Implicit methods of solution to integral formulations in boundary element method based nearfield acoustic holography

Abstract

Two integral equation methods are considered which can be inverted to provide the surface velocity and/or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body. This problem is central to nearfield acoustical holography (NAH). The integral equations are discretized using boundary element methods (BEMs). The integral equation methods considered are (1) an indirect formulation method based on the single layer integral equation and (2) a direct formulation method based on a system of equations derived from the Helmholtz–Kirchhoff integral equation. The formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices, but instead will require this inversion to be done implicitly. Since these methods back-track the sound field from the measurement surface to the surface of the source/vibrator they are ill-posed in nature and Tikhonov regularization are used to stabilize the reconstruction. Problems are noted with the use of higher order shape functions in the BEM discretization which are detrimental to the inversion, and it is shown that the breakdown of the method at the interior resonance frequencies can be mitigated successfully. These two approaches are compared with the conventional direct method using measured experimental data from a point-driven rectangular plate.