Multilevel Fast Multipole Method Research Articles

On Special Functions for Radiation From Sources Close to an Electrically Large Convex Surface

Several high frequency asymptotic methods in electromagnetics include Fock-type functions. Acoustic hard and soft Fock functions are used as special functions in asymptotic solutions for radiation from sources on an electrically large conducting convex surface. Formulations are available for accurate evaluation of these functions. Asymptotic solutions for radiation from sources close to an electrically large conducting convex surface include the derivatives of these Fock functions as well. However, the existing formulations for the functions do not yield accurate and continuous results for their derivatives. In this letter, the existing formulations are modified to adjust the boundaries between separate intervals for efficient and accurate evaluation of the Fock functions and their derivatives, and to assess continuity across the boundaries for different numbers of terms. The modified formulations are used in the extended uniform geometrical theory of diffraction solution to predict the radiation pattern of a quarter wavelength monopole antenna, which is validated in comparison with the results of the multilevel fast multipole method (MLFMM).

Bistatic RCS Measurements of Large Targets in a Compact Range

This communication illustrates the ability to perform bistatic radar cross-section (RCS) measurements at a fixed bistatic angle in a compact range. Literature regarding bistatic RCS measurements in compact ranges is limited. The traditional setup of a compact range was adapted to perform bistatic RCS measurements. These bistatic measurements were conducted on canonical and complex realistic scale airframe models. The targets were illuminated with a plane wave created by an offset parabolic dish reflector. The bistatic scattering of the targets was measured by placing a receive antenna at a fixed bistatic angle and finite distance in the compact range. This communication also investigates the effect of the finite separation between the targets and the receiver on the bistatic scattering measurements of large complex targets. The accuracy of the bistatic RCS measurements is compared to full-wave simulations conducted with FEKO using the multilevel fast multipole method solver. Quantitative comparisons are drawn between the simulations and measurements using the feature selective validation method.

An Accelerated SBR for EM Scattering From the Electrically Large Complex Objects

In this study, an accelerated shooting and bouncing ray (SBR) method was proposed to improve the computational efficiency of electromagnetic (EM) scattering from the electrically large complex objects. Initially, an open graphics library was employed to speed up the shadow faces removal. Then, the neighbor search technique was utilized to accelerate the ray tracing process, which was different from traditional SBR based on the traversal algorithms of Octree. The validation of the proposed method was verified by comparing the simulation results with those of the FEKO multilevel fast multipole method. Moreover, the EM scattering from two electrically large complex ship models was analyzed, and a desirable acceleration ratio was achieved. Therefore, the accelerated SBR method yields a superior performance for scattering analysis of electrically large complex targets, especially at high frequency.

A fast high order algorithm for multiple scattering from large sound-hard three dimensional configurations

We describe an efficient fast high order algorithm for simulating acoustic scattering in an unbounded three dimensional medium exterior to a configuration with a large number of sound-hard particles. Surface integral representations of the scattered and far fields facilitate reformulation of the model for unknown fields on the closed bounded surface of each particle in the configuration. Our surface integral equation based algorithm uses a multiple scattering variant of complex dense system Krylov subspace techniques combined with an efficient matrix vector product scheme. In this article we develop a memory efficient fast algorithm in three steps using appropriate expansion methods (EM). The first step is based on a radiating basis EM (RadBEM) and the third step is based on a regular basis EM (RegBEM). The second step transforms RadBEM to RegBEM and is accelerated using the well known multilevel fast multipole method. The full algorithm has linear complexity with respect to the number of particles to evaluate quadratically large multiple scattering interactions exterior to the configuration. We present numerical results obtained with our fast algorithm for simulations with up to about twenty thousand sound-hard particles and for large dense complex systems with more than twelve million unknowns. We compare our results with a recent high-order accurate accelerated direct solver by Bremer et al. (2015) and demonstrate substantial computational advantages of our RadBEM–RegBEM algorithm.

PEC 또는 PMC 영역을 포함하는 불균일 임피던스 매질의 산란 해석을 위한 수정된 SDIE MLFMM 방법

A coated perfect electric conductor(PEC) interface to reduce scattered fields can be efficiently modeled by using the impedance boundary condition. The self-dual integral equation(SDIE) proposed by Yan et al. may be an efficient multi-level fast multipole method (MLFMM) formulation for the impedance object. This equation can be applied to an inhomogeneous impedance material, but its accuracy can be degenerated when the material contains a PEC or perfect magnetic conductor(PMC) region. In this paper, we modify the original SDIE formulation for an inhomogeneous object containing a PEC or PMC region and numerically verify its accuracy.

Microwave vortex beam launcher design

A novel design for a vectorial vortex beam launcher in the microwave regime is devised. The beam is formed by launching a single guided transverse electric (TE) mode of a metallic circular waveguide into free-space. Excitation is achieved by the mean of an inserted coaxial loop antenna. Modal expansion coefficients are computed, and the resulting electric and magnetic fields are determined. The effect of the antenna location inside the waveguide on its effective input impedance is modelled using transmission-line relations and location for optimal matching is established. The analytical results are confirmed using multi-level fast multipole method full-wave simulations.

A fast numerical framework to compute acoustic scattering by poroelastic plates of arbitrary geometry

We present a fast numerical framework for the computation of acoustic scattering by poroelastic plates of arbitrary geometries. A boundary element method, BEM, is applied to solve the Helmholtz equation subjected to boundary conditions related to structural vibrations. This analysis is performed by rewriting the BEM boundary conditions in terms of a modal basis of the poroelastic plate which is computed by the finite element method, FEM. The current formulation allows a direct solution of the fully coupled fluid-structure interaction problem. In order to accelerate the solution of the large dense linear systems from the BEM formulation in three-dimensional problems, a wideband adaptive multi-level fast multipole method, FMM, is employed. A parametric study is carried out for the trailing-edge scattering of sample acoustic sources, representative of either uncorrelated turbulent eddies or a non-compact turbulent jet. Firstly, the noise scattering by a compact quadrupole source is analyzed for low and high frequencies for square and trapezoidal plates. Results show that geometric features such as trailing-edge sweep and serrations are very effective in the reduction of noise scattering. Moreover, it is shown that finite elastic plates are more effective in reducing the scattered noise at higher frequencies. On the other hand, porosity is more effective in reducing the radiated sound for lower frequencies. Results demonstrate that elasticity and porosity can be combined with trailing-edge sweep and serrations to reduce the scattered noise at a broader range of frequencies for poroelastic plates.

Synthesis and Analysis of Low Profile, Metal-Only Stepped Parabolic Reflector Antenna

This paper describes an intuitive approach for synthesizing metal-only, low-profile, high-gain stepped reflector antennas that can aid emerging technologies in the areas of communications and remote sensing. The required reflector aperture is synthesized using a family of confocal parabolic sections, each of which scatters in phase to provide a focused beam in the far field. The parabolic sections provide a distinct advantage over the conventional Fresnel zone plate (FZP) antenna, since the radiation is inherently unidirectional. Furthermore, the metal-only structure avoids the need for dielectric characterization, which is essential for reflectarrays. Using this technique, it is possible to achieve depths of the order of one wavelength while ensuring high efficiencies. Starting from a symmetric stepped reflector synthesis, two distinct approaches to synthesize offset stepped reflectors are explored in this paper. We highlight several interesting tradeoffs between electromagnetic performance and mechanical complexity that arise for each of these approaches. A thorough analysis of the frequency performance of stepped parabolic reflectors is also presented. Comprehensive simulations using both physical optics and multilevel fast multipole method validate our theoretical predictions. The formulations presented in this paper are especially relevant at mmWave frequencies and beyond, which has attracted significant interest for modern technologies.

MLFMM-based, fast multiple-reflection physical optics for large-scale electromagnetic scattering analysis

The asymptotic, physical optics (PO) approximation is applicable to electromagnetic scattering analysis of electrically large, conducting objects, which is important for various applications, such as radar cross-section (RCS) calculations. It relates the incident magnetic field to the induced surface current at points on a scatterer's surface with line-of-sight visibility to the source. Here, the focus is on mesh-based PO, using a triangle element mesh with standard Rao–Wilton–Glisson basis functions to represent the surface current density. The multiple-reflection PO (MRPO) method involves applying the PO approximation to the field radiated by the present current solution towards the scatterer itself, to account for successive internal reflections. The visibility status between every pair of basis functions is required, taking into account all geometry. In a conventional implementation, internal reflected field calculation and internal shadowing determination runtimes both scale as O(N2); N denotes the number of mesh elements. In this paper an accelerated version of the mesh-based MRPO method is presented: fast MRPO (FMRPO). The multi-level, fast multipole method (MLFMM) is used to accelerate internal reflected field calculation. The key aspect to achieving acceleration for general geometries, is to incorporate internal shadowing into the MLFMM interaction tree. This requires an alteration to the inter-group interaction criterion of the MLFMM, as well as efficient evaluation of inter-group shadowing status flags, to preserve the beneficial cost-scaling property of the MLFMM. Algorithmic parameters are introduced, which control the accuracy of internal shadowing determination. The FMRPO runtime scales as O(NlogN), in the high-frequency limit. Numerical results are presented, demonstrating accuracy and efficiency for general scattering objects.

MLFMM(Multi-Level Fast Multipole Method) 방법에 적용된 BiCGstab(l)반복법의 l값에 따른 연산량 분석 및 효율적인 l값

The method of moments(MoM) is one of the most popular integral-equation-based full-wave simulation methods, and the multi-level fast multipole method(MLFMM) algorithm can be used for its efficient calculation. When calculating the surface current on the large scatterer in the MoM or MLFMM, iterative methods for the final matrix inversion are used. Among them, BiCGstab(l) has been widely adopted due to its good convergence rate. The number of iterations can be reduced when l becomes larger, but the number of operations per iteration is increased. Herein, we analyze the computational complexity of BiCGstab(l) in the MLFMM method and propose an optimum choice of l.

RWG MoM‐via‐locally corrected Nyström method in near‐field to far‐field transformation using very‐near‐field measurement

Locally corrected Nystrom (LCN) method is used to solve electric field integral equations (EFIE) of the equivalent current method in the planar very-near-field measurement of antennas and RF circuits. The exact relationship between the Rao–Wilton–Glisson (RWG) method of moments and first-order and zero-order LCN is established for both magnetic and electric currents to ensure normal current continuity between adjacent triangular patches. The proposed method is a point-based RWG discretisation of EFIE and causes a noticeable decrease in the degree of freedom. It consequently eliminates spurious charges and significantly lowers the condition number of the impedance matrix. Moreover, it is more efficient to be accelerated by fast algorithms such as multi-level fast multipole method. In what follows, the detailed explanation of the proposed method, along with the examples of current reconstruction and antenna far-field calculation, is presented.

Design and Simulation of the Reflector in CATR Based on MLFMM and PO

The design of Compact Antenna Test Range (CATR) is a challenging job. To achieve desired electromagnetic characteristics, choosing an appropriate algorithm for the proper design of the reflector with electrically large size is necessary. The comparison of Multilevel Fast Multipole Method (MLFMM) and Physical Optics (PO), which are the most two common algorithms used in reflector design, are performed in this paper to verify the necessity of MLFMM firstly. Then we design a novel reflector used in CATR with aperture size of 8.7m×6.9m and quiet zone size of Ф2m×2m, noting that all the design and simulation in this paper are on the basis of Boundary Element Method (BEM) module in the RainbowStudio[1].

Adaptive Multipoint Model Order Reduction Scheme for Large-Scale Inductive PEEC Circuits

The model order reduction techniques based on the multipoint projection Krylov methods have become the methods of choice to generate large macromodels of multiport RLC circuits. A well-known difficulty with such methods lies in the need for clever point selection to attain model compactness and accuracy. In this paper, we present an automatic methodology for optimizing sample point selection. This method is general, suitable for circuits obtained by the partial element equivalent circuit method coupled with an adaptive multilevel fast multipole method. Our algorithm has been validated on an industrial example to demonstrate the accuracy and robustness of the approach.

An Alternative Direction Decomposition Scheme and Error Analysis for Parabolic Equation Model

Alternate direction implicit (ADI) method has been widely used to analyze the tunnel propagation or electromagnetic scattering problem for the finite difference method. In this paper, a novel alternate direction decomposition (ADD) scheme is proposed to simplify the split-step Fourier transform (SSFT) algorithm in parabolic equation (PE) model. The spatial matrix of electric field is decoupled into two subarrays by ADD-SSFT algorithm, which is used to decrease the computational burden for 3-D PE problems in a large scale environment after the integration order exchanged. Also, the phase error of this approximation is analyzed to show the algorithm performance, along with several numerical results, including the tunnel cases with traditional modal theory, and they are provided to validate and calibrate the PE model, and demonstrate the efficiency and accuracy of the proposed algorithm. Moreover, the scattering computation is investigated and compared with multilevel fast multipole method to show that ADD-SSFT is appropriate for solving the scattering problems for the electrical large target.

Speed-up of the volumetric method of moments for the approximate RCS of large arbitrary-shaped dielectric targets

A method for the rigorous computation of the electromagnetic scattering of large dielectric volumes is presented. One goal is to simplify the analysis of large dielectric targets with translational symmetries taken advantage of their Toeplitz symmetry. Then, the matrix-fill stage of the Method of Moments is efficiently obtained because the number of coupling terms to compute is reduced. The Multilevel Fast Multipole Method is applied to solve the problem. Structured meshes are obtained efficiently to approximate the dielectric volumes. The regular mesh grid is achieved by using parallelepipeds whose centres have been identified as internal to the target. The ray casting algorithm is used to classify the parallelepiped centres. It may become a bottleneck when too many points are evaluated in volumes defined by parametric surfaces, so a hierarchical algorithm is proposed to minimize the number of evaluations. Measurements and analytical results are included for validation purposes.

Fast multipole method applied to Lagrangian simulations of vortical flows

Lagrangian simulations of unsteady vortical flows are accelerated by the multi-level fast multipole method, FMM. The combination of the FMM algorithm with a discrete vortex method, DVM, is discussed for free domain and periodic problems with focus on implementation details to reduce numerical dissipation and avoid spurious solutions in unsteady inviscid flows. An assessment of the FMM-DVM accuracy is presented through a comparison with the direct calculation of the Biot–Savart law for the simulation of the temporal evolution of an aircraft wake in the Trefftz plane. The role of several parameters such as time step restriction, truncation of the FMM series expansion, number of particles in the wake discretization and machine precision is investigated and we show how to avoid spurious instabilities. The FMM-DVM is also applied to compute the evolution of a temporal shear layer with periodic boundary conditions. A novel approach is proposed to achieve accurate solutions in the periodic FMM. This approach avoids a spurious precession of the periodic shear layer and solutions are shown to converge to the direct Biot–Savart calculation using a cotangent function.

Reliable Approach for Composite Scattering Calculation From Ship Over a Sea Surface Based on FBAM and GO-PO Models

A reliable method based on facet-based asymptotical model, geometrical optics and physical optics (GO-PO) is developed to calculate composite scattering from 3-D complex ship targets over a rough sea surface. The backward ray-tracing is described to identify the visibilities of patches to the incident wave and the reflected wave of other patches on the target and facets on the sea surface. Both the scattering from target, including higher orders of scattering and the interactions between the target and rough sea surface, can be treated handily by the proposed GO-PO method. The accuracy of the target scattering and composite scattering is demonstrated by comparing with Multi-level fast multipole method (MLFMM), method of equivalent currents, and four-path model. Moreover, numerical examples of radar cross section estimation, electric current distribution, and synthetic aperture radar imagery simulation from a real ship target over a randomly rough sea surface are presented and discussed.

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

AbstractA novel fast and accurate algorithm is developed for large‐scale 3‐D gravity and magnetic modeling problems. An unstructured grid discretization is used to approximate sources with arbitrary mass and magnetization distributions. A novel adaptive multilevel fast multipole (AMFM) method is developed to reduce the modeling time. An observation octree is constructed on a set of arbitrarily distributed observation sites, while a source octree is constructed on a source tetrahedral grid. A novel characteristic is the independence between the observation octree and the source octree, which simplifies the implementation of different survey configurations such as airborne and ground surveys. Two synthetic models, a cubic model and a half‐space model with mountain‐valley topography, are tested. As compared to analytical solutions of gravity and magnetic signals, excellent agreements of the solutions verify the accuracy of our AMFM algorithm. Finally, our AMFM method is used to calculate the terrain effect on an airborne gravity data set for a realistic topography model represented by a triangular surface retrieved from a digital elevation model. Using 16 threads, more than 5800 billion interactions between 1,002,001 observation points and 5,839,830 tetrahedral elements are computed in 453.6 s. A traditional first‐order Gaussian quadrature approach requires 3.77 days. Hence, our new AMFM algorithm not only can quickly compute the gravity and magnetic signals for complicated problems but also can substantially accelerate the solution of 3‐D inversion problems.

A Cholesky-Based SGM-MLFMM for Stochastic Full-Wave Problems Described by Correlated Random Variables

In this letter, the multilevel fast multipole method (MLFMM) is combined with the polynomial chaos expansion (PCE)-based stochastic Galerkin method (SGM) to stochastically model scatterers with geometrical variations that need to be described by a set of correlated random variables (RVs). It is demonstrated how Cholesky decomposition is the appropriate choice for the RVs transformation, leading to an efficient SGM-MLFMM algorithm. The novel method is applied to the uncertainty quantification of the currents induced on a rough surface, being a classic example of a scatterer described by means of correlated RVs, and the results clearly demonstrate its superiority compared to the nonintrusive PCE methods and to the standard Monte Carlo method.

Electromagnetic Scattering of Electrically Large Ship above Sea Surface with SBR-SDFM Method

Hybrid scheme combining shooting and bouncing ray with semi-deterministic facet model is proposed to analyze composite scattering from ship-ocean scene in this study. This model can deal with complex electromagnetic interaction between ship and sea surface. Thus, scattering properties of composite ship-ocean scenes with influence of various parameters (such as incident angle and wind speed) can be studied and analyzed efficiently. Studying such properties is of significance for target detection and high-resolution radar imaging in sea environments. Accuracy and performance of this method are validated and evaluated by comparing with multilevel fast multipole method of FEKO for electrically small objects. All simulation results indicate that the proposed method is suitable for providing preliminary radar cross section prediction of electrically large composite model.