Semi-analytical and numerical methods for computing transient waves in 2D acoustic / poroelastic stratified media

Wave propagation in a stratified fluid / porous medium is studied here using analytical and numerical methods. The semi-analytical method is based on an exact stiffness matrix method coupled with a matrix conditioning procedure, preventing the occurrence of poorly conditioned numerical systems. Special attention is paid to calculating the Fourier integrals. The numerical method is based on a high order finite-difference time-domain scheme. Mesh refinement is applied near the interfaces to discretize the slow compressional diffusive wave predicted by Biot's theory. Lastly, an immersed interface method is used to discretize the boundary conditions. The numerical benchmarks are based on realistic soil parameters and on various degrees of hydraulic contact at the fluid / porous boundary. The time evolution of the acoustic pressure and the porous velocity is plotted in the case of one and four interfaces. The excellent level of agreement found to exist between the two approaches confirms the validity of both methods, which cross-checks them and provides useful tools for future researches.Comment: Wave Motion (2012) XX

Dynamic response of a circular tunnel with imperfect surface interaction embedded in an elastic medium

The research work proposed here is part of a global project that aims at better characterizing a specific underground environment in the LSBB (Low Noise Inter-disciplinary Underground Science and Technology), situated in Rustrel, Vaucluse, France. The experimental environment under study is characterized by a system of galleries, several ones with a concrete layer. The first step of the methodology deals with setting up a forward problem to apprehend the geometry of the LSBB. \\In this paper, the 2D transient response of imperfect bonded circular lined pipeline lying in an elastic, homogeneous and infinite medium is studied. At first, the problem is solved in the frequency domain by using the wave function expansion method and imperfect interaction surface between elastic medium and tunnel is modeled as a linear spring. Wave propagation fields in tunnel and soil are expressed in terms of infinite series and stresses and displacements are given based on those series. By implying boundary conditions a linear equations system is obtained and the results of these equations lead to displacement and stress responses of the rock and tunnel.To solve the transient problem, the Laplace transform with respect to time is used which leads to system of linear equations in the Laplace domain. Durbin's numerical Laplace transform inversion method is employed to obtain dynamic responses. To exhibit a behavior of the responses, influences of the different parameters such as wall thickness of the tunnel is investigated. Hoop stresses and the displacements of the tunnel and rock are obtained due to acting load on the inner surface of the tunnel for selected parameters. In order to check the validity of the present work, we pay attention on the convergence of the results and also excellent agreement with previous result is achieved

Contribution to the modeling and the mechanical characterization of the subsoil in the LSBB environment

The present research work aims at better characterizing the specific underground environment of the LSBB (Low Noise Inter-Disciplinary Underground Science and Technology, Rustrel, France) using mechanical wave propagation information. The LSBB experimental environment is characterized by a system of cylindrical galleries, some of them presenting a concrete layer. In the global project, three steps are considered : firstly the construction of an efficient forward mechanical wave propagation model to calculate the displacement vector and stress tensor components; secondly a sensitivity analysis to extract the pertinent parameters in the configurations and models under study (viscoelastic or poroviscoelastic media with potential anisotropy); and lastly an inversion strategy to recover some of the pertinent parameters. In this proposal, the first step, under progress, is described. The work carried out is in the continuity of the work presented by Yi et al. (2016) [1] who studied the harmonic response of a cylindrical elastic tunnel, impacted by a plane compressional wave, embedded in an infinite elastic ground. The interface between the rock mass and the linen is an imperfect contact modeled with two spring parameters, Achenbach and Zhu (1989) [2]. We choose a semi-analytical approach to calculate the two-dimensional displacement and stress fields in order to get a fast tool, from the numerical point of view. The main steps of the theoretical approach are : use of the Helmholtz decomposition, solving the wave equation based on the separation method and the expansion in Bessel function series in the harmonic domain. The harmonic results are validated by comparison with Yi et al. (2016) [1] and new ones are presented. Moreover, the transient regime case obtained with the use of a Fourier transform on the time variable, is under progress

Propagation d'ondes dans un milieu poroélastique multicouches

Cette communication présente une approche semi-analytique permettant l'étude de la propagation d'ondes dans les sols  multicouches poroélastiques en régime transitoire. L'approche théorique est basée sur la méthode de matrice de raideur exacte adaptée à la théorie de Biot, développée dans le domaine des nombres d'ondes et couplée à une technique de conditionnement matriciel. Effectivement, les méthodes classiques peuvent fournir des systèmes matriciels mal conditionnés. La technique de conditionnement appliquée ici au cas poroélastique permet de traiter n'importe quelle configuration de problèmes, y compris ceux impliquant de hautes fréquences, de grandes valeurs de nombres d'ondes ou encore de fortes épaisseurs de couches. Différents types de milieux et d'excitations peuvent alors être étudiés. L'efficacité de la technique ainsi que l'influence de l'hétérogénéité multicouches  sont  présentées dans cette communication. Enfin, les techniques semi-analytiques développées peuvent servir de benchmark aux outils de simulation en poroélasticité

Stress and displacement fields around an arbitrary shape tunnel surrounded by a multilayered elastic medium subjected to harmonic waves under plane strain conditions

International audienceThis study focuses on the propagation of harmonic mechanical waves around both circular and non-circular tunnels. The excavation disturbed zone is explicitly modeled with a multilayered description of the surround-ings of the tunnel. The purpose of this paper is thus to establish closed-form solutions for stress and displacement fields for circular and non-circular geometries taking into account the mechanical heterogeneity of the area adjacent to the tunnel. Particularly, the multilayered non-circular shape configuration is a great step not treated previously. The classical approach based on the wave function expansion method is used to determine the ex-pressions of stresses and displacements around circular geometries. To handle non-circular tunnel shapes, the complex variable method based on the Muskhelishvili approach and conformal mapping functions is introduced. Results indicate that the excavation damaged zone strongly modifies the distribution of stress and displacement fields around the tunnel. Geometrical tunnel shapes also affect stress and displacement polar diagrams

Application du théorème de réciprocité à la détermination de la dérivée de Fréchet de l'opérateur non linéaire de propagation d'ondes mécaniques en milieux viscoélastiques

Nous souhaitons utiliser les ondes mécaniques pour reconstruire les paramètres d'un objet. Ce sujet de recherche trouve des applications dans des domaines tels que le sondage sismique ou le génie civil. Quel que soit le domaine abordé, le traitement des données expérimentales nécessite de résoudre un problème direct et un problème inverse. Résoudre le problème direct consiste à supposer connus les paramètres mécaniques pour ensuite proposer un modèle qui permet de prédire le champ des déplacements et/ou des contraintes, résultant par exemple de l'application d'une force extérieure par un émetteur sur la surface de l'objet. A contrario, nous résolvons le problème inverse lorsque nous estimons au mieux les paramètres mécaniques de l'objet, à partir des champs mécaniques, que nous mesurons par exemple sur un ensemble de détecteurs positionnés sur sa surface. Nous considérons ici un objet (visco)élastique, isotrope et inhomogène. Les paramètres mécaniques tels que la masse volumique et les coefficients de Lamé sont représentés par des fonctions de l'espace. Ce problème inverse est délicat à résoudre car il est mal posé au sens de Hadamard, et que les mesures sont reliées aux paramètres de façon non linéaire. Les techniques de type Newton-Kantorovich font partie des méthodes permettant de le résoudre : leur principe de base consiste à linéariser localement le problème direct en calculant la dérivée de l'opérateur non linéaire décrivant le modèle de prédiction des mesures. Ce calcul conduit à la formulation explicite de l'opérateur de Fréchet, qui est ensuite utilisé pour résoudre itérativement le problème inverse. En effet, à chaque étape de ce processus itératif, nous inversons un problème direct linéarisé, dans lequel, une faible variation des mesures se retrouve reliée à une faible variation des paramètres mécaniques par l'opérateur de Fréchet. Comme ce dernier dépend de l'estimation des paramètres de l'objet faite à l'étape précédente, il doit être réestimé à chaque nouvelle itération. L'approche que nous proposons s'appuie sur le principe de réciprocité, les équations de l'élastodynamique sont réécrites au sens des distributions pour déduire des formes élégantes de l'opérateur de Fréchet. Ces dernières font intervenir le modèle direct qui prédit les données pour la configuration étudiée, et les modèles des configurations réciproques obtenus en permutant fictivement tour à tour l'émetteur avec les différents récepteurs. Ce principe a déjà été appliqué au calcul de l'opérateur de Fréchet dans le contexte des équations de Maxwell [1], puis repris dans différents schémas d'inversion pour des applications en imageries optique et micro-ondes [2]. Pour le problème traité ici, une forme particulière de l'opérateur de Fréchet a été obtenue en régime harmonique pour chaque paramètre mécanique. Ces formes ont ensuite été vérifiées numériquement par un modèle direct de prédiction basé sur une méthode spectrale [3], dans le cas particulier où l'objet est un milieu semi-infini stratifié de type sol. Cette première phase de travail constitue une étape clef dans l'élaboration d'une méthode inverse itérative appliquée à la caractérisation mécanique d'un milieu inhomogène

Contribution to the modeling and the mechanical characterization of the subsoil in the LSBB environment

The present research work aims at better characterizing the specific underground environment of the LSBB (Low Noise Inter-Disciplinary Underground Science and Technology, Rustrel, France) using mechanical wave propagation information. The LSBB experimental environment is characterized by a system of cylindrical galleries, some of them presenting a concrete layer. In the global project, three steps are considered : firstly the construction of an efficient forward mechanical wave propagation model to calculate the displacement vector and stress tensor components; secondly a sensitivity analysis to extract the pertinent parameters in the configurations and models under study (viscoelastic or poroviscoelastic media with potential anisotropy); and lastly an inversion strategy to recover some of the pertinent parameters. In this proposal, the first step, under progress, is described. The work carried out is in the continuity of the work presented by Yi et al. (2016) [1] who studied the harmonic response of a cylindrical elastic tunnel, impacted by a plane compressional wave, embedded in an infinite elastic ground. The interface between the rock mass and the linen is an imperfect contact modeled with two spring parameters, Achenbach and Zhu (1989) [2]. We choose a semi-analytical approach to calculate the two-dimensional displacement and stress fields in order to get a fast tool, from the numerical point of view. The main steps of the theoretical approach are : use of the Helmholtz decomposition, solving the wave equation based on the separation method and the expansion in Bessel function series in the harmonic domain. The harmonic results are validated by comparison with Yi et al. (2016) [1] and new ones are presented. Moreover, the transient regime case obtained with the use of a Fourier transform on the time variable, is under progress

Use of global sensitivity analysis to assess the soil poroelastic parameter influence

In this study, we show how a global sensitivity analysis method can be used to obtain relevant information for the interpretation of the mechanical wave propagation phenomena involved in a poroelastic soil that takes into account the presence of water. The present investigation addresses the issue of the identification and the ranking of the most influential parameters. The sensitivity indices provide a variance-based measure of the uncertainty effects of the input parameters on the mechanical outputs of the model. It allows quantification of, on the one hand, the influence of each parameter and on the other hand, the possible interactions between all the parameters. Numerical simulations are performed in a laboratory-scale configuration: a fluid medium overlying a poroelastic material is submitted to a transient excitation, and the coupling of the acoustic and Biot models is solved using a semi-analytical approach. The analysis of the temporal and spatial evolution of partial variances highlights the most important parameters and the complementary information contained in the signals in function of both the time and the receiver location. In particular, we show that the description of the poroelastic waves is governed by only a restricted number of parameters for the configuration under study

Near subsurface density reconstruction by full waveform inversion in the frequency domain

The work proposed is part of a global project dealing with the characterization of heterogeneous media by both electromagnetic and mechanical full waveform inversions. Indeed Full Waveform Inversion of seismic reflection or Ground Penetrating Radar data is an efficient approach to reconstruct subsurface physical parameters with high resolution. This paper focuses on the mechanical part, and more specifically on quantitative imaging of nearsurface density. Processing field data is challenging because the nature of the source and the sensors used impact the signal-to-noise ratio as well as the frequency range appearing in the recorded data. From then it becomes interesting to process the data in the frequency domain and work on a few representative frequencies of the recorded temporal signal. In this article, field data are simulated by noisy synthetic data. Different frequency strategies are used and their results are compared with each other. The inverse problem consists in assessing the density in the probed medium from the data on the displacement field measured at the detectors. Such a problem is known to be nonlinear and ill-posed. It is solved iteratively by a regularized Gauss-Newton algorithm, which relies on the Fréchet derivatives obtained through the generalized reciprocity principle equivalent to the well-known adjoint method. The numerical results show an optimal strategy, for which the convergence rate and the computation time are reasonable, the spatial resolution is improved and the density is well reconstructed