Multilevel Fast Multipole Algorithm Research Articles

THE EXECUTION OF THE MULTILEVEL FAST MULTIPOLE ALGORITHM USING FOURIER ANALYSIS TO SOLVE PLANE WAVE SCATTERING FROM PERFECT ELECTRIC CONDUCTOR SURFACES

The multilevel fast multipole algorithm (MLFMA) is an algorithm employed to speed up the matrix-vector multiplication required to solve equations using an iterative method. In this algorithm, the Green's function is approximated by an expression that separates the receiver point from the source point. Each problem has a unique Green's function, which means derivations are required to get the approximation. As a result, although the algorithm is the same for all problems, each problem requires a new code. To solve these issues, the MLFMA is executed by approximating the Green's function using a Fourier series. The proposed approach is to finding the approximation is straightforward. In addition, the code will always be the same regardless of the problem. In this study, electromagnetic wave scattering from two- and three-dimensional perfect electric conductor surfaces was solved using the proposed approach, the conventional approach, and the method of moments. The results revealed that the suggested method is precise and faster compared to the method of moments and slightly slower than the MLFMA for a large number of unknowns.

Design, simulation, and measurement of near-zero-index shells for electromagnetic beam generation

We present design and simulation of three-dimensional (3D) shell structures, which generate directional radiation patterns from isotropic sources thanks to their near-zero-index (NZI) characteristics, as well as realizations of these shells via low-cost 3D printing. Throughout the design process of NZI beam generators, both homogenized structures, for which near-zero relative permittivity and/or permeability values are enforced, and actual models involving periodic arrangements of dielectric rods are considered. Solutions of the electromagnetic problems are obtained by using rigorous implementations of the state-of-the-art surface-integral-equation (SIE) formulations in frequency domain. Iterative solutions of matrix equations derived from SIEs are accelerated by different forms of the multilevel fast multipole algorithm (MLFMA) and suitable preconditioners, when necessary. In the design process of NZI shells, alternative strategies are employed to obtain customized radiation patterns. In this context, various cavities with strong resonance behaviors are designed as source regions. At the same time, outer surfaces are modified to either enhance or suppress outgoing electromagnetic fields. In addition to comprehensive simulations and analyses of NZI beam generators, their capabilities are verified by measurements, specifically at 10.3 GHz, on different prototypes fabricated via 3D printing. Measurements of diverse NZI shell structures are presented to demonstrate that NZI properties can successfully be achieved by well-designed arrangements of dielectric rods with proper materials. The results demonstrate the feasibility of efficient, effective, low-cost, and reconfigurable NZI shells to generate alternative beam configurations that can be useful in a plethora of microwave applications.

A Multi-Resolution Preconditioner for Nonconformal Meshes in the MoM Solution of Large Multiscale Structures

The paper presents a multi-resolution preconditioner able to improve the solution convergence, via the method of moments and the multilevel fast multipole algorithm, in the case of non-conformal meshes applying the multi-branch Rao-Wilton-Glisson basis functions. The proposed preconditioner enables, for the first time, an automatic multi-level quasi-Helmholtz decomposition on non-conforming meshes, including also the generation of the topological (global) loop functions. Moreover, the generation of the proposed preconditioning scheme is fully parallelized in a multicore shared-memory enviroment. Numerical results show the great flexibility of this approach for the solution of electrically-large multi-scale objects including h-refinement discretizations.

Massive parallelization of multilevel fast multipole algorithm for 3-D electromagnetic scattering problems on SW26010 many-core cluster

Massive parallelization of multilevel fast multipole algorithm for 3-D electromagnetic scattering problems on SW26010 many-core cluster

An Efficient and Effective Preconditioner for the Discontinuous Galerkin Domain Decomposition Method of Surface Integral Equation

This paper proposes a new preconditioning technique based on a restricted additive Schwarz (RAS) approach to improve the efficiency of the discontinuous Galerkin surface integral equation method. The RAS preconditioning is implemented efficiently using an oct-tree structure derived from the multilevel fast multipole algorithm (MLFMA) and is constructed using near-field matrices associated with boxes at the finest level. Compared to existing domain decomposition preconditioning methods based on subdomain block matrices, the proposed RAS preconditioning significantly reduces computation time and memory requirements, while providing scalable convergence for iterative solutions. Numerical experiments are presented to demonstrate the performance of the proposed cost-effective preconditioning technique.

Investigations on electromagnetic scattering characteristics of aircraft rudder considering electromagnetic discontinuities

This work investigates the scattering characteristics of the rudder structure of aircraft considering electromagnetic discontinuities through multi level fast multipole algorithm (MLFMA) of the computational electromagnetics. Firstly, the scattering characteristics of the rudder seam and its influence on the omnidirectional radar cross section (RCS) are performed. Based on the traditional high-frequency analysis theory, the source and spatial contribution distribution of seam scattering are further studied. Numerical results demonstrate that the sidewall of the rudder seam is an important scattering source, and the radar absorbing material (RAM) coated on the sidewall of the seam has a good RCS reduction effect. In addition, this work presents the influence of the rudder deflection on the stealth performance, analyzes the effect of the rudder movement of the ordinary aileron and the split drag rudder on the scattering characteristics of the whole aircraft. The common aileron has little effect on the stealth performance of the aircraft, and it only causes scattering peak in the direction opposite to the control surface. However, the split drag rudder has great damage to the opposite direction stealth performance of the aircraft. When its deflection angle is 45°, the mean value of the opposite direction RCS increases by about two orders of magnitude compared with the deflection angle 0° state. The results indicate that the structure of aircraft rudder has a significant impact on the RCS characteristics of the whole aircraft.

High-Performance Evaluation of the Interpolations and Anterpolations in the GPU-Accelerated Massively Parallel MLFMA

This communication investigates high-performance computation schemes for local Lagrange interpolation and anterpolation operations in the parallel graphics processing unit (GPU)-accelerated distributed-memory multilevel fast multipole algorithm (MLFMA). Two ELLPACK format-based schemes, namely, block ELLPACK (ELL-B) and hybrid compressed sparse column (CSC)-block ELLPACK (CSC-ELL-B), are proposed for the evaluation of interpolation and anterpolation operations, respectively, which ensure high computational throughput for GPU calculation. Optimization using the GPU hierarchical memory architecture, the mechanism of the stream and the CPU/GPU asynchronous computation pattern are employed to further improve the overall performance. The proposed schemes are proven to be an order of magnitude faster than the conventional schemes for aggregation/disaggregation operations. For an aircraft model involving over 10 billion unknowns, the iteration time is reduced by over half, which is remarkable progress in the development of GPU-accelerated parallelization of MLFMA.

Microwave Imaging of 3-D Dielectric–Magnetic Penetrable Objects Based on Integral Equation Method

Some objects like mineral substances may be both dielectric and magnetic and their high-resolution inner imaging is very desirable when detecting and analyzing their ingredients. An integral equation method is developed for reconstructing or imaging such objects by using microwave illumination. Since the objects are both dielectric and magnetic, full volume integral equations (VIEs) are needed to describe the problem. The objects can be reconstructed by alternatively solving the forward scattering VIEs (FSVIE) and the inverse scattering VIEs (ISVIEs) under the Born iterative method (BIM) or distorted BIM (DBIM). The Nyström method is used to solve the FSVIEs while a Gauss-Newton minimization method (GNMM) with a multiplicative regularization scheme (MRS) is employed to solve the ISVIE. The Nyström method is more suitable for solving the FSVIEs of inverse problems because it is a point-matching method. To accelerate the solution, the multilevel fast multipole algorithm (MLFMA) is incorporated with the Nyström method which is new for solving inverse problems. Also, using the MRS-based GNMM to solve the ISVIEs can facilitate the selection of regularization parameter and reduce the requirements on measured data. Numerical examples are provided to demonstrate the approach and good results have been achieved.

Accurate and Robust Surface MLFMA Solution for Scattering From a Large Plasma Object With Large Negative Permittivity

We present an accurate and robust surface integral equation (SIE) method for electromagnetic analysis of large-scale plasma media with large negative permittivity. To accelerate the matrix-vector multiplication (MVM) in the iterative solution of the matrix equation, the multilevel fast multipole algorithm (MLFMA) is generally preferred. Traditional implementation of MLFMA will lead to inaccurate results or poor convergence due to the numerical instability arising from the large negative permittivity. To tackle this problem, we analyze the formula of MLFMA in detail and propose a simple and efficient truncation strategy to improve its robustness. A practical and elegant truncation bound is derived based on the ratios of matrix elements between an ordinary permittivity background medium and the negative permittivity plasma. Applying the proposed truncation strategy, the performance of several commonly used SIE formulations is investigated comprehensively. Numerical results demonstrate that the proposed method can avoid numerical instability while maintaining high efficiency for large-scale plasma media with large negative permittivity.

An MLFMA-Based Eigenmode Theory for Electromagnetic Scattering Analysis From Electrically Large and Complex Conducting Objects

The eigenmode theory, which uses the modal currents solved from surface differential equations (SDEs) as the basis functions of the method of moment (MoM), has been applied to the analysis of electromagnetic scattering and radiation problems of electrically small and simple perfect electrically conducting (PEC) objects. In this article, a multilevel fast multipole algorithm (MLFMA)-based eigenmode theory is presented for analyzing electromagnetic scattering from electrically large and complex PEC objects. The method applies the MLFMA to accelerate the matrix-vector multiplications involved in filling the dense impedance matrix and therefore is much more efficient than the conventional eigenmode theory, especially for electrically large objects. Meanwhile, the way to determine a priori the number of the required modal currents for arbitrarily shaped objects is presented, which makes the MLFMA-based eigenmode theory applicable to the scattering analysis from complex objects. Numerical results verify the accuracy and efficiency of the proposed method.

A Novel Hybrid Approach for Computing Electromagnetic Scattering from Objects with Honeycomb Structures

We propose in this paper a novel hybrid numerical modeling method for computing electromagnetic scattering from inhomogeneous targets containing honeycomb structures. In the proposed approach, the whole honeycomb structure is divided into the inner and outer two subregions. Each thin wall of a unit cell in the outer subregion is replaced by a zero-thickness surface, with the aid of a resistive sheet boundary condition (RSBC) to describe the electric and magnetic field discontinuities across the surface. Each unit cell in the inner subregion is homogenized by using the Hashin–Shtrikman and the Mori–Tanaka formulae. The two subregions are further divided into smaller subdomains by introducing the Robin-type transmission condition to couple subregion interfaces, as well as subdomain interfaces. The whole solution region is then discretized and solved using the nonconformal domain decomposition-based hybrid finite element–boundary integral–multilevel fast multipole algorithm (FE-BI-MLFMA). The numerical results demonstrate that the proposed approach exhibits a high accuracy, efficiency, and flexibility. Solutions of scattering by a wing-like object and a practical unmanned aerial vehicle (UAV) model with honeycomb radar-absorbing structures are presented, showing the superior performance of the proposed algorithm.

Improving the Efficiency of Three-Dimensional Multilevel Fast Multipole Algorithm by the Triangular Interpolation

The efficiency of the multilevel fast multipole algorithm (MLFMA), which needs the interpolation and anterpolation (I&A) processes between consecutive levels, is improved by using the isosceles triangular interpolation (TI) instead of the commonly used Lagrange interpolation (LI). For the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> th-order interpolation where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p &gt;0$ </tex-math></inline-formula> , the number of sampling data points used for the TI is ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p +1$ </tex-math></inline-formula> )( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p +2$ </tex-math></inline-formula> )/2, much less than the LI, which needs ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p +1$ </tex-math></inline-formula> )2 points. To effectively use the TI, the sampling points on the Ewald sphere should be reselected to form isosceles triangular data grids but not the common rectangular ones. However, the numerical integration in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> -direction over the Ewald sphere is based on the Gauss–Legendre quadrature rule, resulting in nonuniform triangular grids. A novel TI approach suitable for the nonuniform grids is proposed, while the interpolation error in the TI is analyzed in detail. Numerical results of typical electromagnetic (EM) scattering cases are shown to illustrate the accuracy and efficiency of the proposed TI-based MLFMA.

Computational design of nanoantennas with improved power enhancement capabilities via shape optimization

Computational design and analyses of nanoantennas obtained via surface shape optimization are presented. Starting with a kernel geometry, free deformations are applied on selected surfaces to reach optimal designs that can provide improved power enhancement capabilities at desired frequencies. An in-house implementation of genetic algorithms is efficiently combined with the multilevel fast multipole algorithm developed for accurate solutions of plasmonic problems to construct the effective optimization environment. The geometries obtained via optimization do not only represent optimal shapes within the allowed deformation limits but also reveal certain types of modifications on kernel geometries to improve their performances.

Numerical Simulation of the Radar Cross Section of UHV/EHV Power Lines Illuminated by X-, C-, L-, and P-Band Spaceborne SAR

Many spaceborne synthetic aperture radars (SARs), which operate separately in the X, C, and L bands, have been employed to monitor ultra-high-voltage (UHV) and extra-high-voltage (EHV) power lines. However, a full interpretation of the scattering characteristics of these power lines in spaceborne SAR images remains a challenging task. A numerical simulation of the radar cross-section (RCS) of UHV/EHV power lines in the X, C, L, and P bands over a wide range of incidence angles is presented in this paper. A physical model of the UHV/EHV power line is first built. Here, two practical shapes of the UHV/EHV power line are considered, which include the symmetrical catenary when two suspension points have the same height and the inclined catenary when the heights of two suspension points are different. Then, the multilevel fast multipole algorithm (MLFMA) is used for the analysis of the backscatter behavior of UHV/EHV power lines in the X, C, L, and P bands. The effects of several parameters of practical importance on the backscatter RCS of UHV/EHV power lines are also studied, which include the carrier frequency, polarization type, aspect angle of illumination, span length, height difference between two suspension points, conductor sag, and cable diameter. In addition, several comparisons made between simulated and practical observation results demonstrate that the obtained theoretical results can be used to interpret the scattering characteristics of UHV/EHV power lines in spaceborne SAR images and can provide some useful suggestions for monitoring UHV/EHV power lines using spaceborne SAR images.

Hierarchical Pattern Exploitation for Efficient Electromagnetic Analysis of Finite Periodic Array

The hierarchical pattern exploitation (HPE) method is proposed for the efficient electromagnetic analysis of finite periodic arrays. Instead of taking advantage of the Toeplitz symmetry by discretizing integral equations (IEs), HPE partitions the array hierarchically to form considerably larger geometrical repetitions, which result in identical interaction blocks inside the system matrix. These interaction blocks representing patterns are characterized and hashed to generate a directory, which maps the patterns to the references of the submatrices. Due to the large proportion of repetitions in hierarchical matrix ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -matrix), HPE accelerates the matrix assembly and reduces the required storage drastically. Numerical examples show that the HPE outperforms the classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -matrix by a large margin with much less storage and CPU time cost for the analysis of finite periodic arrays. Compared with the multilevel fast multipole algorithm (MLFMA) and the characteristic basis functions method (CBFM), HPE also demonstrates competitive performance, which validates the effectiveness and efficiency of the proposed method.

Design, Fabrication, and Measurement of Efficient Beam-Shaping Reflectors for 5G Applications

An optimization procedure to design passive metallic reflectors for the fifth-generation (5G) mobile network applications, as well as fabrication and measurement of the designed reflectors, are presented. These reflecting surfaces can be used as passive repeaters in both indoor and outdoor applications to change field coverage by redirecting incident beams into desired directions with relatively controllable power distributions. The design procedure involves an efficient combination of a heuristic optimization method [based on genetic algorithms (GAs)] and a full-wave electromagnetic solver [based on the multilevel fast multipole algorithm (MLFMA)]. The optimized geometries are realized using a 3-D printing method that enables low-cost and adaptive fabrication. During optimizations, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10\lambda $ </tex-math></inline-formula> by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10\lambda $ </tex-math></inline-formula> compact and efficient reflectors are implemented as reference designs, and they are fabricated to have 18 GHz center frequency of operation. Measurements in a free-space setup are in good agreement with simulations, demonstrating the success of both the design procedure and the designed reflectors as potential 5G components.

An Efficient Volume Integral Equation Method for Analysis of Boresight Error of a Radome with Minor Ablation

In this paper, an efficient method based on volume integral equation is developed to analyze the effects of ablation of a radome on the boresight error. To avoid recalculating the whole impedance matrix when the permittivity of the radome or the shape of the top portion is slightly changed due to ablation, the radome is divided into unaffected and affected parts and the volume equivalent current instead of the displacement current is used as the unknown. This permits us to reassemble rather than recalculate the impedance matrix when the ablation condition is altered. Moreover, a viable preconditioning technique is introduced and integrated with the multilevel fast multipole algorithm (MLFMA) to cope with the electrically large antenna-radome system (ARS). Simulation results are provided for the boresight error (BSE) and boresight error slope (BSES) of the ARS at some different ablation states. The present approach is considerably faster than using the conventional methods.

Study on Reradiation Interference Characteristics of Steel Towers in Transmission Lines

With the development of ultrahigh-voltage transmission lines in China, the reradiation interference caused by the steel tower used in ultrahigh-voltage transmission becomes increasingly aggravating for nearby radio stations. In this paper, using the multilevel fast multipole algorithm, the reradiation interference of an ultrahigh-voltage transmission steel tower is investigated. Additionally, the reradiation interference characteristics of a transmission steel tower were investigated with various frequencies and azimuth angles of incidence wave. The results show that the frequency and the azimuth angle of incident wave are the impact factors for the reradiation interference of steel towers. It is better to use a detailed angle steel model for the truss structure of steel towers at high frequency, while a simplified structural model can be used at low frequency. Additionally, the diffraction at the edge of the angle steel has a great influence on the reradiation interference, particularly at high frequencies.

Analysis of Oversized Electrodynamic Systems by a Combined Method Based on the Principle of Physical Optics and the Multilevel Fast Multipole Algorithm

Analysis of Oversized Electrodynamic Systems by a Combined Method Based on the Principle of Physical Optics and the Multilevel Fast Multipole Algorithm

A Novel Embedded Domain Decomposition Method for Electromagnetic Simulation of Structures in Inhomogeneous Medium

A novel embedded domain decomposition method (EDDM) is proposed to simulate electromagnetic scattering from structures in an inhomogeneous medium. In this method, the inhomogeneous medium (e.g., layered substrate) is set as the background subdomain, and the integrated structures (e.g., metallic/dielectric components) with properly defined buffer regions (BRs) are set as the embedded subdomains, where the meshes of two can be completely independent and arbitrarily overlapping. The major contribution of this work lies in the following three aspects. First, to allow for arbitrary inclusion of the embedded subdomains in an inhomogeneous background subdomain, an adaptive BR method is developed. The self-coupling equation is rederived to ensure the consistency of the BR and the inhomogeneous background, and a new mutual-coupling equation is introduced to account for the material and/or conductor differences. Second, the boundary element method (BEM) accelerated by the multilevel fast multipole algorithm (MLFMA) is utilized to truncate the finite-element region so that the computation capability can be enhanced and the extra air box setting of the background subdomain in the original EDDM can be avoided. Third, for structure adjustments such as translations and rotations, we further propose an inherited calculation to avoid mesh regeneration and repeated computation of the self-coupling and preconditioning matrices and also reduce the iteration counts and solution time. Numerical examples have shown the accuracy, robustness, and efficiency of the proposed method.