Nilufer Aslihan Ozdemir

On the Relationship Between Multiple-Scattering Macro Basis Functions and Krylov Subspace Iterative Methods

A mathematical link is provided between the Krylov subspace iterative approach based on the full orthogonalization method (FOM) and the macro basis functions (MBF) approach based on a multiple-scattering methodology. The link refers to the subspaces created by those methods as well as to the orthogonality conditions which they satisfy. Both approaches are applied to the same method-of-moments (MoM) system of equations that is preconditioned based on a closest-interaction rule, and where blocks of the MoM impedance matrix are compressed using a rank-revealing method. MBF and FOM approaches are compared numerically, with a special attention given to accuracy, for perfectly conducting objects, comprising an array of tapered-slot antennas, spheres and an aircraft. The respective advantages of both methods are briefly discussed and further prospects are given.

MoM Matrix Generation Based on Frequency and Material Independent Reactions (FMIR-MoM)

A novel and efficient method of moments (MoM) matrix generation technique called the frequency and material independent reactions for the method of moments (FMIR-MoM) technique is presented. This new matrix generation algorithm efficiently calculates impedance matrices while sweeping through frequency, permittivity, conductivity, and/or permeability values. For frequency sweeps it has computational and memory costs comparable to those of interpolation techniques. It has the advantage over interpolation techniques in that it does not divide the frequency range into segments and allows one to dynamically update the precision. The technique expands the exponential of the Greens function into a Taylor series. This allows the problem to be formulated as a summation, where each term consists of a real valued matrix depending only on the geometry (discretization), multiplied by a scalar dependent on the propagation constant. The algorithms efficiency is obtained by calculating the geometry-dependent matrices prior to sweeping through frequency or material parameters.

Harmonics-Based Inhomogeneous Plane-Wave Method (HIPW)

A formulation is presented for fast interactions between subdomains in two-dimensional (2-D) scattering problems. The formulation combines inhomogeneous plane waves with cylindrical harmonic decompositions of fields radiated by the subdomains. It is shown that the complexity of interactions naturally decays with the distance between subdomains and that very few elementary operations are involved at the lowest level. An example of iterative solution for scattering by a collection of cylinders validates the proposed approach.

A near-field preconditioner preserving the low-rank representation of method of moments interaction matrices

A preconditioning methodology is proposed to preserve the low-rank representation of Method-of-Moments interaction matrix blocks in the context of multiple-scattering-based Macro Basis Function methodology. The scatterer is divided into sub-domains that can be connected with each other. A low-rank representation is obtained by employing the incomplete QR factorization without constructing the interaction matrix blocks a priori. Performing compression before preconditioning allows one to change at will the preconditioning technique. The preconditioning considered in this study is based on nearest interactions and involves auxiliary sub-domains which are immediately connected with the sub-domain of interest and partially overlap neighboring sub-domains. Low-rank representation of the interaction matrices is preserved by dealing with the auxiliary sub-domains separately. This approach comes at the cost of an overhead in terms of memory, which is directly connected with the size of the auxiliary sub-domain.

Efficient analysis of periodic structures involving finite dielectric material based on the array scanning method

Efficient numerical analysis of finite and infinite periodic structures involving finite dielectric material by the method of moments (MoM) is important for applications of phased arrays and metamaterials [1]. When periodic structures are excited by nonperiodic sources, one may apply the Array Scanning Method (ASM), which replaces the nonperiodic problem with an integral superposition of periodic problems [2]. The integration is performed over the phase shifts between adjacent elements of the structure from 0 to 2π and requires the numerical solution of periodic subproblems [3]-[4]. However, as the inter-element phase shifts range from 0 to 2π, the convergence of the periodic Greens function -and of its gradient- gets very difficult near Rayleigh-Woods anomaly [5]. This anomaly occurs when dipoles form a beam in the direction of the observation point close to the array plane in the line-by-line formulation of infinite-array Greens function [1]. Therefore, the periodic Greens function needs to be evaluated near Rayleigh-Woods anomaly accurately to apply the ASM properly. Another application of the ASM is the efficient analysis of the finite periodic structures by the Macro Basis Function (MBF) approach [6]-[8]. The MBF approach reduces the number of unknowns describing each element of a finite periodic structure by at least one order of magnitude. The ASM provides a physically based approach for the proper choice of MBFs, which lead to a sufficiently complete basis for the problem of interest [9]. This paper is organized as follows. In Section 2.1, we revisit Levin extrapolation to accelerate the computation of the doubly periodic Greens function near the Rayleigh-Woods anomaly to apply the ASM properly. In Section 2.2 we apply the ASM to electromagnetic scattering from a doubly periodic structure composed of dielectric objects excited by a single point source. We observe that the scattered electric field distribution for uniform phase shifts in the spectral domain has singular behavior at the boundary of the visible space due to the singularity of the periodic Greens function. This behavior is responsible for a highly oscillatory response in the spatial domain. To improve the quality of estimated field near the limits of the visible region, we implement a second order polynomial extrapolation on the (MoM) system matrix. Finally, in Section 2.3, a validation example is shown for finite-array solutions of dielectric objects with the help of the combined infinite array solution and MBF approaches. We observe that 4×4 infinite array solutions in spectral domain remain sufficient to form a sufficiently complete set of MBFs for the analysis of a dielectric sphere array.

Efficient integral equation-based analysis of finite periodic structures in the optical frequency range

The optical response of dense finite arrays of nanoparticles can be efficiently analyzed with the help of macro basis functions obtained by employing the array scanning method. This is demonstrated by analyzing optical collimation in arrays of silver nanorods. The accuracy of the solution obtained with the proposed method has been validated by comparison with solutions obtained employing the Krylov subspace iterative method. The relative error in the electric field distribution on an observation plane above the finite array is of the order of -25 dB, while the number of unknowns is reduced by a factor of 32.

Efficient method of moments analysis of an infinite array of triangular nanoclusters in the optical frequency range

This paper provides an efficient Method-of-Moments solution for an infinite doubly periodic array of magnetic nanoclusters. Multiple-scattering based Macro Basis Function approach is employed to obtain a reduced representation of equivalent current distributions. The incomplete QR algorithm is exploited to reduce the computation time and memory required to solve the reduced system of equations. The accuracy and efficiency of the combined approach are investigated for an infinite doubly periodic array of magnetic nanoclusters composed of eight silver triangular particles at optical frequencies.

Numerically Stable Eigenmode Extraction in 3-D Periodic Metamaterials

A numerical method is presented to compute the eigenmodes supported by 3-D metamaterials using the method of moments. The method relies on interstitial equivalent currents between layers. First, a parabolic formulation is presented. Then, we present an iterative technique that can be used to linearize the problem. In this way, all the eigenmodes characterized by their transmission coefficients and equivalent interstitial currents can be found using a simple eigenvalue decomposition of a matrix. The accuracy that can be achieved is limited only by the quality of simulation, and we demonstrate that the error introduced when linearizing the problem decreases doubly exponentially with respect to the time devoted to the iterative process. We also draw a mathematical link and distinguish the proposed method from other transfer-matrix-based methods available in the literature.

SVD post-compression combined with shielded-block preconditioner

We present a post-compression method based on the Singular Value Decomposition that can be used to retrieve the low-rank representation of the Method of Moments interaction matrix after using the shielded-block preconditioner. A compression ratio of more than 17 has be achieved without significant loss of accuracy. The time needed to compress the preconditioned matrix is negligible with respect to the preconditioning time. We also show that the compression can decrease the solution complexity using iterative solvers or multiple-scattering MBFs, thereby significantly decreasing the total computation time.

Krylov subspace-based and MBF-based analysis of large finite arrays of silver nanorods in the presence of a scatterer

Metamaterials receive increasing attention at optical frequencies due to their potential for subwavelength imaging. Among the possible structures are dense, doubly periodic arrays of silver nanorods where image transmission is achieved via surface plasmon polaritons. However, the numerical simulation of such dense structures may require excessive computational resources without employing an efficient numerical approach. In this paper, the multiple-scattering based Macro Basis Function (MBF) method is applied to the shielded-block preconditioned matrix, which represents interactions between different elements of the array (subdomains). The “rule of thumb” that relates the number of MBFs generated on a subdomain to the number of iterations required by the Full Orthogonalization Method (FOM) for the same error level is investigated. For high levels of error, the number of MBFs are observed to be smaller than the number of iterations required. Conversely, for low levels of error, the FOM converges faster.