High-order finite elements for interior acoustics

Abstract

The paper presents recent developments of high‐order finite‐element methods, for the evaluation of the natural modes of vibration of the air inside a cavity (interior acoustics). Two high‐order methods are addressed. For both, the high‐order accuracy is reached not by adding internal modes, but rather by using higher and higher order derivatives at the eight nodes of topologically hexagonal elements. The distinguishing feature of the methodology is its high accuracy, with the possibility to capture very high spatial frequencies, as required in typical interior acoustics applications. In the first element, the classical Hermite interpolation is extended to three dimensions. For the first order case, the finite‐element unknowns are the nodal values of the unknown function (displacement), of its three first‐order partial derivatives, of its three second‐order mixed derivatives, and of its third‐order mixed derivative. This yields continuous first‐order derivatives (that is, the gradient is continuous). Similar considerations hold for orders higher than three; for instance, for the seventh order Hermite element (that is, using polynomials of order seven) one achieves continuity of the derivatives up to order three. The second method is based upon a hybrid combination of the three‐dimensional extension of the Coons patch technique with the Hermite interpolation. In the hybrid formulation, the unknowns are reduced by a factor two, without reducing the accuracy of the scheme, thereby providing a considerable increase in the eciency, without any reduction of continuity and accuracy.