State-of-the-art finite element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite element methods are described. Stabilized, multiscale and other wavebased discretization methods developed to reduce this error are reviewed. A review of finite element methods for acoustic inverse problems and shape optimization is also given.Running Title: Finite element methods for acoustics

SUMMARYIn this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.

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The authors present numerical results for a systematic parametric study of the effect of honeycomb core geometry on the sound transmission and vibration properties of in-plane loaded honeycomb core sandwich panels using structural acoustic finite element analysis (FEA). Honeycomb cellular structures offer many distinct advantages over homogenous materials because their effective material properties depend on both their constituent material properties and their geometric cell configuration. From these structures, a wide range of targeted effective material properties can be achieved thus supporting forward design-by-tailoring honeycomb cellular structures for specific applications. One area that has not been fully explored is the set of acoustic properties of honeycomb and understanding of how designers can effectively tune designs in different frequency ranges. One such example is the insulation of target sound frequencies to prevent sound transmission through a panel. This work explored the effect of geometry of in-plane honeycomb cores in sandwich panels on the acoustic properties the panel. The two acoustic responses of interest are the general level of sound transmission loss (STL) of the panel and the location of the resonance frequencies that exhibit high levels of sound transmission, or low sound pressure transmission loss. Constant mass honeycomb core models were studied with internal cell angles ranging in increments from −45 deg to +45 deg. Effective honeycomb moduli based on static analysis of honeycomb unit cells are calculated and correlated to the shift in resonance frequencies for the different geometries, with all panels having the same total mass. This helps explain the direction of resonance frequency shift found in the panel natural frequency solutions. Results show an interesting trend of the first resonance frequencies in relation to effective structural properties. Honeycomb geometries with smaller core internal cell angles, under constant mass constraints, shifted natural frequencies lower, and had more resonances in the 1–1000 Hz range, but exhibited a higher sound pressure transmission loss between resonant frequencies.