A wave finite element-based formulation for computing the forced response of structures involving rectangular flat shells

International audienceThe harmonic forced response of structures involving several noncoplanar rectangular flat shells is investigated by using the Wave Finite Element method. Such flat shells are connected along parallel edges where external excitation sources as well as mechanical impedances are likely to occur. Also, they can be connected to one or several coupling elements whose shapes and dynamics can be complex. The dynamic behavior of the connected shells is described by means of numerical wave modes traveling towards and away from the coupling interfaces. Also, the coupling elements are modeled by using the conventional finite element (FE) method. A FE mesh tying procedure between shells having incompatible meshes is considered, which uses Lagrange multipliers for expressing the coupling conditions in wave-based form. A global wave-based matrix formulation is proposed for computing the amplitudes of the wave modes traveling along the shells. The resulting displacement solutions are obtained by using a wave mode expansion procedure. The accuracy of the wave-based matrix formulation is highlighted in comparison with the conventional FE method through three test cases of variable complexities. The relevance of the method for saving large CPU times is emphasized. Its efficiency is also highlighted in comparison with the component mode synthesis technique

New advances in the forced response computation of periodic structures using the wave finite element (WFE) method

International audienceThe wave finite element (WFE) method is investigated to describe the harmonic forced response of onedimensional periodic structures like those composed of complex substructures and encountered in engineering applications. The dynamic behavior of these periodic structures is analyzed over wide frequency bands where complex spatial dynamics, inside the substructures, are likely to occur.Within theWFE framework, the dynamic behavior of periodic structures is described in terms of numerical wave modes. Their computation follows from the consideration of the finite element model of a substructure that involves a large number of internal degrees of freedom. Some rules of thumb of the WFE method are highlighted and discussed to circumvent numerical issues like ill-conditioning and instabilities. It is shown for instance that an exact analytic relation needs to be considered to enforce the coherence between positive-going and negative-going wave modes. Besides, a strategy is proposed to interpolate the frequency response functions of periodic structures at a reduced number of discrete frequencies. This strategy is proposed to tackle the problem of large CPU times involved when the wave modes are to be computed many times. An error indicator is formulated which provides a good estimation of the level of accuracy of the interpolated solutions at intermediate points. Adaptive refinement is carried out to ensure that this error indicator remains below a certain tolerance threshold. Numerical experiments highlight the relevance of the proposed approaches

A wave-based reduction technique for the dynamic behavior of periodic structures

International audienceThe wave finite element (WFE) method is investigated to describe the dynamic behavior of periodic structures like those composed of arbitrary-shaped substructures along a certain straight direction. A generalized eigenproblem based on the so-called S + S −1 transformation is proposed for accurately computing the wave modes which travel in right and left directions along those periodic structures. Besides, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two extra substructures. In doing so, a few wave modes are only required for modeling the central periodic structure. A comprehensive validation of the technique is performed on a 2D periodic structure. Also, its efficiency in terms of CPU time savings is highlighted regarding a 3D periodic structure that exhibits substructures with large-sized FE models

Wave Finite Element based Strategies for Computing the Acoustic Radiation of Stiffened or Non-Stiffened Rectangular Plates subject to Arbitrary Boundary Conditions

20 pagesInternational audienceThe wave finite element method (WFE) is investigated for the computation of the acoustic radiation of stiffened or non-stiffened rectangular plates under arbitrary boundary conditions. The method aims at computing the forced response of periodic waveguides (e.g. rectangular plates that are homogeneous or that contain a periodic distribution of stiffeners) using numerical wave modes. A WFE-based strategy is proposed which uses the method of elementary radiators for expressing the radiation efficiencies of stiffened or non-stiffened baffled rectangular plates immersed in a light acoustic fluid. In addition, a model reduction strategy consisting in using reduced wave bases for computing these radiation efficiencies with small CPU times is proposed. Numerical experiments highlight the relevance of the strategies

Formulation de la réponse dynamique d'une structure maîtresse couplée à un système annexe et formulation locale du comportement énergétique des structures vibrantes

Cette étude comprend deux principales thématiques: D'une part, la formulation théorique de la réponse dynamique d'une structure maîtresse couplée à un système annexe plus ou moins complexe, localement homogène, composé de sous-systèmes annexes élastiques et continus, et en particulier couplée à un flou structural (représentant un système annexe complexe, mal défini du fait de sa complexité) composé de sous-systèmes flous élastiques et continus. Une formulation déterministe d'impédance de frontière du système annexe, qui modélise l'action de ce dernier sur la structure maîtresse, est établie: il apparaît que la formulation proposée est différente de la solution proposée par Soize, établie sur le modèle déterministe d'un oscillateur linéaire excité par son support. Finalement, on développe un modèle probabiliste d'impédance de frontière d'un flou structural composé de barres élastiques dont les longueurs et sections sont aléatoires. D'autre part, la formulation en moyennes et hautes fréquences, à partir d'une équation de diffusion, du comportement énergétique des structures. Le cas d'un système unidimensionnel homogène (barre, poutre) couplé sur sa longueur à un système annexe homogène est analysé: deux problèmes aux limites énergétiques, capables de prédire les densités d'énergies potentielle et cinétique le long du système, sont formulés rigoureusement. Il apparaît que la validité de l'équation de diffusion est liée au caractère très diffusif du système. Finalement, on montre que le comportement énergétique d'une barre hétérogène avec discontinuités de section est équivalent à celui d'une barre homogène

A model reduction strategy for computing the forced response of elastic waveguides using the wave finite element method

International audienceA model reduction strategy is proposed within the framework of the wave finite element method for computing the low- and mid-frequency forced response of single and coupled straight elastic waveguides. For any waveguide, a norm-wise error analysis is proposed for efficiently reducing the size of the wave basis involved in the description of the dynamic behavior. The strategy is validated through the following test cases: single and coupled beam-like structures with thick cross-sections, plates and sandwich structures. The relevance of the model reduction strategy for saving large CPU times is highlighted, considering the computation of the acoustic radiation of plates and Monte Carlo simulations of coupled waveguides

On the low- and mid-frequency forced response of elastic structures using wave finite elements with one-dimensional propagation

International audienceIn this paper, the wave finite element (WFE) method is investigated for computing the low- and mid-frequency forced response of straight elastic structures. The method uses wave modes as representation basis. These are numerically calculated using the finite element model of a typical substructure with a small number of degrees of freedom, and invoking Bloch's theorem. The resulting wave-based boundary value problem is presented and adapted so as to address Neumann-to-Dirichlet problems involving single as well as coupled structures. A regularization strategy is also presented. It improves the convergence of the WFE method when multi-layered systems are specifically dealt with. It employs an alternative form of the wave-based boundary value problem quite stable and easy to solve. The relevance of both classic and regularized WFE formalisms is discussed and numerically established compared with standard finite element solutions

APPROCHE NUMERIQUE POUR LA PROPAGATION MULTI-MODALE GUIDEE

Ces activités de recherche gravitent autour de la Méthode WFE (« Wave Finite Elements ») : cette méthode permet de décrire numériquement la propagation d’ondes en basses et moyennes fréquences (BF & MF) dans les systèmes élancés à section complexe. Cette approche multi-modale a été originellement développée pour décrire les systèmes élastiques; elle présente des avantages intéressants pour le calcul des réponses dynamiques en BF & MF, dans le sens où elle suggère des temps de calculs extrêmement réduits et s’appuie sur des bases de représentation a priori hautement convergentes.Les travaux de recherche présentés dans ce document ont été initiés pour étendre le champ d’application de la méthode WFE. Ces travaux concernent des adaptations de la méthode WFE pour décrire les systèmes élasto-acoustiques, les systèmes multi-couches et la réflexion / transmission d’ondes élastiques au niveau de jonctions complexes. Ils s’inscrivent dans le cadre d’applications industrielles d’actualité telles que l’étude des conduites avec fluide interne (ex : ligne d’échappement, circuit de climatisation), la description des structures sandwichs (ex : coiffe de lanceur spatial) et le contrôle non-destructif des systèmes

Model reduction and perturbation analysis of wave finite element formulations for computing the forced response of coupled elastic systems involving junctions with uncertain eigenfrequencies

International audienceThe wave finite element method is investigated for computing the low- and mid-frequency forced response of coupled elastic systems involving straight structures with junctions. The relevance of the method is discussed when a component mode synthesis procedure is used for modeling the junctions. A norm-wise selection criterion is proposed so as to reduce efficiently the number of junction modes retained in the wave-based formulations. Component-wise perturbation bounds of the wave-based displacement/force solutions are also derived to address slight uncertainties for the junction eigenfrequencies. Numerical comparisons with standard finite element solutions as well as Monte Carlo simulations highlight the relevance of the formulation

Interpolatory model reduction for component mode synthesis analysis of structures involving substructures with frequency-dependent parameters

International audienceAn interpolatory model order reduction (MOR) strategy is proposed to compute the harmonic forced response of structures built up of substructures with frequency-dependent parameters. In this framework, the Craig-Bampton (CB) method is used for modeling each substructure by means of static modes and a reduced number of fixed-interface modes which are interpolated between several master frequencies. Emphasis is on the analysis of several substructures which can vibrate at different scales and, as such, do not need to be modeled with the same sets of interpolation points, depending on whether their modal density is low or high. For this purpose, an error indicator is developed to determine, through greedy algorithm procedure, the optimal number of interpolation points needed for each substructure. Additional investigations concern the selection of the fixed-interface modes which need to be retained for each substructure. Numerical experiments are carried out to highlight the relevance of the proposed approach, in terms of computational saving and accuracy