An Efficient Higher Order MoM-PO Method for EM Scattering From Electrically Large Objects

An efficient hybrid analysis which hybridizes surface-wire integral equations (SWIE) with physical optics (PO) approximation is presented for wire antennas attached to an electrically large aircraft platform. In the analysis, the whole surface of the platform and antennas is divided into three regions, namely, the method of moments (MoM) region, the PO region and a region referred to as the joint PO-MoM (POM) region in this paper. The MoM region, generally, the smallest one among the three regions, includes wires, wire-surface junctions and a small part of the platform surface surrounding the junctions. A large part of the remaining platform surface forms the PO region with the rest being the POM region which is situated between the MoM and PO regions. The POM region is treated as a MoM region on the one hand so that the total MoM region is sufficiently large to well characterize the platform-attached antennas, whereas it is also regarded as a PO region on the other hand so that its interaction with the PO region needs not to be precisely considered so as to drastically reduce computational requirements.

This paper presents an e-cient and accurate hybrid approach of method of moments (MoM) and physical optics (PO) for radiation problems such as antennas mounted on a large platform. The new method employs higher-order hierarchical Legendre basis functions in the MoM region and higher-order Nystrom scheme in the PO region. The two regions are both discretized with large domains. The unknowns can be much less than those in the small-domain MoM- PO solutions, which will lead to a great reduction in computation complexity. Furthermore, with the Nystrom scheme in the PO region, the higher-order accuracy is maintained, and the calculation of the impedances can be more e-cient than that in the existing higher- order MoM-PO procedure. Numerical results show the validity of the proposed method.

A new efficient hybrid technique is presented, which combines the method of moments (MoM) and the shooting and bouncing ray (SBR) for analyzing scattering of complex large structures. By applying the SBR method for the whole structures, a corrected incident field including not only the direct incident field but also the multi-bounce field from the SBR region contributes to the integral equation in the MoM region. For the SBR region, it also considers the interaction from MoM region and then a physical optics (PO) type integration is performed to calculate the scattered field. For complex electrically large electromagnetic problems, the proposed technique is more efficient and simpler to implement in a general-purpose computer code. Numerical results are presented to show the efficiency and accuracy of the method.

ABSTRACTA generalized hybrid scheme combining finite element method (FEM) and the method of moments (MOM) is first presented to investigate scattering from a 3-D dielectric object above a 2-D dielectric rough surface. The hybrid FEM/MOM scheme is divided into the FEM region for the object based on Helmholtz’s equation and the MOM region for the rough surface based on coupled integral function, which can greatly reduce the computed region. With the rough surface getting larger, the unknowns of the rough surface get more. So MLFMA is employed to reduce computation complexity of MOM region itself and coupling interaction matrices between MOM region and FEM region. Consequently, the iterative process can be greatly accelerated. This generalized hybrid FEM/MOM is very suitable for investigating scattering from composite model of a complex material object and electrically large dielectric rough surface. Some examples verify this generalized hybrid FEM/MOM in high efficiency, low storage requirement, high precision and generality.

This paper presents a novel Galerkin-type method of moments solution of the volume surface integral equation (VSIE), which is developed for the analysis of electromagnetic scattering and radiation from composite metallic and dielectric structures. Our new scheme utilizes curved tetrahedral elements and curved triangular patches for geometric modeling, and the associated higher order hierarchical vector (HOHV) basis functions for volume/surface current modeling. In contrast to the previous studies on higher order VSIE, our scheme, which is reported for the first time in VSIE, is much more flexible. In comparison with the conventional low-order scheme, our scheme requires much less memory and computational time for the same accuracy level. Numerical results are provided to demonstrate the accuracy, efficiency, and flexibility of our scheme.

Plane wave scattering by an infinite, three-dimensional wedge whose faces are characterized by periodic anisotropic impedances is investigated. An improved hybrid of the moment method (MM) and the physical optics (PO) is proposed to calculate the equivalent electric current on the wedge surface. As for the infinite periodic wedge surface, a periodic representation of the scattering surface is used to prevent nonphysical edge effects in the scattering evaluation. The surface of the wedge is split into an MM region and a PO region. In the implementation of MM, the evaluation of the interaction matrix elements in the MM region is simplified by deriving six universal series. Moreover, the PO-current is modified by a diffracted current to improve its accuracy. The accuracy and effectiveness of the proposed approach is validated and demonstrated by using series solution, the method in [T. B. A. Senior, “Solution of a class of impedance wedge problem for skew incident,” Radio Sci., vol. 21, no. 2, pp. 185-191, 1986], and the two-dimensional improved MM-PO approach under several canonical two-dimensional wedge cases. At last, the proposed approach is applied to analyze the scattering from the wedges with periodic anisotropic impedance faces, and the numerical results are presented.

A novel higher order large-domain hybrid computational electromagnetic technique based on the finite element method (FEM) and method of moments (MoM) is proposed for three-dimensional analysis of antennas and scatterers in the frequency domain. The geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders. The fields and currents on elements are modeled using curl- and divergence-conforming hierarchical polynomial vector basis functions of arbitrary approximation orders, and the Galerkin method is used for testing. The elements can be as large as about two wavelengths in each dimension. As multiple MoM objects are possible in a global exterior region, the MoM part provides much greater modeling versatility and potential for applications, especially in antenna problems, than just as a boundary-integral closure to the FEM part. The examples demonstrate excellent accuracy, convergence, efficiency, and versatility of the new FEM-MoM technique, and very effective large-domain meshes that consist of a very small number of large flat and curved FEM and MoM elements, with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -refined field and current distributions of high approximation orders. The reduction in the number of unknowns is by two orders of magnitude when compared to available data for low-order FEM-MoM modeling.

An innovative and efficient hybrid technique, which combines the Method of Moments (MoM) with Volume Meshed Shooting and Bouncing Ray (VSBR), is presented to analyze the scattering of metallic–dielectric mixed multi-scale structures in urban scenes. Additionally, the technique can rapidly generate radar images at different angles, which is useful for the video remote sensing community. By dividing the mixed multi-scale targets into sub-regions, different solvers are employed to compute the scattering contributions based on their varying electrical sizes. For large-scale sub-regions, the VSBR method is employed on both the medium and metal parts, leading to a multilevel electromagnetic field including both the direct induced field and the multi-reflection field. This field contributes to the integral equation in the MoM sub-regions. Additionally, interactions from the MoM region are considered within the VSBR region, followed by a surface current integration to compute the scattered field. When addressing mixed multi-scale electromagnetic problems, this technique proves to be more efficient and easier to implement in general-purpose computer codes. Furthermore, this method is accelerated using the local coupling (LC) theory and fast multipole method (FMM). Using this fast computational method, efficient simulation results of radar images at different angles for the scenes can be obtained and the numerical results demonstrate the efficiency and accuracy of the proposed method.

The conventional hybrid method of moments (MoM)-physical optics (PO) technique provides a possible way to handle electrically large objects with affordable computer memory. However, its efficiency is not very good because the evaluation of the PO contribution to the MoM impedance matrix is very time-consuming. An efficient implementation of the iterative MoM-PO hybrid technique is presented in this paper to avoid the calculation of the PO contribution in matrix form. For electrically large objects, the proposed efficient iterative MoM-PO (EI-MoM-PO) method can greatly reduce the computational time and maintain the same or better accuracy with the same number of unknowns compared with the conventional MoM-PO method. Several examples of large-scale structures are analyzed by the EI-MoM-PO method, the conventional MoM-PO method, and the multilevel fast multipole algorithm. The excellent efficiency and accuracy are achieved by the proposed EI-MoM-PO technique.

The method of moments is based on a relation Eθ0(h(Xt, θ)) = 0, from which an estimator of θ is deduced. In many econometric models, the moment restrictions can not be evaluated numerically due to, for instance, the presence of a latent variable. Monte Carlo simulations method make possible the evaluation of the generalized method of moments (GMM) criterion. This is the basis for the simulated method of moments. Another approach involves defining an auxiliary model and finding the value of the parameters that minimizes a criterion based either on the pseudoscore (efficient method of moments) or the difference between the pseudotrue value and the quasi-maximum likelihood estimator (indirect inference). If the auxiliary model is sufficiently rich to encompass the true model, then these two methods deliver an estimator that is asymptotically as efficient as the maximum likelihood estimator.

A newly developed iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. In this fast solver, the weak-form formulation is applied to obtain a less singular discretization of the volume electric field integral equation. Several techniques are utilized to speed up the dyadic function evaluation. To accelerate the operation of the dyadic function on an induced current (i.e., the Green's operation), the function is split into convolutional and correlational components so that FFT can be applied. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns, significantly more efficient than the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic scattering problems in a planarly layered background. A large-scale scattering problem in a layered medium with more than three million unknowns has been solved on a Sun Ultra 60 workstation with 1.2 GBytes memory.

About the Author. Preface. Series Preface. 1 Basic Antenna and Propagation Theory. 1.1 Introduction. 1.2 Characteristics of Electromagnetic Waves. 1.3 Interaction between Two Waves. 1.4 Polarization. 1.5 Characteristics of an Antenna. 1.6 Propagation. References. 2 Aircraft Systems Using Antennas. 2.1 Aircraft Systems. 2.2 Frequencies of the Most Common Aircraft Systems. 2.3 Automatic Direction Finding. 2.4 Distress/SOS. 2.5 Distance Measuring Equipment. 2.6 Electronic Counter Measures. 2.7 Electronic Support Systems. 2.8 Emergency Locator Transmitter/Emergency Position IndicatingRadio Beacon. 2.9 Global Positioning System. 2.10 HF. 2.11 Instrument Landing System. 2.12 In-Flight Telephony. 2.13 Microwave Landing System. 2.14 Radar. 2.15 SatCom Civilian. 2.16 Signals Intelligence. 2.17 Tactical Air Navigation. 2.18 Traffic Collision Avoidance System. 2.19 Telemetry. 2.20 UHF. 2.21 VHF Comms. 2.22 VHF Omnidirectional Ranging. 2.23 Equipment Designation. References. 3 The Antenna Siting Process. 3.1 Introduction. 3.2 New Antenna Layouts. 3.3 Optimum Positions for Blades. 3.4 Design Phase. 3.5 Certification and Qualification Phase. 3.6 Typical Antenna Layouts. References. 4 Frequency and Spatial Coverage Considerations. 4.1 Introduction. 4.2 Effect of the Structure on the Spatial Characteristics ofthe Antenna. 4.3 Combination of Two Waves. 4.4 Measurements on Scaled Test Bodies. 4.5 Effect of Frequency on the Radiation Pattern. 4.6 Effect of Distance from Obstacles. 4.7 Effect of Wings on the Radiation Pattern. 4.8 Effect of the Curved Ground Plane and the ElectricalDimensions of the Fuselage. 4.9 Radiation Patterns on Cylinders in the Absence ofObstacles. References. 5 Antennas Used on Aircraft. 5.1 Introduction. 5.2 Near and Far Fields of an Antenna. 5.3 Antennas on Aerostructures. 5.4 Polar Radiation Patterns. 5.5 Dipoles. 5.6 Monopoles. 5.7 Loops and Notches. 5.8 Helixes. 5.9 Flat/Planar Spirals. 5.10 Patches. 5.11 Aperture Antennas. 5.12 Reflectors. 5.13 Waveguide Fed Antennas. 5.14 Model Numbers Used by Different Manufacturers. References. 6 RF Interoperability. 6.1 Introduction. 6.2 Coupling between Systems on an Aircraft. 6.3 Techniques for Achieving RF Interoperability. 6.4 Modulation. 6.5 Coupling due to Radiated Emissions through the Antennas. 6.6 Coupling between Systems with LOS Antennas. 6.7 Coupling between Systems for Antennas on Opposite Surfacesof the Fuselage. 6.8 Existing Formulas Used for Calculating Coupling between TwoAntennas on Opposite Surfaces. 6.9 Derivation of an Empirical Formula that Correlates with theMeasured Data. References. 7 Computer Modelling Techniques. 7.1 Introduction. 7.2 Overview of Computer Modelling. 7.3 Generic Types of Computer Modelling. 7.4 Method of Moments. 7.5 Finite Difference Time Domain. 7.6 GTD/UTD. 7.7 Physical Optics. 7.8 Hybrid Methods. 7.9 Comparison of Predicted Surface Currents. 7.10 Code-to-code Comparison of Radiation Patterns Predicted onthe Simplistic Airframe. 7.11 Relationship between Number of Unknowns and SurfaceArea. References. 8 Measurements. 8.1 Introduction. 8.2 Positioners. 8.3 Test Facility Antennas. 8.4 Scaled Models. 8.5 Scaled Antennas. 8.6 Absorbers. 8.7 Measurement Facilities. 8.8 Indoor Test Facilities. 8.9 Anechoic Chambers. 8.10 Compact Ranges. 8.11 Near-Field Facilities. 8.12 Outdoor Far-Field Ranges. 8.13 Ground Test Measurements. 8.14 In-Flight Measurements for System and Inter-SystemTesting. References. 9 Reference. 9.1 Centigrade to Fahrenheit Temperature Conversion. 9.2 Conductivity of Common Metals. 9.3 Degrees to Radians and Radians to Degrees. 9.4 Dielectric Constants and Loss Tangents of CommonMaterials. 9.5 Electrochemical Series. 9.6 Electromagnetic Spectrum and Frequency Bands for DifferentNomenclatures. 9.7 Formulas. 9.8 Frequency to Wavelength. 9.9 Gain in dB and Gain as a Linear Ratio. 9.10 Greek Alphabet. 9.11 Imperial to Metric Conversions Distance, Area,Volume, Speed. 9.12 Periodic Table Alphabetically. 9.13 Polarization Matching Matrix. 9.14 Power in dBm and Power in Watts. 9.15 Preferred SI Scientific Prefixes. 9.16 Terms and Definitions. 9.17 VSWR to Return Loss. Appendix: Abbreviations and Acronyms. Index.

This paper performs a Monte Carlo study on efficient method of moments (EMM), generalized method of moments (GMM), quasi-maximum likelihood estimation (QMLE) and maximum likelihood estimation (MLE) for a continuous-time square-root model under two challenging scenarios – high persistence in mean and strong conditional volatility – that are commonly found in estimating the interest rate process. MLE turns out to be the most efficient of the four methods, but its finite sample inference and convergence rate suffer severely from approximating the likelihood function, especially in the scenario of highly persistent mean. QMLE comes second in terms of estimation efficiency, but it is the most reliable in generating inferences. GMM with lag-augmented moments has, overall, the lowest estimation efficiency, possibly due to the ad hoc choice of moment conditions. EMM shows an accelerated convergence rate in the high-volatility scenario, while its over-rejection bias in the mean persistence scenario is unacceptably large. Finally, under a stylized alternative model of the US interest rates, the overidentification test of EMM obtains the ultimate power for detecting misspecification, while the GMM J-test is increasingly biased downwards in finite samples.

The high persistence of interest rates has important implications for the preferred method used to estimate term structure models. We study the finite-sample properties of two standard dynamic simulation methods—efficient method of moments (EMM) and indirect inference—when they are applied to an first order autoregressive (AR[1]) process with Gaussian innovations. When simulated data are as persistent as interest rates, the finite-sample properties of EMM differ both from their asymptotic properties and from the finite-sample properties of indirect inference and maximum likelihood. EMM produces larger confidence bounds than indirect inference and maximum likelihood, yet is much less likely to contain the true parameter value. This is primarily because the population variance of the data plays a much larger role in the EMM conditions than in the moment conditions for either indirect inference or maximum likelihood. These results suggest that, under Gaussian assumptions, indirect inference (if practical) is preferable to EMM when working with persistent data such as interest rates. EMM's emphasis on the population variance strongly enforces stationarity on the underlying process, so this same reasoning suggests that EMM may be preferable in settings where stability and stationarity are important and difficult to impose. Copyright The Author 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

An efficient method of moments (MoM) solution of the volume-surface integral equation (VSIE) is presented for the analysis of electromagnetic (EM) scattering and radiation of composite objects comprising arbitrarily contacted dielectrics and conductors. In contrast to the conventional Schaubert-Wilton-Glisson (SWG) and Rao-Wilton-Glisson (RWG) basis functions, which were widely used in the MoM solutions of the VSIE, a new hybrid pulse-rooftop basis function is proposed to explicitly enforce the boundary condition of vanishing tangential electric field at the dielectric-conductor interface. As a consequence, the displacement currents in the dielectrics can be efficiently modeled by a reduced number of volume bases on the same set of meshes, particularly when thin dielectric slabs are involved in the composite structures. The numerical results of EM scattering and radiation of several composite objects are shown to illustrate the accuracy and efficiency of the proposed scheme. Since the new hybrid basis function can be easily implemented into the existing fast algorithms, the VSIE solution presented in this paper is promising for EM problems of composite targets with large size.