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Item Open AccessAccelerating the multilevel fast multipole algorithm with the sparse-approximate-inverse (SAI) preconditioning(Society for Industrial and Applied Mathematics, 2009) Malas, T.; Gürel, Levent; Gürel, LeventWith the help of the multilevel fast multipole algorithm, integral-equation methods can be used to solve real-life electromagnetics problems both accurately and efficiently. Increasing problem dimensions, on the other hand, necessitate effective parallel preconditioners with low setup costs. In this paper, we consider sparse approximate inverses generated from the sparse near-field part of the dense coefficient matrix. In particular, we analyze pattern selection strategies that can make efficient use of the block structure of the near-field matrix, and we propose a load-balancing method to obtain high scalability during the setup. We also present some implementation details, which reduce the computational cost of the setup phase. In conclusion, for the open-surface problems that are modeled by the electric-field integral equation, we have been able to solve ill-conditioned linear systems involving millions of unknowns with moderate computational requirements. For closed surface problems that can be modeled by the combined-field integral equation, we reduce the solution times significantly compared to the commonly used block-diagonal preconditioner. Item Open AccessAccuracy and efficiency considerations in the solution of extremely large electromagnetics problems(IEEE, 2011) Gürel, Levent; Ergül, Özgür; Gürel, Levent; Ergül, ÖzgürThis study considers fast and accurate solutions of extremely large electromagnetics problems. Surface formulations of large-scale objects lead to dense matrix equations involving millions of unknowns. Thanks to recent developments in parallel algorithms and high-performance computers, these problems can easily be solved with unprecedented levels of accuracy and detail. For example, using a parallel implementation of the multilevel fast multipole algorithm (MLFMA), we are able to solve electromagnetics problems discretized with hundreds of millions of unknowns. Unfortunately, as the problem size grows, it becomes difficult to assess the accuracy and efficiency of the solutions, especially when comparing different implementations. This paper presents our efforts to solve extremely large electromagnetics problems with an emphasis on accuracy and efficiency. We present a list of benchmark problems, which can be used to compare different implementations for large-scale problems. © 2011 IEEE. Item Open AccessAccuracy: The Frequently Overlooked Parameter in the Solution of Extremely Large Problems(IEEE, 2011) Ergul, O.; Gürel, Levent; Gürel, LeventWe investigate error sources and their effects on the accuracy of solutions of extremely large electromagnetics problems with parallel implementations of the multilevel fast multipole algorithm (MLFMA). Accuracy parameters and their effects on the accuracy of MLFMA solutions are studied for large-scale problems discretized with hundreds of millions of unknowns. We show that some error sources are more dominant and should be suppressed for more accurate solutions; identifying less-effective error sources may allow us to derive more efficient implementations. Based on our analysis, we determine a set of benchmark problems that can be used to compare the accuracy of solvers for large-scale computations. A benchmarking tool is provided at www.cem.bilkent.edu.tr/ benchmark. Item Open AccessAccurate modeling of metamaterials with MLFMA(ESA Publications, 2006) Ergül, Özgür; Ünal, Alper; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürElectromagnetic modelling of large metamaterial (MM) structures employing multilevel fast multipole algorithm (MLFMA) is reported. MMs are usually constructed by periodically embedding unit cells, such as split-ring resonators (SRRs), into a host medium. Without utilizing any homogenization techniques, we accurately model large numbers of unit cells that translate into very large computational problems. By considering all of the electromagnetic interactions, the resulting dense matrix equations are solved iteratively with the accelerated matrix-vector products by MLFMA. To increase the efficiency, we also employ parallel computing in the solutions of large SRR problems. Item Open AccessAccurate solutions of extremely large integral-equation problems in computational electromagnetics(IEEE, 2013-02) Ergül, Ö; Gürel, Levent; Gürel, LeventAccurate simulations of real-life electromagnetics problems with integral equations require the solution of dense matrix equations involving millions of unknowns. Solutions of these extremely large problems cannot be achieved easily, even when using the most powerful computers with state-of-the-art technology. However, with the multilevel fast multipole algorithm (MLFMA) and parallel MLFMA, we have been able to obtain full-wave solutions of scattering problems discretized with hundreds of millions of unknowns. Some of the complicated real-life problems (such as scattering from a realistic aircraft) involve geometries that are larger than 1000 wavelengths. Accurate solutions of such problems can be used as benchmarking data for many purposes and even as reference data for high-frequency techniques. Solutions of extremely large canonical benchmark problems involving sphere and National Aeronautics and Space Administration (NASA) Almond geometries are presented, in addition to the solution of complicated objects, such as the Flamme. The parallel implementation is also extended to solve very large dielectric problems, such as dielectric lenses and photonic crystals. Item Open AccessAccurate solutions of scattering problems involving low-contrast dielectric objects with surface integral equations(Institution of Engineering and Technology, 2007) Ergül, Özgür; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürWe present the stabilization of the surface integral equations for accurate solutions of scattering problems involving low-contrast dielectric objects. Unlike volume formulations, conventional surface formulations fail to provide accurate results for the scattered fields when the contrast of the object is small. Therefore, surface formulations are required to be stabilized by extracting the nonradiating parts of the equivalent currents. In addition to previous strategies for the stabilization, we introduce a novel procedure called field-based stabilization (FBS) based on using fictitious incident fields and rearranging the right-hand-side of the equations. The results show that the formulations using FBS provide accurate results even for scattering problems involving extremely low-contrast objects, while the extra cost due to the stabilization procedure is negligible. Item Open AccessAdvanced partitioning and communication strategies for the efficient parallelization of the multilevel fast multipole algorithm(IEEE, 2010) Ergül O.; Gürel, Levent; Gürel, LeventLarge-scale electromagnetics problems can be solved efficiently with the multilevel fast multipole algorithm (MLFMA) , which reduces the complexity of matrix-vector multiplications required by iterative solvers from O(N 2) to O(N logN). Parallelization of MLFMA on distributed-memory architectures enables fast and accurate solutions of radiation and scattering problems discretized with millions of unknowns using limited computational resources. Recently, we developed a hierarchical partitioning strategy , which provides an efficient parallelization of MLFMA, allowing for the solution of very large problems involving hundreds of millions of unknowns. In this strategy, both clusters (sub-domains) of the multilevel tree structure and their samples are partitioned among processors, which leads to improved load-balancing. We also show that communications between processors are reduced and the communication time is shortened, compared to previous parallelization strategies in the literature. On the other hand, improved partitioning of the tree structure complicates the arrangement of communications between processors. In this paper, we discuss communications in detail when MLFMA is parallelized using the hierarchical partitioning strategy. We present well-organized arrangements of communications in order to maximize the efficiency offered by the improved partitioning. We demonstrate the effectiveness of the resulting parallel implementation on a very large scattering problem involving a conducting sphere discretized with 375 million unknowns. ©2010 IEEE. Item Open AccessAlgebraic acceleration and regularization of the source reconstruction method with the recompressed adaptive cross approximation(IEEE, 2014) Kazempour, Mahdi; Gürel, Levent; Gürel, LeventWe present a compression algorithm to accelerate the solution of source reconstruction problems that are formulated with integral equations and defined on arbitrary three-dimensional surfaces. This compression technique benefits from the adaptive cross approximation (ACA) algorithm in the first step. A further error-controllable recompression is applied after the ACA. The numerical results illustrate the efficiency and accuracy of the proposed method. © 2014 IEEE. Item Open AccessAnalysis of dielectric photonic-crystal problems with MLFMA and Schur-complement preconditioners(IEEE, 2011-01-13) Ergül, Özgür; Malas, T.; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürWe present rigorous solutions of electromagnetics problems involving 3-D dielectric photonic crystals (PhCs). Problems are formulated with recently developed surface integral equations and solved iteratively using the multilevel fast multipole algorithm (MLFMA). For efficient solutions, iterations are accelerated via robust Schur-complement preconditioners. We show that complicated PhC structures can be analyzed with unprecedented efficiency and accuracy by an effective solver based on the combined tangential formulation, MLFMA, and Schur-complement preconditioners. Item Open AccessAnalysis of double-negative materials with surface integral equations and the multilevel fast multipole algorithm(IEEE, 2011) Ergül O.; Gürel, Levent; Gürel, LeventWe present a fast and accurate analysis of double-negative materials (DNMs) with surface integral equations and the multilevel fast multipole algorithm (MLFMA). DNMs are commonly used as simplified models of metamaterials at resonance frequencies and are suitable to be formulated with surface integral equations. However, realistic metamaterials and their models are usually very large with respect to wavelength and their accurate solutions require fast algorithms, such as MLFMA. We consider iterative solutions of DNMs with MLFMA and we investigate the accuracy and efficiency of solutions when DNMs are formulated with two recently developed formulations, namely, the combined tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCFIE). Numerical results on canonical objects are consistent with previous results in the literature on ordinary objects. © 2011 IEEE. Item Open AccessAnalysis of Lossy Dielectric Objects with the Multilevel Fast Multipole Algorithm(IEEE, 2011) Ergul, O.; Gurel, LeventRigorous solutions of electromagnetics problems involving lossy dielectric objects are considered. Problems are formulated with two recently developed formulations, namely, the combined-tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCFIE), and solved iteratively using the multilevel fast multipole algorithm (MLFMA). Accuracy and efficiency of solutions are compared for different objects and conductivity values. We show that iterative solutions of CTF are significantly accelerated as the conductivity increases and CTF becomes a good alternative to JMCFIE in terms of efficiency. Considering also its high accuracy, CTF seems to be a suitable formulation for the analysis of lossy dielectric objects. Item Open AccessAnalysis of photonic-crystal problems with MLFMA and approximate Schur preconditioners(IEEE, 2009-07) Ergül, Özgür; Malas, Tahir; Kılınç, Seçil; Sarıtaş, Serkan; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürWe consider fast and accurate solutions of electromagnetics problems involving three-dimensional photonic crystals (PhCs). Problems are formulated with the combined tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCFIE) discretized with the Rao-Wilton-Glisson functions. Matrix equations are solved iteratively by the multilevel fast multipole algorithm. Since PhC problems are difficult to solve iteratively, robust preconditioning techniques are required to accelerate iterative solutions. We show that novel approximate Schur preconditioners enable efficient solutions of PhC problems by reducing the number of iterations significantly for both CTF and JMCFIE. ©2009 IEEE. Item Open AccessApproximate MLFMA as an efficient preconditioner(IEEE, 2007) Malas, Tahir; Ergül, Özgür; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürIn this work, we propose a preconditioner that approximates the dense system operator. For this purpose, we develop an approximate multilevel fast multipole algorithm (AMLFMA), which performs a much faster matrix-vector multiplication with some relative error compared to the original MLFMA. We use AMLFMA to solve a closely related system, which makes up the preconditioner. Then, this solution is embedded in the main solution that uses MLFMA. By taking into account the far-field elements wisely, this preconditioner proves to be much more effective compared to the near-field preconditioners. Item Open AccessBenchmark Solutions of Large Problems for Evaluating Accuracy and Efficiency of Electromagnetics Solvers(IEEE, 2011) Ergul, O.; Gürel, Levent; Gürel, LeventWe present a set of benchmark problems involving conducting spheres and their solutions using a parallel implementation of the multilevel fast multipole algorithm (MLFMA). Accuracy of the implementation is tested by comparing the computational results with analytical Mie-series solutions. Reference solutions are made available on an interactive website to evaluate and compare the accuracy and efficiency of fast solvers. We also demonstrate the capabilities of our solver on real-life problems involving complicated targets, such as the Flamme. Item Open AccessCircular arrays of log-periodic antennas for broadband applications(IEEE, 2006) Ergül, Özgür; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürCircular arrays of log-periodic (LP) antennas are designed for broadband applications. A sophisticated electromagnetic simulation environment involving integral equations and fast solvers is developed to analyze the LP arrays both accurately and efficienuy. The resulting matrix equation obtained by the discretization of the electric field integral equation is solved iteratively via the multilevel fast multipole algorithm (MLFMA). Genetic algorithms interacting with MLFMA is employed to optimize the excitations of the array elements to increase the frequency independence and also to add the beam-steering ability to the arrays. Item Open AccessComparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm(Institute of Electrical and Electronics Engineers, 2009) Ergül, Özgür; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürWe consider fast and accurate solutions of scattering problems involving increasingly large dielectric objects formulated by surface integral equations. We compare various formulations when the objects are discretized with Rao-Wilton-Glisson functions, and the resulting matrix equations are solved iteratively by employing the multilevel fast multipole algorithm (MLFMA). For large problems, we show that a combined-field formulation, namely, the electric and magnetic current combined-field integral equation (JMCFIE), requires fewer iterations than other formulations within the context of MLFMA. In addition to its efficiency, JMCFIE is also more accurate than the normal formulations and becomes preferable, especially when the problems cannot be solved easily with the tangential formulations. Item Open AccessComputational analysis of complicated metamaterial structures using MLFMA and nested preconditioners(IEEE, 2007-11) Ergül, Özgür; Malas, Tahir; Yavuz, Ç; Ünal, Alper; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürWe consider accurate solution of scattering problems involving complicated metamaterial (MM) structures consisting of thin wires and split-ring resonators. The scattering problems are formulated by the electric-field integral equation (EFIE) discretized with the Rao-Wilton- Glisson basis functions defined on planar triangles. The resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications that are required by the iterative solvers are accelerated with the multilevel fast multipole algorithm (MLFMA). Since EFIE usually produces matrix equations that are ill-conditioned and difficult to solve iteratively, we employ nested preconditioners to achieve rapid convergence of the iterative solutions. To further accelerate the simulations, we parallelize our algorithm and perform the solutions on a cluster of personal computers. This way, we are able to solve problems of MMs involving thousands of unit cells. Item Open AccessComputational study of scattering from healthy and diseased red blood cells(Society of Photo Optical Instrumentation Engineers, 2010-08-05) Ergül, Özgür; Arslan-Ergül, A.; Gürel, Levent; Gürel, Levent; Ergül, ÖzgürWe present a comparative study of scattering from healthy red blood cells (RBCs) and diseased RBCs with deformed shapes. Scattering problems involving three-dimensional RBCs are formulated accurately with the electric and magnetic current combined-field integral equation and solved efficiently by the multilevel fast multipole algorithm. We compare scattering cross section values obtained for different RBC shapes and different orientations. In this way, we determine strict guidelines to distinguish deformed RBCs from healthy RBCs and to diagnose various diseases using scattering cross section values. The results may be useful for designing new and improved flow cytometry procedures. Item Open AccessContamination of the accuracy of the combined-field integral equation with the discretization error of the magnetic-field integral equation(Institute of Electrical and Electronics Engineers, 2009) Gürel, Levent; Ergül, Özgür; Gürel, Levent; Ergül, ÖzgürWe investigate the accuracy of the combined-field integral equation (CFIE) discretized with the Rao-Wilton-Glisson (RWG) basis functions for the solution of scattering and radiation problems involving three-dimensional conducting objects. Such a low-order discretization with the RWG functions renders the two components of CFIE, i.e., the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE), incompatible, mainly because of the excessive discretization error of MFIE. Solutions obtained with CFIE are contaminated with the MFIE inaccuracy, and CFIE is also incompatible with EFIE and MFIE. We show that, in an iterative solution, the minimization of the residual error for CFIE involves a breakpoint, where a further reduction of the residual error does not improve the solution in terms of compatibility with EFIE, which provides a more accurate reference solution. This breakpoint corresponds to the last useful iteration, where the accuracy of CFIE is saturated and a further reduction of the residual error is practically unnecessary. Item Open AccessDesign and simulation of circular arrays of trapezoidal-tooth log-periodic antennas via genetic optimization(Electromagnetics Academy, 2008) Gürel, Levent; Ergül, Özgür; Gürel, Levent; Ergül, ÖzgürCircular arrays of log-periodic (LP) antennas are designed and their operational properties are investigated in a sophisticated simulation environment that is based on the recent advances in computational electromagnetics. Due to the complicated structures of the trapezoidal-tooth array elements and the overall array configuration, their analytical treatments are prohibitively difficult. Therefore, the simulation results presented in this paper are essential for their analysis and design. We present the design of a three-element LP array showing broadband characteristics. The directive gain is stabilized in the operation band using optimization by genetic algorithms. We demonstrate that the optimization procedure can also be used to provide beam-steering ability to LP arrays.