Browsing by Subject "surface integral equations"
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Item Open Access Effective preconditioners for iterative solutions of large-scale surface-integral-equation problems(2010) Malas, TahirA popular method to study electromagnetic scattering and radiation of threedimensional electromagnetics problems is to solve discretized surface integral equations, which give rise to dense linear systems. Iterative solution of such linear systems using Krylov subspace iterative methods and the multilevel fast multipole algorithm (MLFMA) has been a very attractive approach for large problems because of the reduced complexity of the solution. This scheme works well, however, only if the number of iterations required for convergence of the iterative solver is not too high. Unfortunately, this is not the case for many practical problems. In particular, discretizations of open-surface problems and complex real-life targets yield ill-conditioned linear systems. The iterative solutions of such problems are not tractable without preconditioners, which can be roughly defined as easily invertible approximations of the system matrices. In this dissertation, we present our efforts to design effective preconditioners for large-scale surface-integral-equation problems. We first address incomplete LU (ILU) preconditioning, which is the most commonly used and well-established preconditioning method. We show how to use these preconditioners in a blackbox form and safe manner. Despite their important advantages, ILU preconditioners are inherently sequential. Hence, for parallel solutions, a sparseapproximate-inverse (SAI) preconditioner has been developed. We propose a novel load-balancing scheme for SAI, which is crucial for parallel scalability. Then, we improve the performance of the SAI preconditioner by using it for the iterative solution of the near-field matrix system, which is used to precondition the dense linear system in an inner-outer solution scheme. The last preconditioner we develop for perfectly-electric-conductor (PEC) problems uses the same inner-outer solution scheme, but employs an approximate version of MLFMA for inner solutions. In this way, we succeed to solve many complex real-life problems including helicopters and metamaterial structures with moderate iteration counts and short solution times. Finally, we consider preconditioning of linear systems obtained from the discretization of dielectric problems. Unlike the PEC case, those linear systems are in a partitioned structure. We exploit the partitioned structure for preconditioning by employing Schur complement reduction. In this way, we develop effective preconditioners, which render the solution of difficult real-life problems solvable, such as dielectric photonic crystals.Item Open Access Solution of electromagnetic scattering problems with the locally corrected Nyström method(2010) Kılınç, SeçilThe locally corrected Nystr¨om (LCN) method is used to solve integral equations with high accuracy and efficiency. Unlike commonly used methods, the LCN method employs high-order basis functions on high-order surfaces. Hence, the number of unknowns in the electromagnetic problem decreases substantially, this also reduces the total solution time of the problem. In this thesis, electromagnetic scattering problems for arbitrary, three-dimensional, and conducting geometries are solved with the LCN method. Both the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE) are implemented. The solution time for Duffy integrals is reduced significantly by modifying the Duffy transform. Then, mixed-order basis functions are implemented to accurately represent the charge density for EFIE. Finally, both the accuracy and the efficiency (in terms of solutions times and the number of unknowns) of the LCN method are compared with the method of moments and the multilevel fast multipole algorithm.