Exterior problems of acoustics by fractal finite element mesh

Abstract

The propagation and attenuation of acoustic waves in an exterior domain is an essential ingredient in the study of acoustic–structure interaction. In this paper the problems of acoustic radiation from an arbitrarily shaped vibrating body in an infinite exterior region are investigated by using a fractal two-level finite element mesh (FEM) with self-similar layers in the media enclosing the conventional FEM for the vibrating body. The fractal two-level FEM has been successfully used in stress intensity factor prediction with self-similar ratio smaller than one so that the mesh converges to the crack tip. In this paper, the similarity ratio is bigger than one so that the mesh extends to infinity. By means of the Hankel functions satisfying automatically Sommerfeld's radiation conditions at infinity, the different unknown nodal pressures in different layers are transformed to some common unknowns of the Hankel coefficients. The final matrix size of the exterior region is equal to the number of terms in the Hankel expansion. The set of infinite number of unknowns of nodal pressure is reduced to a set of small finite number of Hankel's coefficients. All layers have the same unknowns after the transformation. Due to self-similarity, the transformed stiffness matrix of the first layer is proportional to that of the second and so on. Therefore, the stiffness matrices of the infinite layers can be summed by using just one layer. Numerical examples show that this method is efficient and accurate in solving unbounded acoustic problems.