On the location and number of expansion centers of multiple multipole expansion method
Abstract
The multiple multipole expansion, or MME technique is a numerical method for solving electromagnetic scattering problems. Multipole fields are used as basis functions of expansion of the electric and magnetic fields. Specifically, the scattered field is expanded as the sum of multipole fields representing a finite linear combination of fields generated by a certain system of equivalent sources placed in the interior of the scatterer at some distance away from the boundary. The unknown expansion coefficients are determined with a least square procedure from a system of linear equations obtained by enforcing the boundary conditions at some selected points. Numerical efficiency and computational accuracy are decided by the number of sources, N, and matched points, M. A criterion based on the concept of packing number is adopted to provide a variable sampling density that determines the optimal size of the matrix to be inverted. Addition and translation theorems are used to translate the sources from the origin to the desired locations. The locations are automatically determined once M and N are chosen. By varying M and N, the minimum error on the boundary condition is obtained. In this work, oversampling is observed and means to optimally truncate the matrix are found. Numerical efficiency and computational accuracy are simultaneously achieved. The numerical implementation of the technique is demonstrated by considering scattering from an infinitely long conducting circular and elliptical cylinder and an inductive post inside a waveguide. In each case, values of M and N are obtained that yield minimum discrepancy at the boundary. Numerical results obtained here agree with previous work obtained by trial and error means as well as with measured data.
This paper has been withdrawn.