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An approach to singularity cancelation by a variable transformation of Green’s function in 2-D case, having log( $R$ ) singularity, is outlined. It is intended for the method of moments analysis of 2-D structures of both curvilinear and flat cross sections. If the transformation is used with Legendre or Chebyshev polynomials as entire-domain basis functions, all involved integrals can be calculated with high precision using Gauss–Legendre quadrature. The optimal parameter of the transformation, having the minimal number of integration samples needed to reach 14-digit precision, is provided. Error estimations are presented for calculated current distributions on strips up to 400 wavelengths wide, analyzed with up to 1500 entire-domain basis functions.

Nonuniform helical antennas have many degrees of freedom, which makes the search space for the optimal design very challenging. The objective of this paper is to systematically analyze nonuniform helical antennas with linearly varying geometrical parameters and to provide analytical equations that approximate the optimal design and the gain of the designed antennas. Using various optimization algorithms, we made a large database of the optimal nonuniform helical antennas with linearly varying geometrical parameters. Based on these results, we made analytical equations that approximate the optimal design and the gain of the designed antennas. These equations allow for a fast design procedure yielding all necessary parameters needed for the design and fabrication of nonuniform helical antennas that meet specified characteristics. Special attention is devoted to antenna losses. Antennas designed following the presented procedure achieve around 2.5 dB higher gain than uniform helical antennas of the same axial length, while maintaining the bandwidth and axial ratio. As a verification of the proposed design procedure, a helical antenna with the central operating frequency of 1 GHz was designed, simulated, fabricated, and measured. The comparison between measured and simulated results confirms the validity of the presented design procedure.

Machine learning is used to determine the polynomial regression of the coefficients of surface currents approximation for a curvilinear 2-D scatterer. The training data are obtained using method of moments with higher order basis functions and Galerkin testing procedure. The effect of polynomial regression order to the accuracy of calcualted radar cross-section is explored.

We present results for comparison of gain of the optimal uniform and nonuniform helical antennas taking into the account conductor losses. Critical conductivities are observed, for the given operating frequency and the helix axial length. Above the critical conductivity, the gain of the nonuniform helical antennas with linearly varying geometrical parameters is up to 2 dB higher than the gain of the optimal uniform antennas. However, below the critical conductivity, the difference between the gain of the nonuniform and uniform helical antennas is reduced, and the gain of those antennas becomes comparable. Data for the normalized critical conductivities are presented.

This paper proposes a method for efficient identification of instruction-dependent sources on a printed circuit board (PCB) by localizing magnetic field sources from a limited number of measurements around the PCB. We first excite the processor by generating an artificial leakage signal at a specific frequency that is directly related to processor instructions. Then, we collect all three components of the magnetic field, but only at locations around the edge of the board. Furthermore, we model these magnetic field sources and then solve a forward–backward optimization problem using the model and measured data to identify the locations of the magnetic field sources, the magnitudes of the moments, and their orientations. The localization results are first verified using simulations, then tested when noise is added to the simulation results, and finally verified against measurements on field-programmable gate array (FPGA) and internet of things (IoT) development boards. The results show that the number of strong magnetic field sources on a board depends on the instructions used to excite the board. Furthermore, the results show that the proposed localization algorithm can accurately identify those sources, regardless of the frequency at which the measurements are conducted and the instruction pairs that are executed. Finally, the proposed method can significantly reduce the number of measurement points and the time needed to identify magnetic field sources on a PCB.

Excess attenuation of electric field of an electromagnetic wave propagating through deciduous forest is approximated with two novel analytical formulas. The coefficients of those formulas were obtained through a computationally intensive optimization process by fitting the simulation results obtained from stochastic 3D and 2D models of forests. The electric field was either vertically or horizontally polarized, at frequencies 30 MHz, 60 MHz and 100 MHz. According to numerical experiments, these formulas predicted the excess attenuation within several decibels, up to at least 100 MHz. Nevertheless, the proposed formulas matched very well the measured data from literature below 1 GHz, and neatly followed the measurements even up to 10 GHz. Therefore, the frequency range considered in the numerical experiments can be extrapolated at least tenfold.

We present an overview of the recent advances in numerical analysis of 2-D structures using singularity-cancellation transformations in conjunction with entire-domain higher-order analysis. It is demonstrated that the transformation allows high-precision integration. If the orthogonal polynomials are used as basis functions, along with the high-precision integration, the size of curvilinear mesh elements can be several hundreds of wavelengths. Examples of curvilinear 2-D structures are presented to demonstrate the applicability of the approach.

We summarize results for using Gegenbauer polynomials as basis functions in the case of the entire-domain 2-D analysis of strips with curvilinear cross section illuminated by a linearly-polarized plane wave. The involved integrals are calculated with 14-digits precision, due to the use d singularity-cancellation transformation, for arbitrary cross-section shape. We investigated the error estimate of the surface current approximation due to the defining parameter of the Gegenbauer polynomials and the maximal order of the polynomial for the current approximation. The results for the surface-current approximation with the highest degree of the polynomial basis functions of 7000 are presented. The analyzed curvilinear strip had the total length of the cross section of 2000 wavelengths at the operating frequency in the free space.

Proper implementation of max-ortho bases enables effective usage of ultra high order basis functions. Significance of ultra high order bases is demonstrated in the case of p-refinement of 2λ cube scatterer, in which case mean absolute RCS error of 0.001 dB is attained using expansion order of n = 20. In particular, it was shown that taking edge effects into account this level can be further reduced to 0.0001 dB.

We present a design of a high-gain quad array of nonuniform helical antennas. The design is obtained by optimization of a 3-D numerical model of four nonuniform helical antennas placed above a ground plane, including a model of a feeding network, utilizing the method of moments with higher-order basis functions. The gain of one optimal nonuniform helical antenna can be about 2.5 dB higher than the gain of a uniform helical antenna of the same axial length. Creating a 2×2 array further increases the gain up to about 6 dB. The resulting quad array fits into a box whose dimensions are 2.5×3.3×3.3 wavelengths, and the gain in the main radiating direction is about 20.5 dBi in the frequency range from 0.9 GHz to 1.1 GHz. The design is verified by measurements of a prototype of the quad array.

Acceleration of in-core LU decomposition of symmetrical dense MoM matrices by using multiple GPUs in parallel is presented in this paper. Memory limitations of GPUs are overcome by using block LU decomposition, where the entire system matrix is stored in CPU RAM, while only processed blocks are stored in GPU VRAM. The presented algorithm for LU decomposition enables highly efficient utilization of an arbitrary number of GPUs. Comparison of performance of up to 4 GTX 680 GPUs and up to 4 GTX 1080 Ti GPUs is shown. Presented results show that a symmetrical dense MoM matrix with 100 000 complex unknowns in single precision can be LU decomposed in about 3.5 minutes, on a personal computer equipped with 4 GTX 1080 Ti GPUs.

A novel diakoptic method based on volume integral equation (VIE) modeling of subsystems is proposed for 3-D electromagnetic analysis. The theoretical foundation of the method is the surface and volume equivalence principles, as it combines the VIE and surface integral equation (SIE) formulations, in conjunction with the method of moments (MoM). The method breaks the original structure into a number of nonoverlapping closed-region subsystems that contain inhomogeneous dielectric materials and an open-region subsystem. Each subsystem is analyzed completely independently applying the double higher order large-domain Galerkin generalized MoM-VIE-SIE (VSIE) or MoM-SIE solvers. The final solution is obtained from diakoptic matrices expressing linear relations between the electric and magnetic equivalent surface current coefficients on diakoptic surfaces. The proposed VSIE-diakoptic method is validated, evaluated, and discussed in several characteristic examples. The examples demonstrate that the diakoptic method substantially increases the efficiency of the conventional MoM-VIE approach. When compared with the pure MoM-VIE double higher order technique, the diakoptic approach enables very considerable accelerations and memory savings, while fully preserving the accuracy of the analysis.

Recently, we proposed a method for accurately locating instruction-dependent magnetic field sources on printed circuit boards (PCBs). This method first excites the device's processor by executing an alternating pair of two instructions at a specific frequency. Using the measurements of the magnetic field taken around the edges of the PCB and an algorithm based on simplex optimization, the locations of the sources are evaluated. In this paper, we present extensive experimental verification of this method on two devices, a field-programmable gate array (FPGA) development board and an Internet of Things (IoT) device. The results illustrate that sources of instruction-dependent emanations are confined to a small area near the processor and that the positions of these sources are dependent on the specific instructions being executed.

Comparison of various optimization approaches for the design of nonuniform helical antennas is presented. The considered helices have linearly varying radius and pitch. Results show that this design has many similar (or suboptimal) solutions with significant differences in geometry. Combination of particle swarm optimization and Nelder-Mead simplex algorithm proved to be a robust and an efficient optimization approach.

Polynomial higher order basis functions defined over quadrilaterals are regularly used with patches of maximum edge up to 2λ and expansion order up to $n$ = 8. In this paper it is demonstrated that these limits can be significantly extended if max-ortho bases are implemented in a proper way. Numerical results demonstrate effective usage of patches up to 8 λ and expansion orders up to n = 24.

The accuracy of approximation of surface currents at electrically large flat 2-D scatterers is investigated, in TM monostatic mode. Legendre polynomials, of order up to 250, are used as entire domain basis functions. Method of moments is used for numerical analysis with the involved integrals calculated to machine precision. Presented results show that 50 wavelength scatterer can be analyzed as a single (mesh) element.

Acceleration of in-core LU decomposition by using multiple GPUs in parallel and comparison of performance of two different GPU types, is presented in this paper. The presented algorithm for LU decomposition enables highly efficient utilization of an arbitrary number of GPUs. Comparison of performance of up to 8 GTX 680 GPUs and up to 4 GTX 1080 Ti GPUs is shown. The results presented show that a dense MoM matrix with 100 000 complex unknowns in single precision can be LU decomposed in 3.7 minutes, on a personal computer equipped with 4 GTX 1080 Ti GPUs.

We present comparison of higher order basis functions to lower orders, when applied to analysis of 2D electromagnetic structures. To illustrate the performances, several numerical examples have been analyzed. We use method of moments applied to surface integral equations for simulations. The criteria for the comparison are: the accuracy of results, the total simulation time, the total number of unknown coefficients and the memory needed for simulation.

Acceleration of in-core LU decomposition by using multiple GPUs in parallel is presented in this paper. Memory limitations of GPUs are overcome by using block LU decomposition, where the entire system matrix is stored in CPU RAM, while only currently processed blocks are stored in GPU VRAM. The presented algorithm for LU decomposition enables utilization of an arbitrary number of GPUs. High efficiency of the parallelization is achieved by specifically tailored load balancing over the GPUs in the most time consuming parts of the block LU decomposition. Presented results show that a dense MoM matrix with 100 000 complex unknowns in single precision can be LU decomposed in about 8 minutes, on a personal computer equipped with 8 low-cost GTX 680 GPUs.

We present efficient 2D electromagnetic solution of scattering from electrically large and complex platforms. High efficiency of 2D analysis has been achieved by introducing nonuniform segmentation, higher order basis functions, and multi-threading parallelization. Additionally, we illustrate how 2D analysis can be applied in the process of predicting the results of 3D analysis. The method of moments applied to surface integral equations has been used for all simulations.

We present measured results for the parallelization efficiency of a code for numerical 2D electromagnetic analysis. The code is based on method of moments and uses higher-order basis functions with Galerkin testing procedure. The functions for linear algebra within the code are from Intel MKL library, and the code uses OpenMP for parallelization. A workstation with up to 40 hyper-threads is used for the numerical experiments. The calculation of monostatic radar cross-sections of a 2D structure is used as a test example.

We consider the application of multipole based dictionaries, used in the sparse processing framework, for microwave imaging of complex-shaped dielectric targets. We investigate the influence of different subarray configurations on the imaging accuracy. In addition, we examine the images obtained using multipoles of different orders from the prospective of the spectral content of the scattered field.

We present a formula that can approximate the excess attenuation of electric field of electromagnetic wave propagating in the presence of human crowds. The formula is tested on results obtained from full-wave simulations of 2D and 3D stochastic models of human crowds. The electric field is either vertically or horizontally polarized at frequencies 900 MHz and 1800 MHz. We use two stage nonlinear optimizations to find the parameters of the formula.

A new diakoptic method combining the finite-element method and the method of moments is proposed for analysis of inhomogeneous anisotropic dielectric and magnetic scatterers. The method splits the original electromagnetic system into a number of closed-region finite-element method subsystems containing material complexities and an open-region method of moments subsystem, which are analyzed independently. The solution to the original problem is obtained from linear relations between coefficients in expansions of equivalent electric and magnetic surface currents on diakoptic boundary surfaces. Diakoptic electric sources and the magnetic field in finite-element method subsystems are connected using dual sets of higher-order hierarchical basis functions.

This paper presents a novel formula for the complex permittivity of lossy dielectrics, which is valid in a broad frequency range and is ensuring a causal impulse response in the time domain. The application of this formula is demonstrated through the analysis of wet soil, where the coefficients of the formula are tuned to match the measured data from the literature. Additionally, an analytical expression for the impulse response of the relative permittivity is derived. The influence of the frequency dependence of the complex permittivity on the causality of responses is illustrated through the analysis of 1-D, 2-D, and 3-D electromagnetic systems. Being the most complex, the 3-D system is also used as a test bed for comparing the computational limitations of two commercially available solvers, CST and WIPL-D.

Temporal leakage occurs when the time-domain response of a linear electromagnetic system is computed from frequency-domain data using the inverse discrete Fourier transform. Although the temporal leakage occurs in most practical cases, this phenomenon is not recognized in the literature covering electromagnetics, antennas, and microwaves. The purpose of the paper is to demonstrate and explain the temporal leakage.

Synthetic aperture radars (SAR) are widely used for various applications. Efficient and accurate calculation of target scattering in these systems is of interest in both radar system design and target imaging. A full-wave time-domain analysis of target scattering in SAR system is presented. The analysis is performed by using a solver based on method of moments based in combination with fast Fourier transform and far-field excitation. Numerical results for scattering of an electrically large aircraft which is performed for 201 transceiver positions are presented in the paper.

A novel, wideband, symmetrical, printed directional coupler is described, which has tight coupling and high directivity. The coupler consists of four interconnected strips printed in two layers. The dimensions of the strips are optimized to minimize the dispersion of propagating modes. Design data are provided for some typical practical cases. The theoretical results are verified experimentally. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2261–2267, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27051

We present a general method for numerical analysis of electrically large electromagnetic-field problems, which is based on merging diakoptic approach and surface integral-equation formulations. Compared with the classical approach, the proposed method provides significant reduction of c.p.u. time and storage, preserving high accuracy of results.

The project outlined here is to design and construct a portable replica of the apparatus used by Heinrich Hertz in 1888 to demonstrate the existence of electromagnetic waves. The idea is to carry out the experiment using only the primitive tools available to Hertz at the time. He had a powerful induction coil to excite his “primary conductor,” and a micrometer-driven spark gap at the terminals of a loop, his “secondary circuit.” Hertz performed a multitude of experimental steps and kept detailed notes to enable him to build a receiving loop to be resonant at the same frequency as his transmitting dipole. Today we may use modern computer tools to design both dipole and loop. Hopefully, this will result in an agreeably portable set of equipment. This experiment may provide students and engineers with a new level of confidence in their fundamental knowledge of electricity.

We present results for simulations of electromagnetic field propagation in the presence of human crowds in open area. The polarization of the considered incident field is horizontal and vertical, and the operating frequencies are 900 MHz and 1.8 GHz. For the simulations we use 3-D electromagnetic solver based on surface integral-equation formulation, higher-order basis functions and Galerkin testing, accelerated using graphical processing units. The presented results are for the averaged electric field at defined points in space.

Two general approaches to multiminima optimization are considered. The first approach is based on repetition of a single minima method (e.g., the Nelder-Mead simplex applied to the best solution in a set of random trials). The second approach is based on a coarse estimation of local minima using initial set of points and local optimization starting from these local minima (e.g., random search as a generator of the initial set of points and Nelder-Mead simplex as a local optimizer). A comparison of various optimization algorithms has been done on one analytical problem and two well-known examples of antenna design. It is found that: a) the multiminima method based on coarse estimation enables finding more minima with smaller number of iterations than that based on repetition, b) the best multiminima methods are comparable with the best single minima methods in a number of iterations needed for finding the global minima, and c) the multiminima method based on coarse estimation restarted with different weighting coefficients of multiobjective cost function enables efficient Pareto optimization

The method of moments (MoM) has been established as a powerful general tool for discretizing integral equations arising in numerical analysis of radiating and scattering structures. In MoM techniques based on the volume integral equation (VIE) approach, a structure is represented by a distribution of unknown volume polarization and conduction currents radiating in free space. MoM techniques based on the surface integral equation (SIE) approach treat electric and magnetic equivalent surface currents as unknowns in the numerical process. However, one general way to enhance the practical applicability and efficiency of both the MoM-VIE and MoM-SIE techniques is by means of the diakoptic method (E. Chobanyan, D. I. Olcan, M. M. Ilic, and B. M. Notaros, “Combining Diakoptic, VIE-MoM, and SIE-MoM Approaches in Analysis of Dielectric Scatterers,” Proc. 2013 IEEE Antennas and Propagation Society International Symposium).

We present a model of human body for full-wave numerical analysis of contact electrostatic discharge. The model is based on frequency-dependent surface resistance distributed over the surface of the body, which models the measured parameters of human tissue. The numerical analysis is done by method of moments with surface integral-equation formulation in the frequency domain. The results for currents and fields in time domain are obtained using discrete (inverse) Fourier transform and frequency-domain data. The flexibility of the proposed model is tested and discussed by comparing the numerical results to the standard waveform of current in electrostatic discharge. The results for magnetic field, in the vicinity of the discharge, are compared to those found in the literature. The model predicts correctly both waveforms and peak values for currents and magnetic fields.

A diakoptic technique combining the finite element method (FEM) and the method of moments (MoM), and using dual sets of higher order basis functions is proposed for analysis of 3D electromagnetic radiation and scattering structures. These bases enable efficient modeling and significant reductions in the total number of unknowns, while satisfying the natural relation between curl-conforming and divergence-conforming quantities when closing the FEM domain by a MoM surface. The new diakoptic FEM-MoM technique is validated in an example of an array of 4×4 dielectrically coated spherical metallic scatterers.

We outline an algorithm for block-sparse GPU accelerated out-of-core solver. The algorithm takes advantage of the block-sparsity that exists in method of moment analysis of composite electromagnetic systems with multiple materials. The efficiently of the proposed algorithm is demonstrated on the example of Luneburg lenses up to 40 wavelengths diameter. The approach yields speedup of up to 8 times, compared to general-purpose GPU accelerated out-of-core solver.

Full wave analysis of arbitrary structures in frequency domain is presented, as applied in WIPL-D Pro, software for 3D EM simulation. Currents induced over metallic structures and equivalent currents placed over material boundaries are found by solving electric field integral equation (EFIE) and PMCHW integral equations, respectively. High efficiency of the method is achieved by a) combining the Galerkin testing with higher order bases to obtain the symmetrical matrix equation of reduced size, and b) parallelization performed at CPU and/or GPU, for matrix fill and incore or out-of-core matrix solution.

We increase the computation and memory capabilities of GPU in-core solver by using block LU decomposition, RAM for storing the whole matrix and VRAM for storing currently used block(s) of data. Using up to 3 GPUs gives parallelization efficiency of up to 70%. The matrix of EM problems with 50 000 unknowns can be LU decomposed within 4 minutes on a PC with used GPUs.

We compare results for EM transients calculated using frequency domain analysis coupled with FFT with and without frequency-domain rational-function interpolation, in order to reduce the total number of frequency samples needed for accurate calculation of transients. To further reduce the total analysis time, we use zero-padding, for all frequencies where excitation spectrum is negligible. The proposed approach is applied to a problem of finding transient EM fields within metallic resonator with high Q-factor. The presented results show that time-domain response can be calculated more efficiently, with preserving the overall accuracy, if the proposed rational-function interpolation in frequency domain is used.

We present acceleration for numerical solving of electromagnetic (EM) problems by using method of moments (MoM) and NVIDIA graphics processing units (GPU). Three stages of MoM are accelerated: matrix fill, solution of complex linear equations and post-processing. The results show that GPUs can be efficiently used for EM simulations.

The paper presents efficient technique for full wave 3D EM modeling of electrically large arrays. Basically it consists in two steps: 1) the network parameters (Y,Z,S) of the array are calculated together with its radiation patterns when each of the elements is active, while all other elements are short-circuited; 2) radiation patterns are obtained by arbitrary combinations of feeding voltages by post-processing data from the first step. 3D EM modeling is facilitated decomposing the problem having geometrical symmetry and asymmetrical excitation into four fully symmetrical problems of quarter size. Matrix solution is accelerated using graphical processor unit (GPU). Thus, array problems of 400 and 900 microstrip patch antennas become doable in 6 hours and 1 day, respectively.

We present results for frequency-domain MoM simulations of electrically large structures using out-of-core solver accelerated with multiple GPUs on a single personal computer. The structures analyzed in order to demonstrate the efficiency of proposed out-of-core solver are Cassegrain reflector antenna with up to 240 λ reflector diameter and Luneburg lens, up to 16 λ diameter, excited with a half-wavelength dipole. The acceleration of out-of-core solver is up to 10 times with one GPU compared to a standard CPU, or up to 20 times when using 3 GPUs.

The principal objective of the work presented here is to demonstrate the proof-of-concept of coupling a higher-order finite element method (FEM) with a higher-order surface integral equation (SIE) method of moments (MoM) using the diakoptic approach. The diakoptic MoM-SIE approach has been successfully used for the electromagnetic (EM) analysis of microwave transmission lines, large scatterers, and antenna arrays [1], [2]. The diakoptic approach in this work uses coefficients calculated by a higher-order FEM code [3] and a higher-order MoM code [4], [5], and combines them to obtain the solution of the original EM problem. We briefly present the theory behind the concept and results for a case where the proposed approach is successfully applied with some diakoptic subsystems being solved using the FEM and the others using the MoM-SIE formulation.

Diakoptic surface integral-equation formulation is used to efficiently simulate large antenna arrays, modeled as 3D EM structures. The significant accelerations and storage reductions are achieved, when compared to the classical MoM-SIE. Both near field and far field (radiation pattern) of the arrays, calculated using DSIE, match the results calculated using MoM-SIE very well (the maximal relative error is about 1% calculated with respect to peak values). The future work will include diakoptic boundaries with common walls, application of DSIE to optimization of complex EM structures and calculation of EM time-domain responses, and hybridization with other numerical techniques such as finite elements and volume integral-equation formulation.

Modern design procedure of antennas, microwave components and EMC facilities usually includes full wave analysis of composite metallic and dielectric structures in frequency domain. One of the methods that can be used for such full wave analysis is method of moments (MoM) applied to surface integral equations (SIEs).

The canonical particle swarm optimization (PSO) algorithm is known to be very good solution for EM optimization problems. In this paper we present results for a few modifications of the PSO algorithm (fully informed PSO algorithm and repeated PSO algorithm) when they are applied to the optimization of a broadside antenna array. The aim is to find out if and under what circumstances these modifications can give better results than the canonical PSO algorithm. According to obtained results, the well set-up canonical PSO algorithm (with appropriately chosen parameters), outperforms its modifications for EM optimization problems. The number of particles in the swarm seems to be proportional to both the number of dimensions of the optimization problem and the number of iterations.

The improvement of the efficiency of the time-domain analysis using frequency-domain response and the Fourier transform is achieved by adjusting the current expansion at each frequency and by interpolating the results in the frequency domain. The future work will include investigation of the minimal number of frequency samples needed for the interpolation, a non-uniform sampling in the frequency domain and the search for the optimal interpolation technique.

We investigate uniformly-wound helical antennas above ground conductors of various shapes, with aim to determine factors that improve the antenna radiation pattern and gain. We show that the characteristics of helical antennas with large spillover fields can be improved using ground conductors (reflectors). This is achieved by intercepting unwanted radiation and redirecting it favorably. (8 pages)