A study of the effective properties of complex scatterers using multiresolution decomposition

A systematic study of the across scale coupling phenomenology in scattering problems is addressed using the theory of multiresolution decomposition and orthogonal wavelets. By projecting an integral equation formulation of the scattering problem onto a set of sub-spaces that constitutes a multiresolution decomposition of L₂(R), one can derive two coupled formulations. The first governs the macroscale response, and the second governs the microscale response. By substituting the formal solution of the latter into the former, a new self consistent formulation that governs the macroscale response component is obtained. This formulation is written on a macroscale grid, where the effects of the microscale heterogeneity is expressed via an across scale coupling operator. This operator can also be interpreted as representing the effective properties of the microstructure. We study the properties of this operator versus the characteristics of the Green function, and the microstructure for various structural acoustics problems using general asymptotic considerations. Specific examples of scattering from a thin, linearly elastic, fluid-loaded plate with surface mass density, or bending stiffness variations, are provided. We show that mass microscale variation has virtually no effect on the macroscale response, and stiffness microscale variations can significantly affect the macroscale response. Classes of stiffness variations that have an identical macroscale response are derived using a one dimensional local constitutive relation developed to govern the macroscale field components. The results are supported by numerical examples.