Aalto University, School of Arts, Design and Architecture, Department of Arts
Abstract
A variety of electromagnetic field problems can be most elegantly formulated by integral equations. A common way to search for a solution to such an integral equation is the method of moments (MoM) where the equation is discretised by expanding the unknown function in basis functions and forcing the error in the approximation to be orthogonal to test functions. Many times, the method can be enhanced by exploiting wave functions together with addition theorems for them.
The thesis treats three electromagnetic field problems formulated by integral equations: one electrostatic, one magnetostatic and one time-harmonic. The geometry in the static problems consists of ring conductors, and the solution can be constructed by using sophisticated entire-domain basis functions and Galerkin's method. The geometry in the time-harmonic problem is an extremely complex model of a pine tree, and therefore, the solution must be formed by using simple sub-domain basis functions and point matching.
In each of the above solutions, wave functions together with addition theorems for them are exploited. In the static problems, the Green's function is expanded in cylindrical wave functions, the MoM matrix terms are formulated partly in the spectral domain using addition theorems for the cylindrical wave functions, and certain integral results are derived from addition theorems for ultra-spherical wave functions. In the time-harmonic problem, the discretised problem is solved by using an iterative scheme and the calculation is accelerated by using the Multilevel fast multipole algorithm (MLFMA) which is based on addition theorems for spherical wave functions.
The thesis is based upon five publications. The first two publications present an efficient and accurate method for calculating the capacitances and inductances of ring conductors in a layered medium. The third publication gives a unified and transparent derivation of translational addition theorems for spherical wave functions. The last two publications concern the MLFMA. The former one describes a broadband version of the MLFMA with some novel ideas, and the latter one applies the algorithm in calculating the scattering of an electromagnetic plane wave by a pine tree
