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

The wave finite element (WFE) method is investigated to describe the harmonic forced response of one-dimensional 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.