Keywords: wave finite elements, model reduction, mid-frequencies, acoustic radiation.

The wave finite element (WFE) method is investigated for computing the acoustic radiation of stiffened or non-stiffened rectangular plates subject to arbitrary boundary conditions. The method numerically provides an analysis of the waves traveling in positive and negative directions along periodic waveguides, i.e. elastic structures that are assumed to be modelled by means of identical finite element (FE) substructures connected along a main axis (namely, the direction of propagation). Stiffened or non-stiffened rectangular plates that are meshed periodically along their length belong to that class of waveguides. The WFE method uses the FE model of a typical substructure to compute the wave modes. These are to be understood as particular shapes of the displacement and force fields over the system cross-section, "traveling" with different velocities along the waveguide. The WFE method enables the propagating, evanescent and complex wave modes to be captured over the low and mid-frequency range. Using these wave modes as representation bases constitutes an efficient means for computing the forced response of waveguides subject to arbitrary boundary conditions [1].

In this paper, a WFE-based strategy that uses the aforementioned wave modes for expressing the radiation efficiencies of stiffened or non-stiffened rectangular plates is proposed. Such plates are assumed to be surrounded by an infinite baffle while radiating in a light acoustic fluid. The radiation efficiencies are computed using the method of elementary radiators [2], which requires us to discretise the plates into small surfaces while expressing the normal velocities of these elementary surfaces in terms of wave modes. The feature of the proposed WFE formulation is that it exhibits matrices that do not depend on the waveguide boundary conditions, meaning that it can be reused with less computational time (i.e. once these matrices have been computed) to address the changes of those boundary conditions.

In addition, a model order reduction (MOR) strategy consisting of using reduced wave bases for computing these radiation efficiencies is proposed [3]. The motivation behind this work is to reduce significantly the computational times compared to the case when the full wave bases are used in the WFE matrix formulations. It is worth noting that, even in the WFE framework, the CPU times required to compute the radiation efficiency of a plate at many discrete frequencies can be substantial. The proposed MOR strategy constitutes an efficient means for determining precisely the number of wave modes required for computing accurately the forced response of waveguides.

The accuracy and relevance of these strategies for computing the radiation efficiencies of stiffened or non-stiffened rectangular plates are highlighted compared to the conventional finite element method as well as analytical theories.

1

J.M. Mencik, "On the low- and mid-frequency forced response of elastic systems using wave finite elements with one-dimensional propagation", Computers and Structures, 88(11-12), 674-689, 2010.

2

S.J. Elliott, M.E. Johnson, "Radiation modes and the active control of sound power", Journal of the Acoustical Society of America, 94(4), 2194-2204, 1993. doi:10.1121/1.407490

3

J.M. Mencik, "A model reduction strategy for computing the forced response of elastic waveguides using the wave finite element method", Computer Methods in Applied Mechanics and Engineering, 229-232, 68-86, 2012. doi:10.1016/j.cma.2012.03.024

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