Integral equation based computational methods for complex electromagnetic analysis

Abstract

As the one of major solutions to the electromagnetic problems, integral equation methods have been widely explored by computational electromagnetics community. The error controllable and efficient solutions to integral equations are highly desired. Due to today’s challenging complex electromagnetic environments, well-conditioned integral formulations across wide frequencies of interest are also required. These motivated us to develop novel integral equation methods from the fundamental theory point of view to advance complex electromagnetic analysis methodologies in this dissertation. First, a smooth spherical surface instead of the conventionally used cubical surface is employed as the equivalence surface in the equivalence principle algorithm (EPA). As a result, the geometrical singularities of cubical equivalence surfaces can be totally avoided. In order to achieve high order accuracy, a meshfree quadrature rule is used on the spherical surface to allocate the equivalent sources. Meanwhile, a hybrid method consisting of accelerated Cartesian expansion (ACE) and fast multipole algorithm (FMA) is used as the fast solver for the induced current that is required in EPA. Compared to direct solvers, it can reduce the computational time and memory storage significantly especially for the object of interest featuring fine details. Inspired by the aforementioned smooth equivalence surface applied in EPA, a relaxed hierarchical equivalent source algorithm (RHESA) is developed for volume integral equations. Using equivalent sources defined on spherical equivalence surface, the computations of interactions between wellseparated groups can be accelerated. A pertinent grouping scheme is exploited and hence no tap basis function is required that could be cumbersome for EPA when applied to adjoint domains. Compared to FMA, the proposed algorithm is more general and less kernel dependent. The accuracy and the estimated order of the complexity of RHESA are analyzed and validated. Finally, a well-conditioned electric field integral equation (EFIE) based on generalized Debye sources, called GDS-EFIE, is proposed. Different from the traditional EFIE, the proposed GDS-EFIE is formulated in terms of scalar quantities by using surface Helmholtz decomposition and surface Laplacian operator. As a result, GDS-EFIE can be immune from low frequency breakdown unlike the traditional EFIE. The subdivision basis set obtained by subdivision surfaces technique is exploited to expand the unknown scalars and find the inverse of surface Laplacian operator. Hence, the GDS-EFIE can be discretized within the isogeometric analysis (IGA) framework which is an emerging technology for bridging engineering design process and engineering analysis process. The low frequency stability and flexibility of the proposed integral formulation are well-validated through numerical benchmarks.