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

Abstract

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.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 2d Wave Finite Element-Based Superelement Formulation for Acoustic Analysis of Cavities of Arbitrary Shapes

“…Within the framework of the WFE method, the vectors of pressures and acoustic forces of an assembly of N substructures like the one displayed in Figure 1 are expressed in terms of wave shapes, as follows [10]:…”

A Wave-Based Reduction Technique for the Dynamic Behavior of Periodic Structures

“…Within the WFE framework, the vectors of displacements/rotations and forces/moments on the substructure boundary (k) are estimated by means of a wave expansion, as follows [8]:…”

“…Actually, there exist two main WFE-based strategies to compute the forced response of periodic structures. These are labeled as dynamic stiffness matrix (DSM) approach and wave amplitudes (WA) approach, and respectively involve expressing the condensed dynamic stiffness matrix of a periodic structure in terms of wave modes [3], or the vectors of wave amplitudes of the right-going and left-going modes [4,7,8]. The feature of the WFE method is that it makes use of the FE model of one single substructure only, rather than considering the full FE model of a whole periodic structure.…”

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

“…Their computation follows by considering a generalized eigenproblem that is expressed from the mass and stiffness matrices of a particular substructure. Besides, the study of coupled systems involving elastic waveguides and elastic junctions has been proposed in [12], while that of truly periodic structuresi.e., structures composed of complex substructures like those encountered in engineering applications -has been proposed in [13]. Actually, there exist two main WFE-based strategies to compute the forced response of periodic structures.…”