Fast adaptive multipole-algorithm for scattering from inhomogeneous dielectric cylinders of arbitrary cross-section

Abstract

The work reported in this thesis is concerned with the rapid solution of electromagnetic scattering problems involving inhomogeneous dielectric cylinders of arbitrary cross?section. In particular, attention is focused on the scattering from electrically large structures. The electromagnetic problem is formulated in terms of volume integral equations and the solution is sought by numerically solving the integral equations. This involves the discretisation of the integral equation using method of moments and a subsequent solution of the matrix equation using a CG?like iterative method. All CG?like iterative methods require the application of the coefficient matrix to a given vector in every iteration, and this requires an effort proportional to the square of the number of unknowns. Therefore, bulk of the CPU time is spent in computing these products. In the present work, an algorithm is presented for the fast evaluation of matrix–vector products. The proposed algorithm is based on the theory of the fast multipole method and uses an adaptive clustering of the discretisation cells for obtaining the speed?up. It is shown that, for a given maximum linear dimension of the scatterer, in terms of the free?space wavelength, irrespective of the material properties and shape of the scatterer, the proposed algorithm (hereafter referred to as the LTA) computes the matrix–vector products in O(N) operations, where N is the number of cells used in the discretisation. It is further shown that the algorithm does not depend on the size and shape of the cells used for discretising the integral equation. Numerical results from two implementations of the LTA are presented. These results verify the theoretical predictions on the performance of the algorithm. In particular, the speed?up that can be obtained through the use of the LTA, its accuracy and range of applicability are demonstrated through several numerical examples. The ability to handle different types of integral equations and various types of discretisation procedures is also demonstrated through numerical examples. Finally, an algorithm is proposed for efficiently handling multiple incident fields. This algorithm is based on a Fourier–Bessel expansion of the incident field and it attempts to synthesise the solution to a given incident field from a set of known solutions. It is shown that this algorithm is independent of the material properties of the scatterer and is applicable to a wide variety of scatterers. Numerical results are also presented demonstrating the accuracy and applicability of the algorithm.