On the reconstruction of the vibro-acoustic field over the surface enclosing an interior space using the boundary element method

Abstract

The vibrational velocity, sound pressure, and acoustic power on the vibrating boundary comprising an enclosed space are reconstructed by the boundary element method based on the measured field pressures. The singular value decomposition is used to obtain the inverse solution in the least-square sense and to express the acoustic modal expansion between the measurement and source fields. In general, such an inverse operation has been considered an ill-posed problem having a divergence phenomenon involved with extremely small measurement errors. The ill-conditioned nature of the acoustic inverse problem is caused by the singularity of the transfer matrix which produces nonradiating wave components. In order to minimize the singularity and to also reduce the number of measurement points, optimal measurement positions are determined by the effective independence method. Regularization methods are used to stabilize the reconstructed field by suppressing nonradiating components resulting in the singular transfer matrix. In order to enhance the resolution of the reconstructed field, the optimal regularization order for yielding the minimum mean-square error is estimated from the known measurement noise variance by virtue of the statistical analysis. A half-scaled automotive cabin is considered an example for validating and demonstrating the proposed reconstruction process. It is noted that the present method can improve the resolution of the reconstructed field; thus vibro-acoustic parameters of the vibrating boundary can be estimated in reasonably good precision.

On the in situ estimation of surface acoustic impedance in interiors of arbitrary shape by acoustical inverse methods

“…This method consists on solving the BEM equations for one unknown boundary parameter at a time. In addition, truncation of some singular values (known as truncated-SVD or T-SVD [8,12,13],) is done to reduce the sensitivity to noise. This approach is briefly discussed here for the purpose of comparisons in further numerical examples.…”

Single layer planar near-field acoustic holography for compact sources and a parallel reflector

“…From a data-acquisition point of view, it is a common practice to measure pressure or particle velocity in two closely spaced surfaces, or to measure Cauchy data in a single layer. Among these two categories, some studies formulate the problem with inverse boundary element and Helmholtz equation least-squares methods via the singular value decomposition [3,4,5,6,7,8], while others use the spherical wave superposition method [9,10,11], the equivalent source method [12,13,14,15,16,17] or the statistically optimized NAH method [18,19,20,21,22,23]. There is also a recent application of the equivalent source method in which the (double-layer) measurements completely cover the source region, allowing acoustic visualization of the field in small cavities [24].…”

Sound source reconstruction using inverse boundary element calculations

“…Whereas setting up the transfer matrix and making use of its SVD is well documented, the final step of choosing the correct amount of regularization is not fully understood. The discrepancy principle, which uses a priori knowledge of the measurement errors, has been used in connection with Tikhonov regularization 7 and Landweber iteration. 10 The generalized cross-validation ͑GCV͒ method has also been used by several researchers in conjunction with Tikhonov regularization, and it has been shown to produce useful results for cases where spatially white noise contaminates the field data.…”

“…Both interior and exterior noise problems have been studied. [6][7][8][9] Some parts of the sound field radiated by a vibrating structure die out very quickly away from the source and therefore contribute very little at the field microphone positions; these sound field components are often referred to as the evanescent waves. The reconstruction of the particular vibration patterns that create the evanescent waves will involve a strong amplification of very small signal components, and as a consequence, the inverse problem is very sensitive to the noise and errors in the measured data.…”