Couplage entre la MEF et la DGM avec ondes planes pour l'acoustique

L’objectif de ce travail est de coupler pour des problèmes acoustiques simples la Méthode des Eléments Finis (MEF) et la Discontinuous Galerkin Method (DGM) avec ondes planes. Le travail consiste principalement à réécrire les opérateurs de surface de la MEF en les décomposant sur les caractéristiques entrantes et sortantes de l’interface. Cela est effectué à l’aide des techniques classique en DGM de décomposition des flux et qui doivent dans le cas présent être discrétisés à l’aide des fonctions de forme du problème éléments-finis. Il sera montré que cela entrainera, en raison de la dérivation des fonctions de forme, la perte d’un ordre convergence si une interpolation de Lagrange est utilisées et par une conservation de l’ordre de convergence pour une interpolation de type Hermite. Plusieurs problèmes acoustiques simples académiques avec et sans matériaux poreux seront présentés

Numerical investigations on the resonance errors of multiscale discontinuous Galerkin methods for one-dimensional stationary Schr\"{o}dinger equation

In this paper, numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high order multiscale discontinuous Galerkin (DG) methods [6, 7] for one-dimensional stationary Schr\"{o}dinger equation. Previous work showed that penalty parameters were required to be positive in error analysis, but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes. In this work, by performing extensive numerical experiments, we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods, and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers

Trefftz -Discontinuous Galerkin Approach for Solving Elastodynamic Problem

International audienceMethods based on Discontinuous Finite Element approximation (DG FEM) are basically well-adapted to specifics of wave propagation problems in complex media, due to their numerical accuracy and flexibility. However, they still lack of computational efficiency, by reason of the high number of degrees of freedom required for simulations. The Trefftz-DG solution methodology investigated in this work is based on a formulation which is set only at the boundaries of the mesh. It is a consequence of the choice of test functions that are local solutions of the problem. It owns the important feature of involving a space-time approximation which requires using elements defined in the space-time domain. Herein, we address the Trefftz-DG solution of the Elastodynamic System. We establish its well-posedness which is based on mesh-dependent norms. It is worth noting that we employ basis functions which are space-time polynomial. Some numerical experiments illustrate the proper functioning of the method

Singular enrichment functions for Helmholtz scattering at corner locations using the Boundary Element Method

In this paper we use an enriched approximation space for the efficient and accurate solution of the Helmholtz equation in order to solve problems of wave scattering by polygonal obstacles. This is implemented in both Boundary Element Method (BEM) and Partition of Unity Boundary Element Method (PUBEM) settings. The enrichment draws upon the asymptotic singular behaviour of scattered fields at sharp corners, leading to a choice of fractional order Bessel functions that complement the existing Lagrangian (BEM) or plane wave (PUBEM) approximation spaces. Numerical examples consider configurations of scattering objects, subject to the Neumann ‘sound hard’ boundary conditions, demonstrating that the approach is a suitable choice for both convex scatterers and also for multiple scattering objects that give rise to multiple reflections. Substantial improvements are observed, significantly reducing the number of degrees of freedom required to achieve a prescribed accuracy in the vicinity of a sharp corner

An integral formulation for wave propagation on weakly non-uniform potential flows

An integral formulation for acoustic radiation in moving flows is presented. It is based on a potential formulation for acoustic radiation on weakly non-uniform subsonic mean flows. This work is motivated by the absence of suitable kernels for wave propagation on non-uniform flow. The integral solution is formulated using a Green's function obtained by combining the Taylor and Lorentz transformations. Although most conventional approaches based on either transform solve the Helmholtz problem in a transformed domain, the current Green's function and associated integral equation are derived in the physical space. A dimensional error analysis is developed to identify the limitations of the current formulation. Numerical applications are performed to assess the accuracy of the integral solution. It is tested as a means of extrapolating a numerical solution available on the outer boundary of a domain to the far field, and as a means of solving scattering problems by rigid surfaces in non-uniform flows. The results show that the error associated with the physical model deteriorates with increasing frequency and mean flow Mach number. However, the error is generated only in the domain where mean flow non-uniformities are significant and is constant in regions where the flow is uniform

Implementation of an interior point source in the ultra weak variational formulation through source extraction

The Ultra Weak Variational Formulation (UWVF) is a powerful numerical method for the approximation of acoustic, elastic and electromagnetic waves in the time-harmonic regime. The use of Trefftz-type basis functions incorporates the known wave-like behaviour of the solution in the discrete space, allowing large reductions in the required number of degrees of freedom for a given accuracy, when compared to standard finite element methods. However, the UWVF is not well disposed to the accurate approximation of singular sources in the interior of the computational domain. We propose an adjustment to the UWVF for seismic imaging applications, which we call the Source Extraction UWVF. Differing fields are solved for in subdomains around the source, and matched on the inter-domain boundaries. Numerical results are presented for a domain of constant wavenumber and for a domain of varying sound speed in a model used for seismic imaging

Vers un couplage plus naturel des méthodes de Galerkin Discontinue avec ondes planes et Élements Finis pour l'acoustique

Une des voies possibles vers la réduction des coûts de simulation passe par le développement de méthodes hybrides. Dans le but de construire une hybridation efficace, le choix des méthodes est primordial. Ce travail porte ainsi sur l'association de la Méthode des Éléments Finis (MEF) et la Méthode de Galerkin Discontinue avec Ondes Planes (PWDGM) avec la volonté de pouvoir tirer parti de leur spécificités. D'une part, la méthode des Éléments Finis (MEF), malgré sa versatilité, demande un maillage fin du domaine qui s'avère coûteux lorsque celui-ci devient grand. Ce fonctionnement, basé sur une discrétisation spatiale, permet toutefois à la méthode de s'adapter à des géométries complexes pouvant entre autres contenir diffracteurs et éléments résonnants. D'autre part, la méthode de Galerkin Discontinue avec ondes planes (PWDGM), approxime les champs en les décomposant sur une base d'ondes planes. Basée également sur un maillage, cette méthode ne requiert toutefois pas qu'il soit fin. En effet, elle fonctionne de manière optimale dans le cas de grands domaines. Une proposition de couplage entre ces méthodes a déjà été présenté. Ce couplage, s'appuyant sur la réécriture des champs MEF à l'interface dans le formalisme requis par la PWDGM, poussait à dériver les fonctions de forme et s'accompagnait de la perte d'un ordre de convergence. Ce travail vise à corriger cette faille de la technique de couplage en proposant une approche repose sur l'écriture de conditions de continuité entre les caractéristiques de la PWDGM et les champs de la MEF. Cette nouvelle proposition permet de conserver l'ordre de convergence et les résultats numériques ont montré un excellent accord entre la méthode hybride et des références. Dans le cadre de cette réécriture, la théorie a notamment été étendue à d'autres physiques. En particulier, les opérateurs pour des matériaux poroélastiques (théorie de Biot) sont introduits et mis en ?uvre dans les exemples. Cette méthode sera présentée d'abord sur des exemples académiques et ses propriétés seront discutées. Une attention particulière sera portée à l'impact que le couplage a sur les propriété de convergence et de dispersion, les résultats pour la méthode hybride seront mis en regard avec ceux pour le méthodes seules. Dans un second temps, des cas plus proches des cas d'utilisation réels seront présentés. Il sera montré notamment le cadre dans lequel le recours à la méthode hybride permet de réduire significativement la taille du système linéaire final tout en minimisant la perte de précision

A Plane Wave Virtual Element Method for the Helmholtz Problem

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: i) a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; ii) a proper local projection operator onto the high-frequency space, made of plane waves; iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented

Coupling of Finite-Element and Plane Waves Discontinuous Galerkin methods for time-harmonic problems

A coupling approach is presented to combine a wave-based method to the standard finite element method. This coupling methodology is presented here for the Helmholtz equation but it can be applied to a wide range of wave propagation problems. While wave-based methods can significantly reduce the computational cost, especially at high frequencies, their efficiency is hindered by the need to use small elements to resolve complex geometric features. This can be alleviated by using a standard Finite-Element Model close to the surfaces to model geometric details and create large, simply-shaped areas to model with a wave-based method. This strategy is formulated and validated in this paper for the wave-based discontinuous Galerkin method together with the standard finite element method. The coupling is formulated without using Lagrange multipliers and results demonstrate that the coupling is optimal in that the convergence rates of the individual methods are maintained

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived