Convergent Solvers for High-Frequency Wave Scattering

Case Western Reserve University, Cleveland OH

Investigators

Abstract

The calculation of wave scattering from acoustically or electromagnetically large objects is one of the most interesting and challenging problems in computational science. Nevertheless, current state-of-the-art simulation technology is limited by the competing demands of accuracy (which requires an increasing number of degrees of freedom to resolve the fields on the scale of the wavelength) and efficiency (which favors coarse discretizations). The PI proposes to develop and rigorously analyze new integral equation methods that overcome these limitations by avoiding the need to discretize the fields on the scale of the wavelength, while retaining error-controllability. The PI proposes to develop such methods for acoustic scattering by impenetrable bodies in two and three dimensions, with the longer-term goal of developing methods for full electromagnetic scattering in three dimensions. The methods the PI proposes to develop will combine existing, high-order discretizations techniques based on smooth partitions of unity and FFTs with new, asymptotically-derived integral equations that can be solved numerically with a prescribed error tolerance for arbitrarily high frequencies, within a fixed computational time. The development of these methods calls for a number of intellectual innovations. The PI proposes to (i) further advance his work on high-order high-frequency integral equation solvers for surface scattering by smooth, convex obstacles. Here the effort will concentrate on an extension of the convergent, localized singular integrators developed in the two-dimensional setting to three dimensions; (ii) develop methods suitable for non-convex obstacles, using the infinite frequency limit of the problem (i.e., geometrical optics) to approximate the phase of the highly oscillatory unknowns; and (iii) introduce new high-order accurate integration schemes in small regions near geometrical singularities of the scatterers, in order to deal with complex obstacles with sharp features such as corners and edges. The proposed research has the potential to impact numerous scientific and engineering disciplines dealing with acoustic and electromagnetic wave propagation phenomena, which arise in a wide range of applications. Examples include antenna design (from consumer electronics devices like cell phones to satellites and space exploration probes), non-destructive testing, medical and biological applications (using ultrasound, tomography), electromagnetic compatibility testing and certification, evaluation of propagation effects on communication systems, design of low observable military hardware, automatic target recognition, etc. The ever increasing frequencies involved in these problems places many of them beyond the capabilities of currently available computational methods, thus requiring new and innovative mathematical approaches. This proposal is focused on the development of such an approach, leading to solvers whose computational cost is virtually independent of frequency, and which are thus able to accurately solve such problems for a tremendously larger range of frequencies than current techniques.