Fractal two-level finite element mesh for acoustic scattering problems

Abstract

The problems of acoustic waves scattered by scatterer immersed in unbounded domain is an essential ingredient in the study of acoustic-structure interaction. In this paper the problems of acoustic scattering in an infinite exterior region are investigated by using a fractal two-level finite element mesh with self-similar layers in the media which encloses the conventional finite element mesh for the cavity. The similarity ratio is bigger than one so that the fractal mesh extends to infinity. Because of the self-similarity, the equivalent stiffness (mass) matrix of one layer is proportional to the others. By means of the Hankel functions automatically satisfying Sommerfeld's radiation conditions at infinity, the different unknown nodal pressures on different layers are transformed to some common unknowns of the Hankel coefficients. The set of infinite number of unknowns of nodal pressure is reduced to the set of finite number of Hankel's coefficients. All layers have the same matrix dimension after the transformation and the respective matrices of each layer are summed. Due to the proportionality, the infinite number of layers can be summed in closed form as the entries of each matrix are in geometric series. That is, processing one layer is enough to virtually represent a set of infinite number of layers covering an infinity domain. No new elements are created. Numerical examples show that this method is efficient and accurate in solving unbounded acoustic problems.