A High-order Discontinuous Galerkin Scheme for Elastic Wave Propagation

In this paper, we introduce a fourth-order leap-frog time scheme combined with a high-order discontinuous Galerkin method for the solution of the elastodynamic equations. The time discretization, obtained via a simple construction based on Taylor developments, provides an accurate scheme for the numerical simulation of seismic wave propagation. Results of the propagation of an eigenmode allow a numerical study of stability and convergence of the scheme for both uniform and non structured meshes proving the high level of accuracy of the method. The robustness of the scheme in the heterogeneous case is studied and we also examine the propagation of an explosive source in a homogeneous half-space.On présente un schéma saute-mouton en temps d'ordre quatre combiné a une méthode de type Galerkin discontinu d'ordre élevé en espace pour la résolution des équations de l'élastodynamique. La discrétisation temporelle, simplement déduite de développements de Taylor, permet d'obtenir un schéma précis pour la simulation numérique de la propagation d'ondes sismiques. Une étude numérique de la stabilité et de la convergence du schéma, via l'étude de la propagation d'un mode propre utilisant des maillages uniformes et non structurés, prouve la précision de la méthode. La robustesse du schéma est étudiée dans le cas d'un milieu héterogène et l'on s'intéresse également à la propagation d'une source explosive dans un demi-espace homogène

Une nouvelle méthode de relaxation pour les équations de Navier-Stokes compressibles. II : validation numérique

On propose un nouveau schéma de relaxation pour résoudre les équations de Navier-Stokes compressibles munies de lois de pression et de température générales. L'accent est mis sur l'étude de la précision et de l'efficacité numérique de cette nouvelle méthode de relaxation. Deux cas tests sont étudiés : l'advection d'un réseau périodique de vortex puis l'interaction d'un choc faible et d'un spot de température

Semi-Implicit Roe-Type Fluxes for Low-Mach Number Flows

Two semi-implicit methods based on the splitting of the Euler equations flux into fluid and acoustic parts applied to low Mach number flows are presented. The first method is based on the splitting of slow and fast eigenvalues of the jacobian matrix of the fluxes and a semi-implicit scheme is constructed by introducing only the fast eigenvalues in the implicit matrices. The second method is based on the splitting of the Euler flux by separating the terms in velocity and the terms in pressure ; this system is solved by a fractional step method. A semi-implicit scheme is obtained by using a linearised implicit scheme for the acoustic step only. These two methods are applied to the convection of a density pulse for Mach numbers equal to 0.1 and 0.01. Accuracy and efficiency of the different schemes are compared

Seismic hazard assessment in Menton, France: Topographical site effect zoning considering a semi-empirical approach and a Machine Learning scheme

The presence of topography influences the seismic ground motion and may result in strong amplifications, generally at the top of hills and reliefs. The increasing urbanization of hills requires an accurate estimation of these effects even in areas of moderate seismicity. The simplified coefficients provided by the Eurocodes8 do not depend on the frequency and underestimate the amplification in many situations, which justifies the development of new methods based on easily accessible data. The city of Menton, located in the southeast of France, between the Alps and the Ligurian basin, is one of the most exposed metropolitan cities. We propose a study of topographic effects applied to the Menton area. Topographic amplification is calculated, on a wide frequency band, using the Frequency-Scaled Curvature method, from a DEM and an average value of the shear wave velocity. We then propose to apply an automatic clustering approach to classify the amplification curves into five groups with similar properties. We then deduce a first microzonation map of the topographic effects in the Menton area

A nodal high-order discontinuous Galerkin method for elastic wave propagation in arbitrary heterogeneous media

International audienceWe present an extension of the nodal discontinuous Galerkin method for elastic wave propagation to high interpolation orders and arbitrary heterogeneous media. The high-order lagrangian interpolation is based on a set of nodes with excellent interpolation properties in the standard triangular element. In order to take into account highly variable geological media, another set of suitable quadrature points is used where the physical and mechanical properties of the medium are defined. We implement the methodology in a 2-D discontinuous Galerkin solver. First, a convergence study confirms the hp-convergence of the method in a smoothly varying elastic medium. Then, we show the advantages of the present methodology, compared to the classical one with constant properties within the elements, in terms of the complexity of the mesh generation process by analysing the seismic amplification of a soft layer over an elastic half-space. Finally, to verify the proposed methodology in a more complex and realistic configuration , we compare the simulation results with the ones obtained by the spectral element method for a sedimentary basin with a realistic gradient velocity profile. Satisfactory results are obtained even for the case where the computational mesh does not honour the strong impedance contrast between the basin bottom and the bedrock

A High-order Discontinuous Galerkin Scheme for Elastic Wave Propagation

In this paper, we introduce a fourth-order leap-frog time scheme combined with a high-order discontinuous Galerkin method for the solution of the elastodynamic equations. The time discretization, obtained via a simple construction based on Taylor developments, provides an accurate scheme for the numerical simulation of seismic wave propagation. Results of the propagation of an eigenmode allow a numerical study of stability and convergence of the scheme for both uniform and non structured meshes proving the high level of accuracy of the method. The robustness of the scheme in the heterogeneous case is studied and we also examine the propagation of an explosive source in a homogeneous half-space.On présente un schéma saute-mouton en temps d'ordre quatre combiné a une méthode de type Galerkin discontinu d'ordre élevé en espace pour la résolution des équations de l'élastodynamique. La discrétisation temporelle, simplement déduite de développements de Taylor, permet d'obtenir un schéma précis pour la simulation numérique de la propagation d'ondes sismiques. Une étude numérique de la stabilité et de la convergence du schéma, via l'étude de la propagation d'un mode propre utilisant des maillages uniformes et non structurés, prouve la précision de la méthode. La robustesse du schéma est étudiée dans le cas d'un milieu héterogène et l'on s'intéresse également à la propagation d'une source explosive dans un demi-espace homogène

Une méthode Galerkin discontinue d'ordre élevé pour la propagation d'ondes sismiques en milieu viscoélastique

We present a high-order discontinuous Galerkin method for the simulation of P-SV seismic wave propagation in heterogeneous media and two dimensions of space. The first-order velocity-stress system is obtained by assuming that the medium is linear, isotropic and viscoelastic, thus considering intrinsic attenuation. The associated stress-strain relation in the time domain being a convolution, which is numerically intractable, we consider the rheology of a generalized Maxwell body replacing the convolution by differential equations. This results in a velocity-stress system which contains additional equations for the anelastic functions including the strain history of the material. Our numerical method, suitable for complex triangular unstructured meshes, is based on a centered numerical flux and a leap-frog time-discretization. The extension to high order in space is realized by Lagrange polynomial functions, defined locally in each element. The inversion of a global mass matrix is avoided since an explicit scheme in time is used and because of the local nature of the discontinuous Galerkin formulation. The method is validated through numerical simulations including comparisons with a finite difference scheme.Nous présentons une méthode Galerkin discontinue d'ordre élévé pour la simulation de la propagation d'ondes sismiques P-SV en milieu hétérogène et en deux dimensions d'espace. Le système vitesse-contraintes du premier ordre est obtenu en supposant un milieu linéaire, isotrope et viscoélastique, prenant ainsi en compte l'atténuation intrinsèque du milieu. La relation contraintes-déformations dans le domaine temporel étant une convolution, qui nécessiterait une approximation numérique très coûteuse, nous considérons la rhéologie d'un "generalized Maxwell body" (GMB) remplaçant la convolution par un jeu d'équations différentielles. Il en résulte un système vitesse-contraintes contenant des équations supplémentaires pour les fonctions anélastiques qui traduisent l'historique de déformation du matériau. Notre méthode numérique, applicable à des maillages triangulaires non structurés, est basée sur des flux centrés et un schéma saute-mouton en temps. L'extension en espace à l'ordre élevé est obtenue grâce à des polynômes de Lagrange, définis localement dans chaque élément. La méthode étant explicite, elle ne nécessite pas d'inversion de matrice de masse globale. La méthode est validée via des simulations numériques, notamment des comparaisons avec un schéma aux différences finies

A Mouse Stromal Response to Tumor Invasion Predicts Prostate and Breast Cancer Patient Survival

Primary and metastatic tumor growth induces host tissue responses that are believed to support tumor progression. Understanding the molecular changes within the tumor microenvironment during tumor progression may therefore be relevant not only for discovering potential therapeutic targets, but also for identifying putative molecular signatures that may improve tumor classification and predict clinical outcome. To selectively address stromal gene expression changes during cancer progression, we performed cDNA microarray analysis of laser-microdissected stromal cells derived from prostate intraepithelial neoplasia (PIN) and invasive cancer in a multistage model of prostate carcinogenesis. Human orthologs of genes identified in the stromal reaction to tumor progression in this mouse model were observed to be expressed in several human cancers, and to cluster prostate and breast cancer patients into groups with statistically different clinical outcomes. Univariate Cox analysis showed that overexpression of these genes is associated with shorter survival and recurrence-free periods. Taken together, our observations provide evidence that the expression signature of the stromal response to tumor invasion in a mouse tumor model can be used to probe human cancer, and to provide a powerful prognostic indicator for some of the most frequent human malignancies