Approximations of Integral Equations for WaveScattering

Abstract: Wave scattering is the phenomenon in which a wave field interacts with physical objects. An incoming wave is scattered at the surface of the object and a scattered wave is produced. Common practical cases are acoustic, electromagnetic and elastic wave scattering. The numerical simulation of the scattering process is important, for example, in noise control, antenna design, prediction of radar cross sections and nondestructive testing.Important classes of numerical methods for accurate simulation of scattering are based on integral representations of the wave fields and theses representations require the knowledge of potentials on the surfaces of the scattering objects. The potential is typically computed by a numerical approximation of an integral equation that is defined on the surface. We first develop such numerical methods in time domain for the scalar wave equation. The efficiency of the techniques are improved by analytic quadrature and in some cases by local approximation of the potential.Most scattering simulations are done for harmonic or single frequency waves. In the electromagnetic case the corresponding integral equation method is called the method of moments. This numerical approximation is computationally very costly for high frequency waves. A simplification is suggested by physical optics, which directly gives an approximation of the potential without the solution of an integral equation. Physical optics is however only accurate for very high frequencies.In this thesis we improve the accuracy in the physical optics approximation of scalar waves by basing the computation of the potential on the theory of radiation boundary conditions. This theory describes the local coupling of derivatives in the wave field and if it is applied at the surface of the scattering object it generates an expression for the unknown potential. The full wave field is then computed as for other integral equation methods.The new numerical techniques are analyzed mathematically and their efficiency is established in a sequence of numerical experiments. The new on surface radiation conditions give, for example, substantial improvement in the estimation of the scattered waves in the acoustic case. This numerical experiment corresponds to radar cross-section estimation in the electromagnetic case.