Experimental study of the mapping relationship based near-field acoustic holography with spherical fundamental solutions

Abstract This paper is a consequent work of the previously proposed mapping relationship based near-field acoustic holography (MRS-based NAH), [H.J.Wu W.K. Jiang and H.B. Zhang, JSV, 373:66-88, 2016]. It is devoted to the performance study of its practical application with error analysis and experimental validation. Two types of errors, the truncation errors due to the limited number of participant modes, and the inevitable measurement errors caused by uncertainties in the experiment, are considered in the analysis. The influences of the errors on the performance of MRS-based NAH are systematically investigated. First of all, expression of the relative reconstruction error of the pressure energy is derived based on the two types of errors. An approach is developed to estimate the lower and upper bounds of the relative error. It gives a guide to predict the error for a reconstruction under the condition that the truncation error and the signal-to-noise ratio are given. Then, the condition number of the inverse operator is investigated to measure the sensitivity of the reconstruction to the input errors. Asymptotic expressions of the condition number for a special case, conformal spherical model and hologram, are obtained, which indicates the condition number has a geometric growth with the number of participant modes. Numerical examples with different kinds of errors are elaborately designed to validate the stability as well as the correctness of the error analysis. At last, the MRS-based NAH is further examined and verified by a physical experiment, a vibrating cubic model reconstructed from measurement on a spherical hologram. A satisfied agreement with the directly measured pressure on a validation surface is observed for both quantity and distribution of the reconstructed pressure.

[1]  David Thompson,et al.  The Quantification of Structure-Borne Transmission Paths by Inverse Methods - Part 2: Use of Regularization Techniques , 2003 .

[2]  Zhang,et al.  Novel Method for Selection of Regularization Parameter in the Near-field Acoustic Holography , 2011 .

[3]  Haijun Wu,et al.  A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems , 2013 .

[4]  P. Nelson,et al.  ESTIMATION OF ACOUSTIC SOURCE STRENGTH BY INVERSE METHODS: PART I, CONDITIONING OF THE INVERSE PROBLEM , 2000 .

[5]  Ding-Yu Hu,et al.  Reconstruction of the free-field radiation from a vibrating structure based on measurements in a noisy environment. , 2013, The Journal of the Acoustical Society of America.

[6]  Wenbo Lu,et al.  A method to compute the radiated sound power based on mapped acoustic radiation modes. , 2014, The Journal of the Acoustical Society of America.

[7]  David Thompson,et al.  The quantification of structure-borne transmission paths by inverse methods. Part 1: Improved singular value rejection methods , 2003 .

[8]  E. Williams.  Regularization methods for near-field acoustical holography. , 2001, The Journal of the Acoustical Society of America.

[9]  Sean F. Wu,et al.  Reconstruction of vibroacoustic responses of a highly nonspherical structure using Helmholtz equation least-squares method. , 2009, The Journal of the Acoustical Society of America.

[10]  J. Stuart Bolton,et al.  An equivalent source technique for recovering the free sound field in a noisy environment. , 2012, The Journal of the Acoustical Society of America.

[11]  Earl G Williams,et al.  Approximations of inverse boundary element methods with partial measurements of the pressure field. , 2008, The Journal of the Acoustical Society of America.

[12]  J. Antoni.  A Bayesian approach to sound source reconstruction: optimal basis, regularization, and focusing. , 2012, The Journal of the Acoustical Society of America.

[13]  Gene H. Golub,et al.  Generalized cross-validation as a method for choosing a good ridge parameter , 1979, Milestones in Matrix Computation.

[14]  Philip A. Nelson,et al.  Optimal regularisation for acoustic source reconstruction by inverse methods , 2004 .

[15]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[16]  Philip A. Nelson,et al.  ESTIMATION OF ACOUSTIC SOURCE STRENGTH BY INVERSE METHODS: PART II, EXPERIMENTAL INVESTIGATION OF METHODS FOR CHOOSING REGULARIZATION PARAMETERS , 2000 .

[17]  L. G. Copley,et al.  Integral Equation Method for Radiation from Vibrating Bodies , 1967 .

[18]  Haijun Wu,et al.  A mapping relationship based near-field acoustic holography with spherical fundamental solutions for Helmholtz equation , 2016 .

[19]  A. Seybert,et al.  An advanced computational method for radiation and scattering of acoustic waves in three dimensions , 1985 .

[20]  Jussi Rahola,et al.  Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems , 1995 .

[21]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[22]  J. Antoni,et al.  Empirical Bayesian regularization of the inverse acoustic problem , 2015 .