On the use of Higher Order and Irregularly Shaped Boundary Elements in Nearfield Acoustical Holography

Nearfield acoustical holography (NAH) is used to find the regions of acoustic activity on the surface of a sound radiating object. One of the most general NAH approaches is the inverse frequency response function (IFRF) technique, since it imposes no limitations on the geometry of the radiating boundary. In the IFRF method acoustic measurements on a grid in the nearfield of the object are used to determine the corresponding normal velocity distribution on the surface of the object. The measured sound field is related to the surface vibrations via a transfer matrix, which is calculated using a boundary element method. The required inversion of the transfer matrix is not that simple, because the system is ill-conditioned. Hence, a physically meaningful solution can only be obtained by applying regularization techniques. In the literature, the boundary surface of the source is usually meshed with constant or linear elements of equal shape and size. Most likely, this is done to avoid the effect that the regularized inversion process favors nodes that are associated with a high mean square surface normal velocity. This effect is due to the fact that such nodes have a more effective contribution to the sound field. In this paper it is demonstrated that the problem in which the inverse solution is affected by the topology of the mesh gets even worse when quadratic elements are applied. A new technique will be described that circumvents this problem completely by the introduction of an appropriate smoothing operator. As a result all boundary nodes are treated in an equal way, irrespective of their associated area or type of shape function. With the presented approach, irregular meshes and/or higher order boundary elements can be successfully used in NAH applications.