Complex Image Method Research Articles

Closed-Form Uniform Asymptotic Expansions of Green's Functions in Layered Media

Far-field closed-form expressions are derived for spatial domain multilayered Green's functions (GF). For the derivation of these far-field expressions, the spectral domain multilayered GF are approximated by means of the total least square algorithm (TLSA) in terms of the spectral variable u0=(kρ2-k02)1/2, and uniform asymptotic expansions are determined for the Sommerfeld integrals of the resulting TLSA approximations. Numerical results show that the far-field asymptotic expressions are accurate within 0.1% for distances between source and field points larger than one free-space wavelength. Also, it is shown that the hybrid use of the discrete complex image method and the TLSA in terms of the spectral variable kρ leads to near-field closed-form expressions of multilayered GF that are typically accurate within 0.1% for distances between source and field points smaller than one free-space wavelength. Therefore, the combination of the novel far-field asymptotic expressions and the well known near-field expressions makes it possible to compute multilayered GF with a great accuracy in the whole range of distances between source and field points, and in a wide range of frequencies.

Comparison of Two Approaches to Evaluate Nonsymmetrical Components of Green's Functions for Multi-Layered Media in the Near-Field Region

The numerical implementation of Green's functions is a fundamental step for analyzing multi-layered media applications. The computation time and the accuracy of results, which should be maintained for all field regions, are two crucial issues. The discrete complex image method is one of the methods that proved its efficiency in accelerating the computation time and yielding accurate results in the near- and intermediate-field regions. In order to retain this accuracy for the far-field region, the surface-wave contribution should be extracted. However, this procedure results in erroneous results in the near field. This article is concerned with the nonsymmetrical components of the magnetic vector potential. Two different approaches to handle this problem are analyzed. In both approaches, additional artificial poles are extracted in order to eliminate the problem of contaminated results in the near field. The results of both approaches are presented, and the effect of these artificial poles is explored. The results are compared with the numerical integration technique.

Dual GPOF/DCIM for Fast Computation of Sommerfeld Integrals and EM Scattering From an Object Partially Embedded in Dielectric Half-Space

A novel approach of dual GPOF (general pencil of functions) combined with the DCIM (the discrete complex image method) is developed to accelerate the Sommerfeld integral evaluation, when the mixed potential integral equation (MPIE) is applied to numerically calculate scattering from a conductive object partially embedded in a dielectric half-space, where the field and source points are located in different media. Firstly, the factors related with the field point are separated from the spectrum function, and the GPOF is used to find the complex image parameters at finite discrete source points. Secondly, the GPOF is used again to fit the functional relationship of each complex image parameter with the source positions to evaluate the complex image parameters for any source position in a closed functional summation. Comparing numerical results of dual GPOF and direct numerical integration, this dual GPOF process is proved highly effective. Finally, dual GPOF/DCIM is applied to computation of electromagnetic scattering from a PEC spherical object partially embedded in a dielectric half-space medium, and numerical scattering results upon different model parameters are presented and analyzed.

The BCGS-FFT Method Combined With an Improved Discrete Complex Image Method for EM Scattering From Electrically Large Objects in Multilayered Media

This paper presents an efficient algorithm combining the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method with an improved discrete complex image method (DCIM) for electromagnetic scattering from electrically large objects in both lossless and lossy multilayered media. The required spatial Green's functions obtained by the improved DCIM are accurate both in the near- and far-field regions without any quasi-static and surface-wave extraction. Then, the scattering by buried objects is considered using the BCGS-FFT method combined with the improved DCIM. Numerical results show the improved DCIM can save tremendous CPU time in scattering involving buried objects.

Closed-Form Green's Functions in Planar Layered Media for All Ranges and Materials

An important extension of the two-level discrete complex image method is proposed to eliminate any concerns on and shortcomings of the approximations of the spatial-domain Green's functions in closed form in planar multilayered media. The proposed approach has been devised to account for the possible wave constituents of a dipole in layered media, such as spherical, cylindrical, and lateral waves, with the aim of obtaining accurate closed-form approximations of Green's functions over all distances from the source. This goal has been achieved by judiciously introducing an additional level into the two-level approach to pick up the contributions of lateral waves in the spatial domain. As a result, three different three-level algorithms have been proposed, investigated, and shown that they work properly over all ranges of distances from the source. In addition to the accuracy of the results at all distances, these approaches also proved to be robust and computationally efficient as compared to the previous algorithms, which can be attributed to the fact that the sampling of the spectral-domain Green's functions in the proposed approaches gives proper emphasis to the associated singularities of the wave types in the spectral domain. However, the judicious choices of the sampling paths may not be enough to get accurate results from the approximations unless the approximating functions in the spectral domain can provide similar wave natures in the spatial domain. To address this issue, the proposed algorithms employ two different approximations; the rational function fitting methods to capture the cylindrical waves (surface waves), and exponential fitting methods to capture both spherical and lateral waves. It is shown and numerically verified that a linear combination of exponential functions in the spectral domain represent the lateral waves at the interface of the involved layers.

Modeling of Acoustic Scattering by Spherical Targets Above a Planar Seafloor

AbstractA model of acoustic scattering from spherical targets above a planar seafloor is established on a condition of irradiation by a point source, with both of T matrix method and complex images method. Compared to the other model of Fawcett and Lim, this model can be used to calculate the scattering field from targets above a planar interface in three‐dimensional space. Comparison of some coefficients by the complex images method with that by analytical formula or numerical quadrature, the computation method of the model is examined to be efficient and accurate. In addition, numerical examples of a rigid sphere and an elastic spherical shell are presented for the scattering field from the target on a fluid seabed, and the scattering field dependent angle is investigated from the different grazing angles of incident wave. In order to verify the model, an experiment is carried out to measure the scattered field received by a vertical array in laboratory. Data are analyzed and compared with calculated results.

Simulation on DC Current Distribution in AC Power Grid under HVDC Ground-Return-Mode

This paper focus on the Modeling and Calculation of DC current distribution in AC power grid induced under HVDC Ground-Return-Mode. Applying complex image method and boundary element method, a new field-circuit coupling model was set up. Based on the calculation result with complex image method, this paper derived the modification factor for induced earth potential from practical measurement, which increased the accuracy of calculation. The modification method is helpful for evaluation on the effect of means used for blocking the dc-bias current in transformer neutral and also useful for the forecast of the DC current distribution when the power grid is in different line connection mode. The DC distribution character in Guangdong power grid is shown and suggestion is proposed that the mitigation of dc-bias should start from those substations whose earth-potential is highest.

Direct use of discrete complex image method for evaluating electric field expressions in a lossy half space

A modelling technique is proposed for direct use of the discrete complex image method (DCIM) to derive closed-form expressions for electric field components encountered in the electric field integral equation (EFIE) representing a lossy half space problem. The technique circumvents time consuming numerical computation of Sommerfeld integrals by approximating the kernel of the integrals with appropriate mathematical functions. This is done by appropriate use of either the least-square Prony (LS-Prony) method or the matrix pencil method (MPM) to represent electric field expressions in terms of spherical waves and their derivatives. A comparison is made between the two methods based on the computation time and accuracy and it is shown that the LS-Prony method performs two–three times faster than the MPM in approximating the integral kernels depending on the platform. The main feature of the proposed technique is its ability for direct inclusion in the kernel of computational tools based on the method of moments solution of the EFIE. This can be viewed as an advantage over the conventional DCIM approximation of spatial Green's functions for mixed potential integral equation for cases where the problem in hand can be more efficiently represented by the EFIE (e.g. the thin-wire EFIE). The accuracy of the proposed technique is validated against numerical integration of Sommerfeld integrals for an arbitrary electric dipole inside a lossy half space.

Multi-GPOF Method in the Calculation of the Rectangular Waveguide Layered Green's Functions

In this letter, we adopt a new procedure in the calculation of the layered Green's functions in the rectangular waveguide structure. In this procedure, the surface-wave poles are extracted first, and then the generalized pencil-of-function (GPOF) method is applied to approximate the spectral-domain Green's function along the real axis of the plane directly. Because no branch point is introduced in this procedure, improvement can be achieved compared to the traditional discrete complex image method (DCIM). In order to make this procedure more accurate and robust in both near- and far-field regions, a specially devised multiple-times GPOF procedure is applied to approximate the spectral-domain Green's function. Numerical results in this letter demonstrate the accuracy of our method.

ELECTROMAGNETIC SCATTERING BY CONDUCTING BOR COATED WITH CHIRAL MEDIA ABOVE A LOSSY HALF-SPACE

Electromagnetic scattering by conducting bodies of revolution (BOR) coated with homogeneous chiral media above a lossy half-space is formulated in terms of the Poggio-Miller-ChangHarrington-Wu surface integral equation combined with combined field integral equation. A field decomposition scheme is utilized to split a chiral media into two equivalent homogeneous media. The spatial domain half-space Green’s functions are obtained via the discrete complex image method. Due to the rotational symmetry property of BOR, the method of moment for BOR (BORMoM) is applied to the linear system solved by the multifrontal direct solver. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Locating All the Modes of Green's Function for a Three-Layered Medium Based on the Path Tracking Algorithm

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Through a mathematical analysis on the relationship between the characteristic equations of the microstrip geometry and a configuration with a three-layered medium, a method based on the path tracking algorithm is proposed for precisely locating all the modes for a three-layered medium with moderate thickness, including surface wave modes, leaky wave modes and improper real modes. With this method, the difficulty caused by branch points and cuts in the contour integral method could be avoided. This method can be applied not only in the discrete complex image method (DCIM), but also in the steepest descent path (SDP) method. The proposed method can be extended to more complicated configurations. </para>

Modeling of acoustic scattering by sphere on a planar seabed.

A model of acoustic scattering from spherical target above a planar seabed is advanced to a condition of irradiation by a point source, with both of T matrix method and complex images method. Compared to the other model [J. A. Fawcett and R. Lim, J. Acoust. Soc. Am. 114, 1406–1415 (2003)], this model could be used to calculate scattering field from target above a planar interface in three dimension space or by bounded source beam. Comparing some coefficients by complex images method with that by analytical formula or numerical quadrature, the computing method of the model is examined to be efficient and accurate. In addition, numerical examples of a rigid sphere and an elastic spherical shell are compared for scattering field from the target on a fluid seabed, and the scattering field dependent angle is investigated from the different of grazing angle of incident wave. [Work supported by the National Natural Science Foundation of China (Grant No. 10704068).]

Closed-Form Green's Function Representations in Cylindrically Stratified Media for Method of Moments Applications

Closed-form Green's function (CFGF) representations for cylindrically stratified media, which can be used as the kernel of an electric field integral equation, are developed. The developed CFGF representations can safely be used in a method of moments solution procedure, as they are valid for almost all possible source and field points that lie on the same radial distance from the axis of the cylinder (such as the air-dielectric and dielectric-dielectric interfaces) including the axial line (rho = rho' and phi = phi' ), which has not been available before. In the course of obtaining these expressions, the conventional spectral domain Green's function representations are rewritten in a different form so that: i) we can attack the axial line problem and ii) the method can handle electrically large cylinders. Available acceleration techniques that exist in the literature are implemented to perform the summation over the cylindrical eigenmodes efficiently. Lastly, the resulting expressions are transformed to the spatial domain using the discrete complex image method with the help of the generalized pencil of function method, where a modified two-level approach is used. Numerical results are presented in the form of mutual coupling between two current modes to assess the accuracy of the final spatial domain CFGF representations.

Accurate evaluation of Green’s functions in a layered medium by SDP-FLAM

Based on local Taylor expansions on the complex plane, a method for fast locating all modes (FLAM) of spectral-domain Green’s Functions in a planar layered medium is developed in this paper. SDP-FLAM, a combination of FLAM with the steepest descent path algorithm (SDP), is employed to accurately evaluate the spatial-domain Green’s functions in a layered medium. According to the theory of complex analysis, the relationship among the poles, branch points and Riemann sheets is also analyzed rigorously. To inverse the Green’s functions from spectral to spatial domain, SDP-FLAM method and discrete complex image method (DCIM) are applied to the non-near field region and the near filed region, respectively. The significant advantage of SDP-FLAM lies in its capability of calculating Green’s functions in a layered medium of moderate thickness with loss or without loss. Some numerical examples are presented to validate SDP-FLAM method.

Efficient implementation of discrete complex images method in spatial domain moment method to analyze microstrip transmission lines

Planar guiding structures are analyzed using a mixed-potential integral-equation (MPIE) formulation solved by the method of moments (MoM). Closed-form two-dimensional Green's functions for the scalar and vector potentials are analytically obtained in the space domain due to the approximation of its spectral domain version with discrete complex images method (DCIM), thus avoiding lengthy numerical evaluations. It will be shown that a suitable choice of the sampling path is key to reducing the number of images and to avoid the necessity of extracting spectral poles. The advantages of this approach will be illustrated by numerical results.

An accurate evaluation of the nonsymmetrical components of Green's functions for multilayered media in the near‐field region

AbstractThe discrete complex image method has proved to be one of the most efficient techniques to evaluate Green's functions for multilayered media, particularly in the near‐ and intermediate‐field regions. Although the extraction of surface waves extends its validity to the far field, it suffers from erroneous results that contaminate the computation of the near field. To handle this problem for the nonsymmetrical components of Greens functions, we follow the same approach adopted by the author and others for the symmetrical components where a number of artificial poles are extracted for each physical surface‐wave pole occurrence. This approach has proved its validity and yielded good results in all field regions. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 881–885, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24201

ELECTROMAGNETIC SOURCE EQUIVALENCE AND EXTENSION OF THE COMPLEX IMAGE METHOD FOR GEOPHYSICAL APPLICATIONS

In this work, source equivalence and computation of the re∞ected (induced) electromagnetic fleld in geophysical situations are studied. It is shown that the application of Huygens' principle allows for full generalization of Fukushima's equivalence theorem that applies only for magnetic fleld. The source equivalence is revisited for a vertical line current element, and it is shown that the equivalent charge required to replace the original source by a planar equivalent source together with the surface charge associated with the re∞ected fleld generates a purely vertical total electric fleld on the ground. Consequently, if the magnetic fleld and horizontal components of the total electric fleld on the ground are of interest, only equivalent currents need to be considered. The classical Complex Image Method (CIM) is derived from the exact image theory for planar impedance surfaces. The classical CIM is extended by considering a divergence-free source current that may have components also perpendicular to the ground plane. The extension is seen to generate a complex image charge not present in the classical CIM. Further, a generalized application of the extended CIM to geophysical situations having divergence-free volume source currents is introduced. The application involves decomposition of the source into linear current elements and rotations, translations and re∞ections of the electromagnetic fleld expressions associated with each element. The validity of the new approach is verifled for an

A novel approach of dual GPOF /DCIM for fast computation of the sommerfeld integrals and electromagnetic scattering from an object partially embedded in dielectric half-space

The mixed-potential integral equation （MPIE） has been usually employed in numerical approach of electromagnetic scattering of the object, such as the method of moment （MoM）, due to its low-level singularity of vector and scalar potential Green functions. When an object is embedded partially in dielectric half-space medium, the Green’s function contains the Sommerfeld-type integrals, which embody the effect of the dielectric interface on scattering fields. Using the discrete complex image method （DCIM） and the Sommerfeld identity, the Sommerfeld integrals can be evaluated as the summation of finite complex image functions without directly numerical integration which always consumes large CPU time. As the points of the field and source are co-located in the same side of the interface, the spectrum function g（kzp） is not related with the field or source positions, and the complex image parameters fitted with the general pencil of functions （GPOF） method are approximate for all positions. However, if the points of the field and source are located, separately, in different sides of the interface, the spectral function is now related to z and z′. Generally, the GPOF is repetitiously used to find the complex image parameters for every z and z′, which consumes large CPU time and memory. This paper presents a novel method of Dual GPOF combining with DCIM for fast computation of the Sommerfeld integral. Firstly, the factors related with the variable z are separated, and the GPOF is used to find the complex image parameters for finite discrete source points z′l. Secondly, GPOF is used again to fit the relationship of each complex image parameter with variable z′. Then, the complex image parameter of any z′ can be evaluated as the direct function summation, and there is no need to perform GPOF for all z′ points. Comparing numerical values of Dual GPOF, point-by-point GPOF, and direct numerical integration, the Dual GPOF method is proved effective and efficient. Finally, Dual GPOF is applied to computation of electromagnetic scattering from a P.E.C. sphere object partially embedded in dielectric half-space, and the scattering patterns are presented and analyzed.

Numerical calculation of grounding system buried in vertical earth model in low frequency domain based on the boundary element method

AbstractTo solve a grounding problem which describes a grounding system with floating metallic conductors buried in vertical multilayer earth model under a high voltage a.c. substation, a new mathematical model for accurately computing currents flowing along conductors of the grounding system has been developed in this paper, which is hybrid of Galerkin's type Boundary Element Method and conventional nodal analysis method. Propagation effect of electromagnetic wave within the substation has been neglected due to its limited area. A quasi‐static complex image method and new closed form of Green's function for the vertical multilayer earth model are introduced into this model to accelerate mutual impedance and induction coefficients calculation. The model is then implemented in a computer program, which can be used to calculate currents distribution of any configuration of grounding system, and with or without floating metallic conductors buried in the vertical multilayer earth model. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Distribution of Currents in the Lightning Protective System of a Residential Building—Part II: Numerical Modeling

The distribution of lightning current in the lightning protective system (LPS) of a test residential building was experimentally studied in 2004 and 2005 at Camp Blanding, FL. Lightning was initiated using a rocket-and-wire technique and its current was injected into the LPS. Results are presented in the companion paper. The current distribution is modeled here using a model based on the lumped circuit theory in the frequency domain. The injected lightning current is represented in the model by an ideal current source. The effect of electromagnetic field radiated by the lightning channel is also accounted in the model (for one of the configurations tested). In field calculations, the lightning channel and the wire connecting the rocket launcher and the test house are modeled as vertical and horizontal electric dipoles above lossy ground. The discrete complex images method is used to calculate the electromagnetic field radiated by the electric dipoles. The time-domain current waveforms are obtained by means of the inverse Fourier transform. The calculated currents in the LPS are compared with those measured.