Discrete Complex Image Method Research Articles

Inversion of one‐dimensional parameters of horizontal multi‐layer soil model based on dynamic state electromagnetic field theory and ant colony optimization algorithm

AbstractBuilding a reliable soil model is a prerequisite for accurately calculating the ground potential distribution of the grounding grid. Research shows that the soil electrical parameters are frequency‐dependent, so frequency domain inversion of soil parameters should consider the soil inhomogeneity and the frequency characteristic of parameters. A method for retrieving parameters of horizontal multilayer soil model is proposed. The theoretical formula of apparent complex resistivity with considering dynamic state electromagnetic field theory is derived from Green's function. Discrete dynamic state complex image method is introduced to solve the infinite integral in Green's function. Ant colony optimization algorithm is applied to optimize the inversion process of soil parameters. In the first stage, applicability of this method is confirmed by interpreting field data under DC field. DC parameters including DC resistivity, thickness and number of layer are determined. The performance of different optimization algorithms in terms of computational efficiency is compared at this stage. In the second stage, soil resistivity and relative dielectric constant are retrieved by inverting frequency domain experimental data. Different representations based on quasi‐static electromagnetic field and dynamic state electromagnetic field theories are taken into account. The frequency characteristic of soil parameters is concluded by discussing the inversion results. In the study of parameter inversion of horizontal multi‐layer soil model, our method has shown strong performance in computational efficiency and accuracy of inversion results.

PEEC Modeling in 3D IC/Packaging Applications Based on Layered Green's Functions

A circuit modeling application for 3D IC/packages is proposed in this paper. The method is based on the partial element equivalent circuit (PEEC) method and layered Green's functions (LGF). The LGFs are calculated from the discrete complex image method with three terms, direct coupling, complex images, and surface wave, extracted to analyze the wave behaviors. The dominant terms for the LGFs are analyzed for four canonical stack-ups in 3D IC/packaging systems. Analytical formulas that include the contribution of the complex images calculated from the LGFs are used for the partial capacitance calculation. A fast-modeling approach is proposed by applying the LGF in PEEC using three acceleration treatments to handle the 3D IC/packaging geometry without sacrificing accuracy. An on-chip power distribution network (PDN) geometry is used to illustrate and validate the method.

Closed-Form Evaluation of Mixed Potential Shielded Layered Media Green’s Functions With Spectral Differential Equation Approximation Method

A novel, robust, and computationally efficient approach is proposed for the evaluation of the Michalski–Zheng’s mixed-potential Green’s functions of general shielded layered media. A high-order variant of the spectral differential equation approximation method (SDEAM) is used to cast the spectra of Green’s functions’ components into pole-residue forms. The latter allows for closed-form evaluation of the Sommerfeld integrals (SIs) defining spatial components of the mixed-potential Green’s functions required by Method-of-Moment (MoM) discretization of different types of integral equations. The method produces Green’s functions for all elevations of interest at a fixed cost, making it substantially faster than popular alternative approaches. Numerical results validate the developed high-order SDEAM scheme and demonstrate its extended range of validity compared to the discrete complex image method (DCIM).

Integral Equation Formulation for Planar Plasmonic Structures With Finite Thickness in Layered Media

A detailed Volume Integral Equation (VIE) formulation for planar plasmonic nano structures with finite thickness in flat multi-layers medium is presented. The boundary condition along the localized metallic objects is expressed in terms of the unknown polarization current flowing through these objects in the form of an integral equation, which is solved using the Method of Moments (MoM). The Green's functions associated with a layered medium of practical importance are expressed in the spectral domain. The corresponding spatial domain Green's functions are obtained using the Discrete Complex Images Method (DCIM). Special treatment for the spectral function's asymptote at high spectral values is performed. The presented formulation is applied on different plasmonic structures immersed inside layered media. The structures include nano-rod and nano-patch excited by an incident plane wave. In addition, a simple band-stop filter based on quarter-wavelength stubs is considered. This filter is fed with a couple of plasmonic transmission lines. The obtained current distributions and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -parameters are compared with those obtained using a commercial full-wave electromagnetic simulator, namely CST Microwave Studio. A very good agreement is observed. The proposed integral equation formulation enjoys high degree of stability, numerical efficiency, and accuracy.

Direct discrete complex image method for sound field evaluation above a non-locally reacting layer.

Thin layers of porous media are widely adopted in sound absorption and noise control applications due to their compact arrangement. Particularly, porous media with low flow resistivity exhibit complex, non-local reaction behavior. Therefore, sound field prediction above these media is computationally challenging. This is due to singularities and branch points in the expression for the reflection coefficient. This paper introduces a framework based on the direct discrete complex image method to analyze the sound field above a rigid-backed, non-locally reacting porous sample. In contrast to traditional complex image applications, the proposed framework avoids the extraction of the quasi-static term and the poles from the reflection coefficient. Instead, the reflection coefficient-including its singularities-is directly approximated in terms of a series of complex exponentials, whose coefficients are determined with the matrix pencil method. This study analyzes the sound field above two test samples made of melamine and rockwool. The sound field computation is efficient and accurate in the near- and in the far-field. Moreover, predicted specific impedances agree well with experimental in situ impedance measurements. The proposed framework serves for more applications including object detection in multilayered porous grounds or sound propagation prediction in layered atmosphere.

Fast Computation of Green Function for Layered Seismic Field via Discrete Complex Image Method and Double Exponential Rules

A novel computational method to evaluate the Sommerfeld integral (SI) efficiently and accurately is presented. The method rewrites the SI into two parts, applying discrete complex image method (DCIM) to evaluate the infinite integral while using double exponential quadrature rules (DE rules) for the computation of the finite part. Estimation of signal parameters via rotational invariance techniques (ESPRIT) is used to improve the accuracy and efficiency of extracting DCIM compared to the generalized pencil of function (GPOF). Due to the symmetry of the horizontal layered media, the Green function, representing the seismic fields due to a point source, can be written in the form of Sommerfeld integral in cylindrical coordinate system and be calculated by the proposed method. The performance of the method is then compared to the DE rules with weighted average partition extrapolation (WA), which shows a good agreement, with computational time reduced by about 40%.

The optimization of soil parameters of horizontally multilayered soil in frequency domain by genetic algorithm with considering dynamic state field

A inversion method to determine frequency domain soil parameters of horizontally multilayered earth structure is developed. The inversion of frequency domain soil parameters is divided into two stages. At the first stage, layered structure of soil model is determined under the DC field. The estimated parameters in the first stage include the number of soil layers, DC resistivity and thickness of each layer. These parameters are obtained by interpreting resistivity sounding data. At the second stage, thickness of soil layers in frequency domain is equal to that in the DC field. Soil resistivity and permittivity are estimated by inverting complex apparent resistivity in frequency domain. Theoretical formulas of complex apparent resistivity are derived from Green’s function by considering dynamic state field theory. Discrete complex image method is applied to calculate Sommerfeld integral. Genetic algorithm is chosen to optimize soil parameters. The availability of the first stage of the proposed method is verified by applying the method to interpret field data. Both synthetic data and real measurement data are used to validate the second stage of the proposed method, and the synthetic data are generated by using the soil model obtained in the first stage. The application cases show the accuracy and reliability of the proposed method.

Research on Fast Algorithm of Radio Wave Propagation in Low-Lossy Obstacles Environment

To predict the propagation of radio waves in the environment of dielectric ground and dielectric obstacles, a new two-way parabolic equation (2W-PE) method based on the domain decomposition principle and surface impedance boundary conditions (SIBC) is proposed. First, we decompose the obstacle area into different subdomains and derive the SIBC in each subdomain in detail; then, the discrete hybrid Fourier transform (DMFT) in the upper subdomain and finite difference (FD) algorithm in the lower subdomain is used to solve 2W-PE combined with SIBC, respectively. After that, we explain the algorithm steps in the process of calculating the total field, compared with the traditional 2W-PE, and then finally introduce the method of moments (MoM) combined with the enhanced discrete complex image (E-DCIM) method for accuracy verification of the new 2W-PE algorithm. The simulation results show that no matter how the obstacle medium parameters change, the results of 2W-PE method proposed in this paper and MoM are always in good agreement, which proves the accuracy of 2W-PE and its superiority in speed. Therefore, this paper provides a reliable and efficient method for solving the problem of radio wave propagation in the presence of obstacles, especially in the case of low-lossy obstacles.

Singularity Cancellation for Accurate MoM Analysis of Periodic Planar Structures in Layered Media

One of the singularity cancellation schemes proposed in the literature is applied to calculate singular integrals arising in the method of moments (MoM) analysis of 2-D periodic planar structures in multilayered media. Discrete complex image method is utilized for the accurate approximation of Green's function which also makes possible the application of the Ewald transformation for the efficient computation of the series associated with the periodic structure. This approximation and transformation modifies the kernel of the integral appearing in the computation of the MoM matrix entries. Formulation of the problem for this new kernel is presented together with numerical results demonstrating the achieved accuracy.

Discrete complex image method for periodic Green’s function

The electromagnetic scattering of periodic structure may be expressed by integral equation with periodic Green function as the core. The periodic Green’s function can be expressed as infinite sum in spectral domain. In order to avoid the numerical calculation of infinite sum, the discrete complex image method is proposed. By using the Generalized pencil-of-function method, the closed form can be obtained without any numerical integration. It can be expressed as one group of real sources and two groups of complex sources with complex amplitude and complex position. The accuracy of the method is demonstrated through numerical examples.

Electromagnetic scattering from perfectly conducting periodic rough surfaces using discrete two-level complex images method

The one-level complex image method is only useful in deriving the spectral-domain Green’s function at low frequencies. In this paper, a two-level complex image method is proposed to derive closed form periodic Green’s function at high frequencies for problem of scattering from perfectly conducting periodic surfaces. The remaining quotients in the spectral domain are approximated by a finite series of exponential using the Generalized pencil-of- function method. The approximation is performed along the two-level path, with this scheme the contribution beyond T0 has been considered and it will give accurate results over a wide frequency range. The results are compared with those obtained by the spectral Kummer- Poisson’s method and good agreements are observed. Also the complex image method is more computational efficient.

Efficient numerical analysis of finite FSS with multilayered media by MLACA

In this study, an efficient numerical technique for transmission characteristic analysis of finite planar frequency selective surface (FSS) with multilayered media is presented by using the multilevel adaptive cross approximation (MLACA). The computational scheme is based on the mixed-potential electric field integral equation (MPIE) with multilayered media Green's functions (MLGFs). The three-level modified discrete complex images method (DCIM) is presented for the spatial-domain MLGFs without any quasi-static and surface-wave extraction. For electrically large-size FSS structures, the MLACA method is then employed to alleviate the high computational complexity and memory requirement of method of moments. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Estimation of frequency domain soil parameters of horizontally multilayered earth by using Cole–Cole model based on the parallel genetic algorithm

A method to estimate frequency domain soil parameters of horizontally multilayered earth is developed. Cole–Cole model of complex conductivity form is adopted to describe the frequency dependence of soil parameters. The estimation of frequency domain soil parameters includes two stages. In the first stage, soil-layered structure is determined under the DC field by the interpretation of resistivity sounding data. The obtained model parameters in this stage include the DC soil conductivity and thickness of soil layers. In the second stage, Cole–Cole model parameters are estimated based on the soil-layered model obtained in the first stage. The theoretical formula of complex apparent resistivity is derived from Green's function with considering dynamic-state field theory. Since dynamic-state field is considered in the inversion algorithm, this method can be extended into high-frequency domain. Discrete complex image method is used to calculate Sommerfeld integral quickly. In order to remove electromagnetic (EM) coupling between cables at high frequencies, a new electrode configuration method is proposed. Genetic algorithm is improved by combining with parallel computing. The parallel genetic algorithm is applied to the optimisation of model parameters.

Surface-Volume-Surface EFIE for Electromagnetic Analysis of 3-D Composite Dielectric Objects in Multilayered Media

The surface-volume-surface electric field integral equation (SVS-EFIE) is generalized for the case of scattering problems on the composite nonmagnetic dielectric objects situated in planar nonmagnetic layered medium. The piece-wise homogeneous regions of the scatterer can be arbitrarily positioned with respect to the layers of stratification. The SVS-EFIE being a class of single-source integral equations is formed by restricting the surface single-source electric field representation in each distinct region of the scatterer through the volume-EFIE (V-EFIE) enforced on the boundary of that region for only the tangential component of the total field. As a result, the SVS-EFIE utilizes only the electric field dyadic Green's functions. This allows for its cast into the mixed-potential form using classical Michalski-Zheng's formulation and method of moments (MoM) discretization featuring easily computable integrals with singularities no stronger than 1/R, R being the distance from the source to the observation point in such integrals. The matrices of MoM discretization are represented inhierarchical form (as H-matrices) enabling solution of the scattering problems in multilayered media with O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> log N) CPUtimeand memory complexities, where α is a geometry-dependent constant ranging from 1 to 1.5 depending on the shape of the scatterer. While the MoM surface and volume meshes discretizing the regions of the scatterer are constructed to ensure that no mesh element crosses interfaces between the layers, the clusters of both the surface and volume elements in their respective recursive partitionings in the process of H-matrix construction are allowed to span multiple layers of the medium. Upon computation of the layered medium Green's function kernels with the discrete complex image method allowing clusters of elements to cross dielectric interfaces between the layers is shown to preserve compressibility of the corresponding H-matrix blocks.

Model Order Reduction of Large-Scale Metasurfaces Using a Hierarchical Dipole Approximation

Advances in the field of metasurfaces require simulation of large-scale metasurfaces that extend over many light wavelengths. Adopting standard numerical methods leads to models featuring a large number of degrees of freedom, which are prohibitive to solve within a time window compatible with the design workflow. Therefore, this demands developing the techniques to replace large-scale computational models with simpler ones, still capable of capturing the essential features but imposing a fraction of the initial computational costs. In this work, we present a simulation approach in order to handle reduced order analyses of large-scale metasurfaces of arbitrary elements. We use the discrete dipole approximation in conjunction with the discrete complex image method and hierarchical matrix construction as a common theoretical framework for dipole approximation in the hierarchy of individual elements and the array scale. We extract the contributions of multipoles in the scattering spectra of the nanoantennas f...

A Complex Image Reduction Technique Using Genetic Algorithm for the MoM Solution of Half-Space MPIE

The discrete complex image method (DCIM) has been widely used to analytically approximate the Sommerfeld integrals (SIs) of Green's functions in layered media. This method has been greatly improved in accuracy and generality in the past years, but has resulted in a large amount of complex images. It is found that using so many complex images for SI approximation in method of moment (MoM) applications is redundant and inefficient. To reduce the number of complex images, a technique by means of genetic algorithm is proposed and applied to the MoM solution of the half-space mixed-potential integral equation (MPIE).

Analysis of finite frequency selective surfaces backed by dielectric substrate using sub-entire-domain basis function method

Purpose – Analysis of the frequency selective surface (FSS) is of great significance. In the method of moments, when the electric size of the FSS increases, huge in-core memory and CPU time are required. The purpose of this paper is to efficiently analyze the finite FSS backed by dielectric substrate utilizing sub-entire-domain (SED) basis function method. Design/methodology/approach – Different types of SED basis functions are generated according to the locations of the cells in the entire structure, and a reduced system is constructed and solved. The couplings of all cells of the FSS are taken into account by using Green’s function and Galerkin’s test procedure. The spatial Green’s function is obtained with the discrete complex image method. The reflection and transmission coefficients of the FSS are calculated using the far field of the FSS and the metallic plate with the same size. Findings – Moderate problems of the finite FSS backed by dielectric substrate are solved with the SED basis function method. The original problem can be simplified to two smaller problems. It enables a significant reduction to the matrix size and storage, and efficient analysis of FSS can be performed. The band-stop type of FSS can be composed of periodic conductive patch cells on the dielectric substrate, and shows total reflection property at the resonant frequency. Originality/value – The SED basis function method is mostly used to analyze periodic PEC structures in free space. The layered medium Green’s function is successfully employed and several dielectric substrate backed finite FSSs are discussed in this paper. The calculation of reflection and transmission coefficients, which are more effective rather than far field scattering of the FSS, are described.

An Efficient Method to Extract Surface-Wave Poles of Green's Functions Near Branch Cut in Lossy Layered Media

Calculating the Green's functions in lossy layered media using the discrete complex image method (DCIM) is challenging, due to the difficulties in extracting the surface-wave poles that are very close to a branch cut. An efficient algorithm based on the contour method is proposed in this communication to locate these poles and calculate the residues. The proposed method is robust for both the lossless and lossy media. With the proposed approach, it is shown in numerical examples that some poles, very close to a branch cut, are successfully extracted in lossy media. The accurate calculation of the Green's functions in lossy layered media enables the accurate and efficient modeling of complex structures in lossy semiconductor substrates and new 3D IC structures including through-silicon vias (TSVs).

An Improved Discrete Complex Image Method Based on a Special Sampling Path

A special sampling path is proposed to improve the accuracy of the discrete complex image method (DCIM) in evaluating the lateral wave. Different from all the previous DCIM implementations, the proposed sampling path goes through the branch point directly, which helps to evaluate its contribution to the lateral wave more accurately. In this communication, different sampling paths of DCIM have been implemented and compared. It is found that in the situations where the lateral wave dominates the far-field region, DCIM with the new path gives more accurate and stable results.

A Nonlinear Microwave Breast Cancer Imaging Approach Through Realistic Body–Breast Modeling

A nonlinear microwave tomography approach which suggests monitoring of differences inside the breast with respect to a reference breast model that adequately represents the healthy breast is presented. This approach simplifies the breast cancer imaging problem through realistic modeling of the body-breast configuration and does not require the imaging of the entire breast tissues. The tumor is considered as a buried object inside the breast which is assumed to be located on the interface of two-half spaces media composed of air or a coupling liquid and the chest wall or the base of the bed on which patient lies in a prone position. Then, the tumor, as well as the deviations of the actual breast from the breast model, is imaged by using the contrast source inversion method at a single frequency. The Green's function of the inhomogeneous background required in the application of the inversion algorithm is obtained through the buried object approach and accelerated by an adaptation of the discrete complex images method. Through the simulations it has been shown that, the proposed modeling and nonlinear inversion algorithm is capable of reconstructing tumors as small as 4 mm in radius for 3-D scenarios.