Browsing by Subject "Multilevel fast multipole algorithm (MLFMA)"

Now showing 1 - 18 of 18

Open Access
Analysis of dielectric photonic-crystal problems with MLFMA and Schur-complement preconditioners
(IEEE, 2011-01-13) Ergül, Özgür; Malas, T.; Gürel, Levent
We 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.
Open Access
Broadband analysis of multiscale electromagnetic problems: Novel incomplete-leaf MLFMA for potential integral equations
(IEEE, 2021-06-24) Khalichi, Bahram; Ergül, Ö.; Takrimi, Manouchehr; Ertürk, Vakur B.
Recently introduced incomplete tree structures for the magnetic-field integral equation are modified and used in conjunction with the mixed-form multilevel fast multipole algorithm (MLFMA) to employ a novel broadband incomplete-leaf MLFMA (IL-MLFMA) to the solution of potential integral equations (PIEs) for scattering/radiation from multiscale open and closed surfaces. This population-based algorithm deploys a nonuniform clustering that enables to use deep levels safely and, when necessary, without compromising the accuracy resulting in an improved efficiency and a significant reduction for the memory requirements (order of magnitudes), while the error is controllable. The superiority of the algorithm is demonstrated in several canonical and real-life multiscale geometries.
Open Access
Comparison 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
We 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.
Open Access
Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems
(Institute of Electrical and Electronics Engineers, 2008-08) Ergül, Özgür; Gürel, Levent
We present fast and accurate solutions of large-scale scattering problems involving three-dimensional closed conductors with arbitrary shapes using the multilevel fast multipole algorithm (MLFMA). With an efficient parallelization of MLFMA, scattering problems that are discretized with tens of millions of unknowns are easily solved on a cluster of computers. We extensively investigate the parallelization of MLFMA, identify the bottlenecks, and provide remedial procedures to improve the efficiency of the implementations. The accuracy of the solutions is demonstrated on a scattering problem involving a sphere of radius 110 discretized with 41 883 638 unknowns, the largest integral-equation problem solved to date. In addition to canonical problems, we also present the solution of real-life problems involving complicated targets with large dimensions
Open Access
Efficient solution of the combined-field integral equation with the parallel multilevel fast multipole algorithm
(IEEE, 2007-08) Gürel, Levent; Ergül, Özgür
We present fast and accurate solutions of large-scale scattering problems formulated with the combined-field integral equation. Using the multilevel fast multipole algorithm (MLFMA) parallelized on a cluster of computers, we easily solve scattering problems that are discretized with tens of millions of unknowns. For the efficient parallelization of MLFMA, we propose a hierarchical partitioning scheme based on distributing the multilevel tree among the processors with an improved load-balancing. The accuracy of the solutions is demonstrated on scattering problems involving spheres of various radii from 80λ to 110λ. In addition to canonical problems, we also present the solution of real-life problems involving complicated targets with large dimensions. © 2007 IEEE.
Open Access
Efficient solution of the electric and magnetic current combined‐field integral equation with the multilevel fast multipole algorithm and block‐diagonal preconditioning
(Wiley-Blackwell Publishing, Inc., 2009-12) Ergül, Özgür; Gürel, Levent
We consider the efficient solution of electromagnetics problems involving dielectric and composite dielectric-metallic structures, formulated with the electric and magnetic current combined-field integral equation (JMCFIE). Dense matrix equations obtained from the discretization of JMCFIE with Rao-Wilton-Glisson functions are solved iteratively, where the matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm. JMCFIE usually provides well conditioned matrix equations that are easy to solve iteratively. However, iteration counts and the efficiency of solutions depend on the contrast, i.e., the relative variation of electromagnetic parameters across dielectric interfaces. Owing to the numerical imbalance of off-diagonal matrix partitions, solutions of JMCFIE become difficult with increasing contrast. We present a four-partition block-diagonal preconditioner (4PBDP), which provides efficient solutions of JMCFIE by reducing the number of iterations significantly. 4PBDP is useful, especially when the contrast increases, and the standard block-diagonal preconditioner fails to provide a rapid convergence.
Open Access
Error analysis of MLFMA with closed-form expressions
(IEEE, 2021-04-06) Kalfa, Mert; Ertürk, Vakur B.; Ergül, Özgür
The current state-of-the-art error control of the multilevel fast multipole algorithm (MLFMA) is valid for any given error threshold at any frequency, but it requires a multiple-precision arithmetic framework to be implemented. In this work, we use asymptotic approximations and curve-fitting techniques to derive accurate closed-form expressions for the error control of MLFMA that can be implemented in common fixed-precision computers. Moreover, using the proposed closed-form expressions in conjunction with the state-of-the-art scheme, we report novel design curves for MLFMA that can be used to determine achievable error limits, as well as the minimum box sizes that can be solved with a given desired error threshold for a wide range of machine precision levels.
Open Access
Fast and accurate solutions of scattering problems involving dielectric objects with moderate and low contrasts
(IEEE, 2007-08) Ergül, Özgür; Gürel, Levent
We consider the solution of electromagnetic scattering problems involving relatively large dielectric objects with moderate and low contrasts. Three-dimensional objects are discretized with Rao-Wilton-Glisson functions and the scattering problems are formulated with surface integral equations. The resulting dense matrix equations are solved iteratively by employing the multilevel fast multipole algorithm. We compare the accuracy and efficiency of the results obtained by employing various integral equations for the formulation of the problem. If the problem size is large, we show that a combined formulation, namely, electric-magnetic current combined-field integral equation, provides faster iterative convergence compared to other formulations, when it is accelerated with an efficient block preconditioner. For low-contrast problems, we introduce various stabilization procedures in order to avoid the numerical breakdown encountered in the conventional surface formulations. © 2007 IEEE.
Open Access
Incomplete-leaf multilevel fast multipole algorithm for multiscale penetrable objects formulated with volume integral equations
(Institute of Electrical and Electronics Engineers Inc., 2017) Takrimi, M.; Ergül, Ö.; Ertürk, V. B.
Recently introduced incomplete-leaf (IL) tree structures for multilevel fast multipole algorithm (referred to as IL-MLFMA) is proposed for the analysis of multiscale inhomogeneous penetrable objects, in which there are multiple orders of magnitude differences among the mesh sizes. Considering a maximum Schaubert-Wilton-Glisson function population threshold per box, only overcrowded boxes are recursively divided into proper smaller boxes, leading to IL tree structures consisting of variable box sizes. Such an approach: 1) significantly reduces the CPU time for near-field calculations regarding overcrowded boxes, resulting a superior efficiency in comparison with the conventional MLFMA where fixed-size boxes are used and 2) effectively reduces the computational error of the conventional MLFMA for multiscale problems, where the protrusion of the basis/testing functions from their respective boxes dramatically impairs the validity of the addition theorem. Moreover, because IL-MLFMA is able to use deep levels safely and without compromising the accuracy, the memory consumption is significantly reduced compared with that of the conventional MLFMA. Several examples are provided to assess the accuracy and the efficiency of IL-MLFMA for multiscale penetrable objects.
Open Access
Iterative near-field preconditioner for the multilevel fast multipole algorithm
(Society for Industrial and Applied Mathematics, 2010-07-06) Gürel, Levent; Malas, T.
For iterative solutions of large and difficult integral-equation problems in computational electromagnetics using the multilevel fast multipole algorithm (MLFMA), preconditioners are usually built from the available sparse near-field matrix. The exact solution of the near-field system for the preconditioning operation is infeasible because the LU factors lose their sparsity during the factorization. To prevent this, incomplete factors or approximate inverses can be generated so that the sparsity is preserved, but at the expense of losing some information stored in the near-field matrix. As an alternative strategy, the entire near-field matrix can be used in an iterative solver for preconditioning purposes. This can be accomplished with low cost and complexity since Krylov subspace solvers merely require matrix-vector multiplications and the near-field matrix is sparse. Therefore, the preconditioning solution can be obtained by another iterative process, nested in the outer solver, provided that the outer Krylov subspace solver is flexible. With this strategy, we propose using the iterative solution of the near-field system as a preconditioner for the original system, which is also solved iteratively. Furthermore, we use a fixed preconditioner obtained from the near-field matrix as a preconditioner to the inner iterative solver. MLFMA solutions of several model problems establish the effectiveness of the proposed nested iterative near-field preconditioner, allowing us to report the efficient solution of electric-field and combined-field integral-equation problems involving difficult geometries and millions of unknowns.
Open Access
MLFMA solutions of transmission problems Involving realistic metamaterial walls
(IEEE, 2007-08) Ergül, Özgür; Ünal, Alper; Gürel, Levent
We present the solution of multilayer metamaterial (MM) structures containing large numbers of unit cells, such as split-ring resonators. Integral-equation formulations of scattering problems are solved iteratively by employing a parallel implementation of the multilevel fast multipole algorithm. Due to ill-conditioned nature of the problems, advanced preconditioning techniques are used to obtain rapid convergence in the iterative solutions. By constructing a sophisticated simulation environment, we accurately and efficiently investigate large and complicated MM structures. © 2007 IEEE.
Open Access
Multiple-precision arithmetic implementation of the multilevel fast multipole algorithm
(IEEE, 2024-01-01) Kalfa, Mert; Ergül, Özgür; Ertürk, Vakur Behçet
We propose and demonstrate a multiple-precision arithmetic (MPA) framework applied to the inherent hierarchical tree structure of the multilevel fast multipole algorithm (MLFMA), dubbed the MPA-MLFMA that provides an unconventional but elegant treatment to both the low-frequency breakdown (LFB) and the efficiency limitations of MLFMA for electrically large problems with fine geometrical details. We show that a distinct machine precision (MP) can be assigned to each level of the tree structure of MPA-MLFMA, which, in turn, enables controlled accuracy and efficiency over arbitrarily large frequency bandwidths. We present the capabilities of MPA-MLFMA over a wide range of broadband and multiscale scattering problems. We also discuss the implications of a multiple-precision framework implemented in software and hardware platforms.
Open Access
Parallel preconditioners for solutions of dense linear systems with tens of millions of unknowns
(2007-11) Malas, Tahir; Ergül, Özgür; Gürel, Levent
We propose novel parallel preconditioning schemes for the iterative solution of integral equation methods. In particular, we try to improve convergence rate of the ill-conditioned linear systems formulated by the electric-field integral equation, which is the only integral-equation formulation for targets having open surfaces. For moderate-size problems, iterative solution of the near-field system enables much faster convergence compared to the widely used sparse approximate inverse preconditioner. For larger systems, we propose an approximation strategy to the multilevel fast multipole algorithm (MLFMA) to be used as a preconditioner. Our numerical experiments reveal that this scheme significantly outperforms other preconditioners. With the combined effort of effective preconditioners and an efficiently parallelized MLFMA, we are able to solve targets with tens of millions of unknowns, which are the largest problems ever reported in computational electromagnetics. ©2007 IEEE.
Open Access
PO-MLFMA hybrid technique for the solution of electromagnetic scattering problems involving complex targets
(Institution of Engineering and Technology, 2007) Gürel, Levent; Manyas, Alp; Ergül, Özgür
The multilevel fast multipole algorithm (MLFMA) is a powerful tool for efficient and accurate solutions of electromagnetic scattering problems involving large and complicated structures. On the other hand, it is still desirable to increase the efficiency of the solutions further by combining the MLFMA implementations with the high- frequency techniques such as the physical optics (PO). In this paper, we present our efforts in order to reduce the computational cost of the MLFMA solutions by introducing PO currents appropriately on the scatterer. Since PO is valid only on smooth and large surfaces that are illuminated strongly by the incident fields, accurate solutions require careful choices of the PO and MLFMA regions. Our hybrid technique is useful especially when multiple solutions are required for different frequencies, illuminations, and scenarios, so that the direct solutions with MLFMA become expensive. For these problems, we easily accelerate the MLFMA solutions by systematically introducing the PO currents and reducing the matrix dimensions without sacrificing the accuracy.
Open Access
Solution of extremely large integral-equation problems
(IEEE, 2007) Ergül, Özgür; Malas, Tahir; Gürel, Levent
We report the solution of extremely large integral-equation problems involving electromagnetic scattering from conducting bodies. By orchestrating diverse activities, such as the multilevel fast multipole algorithm, iterative methods, preconditioning techniques, and parallelization, we are able to solve scattering problems that are discretized with tens of millions of unknowns. Specifically, we report the solution of a closed geometry containing 42 million unknowns and an open geometry containing 20 million unknowns, which are the largest problems of their classes, to the best of our knowledge.
Open Access
Solution of large-scale scattering problems with the multilevel fast multipole algorithm parallelized on distributed-memory architectures
(IEEE, 2007) Ergül, Özgür; Gürel, Levent
We present the solution of large-scale scattering problems involving three-dimensional closed conducting objects with arbitrary shapes. With an efficient parallelization of the multilevel fast multipole algorithm on relatively inexpensive computational platforms using distributed-memory architectures, we perform the iterative solution of integral-equation formulations that are discretized with tens of millions of unknowns. In addition to canonical problems, we also present the solution of real-life problems involving complicated targets with large dimensions.
Open Access
Stabilization of integral-equation formulations for the accurate solution of scattering problems involving low-contrast dielectric objects
(Institute of Electrical and Electronics Engineers, 2008) Ergül, Özgür; Gürel, Levent
The solution of scattering problems involving low-contrast dielectric objects with three-dimensional arbitrary shapes is considered. Using the traditional forms of the surface integral equations, scattered fields cannot be calculated accurately if the contrast of the object is low. Therefore, we consider the stabilization of the formulations by extracting the nonradiating parts of the equivalent currents. We also investigate various types of stable formulations and show that accuracy can be improved systematically by eliminating the identity terms from the integral-equation kernels. Traditional and stable formulations are compared, not only for small scatterers but also for relatively large problems solved by employing the multilevel fast multipole algorithm. Stable and accurate solutions of dielectric contrasts as low as 104 are demonstrated on problems involving more than 250000 unknowns.
Open Access
Two-step lagrange interpolation method for the multilevel fast multipole algorithm
(Institute of Electrical and Electronics Engineers, 2009) Ergül, Z.; Bosch, I.; Gürel, Levent
We present a two-step Lagrange interpolation method for the efficient solution of large-scale electromagnetics problems with the multilevel fast multipole algorithm (MLFMA). Local interpolations are required during aggregation and disaggregation stages of MLFMA in order to match the different sampling rates for the radiated and incoming fields in consecutive levels. The conventional one-step method is decomposed into two one-dimensional interpolations, applied successively. As it provides a significant acceleration in processing time, the proposed two-step method is especially useful for problems involving large-scale objects discretized with millions of unknowns.