sound simulations to derive objective functions. The novelties are the obtention of optimal bores for objective functions representative of the intonation but also of the spectrum of the instrument, and the possibility to include constraints in the optimization problem. A complete physics-based model, taking into account the instrument and the musician's embouchure, is used, in order to simulate sounds' permanent regimes using the harmonic balance technique. The instrument is modeled by its input impedance computed with the transfer matrix method under plane wave propagation and visco-thermal losses. Some embouchure's parameters remain variable during the optimization procedure in order to get the average behavior of the instrument. The design variables are the geometrical dimensions of the resonator. Given the computationally expensive function evaluation and the unavailability of gradients, a surrogate-assisted optimization framework is implemented using the mesh adaptive direct search algorithm (MADS). Two optimization examples of a Bb trumpet’s bore (with two and ten design optimization variables) demonstrate the effectiveness of the approach. Results show that solvers can deal flawlessly with high dimensional problems, under constraints, improving significantly the value of the objective functions.

Energy based model and simulation in the time domain of linear acoustic waves in a radiating pipe

Participants : Juliette Chabassier, Robin Tournemenne.

We model in the time domain linear acoustic waves in a radiating pipe without damping. The acoustic equations system in formulated in flow and pressure, which leads to a first order space time equations system. The radiation condition is also written as a first order in time equation, and is parametrized by two real coefficients. Moreover, an auxiliary variable is introduced at the radiating boundary. The choice of this variable is adapted to the considered source type in order to ensure the model stability by energy techniques, under some conditions on the radiating condition. We then propose a stable space time explicit discretization, which ensures the dissipation of a discrete energy. The novelty of the discretization lies, on the one hand, in the variational nature of the space approximation ( which leads to arbitrary order finite elements with no required matrix inversion), and on the other hand, on the definition of the auxiliary variable for any acoustic source type (which leads to the decay of a well defined energy). Finally, we quantify the frequential domain of validity of the used radiation condition by comparison with theoretical and experimental models of the literature. This is a collaboration with Morgane Bergot (Université Claude Bernard, Lyon 1). An article has been written and will be submitted soon. This work has been presented to the Congrès Français d'Acoustique [25].

Computation of the entry impedance of a dissipative radiating pipe

Participants : Juliette Chabassier, Robin Tournemenne.

Modeling the entry impedance of wind instruments pipes is essential for sound synthesis or instrument qualification. We study this modeling with the finite elements method in one dimension (FEM1D) and with the more classically used transfer matrix method (TMM). The TMM gives an analytical formula of the entry impedance depending on the bore (intern geometry of the instrument) defined as a concatenation of simple elements (cylinders, cones, etc). The FEM1D gives the entry impedance for any instrument geometry. The main goals of this work are to assess the viability of the FEM1D and to study and analyse the approximations necessary for the TMM in dissipative pipes. First, lossless Webster's equation in one dimension is studied with arbitrary radiation conditions. In this context and for cylinders or cones, the TMM is exact. We verify that the error made with FEM1D for fine enough elements is as small as desired. When we consider viscothermal losses, the TMM does not solve the classical Kirchhoff model because two terms are supposed constant. In order to overcome this model approximation, simple segments, on which are based the TMM, are decomposed into much smaller segments. We show that using the TMM actually amounts to solving a different equation than the original one, on each small segment. The FEM1D does not necessitate any model approximation, and it is possible to show that it solves the dissipative equation with any arbitrarily small error. With this in hand, we can quantify the TMM model approximation error. The methods are compared in terms of accuracy and computational burden. On realistic cases as the case of a trumpet, the FEM show a better efficiency. Moreover, unusual phenomena as a non constant air temperature can easily be tackle with the FEM. An article has been written and will be submitted soon. This work has been presented to the Congrès Français d'Acoustique [27].

Seismic wave propagation in carbonate rocks at the core scale

Participants : Julien Diaz, Florian Faucher, Chengyi Shen.

Reproduction of large-scale seismic exploration at lab-scale with controllable sources is a promising approach that could not only be applied to study small-scale physical properties of the medium, but also contribute to significant progress in wave-propagation understanding and complex media imaging at exploration scale via upscaling methods. We propose to apply a laser-generated seismic source for lab-scale new geophysical experiments. This consists in generating seismic waves in various media by well-calibrated pulsed-laser impacts and measuring precisely the wavefield (displacement) by Laser Doppler Vibrometer. Parallel 2D/3D simulations featuring the Discontinuous Galerkin discretization method with Interior Penalties (IPDG) are done to match the experimental data. The IPDG method is of particular interest when it comes to solve wave propagation problems in highly heterogeneous media, such as the limestone cores that we are studying.

Current seismic data allowed us to retrieve $V_p$ tomography slices. Further more, qualitative/quantitative comparisons between simulations and experimental data validated the experiment protocol and vice-versa the numerical schemes, opening the possibility of performing FWI on these high resolution data.

This work is in collaboration with Clarisse Bordes, Daniel Brito and Deyuan Zhang (LFCR, UPPA) and with Stéphane Garambois (ISTerre). It was presented at conference AGU [42].