Performance study of plane wave finite element methods with a Padé-type artificial boundary condition in acoustic scattering

International audienceThe aim of this paper is to propose and numerically study the performance of coupling a high-order Pad\'{e}-type non-reflecting boundary condition with plane wave finite element formulations for solving high-frequency scattering problems involving elongated scatterers. It is shown on some numerical examples that the approximate solution can be obtained using a small number of degrees of freedom for a suitable accuracy

Une méthode de couplage éléments finis-conditions absorbantes de type-padé pour les problèmes de diffraction acoustique

Nous nous intéressons aux problèmes harmoniques de diffraction acoustique en milieu infini régis par l'équation de Helmholtz. La simulation numérique de ces phénomènes est complexe notamment lorsqu'il est question de fréquences élevées et d'obstacles de forme allongée tel qu'un sous-marin. Les codes éléments finis commerciaux sont incapables de cerner tous les aspects liés à ce type de problèmes. De plus, ce genre d'applications fait appel à de grandes ressources de calcul. En effet, la taille du système d'équations à résoudre (plusieurs millions de ddl) engendre souvent l'épuisement des ressources des calculateurs traditionnels. Notre objectif est de solutionner ce type de problèmes avec une précision pratique en utilisant le minimum de ressources. Nous proposons ainsi une méthode de couplage éléments finis de type Lagrange et à base d'ondes planes avec les conditions absorbantes d'ordre élevé basées sur les approximants complexes de Padé. A travers une série d'expériences numériques, nous montrons l'efficacité de ces conditions absorbantes en comparaison avec les conditions absorbantes de Bayliss-Gunzburger-Turkel d'ordre deux implémentées dans les codes commerciaux. La méthodologie proposée permet non seulement une réduction de la taille du domaine de calcul sans dégradation de la précision mais conduit également à la résolution de systèmes d'équations de taille relativement réduite

Dynamic soil-structure interaction analysis using the scaled boundary finite-element method.

This thesis presents the development of a reliable and efficient technique for the numerical simulation of dynamic soil-structure interaction problems in anisotropic and nonhomogeneous unbounded soils of arbitrary geometry. Such a technique is indispensable in the seismic analysis of large-scale engineering constructions and, to my best knowledge, does not exist at present. The theoretical framework of the research is based on the scaled boundary finite-element method. The following advances are achieved: The scaled boundary finite-element method is extended to simulate the dynamic response of non-homogeneous unbounded domains. The scaled boundary finite element equations in the frequency and time domains are derived for power-type non-homogeneity frequently employed in geotechnical engineering. A high-frequency asymptotic expansion of the dynamic-stiffness matrix is developed. The frequency domain analysis is performed by integrating the scaled boundary finite-element equation in dynamic stiffness. In the time domain, the scaled boundary finite-element equation including convolution integrals is solved for the unit-impulse response at discrete time stations. A Padé series solution for the scaled boundary finite-element equation in dynamic stiffness is developed. It converges over the whole frequency range as the order of the approximation increases. The computationally expensive task of numerically integrating the scaled boundary finite-element equation is circumvented. Exploiting the sparsity of the coefficientmatrices in the scaled boundary finite-element equation leads to a significant reduction in computer time and memory requirements for solving large-scale problems. Furthermore, lumped coefficient matrices are obtained by adopting the auss-Lobatto-Legendre shape functions with nodal quadrature, which avoids the eigenvalue problem in determining the asymptotic expansion. A high-order local transmitting boundary constructed from a continued-fraction solution of the dynamic-stiffness matrix is developed. An equation of motion as occurring in standard structural dynamics with symmetric and frequency-independent coefficient matrices is obtained. This transmitting boundary condition can be coupled seamlessly with standard finite elements. Transient responses are evaluated by using a standard timeintegration scheme. The expensive task of evaluating convolution integrals is circumvented. The advances developed in this thesis are applicable in other disciplines of engineering and science to the analysis of scalar and vector waves in unbounded media

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

A quasi-optimal non-overlapping domain decomposition method for two-dimensional time-harmonic elastic wave problems

peer reviewedThis article presents the construction of a new non-overlapping domain decomposition method (DDM) for two-dimensional elastic scattering problems. The method relies on a high-order Transmission Boundary Condition (TBC) between sub-domains, which accurately approximates the exact Dirichlet-to-Neumann map. First, we explain the derivation of this new TBC in the context of a non-overlapping DDM. Next, a mode-by-mode convergence study for a model problem is presented, which shows the new method to be quasi-optimal, i.e. with an optimal convergence rate for evanescent modes and an improved convergence rate for the other modes compared to the standard low-order Lysmer-Kuhlemeyer TBC. Finally, the effectiveness of the new DDM is demonstrated in a finite element context by analyzing the behavior of the method on high-frequency elastodynamic simulations. © 201

Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves

International audienceIt has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies

Optimized weak coupling of boundary element and finite element methods for acoustic scattering

International audienceIn this paper we present an optimized weak coupling of boundary element and finite element methods to solve acoustic scattering problems. This weak coupling is formulated as a non-overlapping Schwarz domain decomposition method, where the transmission conditions are constructed through Padé localized approximations of the Dirichlet-to-Neumann map. The performance of the resulting formulations is analyzed on several three-dimensional examples, with both homogeneous and inhomogeneous scatterers

Benchmarking preconditioned boundary integral formulations for acoustics.

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretization of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimized for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves

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

Propogation Of Wind Turbine Noise Through Wakes And Turbulent Atmosphere

It is well known that the atmospheric inhomogeneities have great impact on sound propagation over long ranges. For the application of predicting wind turbine noise, either the flow wakes generated by rotating turbine blades or small-scale atmospheric turbulence can affect the propagation of sound over ground surfaces from individual turbines. In this thesis, the effects of wake and atmospheric turbulence on the propagation of wind turbine noise are investigated. By introducing the Parabolic Equation (PE) method, the effects of atmospheric changes in sound speed can be incorporated at each marching step as the prediction of sound field advances in the horizontal ranges. With a simulated wake profile near the wind turbine, more accurate predictions in the sound field can be achieved for realistic atmospheric conditions. This work aims to improve current prediction schemes for assessing the impact of wind turbine noise on the neighborhood communities