Helmholtz Equation Least Squares Research Articles

Reconstructing interior acoustic pressure fields via Helmholtz equation least-squares method

This paper extends the Helmholtz equation least-squares (HELS) method previously developed by Wang and Wu [J. Acoust. Soc. Am. 102, 2020–2032 (1997)] to reconstruction of acoustic pressure fields inside the cavity of a vibrating object. The acoustic pressures are reconstructed through an expansion of the acoustic modes generated by the Gram–Schmidt orthonormalization with respect to the particular solutions to the Helmholtz equation. Such an expansion is uniformly convergent because the selected acoustic modes consist of a uniformly convergent series of Legendre functions. The coefficients associated with these acoustic modes are determined by requiring the assumed-form solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Numerical examples of partially vibrating spheres and cylinders with various half-length to radius aspect ratios subject to different frequency excitations are demonstrated. The reconstructed acoustic pressures are compared with the analytic solutions and numerical ones obtained by using the standard boundary element method (BEM) codes.

On reconstruction of acoustic pressure fields using the HELS method

The Helmholtz equation-least squares (HELS) method was recently developed by Wang and Wu [J. Acoust. Soc. Am. 102, 2020–2032 (1997)] to reconstruct the acoustic pressures generated by an arbitrary vibrating structure. In this method the acoustic pressures are expressed in terms of an expansion of ‘‘acoustic modes.’’ The coefficients associated with these acoustic modes are determined by requiring the assumed-form solution to satisfy the pressure boundary condition at measurement points. The errors incurred in this process are minimized by the least-squares method. The advantages of this method are that (1) acoustic pressures can be reconstructed over the entire surface; (2) solutions thus obtained are unique; and (3) efficiency of numerical computation is high, because the number of measurements is determined by that of expansion terms, which is small when a right coordinate system is selected for a particular source geometry under consideration. Such an approach is applicable to both exterior and interior regions. For exterior problems, the reconstructed acoustic pressures can be accurate when the points fall outside the minimum circumscribing spheroidal surface which encloses the vibrating surface, but approximate when they are inside. For interior problems, the accuracy of reconstruction is high because the entire region is inside the minimum circumscribing spheroidal surface. In both cases, the accuracy of reconstruction is limited by excitation frequencies, which is inherent in all expansion theories. [Work supported by NSF.]

Reconstruction of acoustic pressure fields inside a vibrating cavity

The Helmholtz equation–least-squares (HELS) method developed by Wang and Wu [J. Acoust. Soc. Am. (to be published)] is extended to reconstruction of acoustic pressure fields inside a vibrating cavity. In this method, the acoustic pressures are reconstructed through an expansion of acoustic modes, or a set of independent functions generated via an orthonormalization with respect to the particular solutions to the Helmholtz equation on the particular surface under consideration. The coefficients associated with these acoustic modes are determined by requiring the assumed form solution to satisfy the pressure boundary condition at the measurement points. The errors incurred in this process are minimized by the least-squares method. Numerical examples of partially vibrating spheres and cylinders with various half-length to radius ratios and dimensionless frequencies are demonstrated. The results thus obtained are compared with the numerical solutions obtained by using a direct boundary-element method (BEM). Results show that the HELS method enables one to reconstruct the radiated acoustic pressure fields much more effectively and accurately than BEM does. [Work supported by the Institute for Manufacturing Research at Wayne State University.]