Modal finite element synthesis of acoustic boundary receptances

Abstract

The main issue in the present paper is to comment and illustrate on new acoustic examples the method of analysis of modal series, termed “method of orthocomplement”, that has been recently proposed by the authors to improve the convergence of such series. The general method consists in a direct analysis and transformation of the remainders of ordinary series. It results in a family of “hybrid” modal representations involving an ordinary modal sum of order N, a “quasi-static” term based on the N first modes, and an “accelerated” modal series. Using the transformed modal formulae eliminates the Gibbs oscillations – that are attached in infinite dimensional models to modal boundary discontinuities – and also the consequences of such phenomena on finite element approximations. The method is applied in the present paper to plane waves in acoustic tubes and to 3D acoustic fields inside a car compartment, in view of the synthesis of acoustic receptances or impedances to be used in practical acoustic design. The main technical difficulty being the treatment of singular linear boundary problems or systems of linear equations that arise during the study of closed rigid cavities or tubes, a whole section of the paper had thus to be devoted to pseudo-inversion.