On the choice of expansion functions in the Helmholtz equation least-squares method

This paper examines the performance of Helmholtz equation least-squares (HELS) method in reconstructing acoustic radiation from an arbitrary source by using three different expansions, namely, localized spherical waves (LSW), distributed spherical waves (DSW), and distributed point sources (DPS), under the same set of measurements. The reconstructed acoustic pressures are validated against the benchmark data measured at the same locations as reconstruction points for frequencies up to 3275 Hz. Reconstruction is obtained by using Tikhonov regularization or its modification with the regularization parameter selected by error-free parameter-choice methods. The impact of the number of measurement points on the resultant reconstruction accuracy under different expansion functions is investigated. Results demonstrate that DSW leads to a better-conditioned transfer matrix, yields more accurate reconstruction than both LSW and DPS, and is not affected as much by the change in measurement points. Also, it is possible to obtain optimal locations of the auxiliary sources for DSW, LSW, and DPS by taking an independent layer of measurements. Use of these auxiliary sources and an optimal combination of regularization and error-free parameter choice methods can yield a satisfactory reconstruction of acoustic quantities on the source surfaces as well as in the field in the most cost-effective manner.