Development and application of the FETI-DPEM algorithm for analysis of three-dimensional large-scale electromagnetic problems

The work in this dissertation is primarily concerned with the development of a fast and accurate algorithm for the numerical analysis of three-dimensional (3D) large-scale electromagnetic problems. The proposed algorithm is named the electromagnetic dual-primal finite element tearing and interconnecting (FETI-DPEM) algorithm. The FETI-DPEM hybridizes the finite element method (FEM) and the Lagrange multiplier based nonoverlapping domain decomposition method (DDM). The general principle of the FETI-DPEM is first to divide the entire computational domain into nonoverlapping subdomains, where an incomplete solution of the fields in the subdomain is first evaluated using a direct solver. Next, tangential field continuities are enforced at the subdomain interfaces by using the Lagrange multipliers. This yields an equivalent reduced-order interface equation, which can be solved using an iterative algorithm. This iterative solution using the Krylov subspace method is then accelerated by the construction of a global coarse system associated with the degrees of freedom (DOFs) defined at the subdomain corner edges, which propagates the residual error to the whole computational domain at each iteration. The solution to the interface equation serves as the boundary condition (BC) for individual subdomain problems to evaluate the fields inside the subdomains There are two different approaches to enforcing the field continuities at the subdomain interfaces. The first one imposes the tangential field continuities through the Dirichlet transmission condition and results in the first version of the FETI-DPEM algorithm, which is referred to as the FETI-DPEM1 in this dissertation. The FETI-DPEM1 is shown to be numerically scalable with respect to (w.r.t.) the size of finite elements and number of subdomains, and it is conditionally scalable w.r.t. the working frequency. A slowdown in the convergence of the iterative solution of the interface equation, due to the use of the Neumann BC at the subdomain interfaces, has been observed as the electrical size of the subdomain increases beyond the cutoff frequency of the lowest-order resonant mode of the subdomain with its surfaces enclosed by perfect magnetic conducting (PMC) surfaces. The problem of the slowed convergence at higher frequencies is then alleviated by using a different set of interface conditions, which leads to the FETI-DPEM2 algorithm. The FETI-DPEM2 combines the dual-primal (DP) idea with two Lagrange multipliers and implements the Robin-type transmission condition at the subdomain interfaces to significantly improve the convergence of the interface solution in the high-frequency region. The numerical scalability of the algorithm ensures the achievable parallel efficiency and is successfully implemented on massively parallel systems. As in the frequency domain, a volumetric problem can be converted into a surface problem by using the electromagnetic surface equivalence principle, and when applying the FETI-DPEM algorithm to the analysis of large antenna arrays with repeated unit tells, the geometrical repetition is fully utilized by performing the subdomain volume-to-surface conversion only for geometrically different unit tells. Therefore, the FETI-DPEM algorithm becomes ideal for fast analysis of finite arrays consisting of only a few different unit tells, such as antenna arrays and photonic crystals (PhCs). For applications involving broadband simulations, the FETI-DPEM algorithm is further accelerated by incorporating with fast frequency sweep techniques, such as the asymptotic waveform evaluation (AWE) and complex frequency hopping (CFH) method. When dealing with 3D open-region radiation and scattering problems, the computational accuracy is improved by using the second-order absorbing boundary condition (ABC) as a truncation. The FETI-DPEM implementation of such an ABC using auxiliary variables eliminates the limitation of the domain partition strategy. The resultant parallel FETI-DPEM solver is applied to the analysis of antenna array mutual coupling in the presence of a large and complex platform. Furthermore, the FETI-DPEM algorithm is incorporated with the implicit restarted Arnoldi method (IRAM) and applied to the analysis of eigenvalue problems for a selected number of eigenvalues and eigenvectors using the shift-and-invert spectral transformation.;The dissertation first briefly introduces the basics of the frequency-domain (FD) FEM in electromagnetics (EM), which includes the discussion of ABCs for the open-region analysis. It is then followed by a detailed study of the FETI-DPEM1 algorithm to show its efficiency and capability. The applications of the FETI-DPEM1 are then further extended to the analysis of complex PhC devices, which provides a fast and accurate solution to the numerical analysis of 3D PhCs. The FETI-DPEM2 algorithm with an excellent numerical scalability is then presented together with the details of its parallel implementation using the message passing interface (MPI). (Abstract shortened by UMI.)...