Application of the multi-level time-harmonic fast multipole BEM to 3-D visco-elastodynamics

Engineering Analysis with Boundary elements (accepted, to appear)International audienceThis article extends previous work by the authors on the single- and multi-domain time-harmonic elastodynamic multi-level fast multipole BEM formulations to the case of weakly dissipative viscoelastic media. The underlying boundary integral equation and fast multipole formulations are formally identical to that of elastodynamics, except that the wavenumbers are complex-valued due to attenuation. Attention is focused on evaluating the multipole decomposition of the viscoelastodynamic fundamental solution. A damping-dependent modification of the selection rule for the multipole truncation parameter, required by the presence of complex wavenumbers, is proposed. It is empirically adjusted so as to maintain a constant accuracy over the damping range of interest in the approximation of the fundamental solution, and validated on numerical tests focusing on the evaluation of the latter. The proposed modification is then assessed on 3D single-region and multi-region visco-elastodynamic examples for which exact solutions are known. Finally, the multi-region formulation is applied to the problem of a wave propagating in a semi-infinite medium with a lossy semi-spherical inclusion (seismic wave in alluvial basin). These examples involve problem sizes of up to about $3\,10^{5}$ boundary unknowns

Nonreflecting boundary conditions for time-dependent wave propagation

Many problems in computational science arise in unbounded domains and thus require an artificial boundary B, which truncates the unbounded exterior domain and restricts the region of interest to a finite computational domain, . It then becomes necessary to impose a boundary condition at B, which ensures that the solution in coincides with the restriction to of the solution in the unbounded region. If we exhibit a boundary condition, such that the fictitious boundary appears perfectly transparent, we shall call it exact. Otherwise it will correspond to an approximate boundary condition and generate some spurious reflection, which travels back and spoils the solution everywhere in the computational domain. In addition to the transparency property, we require the computational effort involved with such a boundary condition to be comparable to that of the numerical method used in the interior. Otherwise the boundary condition will quickly be dismissed as prohibitively expensive and impractical. The constant demand for increasingly accurate, efficient, and robust numerical methods, which can handle a wide variety of physical phenomena, spurs the search for improvements in artificial boundary conditions. In the last decade, the perfectly matched layer (PML) approach [16] has proved a flexible and accurate method for the simulation of waves in unbounded media. Standard PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as first-order systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step we propose instead a simple PML formulation directly for the wave equation in its second-order form. Our formulation requires fewer auxiliary unknowns than previous formulations [23, 94]. Starting from a high-order local nonreflecting boundary condition (NRBC) for single scattering [55], we derive a local NRBC for time-dependent multiple scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NRBC for time-dependent multiple scattering. Remarkably, the information transfer (of time retarded values) between sub-domains will only occur across those parts of the artificial boundary, where outgoing rays intersect neighboring sub-domains, i.e. typically only across a fraction of the artificial boundary. The accuracy, stability and efficiency of this new local NRBC is evaluated by coupling it to standard finite element or finite difference methods

Elastic wave scattering from randomly rough surfaces

Elastic wave scattering from randomly rough surfaces and a smooth surface are essentially different. For ultrasonic nondestructive evaluation (NDE) the scattering from defects with smooth surfaces has been extensively studied, providing fundamental building blocks for the current inspection techniques. However, all realistic surfaces are rough and the roughness exists in two dimensions. It is thus very important to understand the rough surface scattering mechanism, which would give insight for practical inspections. Knowledge of the stochastics of scattering for different rough surfaces would also allow the detectability of candidate rough defects to be anticipated. Hence the main motivation of this thesis is to model and study the effect of surface roughness on the scattering field, with focus on elastic waves. The main content of this thesis can be divided into three contributions. First of all, an accurate numerical method with high efficiency is developed in the time domain, for computing the scattered waves from obstacles with arbitrary shapes. It offers an exact solution which covers scenarios where approximation-based algorithms fail. The method is based on the hybrid idea to combine the finite element (FE) and boundary integral (BI) methods. The new method efficiently couples the FE equations and the boundary integral formulae for solving the transient scattering problems in both near and far fields, which is implemented completely in the time domain. Several numerical examples are demonstrated and sufficiently high accuracy is achieved with different defects. It enables the possibility for Monte Carlo simulations of the elastic wave scattering from randomly rough surfaces in both 2D and 3D. The second contribution relates to applying the developed numerical method to evaluate the widely used Kirchhoff approximation (KA) for rough surface scattering. KA is a high-frequency approximation which limits the use of the theory for certain ranges of roughness and incidence/scattering angles. The region of validity for elastic KA is carefully examined for both 1D and 2D random surfaces with Gaussian spectra. Monte Carlo simulations are run and the expected scattering intensity is compared with that calculated by the accurate numerical method. An empirical rule regarding surface parameters and angles is summarized to establish the valid region of both 2D and 3D KA. In addition, it is found that for 3D scattering problems, the rule of validity becomes stricter than that in 2D. After knowing the region of validity, KA is applied to investigate how the surface roughness affects the statistical properties of scattered waves. An elastodynamic Kirchhoff theory particularly for the statistics of the diffused field is developed with slope approximations for the first time. It provides an analytical expression to rapidly predict the expected angular distribution of the scattering intensity, or the scattering pattern, for different combinations of the incidence/scattering wave modes. The developed theory is verified by comparison with numerical Monte Carlo simulations, and further validated by the experiment with phased arrays. In particular the derived formulae are utilized to study the effects of the surface roughness on the mode conversion and the 2D roughness caused depolarization, which lead to unique scattering patterns for different wave modes.Open Acces

Efficient finite element methods for solving high-frequency time-harmonic acoustic wave problems in heterogeneous media

This thesis focuses on the efficient numerical solution of frequency-domain wave propagation problems using finite element methods. In the first part of the manuscript, the development of domain decomposition methods is addressed, with the aim of overcoming the limitations of state-of-the art direct and iterative solvers. To this end, a non-overlapping substructured domain decomposition method with high-order absorbing conditions used as transmission conditions (HABC DDM) is first extended to deal with cross-points, where more than two subdomains meet. The handling of cross-points is a well-known issue for non-overlapping HABC DDMs. Our methodology proposes an efficient solution for lattice-type domain partitions, where the domains meet at right angles. The method is based on the introduction of suitable relations and additional transmission variables at the cross-points, and its effectiveness is demonstrated on several test cases. A similar non-overlapping substructured DDM is then proposed with Perfectly Matched Layers instead of HABCs used as transmission conditions (PML DDM). The proposed approach naturally considers cross-points for two-dimensional checkerboard domain partitions through Lagrange multipliers used for the weak coupling between subproblems defined on rectangular subdomains and the surrounding PMLs. Two discretizations for the Lagrange multipliers and several stabilization strategies are proposed and compared. The performance of the HABC and PML DDM is then compared on test cases of increasing complexity, from two-dimensional wave scattering in homogeneous media to three-dimensional wave propagation in highly heterogeneous media. While the theoretical developments are carried out for the scalar Helmholtz equation for acoustic wave propagation, the extension to elastic wave problems is also considered, highlighting the potential for further generalizations to other physical contexts. The second part of the manuscript is devoted to the presentation of the computational tools developed during the thesis and which were used to produce all the numerical results: GmshFEM, a new C++ finite element library based on the application programming interface of the open-source finite element mesh generator Gmsh; and GmshDDM, a distributed domain decomposition library based on GmshFEM.Cette thèse porte sur la résolution numérique efficace de problèmes de propagation d'ondes dans le domaine fréquentiel avec la méthode des éléments finis. Dans la première partie du manuscrit, le développement de méthodes de décomposition de domaine est abordé, dans le but de surmonter les limitations des solveurs directs et itératifs de l'état de l'art. À cette fin, une méthode de décomposition de domaine sous-structurée sans recouvrement avec des conditions absorbante d'ordre élevé utilisées comme conditions de transmission (HABC DDM) est d'abord étendue pour traiter les points de jonction, où plus de deux sous-domaines se rencontrent. Le traitement des points de jonction est un problème bien connu pour les HABC DDM sans recouvrement. La méthodologie proposée mène à une solution efficace pour les partitions en damier, où les domaines se rencontrent à angle droit. La méthode est basée sur l'introduction de variables de transmission supplémentaires aux points de jonction, et son efficacité est démontrée sur plusieurs cas-tests. Une DDM sans recouvrement similaire est ensuite proposée avec des couches parfaitement adaptées au lieu des HABC (DDM PML). L'approche proposée prend naturellement en compte les points de jonction des partitions de domaine en damier par le biais de multiplicateurs de Lagrange couplant les sous-domaines et les couches PML adjacentes. Deux discrétisations pour les multiplicateurs de Lagrange et plusieurs stratégies de stabilisation sont proposées et comparées. Les performances des DDM HABC et PML sont ensuite comparées sur des cas-tests de complexité croissante, allant de la diffraction d'ondes dans des milieux homogènes bidimensionnelles à la propagation d'ondes tridimensionnelles dans des milieux hautement hétérogènes. Alors que les développements théoriques sont effectués pour l'équation scalaire de Helmholtz pour la simulation d'ondes acoustiques, l'extension aux problèmes d'ondes élastiques est également considérée, mettant en évidence le potentiel de généralisation des méthodes développées à d'autres contextes physiques. La deuxième partie du manuscrit est consacrée à la présentation des outils de calcul développés au cours de la thèse et qui ont été utilisés pour produire tous les résultats numériques : GmshFEM, une nouvelle bibliothèque d'éléments finis C++ basée sur le générateur de maillage open-source Gmsh ; et GmshDDM, une bibliothèque de décomposition de domaine distribuée basée sur GmshFEM

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles

Advanced BEM-based methodologies to identify and simulate wave fields in complex geostructures

To enhance the applicability of BEM for geomechanical modeling numerically optimized BEM models, hybrid FEM-BEM models, and parallel computation of seismic Full Waveform Inversion (FWI) in GPU are implemented. Inverse modeling of seismic wave propagation in inhomogeneous and heterogeneous half-plane is implemented in Boundary Element Method (BEM) using Particle Swarm Optimization (PSO). The Boundary Integral Equations (BIE) based on the fundamental solutions for homogeneous elastic isotropic continuum are modified by introducing mesh-dependent variables. The variables are optimized to obtain the site-specific impedance functions. The PSO-optimized BEM models have significantly improved the efficiency of BEM for seismic wave propagation in arbitrarily inhomogeneous and heterogeneous media. Similarly, a hybrid BEM-FEM approach is developed to evaluate the seismic response of a complex poroelastic soil region containing underground structures. The far-field semi-infinite geological region is modeled via BEM, while the near-field finite geological region is modeled via FEM. The BEM region is integrated into the global FEM system using an equivalent macro-finite-element. The model describes the entire wave path from the seismic source to the local site in a single hybrid model. Additionally, the computational efficiency of time domain FWI algorithm is enhanced by parallel computation in CPU and GPU

Finite element computation of absorbing boundary conditions for time-harmonic wave problems

International audienceThis paper proposes a new method, in the frequency domain, to define absorbing boundary conditions for general two-dimensional problems. The main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general. It is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software. Boundary conditions at various orders of accuracy can be obtained in a simple way. This is then applied to study some examples for which analytical or numerical results are available. Good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method

Seismic Waves

The importance of seismic wave research lies not only in our ability to understand and predict earthquakes and tsunamis, it also reveals information on the Earth's composition and features in much the same way as it led to the discovery of Mohorovicic's discontinuity. As our theoretical understanding of the physics behind seismic waves has grown, physical and numerical modeling have greatly advanced and now augment applied seismology for better prediction and engineering practices. This has led to some novel applications such as using artificially-induced shocks for exploration of the Earth's subsurface and seismic stimulation for increasing the productivity of oil wells. This book demonstrates the latest techniques and advances in seismic wave analysis from theoretical approach, data acquisition and interpretation, to analyses and numerical simulations, as well as research applications. A review process was conducted in cooperation with sincere support by Drs. Hiroshi Takenaka, Yoshio Murai, Jun Matsushima, and Genti Toyokuni