A wave-based model reduction technique for the description of the dynamic behavior of periodic structures involving arbitrary-shaped substructures and large-sized finite element models (original) (raw)
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A WAVE-BASED REDUCTION TECHNIQUE FOR THE DYNAMIC BEHAVIOR OF PERIODIC STRUCTURES
The 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.
The wave finite element (WFE) method is investigated to describe the dynamic behavior of finite-length periodic structures with local perturbations. The structures under concern are made up of identical substructures along a certain straight direction, but also contain several perturbed substructures whose material and geometric characteristics undergo arbitrary slight variations. Those substructures are described through finite element (FE) models in time-harmonic elasticity. Emphasis is on the development of a numerical tool which is fast and accurate for computing the related forced responses. To achieve this task, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two unperturbed substructures, and considering perturbed parts which are composed of perturbed substructures surrounded by two unperturbed ones. In doing so, a few wave modes are only required for modeling the central periodic structure, outside the perturbed parts. For forced response computation purpose, a reduced wave-based matrix formulation is established which follows from the consideration of transfer matrices between the
2016
To cite this version: J.-M. Mencik, Denis Duhamel. A wave finite element-based approach for the modeling of periodic structures with local perturbations. Abstract The wave finite element (WFE) method is investigated to describe the dynamic behavior of finite-length periodic structures with local perturbations. The structures under concern are made up of identical substructures along a certain straight direction, but also contain several perturbed substructures whose material and geometric characteristics undergo arbitrary slight variations. Time-harmonic elasticity is considered. Emphasis is on the development of a numerical tool which is fast and accurate for computing the related forced responses. To achieve this task, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two unperturbed sub-structures, and considering perturbed parts which are composed of perturbed sub-structures surrounded by two u...
2016
International audienceA wave finite element (WFE) based approach is proposed to analyze the dynamic behavior of finite-length periodic structures which are made up of identical substructures but also contain several substructures whose material and geometric characteristics are slightly perturbed. Within the WFE framework, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two unperturbed substructures, and considering perturbed parts which are composed of perturbed substructures surrounded by two unperturbed ones. In doing so, a few wave modes are only required for modeling the central periodic structure, outside the perturbed parts. For forced response computation purpose, a reduced wave-based matrix formulation is established which follows from the consideration of transfer matrices between the right and left sides of the perturbed parts. Numerical experiments are carried out on a periodic 2D str...
Computational Mechanics
The wave finite element method has proved to be an efficient and accurate numerical tool to perform the free and forced vibration analysis of linear reciprocal periodic structures, i.e. those conforming to symmetrical wave fields. In this paper, its use is extended to the analysis of rotating periodic structures, which, due to the gyroscopic effect, exhibit asymmetric wave propagation. A projection-based strategy which uses reduced symplectic wave basis is employed, which provides a wellconditioned eigenproblem for computing waves in rotating periodic structures. The proposed formulation is applied to the free and forced response analysis of homogeneous, multi-layered and phononic ring structures. In all test cases, the following features are highlighted: well-conditioned dispersion diagrams, good accuracy, and low computational time. The proposed strategy is particularly convenient in the simulation of rotating structures when parametric analysis for several rotational speeds is usually required, e.g. for calculating Campbell diagrams. This provides an efficient and flexible framework for the analysis of rotordynamic problems.
Journal of Sound and Vibration, 2018
A numerical approach is proposed to compute the dynamic response of periodic structures with cyclic symmetry, and assemblies made up of these periodic structures. The wave finite element (WFE) method is used to describe the wave modes which occur around the circumferential direction of these periodic structures. Emphasis is placed on assessing the dynamic flexibility modes of a periodic structure by considering unit forces which are successively applied to the boundary degrees of freedom. It is shown that the matrices of dynamic flexibility modes can be quickly computed. This yields an efficient dynamic substructuring technique to analyze the dynamic behavior of assemblies made up of several periodic structures. Numerical experiments are carried out which concern the analysis of a single periodic structure as well as assemblies made up of two and three structures.
Computational Mechanics, 2014
The 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 the WFE 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.
Finite Elements in Analysis and Design, 2021
The dynamic analysis of 2D nearly periodic structures of finite dimensions, subject to harmonic excitations, is addressed. Such structures are often made up of slightly different locally resonant layered substructures whose geometrical properties randomly vary in space and which are described here by means of distorted finite element (FE) meshes. It is well known that purely periodic structures with resonant substructures possess band gap properties, i.e., frequency bands where the vibration levels are low. The question arises whether nearly periodic structures provide additional features, e.g., the fact that the vibrational energy remains localized around the excitation points. Predicting the harmonic responses of such structures via efficient numerical approaches is the motivation behind the present paper. Usually, the Craig Bampton (CB) method is used to model the substructures in terms of reduced mass and stiffness matrices, which can be further assembled together to model a whole structure. The issue arises because the reduced mass and stiffness matrices of the substructures need to be computed several times-i.e., for several substructures whose properties differ to each other-, which is computationally cumbersome. To address this issue, a strategy is proposed which involves computing the reduced matrices of the substructures for some particular distorted FE meshes (a few number), and interpolating these matrices between these "interpolation points" for modeling substructures with random FE meshes. The relevance of the interpolation strategy, in terms of computational time saving and accuracy, is highlighted through comparisons with the FE and CB methods. Three structures are analyzed, i.e., (1) a plate with 8 × 8 substructures, (2) a plate with 15 × 15 substructures, and (3) a plate with 8 × 4 substructures embedded in a floor panel. Results show that, at high frequencies, the vibration levels of the nearly periodic structures undergo an overall reduction compared to the purely periodic cases.
A Finite Element Based Eigenanalysis of Periodic Structures
A FEM-based simulation tools is developed for the eigenanalysis of 3D periodic structures. The first task refers to the analysis of closed periodic waveguiding structures with periodicity in the propagation direction. For the next step, the analysis is extended to open structures periodical in two directions, while finally arbitrary three dimensional periodic structures will be elaborated.
A wave-based numerical approach is proposed for modeling periodic structures with cyclic symmetry. Wave modes which travel around the circumferential direction of those structures are calculated with the wave finite element method. Emphasis is placed on building the matrices of dynamic flexibility modes of the periodic structures by considering unit forces which are successively applied to the degrees of freedom of their boundaries. As it turns out, the matrices of dynamic flexibility modes may be quickly computed, leading the way to efficient domain decomposition techniques to analyze assemblies made up of several periodic structures.