Volume-surface Integral Equation Research Articles

Surface–Volume–Surface Electric Field Integral Equation for Magneto-Quasi-Static Analysis of Complex 3-D Interconnects

A novel single-source surface–volume–surface integral equation is proposed for accurate broadband resistance and inductance extraction in 3-D interconnects. The new equation originates in the volume integral equation (VIE) traditionally used for magneto-quasi-static modeling of current flow in 3-D wires. The latter is reduced to a surface integral equation by representing the electric field inside each conductor segment as a superposition of cylindrical waves emanating from the conductor's boundary. As no approximation is utilized and all underlying boundary conditions and pertinent equations are satisfied in the reduction, the new integral equation is rigorously equivalent to the solution of the traditional volume electric field integral equation. The rigorous nature of the proposed single-source surface integral equation is corroborated numerically. In this paper, a detailed description of the method of moments discretization for the proposed surface integral equation is also presented. Numerical solution of the proposed surface integral equation for a 12-conductor bond-wire package is used to demonstrate the accuracy of the method and its computational benefits compared to the traditional solution based on the VIE.

An Efficient Integral Equation/Modified Surface Integration Method for Analysis of Antenna-Radome Structures in Receiving Mode

An efficient approach that combines the coupled volume surface integral equation (VSIE) accelerated by multilevel fast multipole algorithm (MLFMA) and the modified surface integration (MSI) method is proposed to analyze the antenna-radome structures (ARS) in receiving mode. The hybrid method not only reduces the computational time and memory requirement as compared to the purely full-wave solutions, but also more importantly overcomes the difficulties in conventional high-frequency approximate methods for analyzing the properties of the radome-enclosed receiving antennas or arrays. Numerical results are shown to illustrate the validity, efficiency, and accuracy of the proposed method.

A surface integral equation formulation for electromagnetic scattering from a conducting cylinder coated with multilayers of homogeneous materials

A surface integral equation formulation is presented for electromagnetic scattering by a conducting cylinder coated with multilayers of homogeneous materials. Each layer may have a nonunity relative permittivity and permeability. Both the TM and TE polarizations are considered. The surface equivalence principle is utilized to model the problem where each layer is replaced by equivalent surface currents residing on the enclosing boundaries. A systematic procedure is developed to generate a set of coupled integral equations for an arbitrary number of layers. The method of moments is invoked to convert these equations into a sparse matrix equation which can be solved using sparse matrix routines. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method. The performance of the method is compared with that of the volume-surface integral equation formulation where a great saving in memory storage and computation time is achieved.

Electromagnetic TE scattering by a conducting cylinder coated with an inhomogeneous dielectric/magnetic material

The problem of electromagnetic TE scattering by an infinitely long conducting cylinder coated with an inhomogeneous dielectric/magnetic material is analyzed. A volume-surface integral equation (VSIE) approach is utilized to model the problem. By imposing the boundary conditions on the conducting surface, a surface magnetic field integral equation (MFIE) is developed. Inside the volume of the coating region, volume MFIEs are applied. The resultant integral equations are solved using the moment method. Numerical results for the bistatic radar cross section for different structures are presented. The results are validated using the exact series solution for a conducting circular cylinder coated with multilayers of homogeneous materials. Two types of coating materials are studied: the conventional or double positive (DPS) materials and the double negative (DNG) materials.

AN INTEGRAL EQUATION FORMULATION FOR TM SCATTERING BY A CONDUCTING CYLINDER COATED WITH AN INHOMOGENEOUS DIELECTRIC/MAGNETIC MATERIAL

A volume-surface integral equation (VSIE) formulation is developed for determining the electromagnetic TM scattering by a two-dimensional conducting cylinder coated with an inhomogeneous dielectric/magnetic material. The electric fleld integral equations (EFIEs) are utilized to derive the VSIE. The surface EFIE is applied to the conducting surface, while the volume EFIE is applied to the coating region. By employing the surface and equivalence principles, the problem is reduced into a set of coupled integral equations in terms of equivalent electric and magnetic currents radiating into unbounded space. The moment method is used to solve the integral equations. Numerical results for the bistatic radar cross section for difierent structures are presented. The well-known exact series-solution for a conducting circular cylinder coated with multilayers of homogeneous materials is used along with the available published data to validate the results. The in∞uence of using coatings with double-positive (DPS) and/or double-negative (DNG) materials on the radar cross section is investigated.

Analysis on spherical conformal microstrip antenna array by characteristic basis function method

Spherical conformal microstrip antenna array which is used widely in the field of aeronautics is analyzed by the full wave analysis in this paper. Using the surface integral equation with spherical dyadic Green's function can decrease the number of unknowns remarkably, and also reduce the demand of memory as compared with the method using volume-surface integral equation. The curvilinear triangle is proposed to mesh the surface of the microstrip antenna array, which can simulate the characteristic of spherical surface accurately. Firstly, the problem about probe feed model is solved successfully by introducing the half Rao-Wilton-Glisson function and boundary charge, and the image method is used to treat the line integral singularity problem. Then, the characteristic basis function method is employed to further save memory and computation time further by reducing the rank of impedance matrix. Finally, the input impedance and far-field spherical conformal microstrip antenna array of different sizes are analyzed. Results are in good agreement with those in the literature and simulation software, and thus the validity and effectiveness of the analysis method are demonstrated.

Efficient Nyström Solutions of Electromagnetic Scattering by Composite Objects With Inhomogeneous Anisotropic Media

Analysis of electromagnetic (EM) scattering by composite objects with inhomogeneous or anisotropic media requires an efficient solution of volume integral equations (VIEs) or volume-surface integral equations (VSIEs) in the integral equation approach. The traditional method of moments (MoM) with the Rao-Wilton-Glisson (RWG) basis function and the Schaubert-Wilton-Glisson (SWG) basis function may not be convenient because the basis functions are defined over a triangular or tetrahedral element pairs and the SWG basis function could cause a surface charge density on the common faces separating dissimilar media. In this work, we develop an efficient Nystrom scheme as an alternative to solve the VIEs or VSIEs for those objects. The advantages of Nystrom method include the simple mechanism of implementation, allowance of nonconforming meshes, and removal of basis or testing function, making the method more suitable for solving such problems. Numerical examples for EM scattering by typical objects are presented to demonstrate the method.

Electromagnetic Analysis for Inhomogeneous Interconnect and Packaging Structures Based on Volume-Surface Integral Equations

Electromagnetic analysis for interconnect and packaging structures usually relies on the solutions of surface integral equations (SIEs) in integral equation solvers. Though the SIEs are necessary for the conductors in the structures, one has to assume a homogeneity of material for each layer of a substrate if SIEs are used for the substrate. When the inhomogeneity of materials in the substrate has to be taken into account, then volume integral equations (VIEs) are indispensable. In this paper, we consider the inhomogeneous materials of substrate and replace the SIEs with the VIEs to form volume-surface integral equations (VSIEs) for the entire structures. Also, the use of VIEs could alleviate low-frequency effects, remove the need of selecting a basis function for magnetic current, and facilitate geometric discretization in some scenarios. The VSIEs are solved with the method of moments by using the Rao-Wilton-Glisson basis function to represent the surface current on the conductors and Schaubert-Wilton-Glisson basis function to expand the volume current inside the substrate. To avoid the inconvenience of charge density in the traditional implementation of the VIEs, we suggest that the dyadic Green's function be kept in its original form without moving the gradient operator onto the basis and testing functions. Numerical examples are presented to demonstrate the effectiveness of the approach.

Theory of aperture-coupled hemispherical dielectric resonator antennas with radiating elements

A new approach based on hybrid volume-surface integral equation (VSIE) formulation in combination with spherical dyadic Green's function (DGF) is presented in this paper to analyze aperture-coupled multilayer hemispherical dielectric resonator antennas (DRA) with conformal conducting patches. Hybrid VSIE is used for the planar part of the structure and is solved with the aid of spatial-domain method of moments (MoM) in order to compute magnetic surface current in a slot cut in a finite planar perfect electric conductor (PEC) sheet. Multilayer spherical electromagnetic DGFs are used to compute loading effects of hemispherical dielectric resonators and conformal patches on antenna characteristics. The effects of variation in some parameters of the structure on the return loss of the antenna are studied. Accuracy of the presented method is validated by comparing the results obtained from the proposed method with those of CAD simulations.

An Efficient Solution of the Volume-Surface Integral Equation for Electromagnetic Scattering and Radiation of the Composite Dielectric-Conductor Objects With Reduced Number of Unknowns

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.

Analysis and synthesis of conformal conical surface linear phased array with volume surface integral equation+AEP (Active Element Pattern) and INSGA-II

In this study, for the first time, volume-surface integral equation (VSIE) and improved NSGA-II (INSGA-II) algorithms analyse and synthesise a conformal conical surface linear phased array. The antenna element patterns and port characteristic with mutual coupling have been simulated with the VSIE algorithm. Then, the patterns of all the elements terminated by their characteristic impedances, which are called active element patterns, are measured in a microwave chamber for comparison. The results show a good match between the simulated and the measured results. The pattern of the conformal antenna array is a superposition of all these active patterns of the elements with various weights. However, the design objects containing the 3-dB beam width, side lobe levels, and beam direction must be taken into account simultaneously. So, the INSGA-II algorithm is used for a multi-object optimisation here. Moreover, by using this algorithm, the entire optimum solutions, that is, the optimum solutions of the main beam of the conformal-phased array directing to different angles, can be solved by one optimisation. Comparisons between the experimental and the simulated results show the effectiveness of the proposed approach for conformal-phased arrays.

Numerical Solutions of the Integral Equation for Excitation-Transmission-Radiation of the Beam Waveguide

The numerical method of the Integral Equation has been used to model the electromagnetic excitation, transmission, and radiation problems of the beam waveguide. The Mode Matching Method has been used to describe the impedance matching situation at the exciting aperture of the waveguide. The excitation conditions were established based on the expansion of the waveguide modes and the continuity of the tangential components of the fields. The volume-surface integral equations combined with the equavalence principle have been used to model the wave transmission in the beam waveguide. The numerical solutions of the electromagnetic transmission and aperture radiation of the waveguide have been given by using the Multilevel Fast Multipole Algorithm (MLFMA). It has been demonstrated that the method proposed by this paper is able to provide the efficient and accurate numerical solution for the excitation, propagation and radiation problems of the beam waveguide with arbitrary shape and electrically large size.

Numerical Solutions of the Integral Equation for Excitation-Transmission-Radiation in Aperture Antenna

The numerical method of the volume-surface integral equation has been used to model the electromagnetic excitation and propagation problems of the waveguide-fed aperture antenna. The mode-matching method has been used to describe the impedance marching situation at the exciting aperture. The excitation conditions were established based on the expansion of the waveguide modes and the equality principle of electromagnetics. The multi-level fast multi-pole algorithm is applied to speed up the matrix-vector multiplication and reduce the memory requirement of the method of moment. The numerical solutions of the electromagnetic transmission and radiation of an offset paraboloidal reflector antenna have been given. It has been demonstrated that the method proposed by this article is able to provide the valid and accurate numerical solution for the electromagnetic propagation and radiation problems of an aperture antenna with an arbitrary shape, composite metallic and material structure, and electrically large size.

Fast solution of volume-surface integral equation for scattering from composite conducting-dielectric targets using multilevel fast dipole method

This article presents a fast solution to the volume–surface integral equation for electromagnetic scattering from three-dimensional (3D) targets comprising both conductors and dielectric materials by using the multilevel fast dipole method (MLFDM). This scheme is based on the concept of equivalent dipole-moment method (EDM) that views the Rao–Wilton–Glisson and the Schaubert–Wilton–Glisson basis functions as dipole models with equivalent dipole moments. In the MLFDM, a simple Taylor's series expansion of the terms Rα (α = 1, −1, −2, −3) and R̂R̂ in the formulation of the EDM transforms the interaction between two equivalent dipoles into an aggregation–translation–disaggregation form naturally. Furthermore, benefiting from the multilevel grouping scheme, the matrix-vector product can be accelerated and the memory cost is reduced remarkably. Simulation results are presented to demonstrate the efficiency and accuracy of this method. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2012.

A Volume-Surface Integral Equation Solver for Radiation from Microstrip Antenna on Anisotropic Substrate

A volume-surface integral equation (VSIE) solver is presented for the calculation of electromagnetic radiation from arbitrary shaped microstrip antenna on anisotropic substrate. The method of moments (MoM) is used to convert the integral equation into a matrix equation, where the equivalent volume current and surface current are expanded into a finite series of SWG and RWG basis function, respectively. A simple strip model is incorporated in the VSIE to simplify the analysis of the probe-fed microstrip antenna. The present approach is sufficiently versatile in handling microstrip antenna with arbitrary shaped anisotropic dielectric substrate. Numerical results indicate the reliability and accuracy of the proposed method.

Electromagnetic Radiation from Arbitrarily Shaped Microstrip Antenna Using the Equivalent Dipole-Moment Method

The equivalent dipole-moment method (EDM) is extended and applied in the analysis of electromagnetic (EM) radiation by arbitrarily shaped microstrip antenna in this paper. The method of moments (MoM) is used to solve the volume-surface integral equation (VSIE). A strip model is applied in the treatment of the feeding probe of the microstrip antenna, in which the discretized triangular elements of the excitation source are equivalent as dipole models. The proposed approach is sufficiently versatile in handling arbitrarily shaped microstrip antenna and is easily constructed through a simple procedure. Numerical results are given to demonstrate the accuracy and efficiency of this method.

A FAST VOLUME-SURFACE INTEGRAL EQUATION SOLVER FOR SCATTERING FROM HIGH-CONTRAST MATERIALS

This paper presents a generalized volume-surface integral equation (GVSIE) to solve electromagnetic (EM) scattering of high contrast inhomogeneous materials. Then the method of moments (MoM) is employed to solve the GVSIE. The GVSIE technique where the domain is represented by a corresponding uniform background medium coupled with a variation, together representing the overall inhomogeneity, is solve by the method of moments (MoM) using Schaubert-Wilton-Glisson (SWG) and Rao-Wilton-Glisson (RWG) basis functions. The adaptive cross approximation (ACA) algorithm combined with the equivalent dipole-moment (EDM) method are extended to reduce memory and CPU time. A highly efiective preconditioning strategy is presented to solve the system of equations without any increase in the computational complexity. Experiments on several problems representative of scattering simulations are given to illustrate the potential of the above proposed techniques for solving EM scattering involving high contrast applications.

Marching-on-in-order time-domain volume-surface integral equations in analysis of transient electromagnetic scattering from composite conducting-dielectric objects

In this letter, the matching-on-in-order time-domain volume-surface integral equation method is used to analyze transient electromagnetic scattering from objects comprising both conductor and dielectric material. SWG and RWG basis functions are used as the spatial basis functions in dielectric region and on the conducting surface, respectively. Different orders of weighted Laguerre polynomials are used as temporal basis functions, and the relative expansion coefficients are solved recursively order by order. The spatial and temporal testing procedures are separate, and both of them are carried on with the Galerkin's method. Stable solutions are ensured, and several numerical results are given to verify the proposed method. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1315–1319, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25987

COMPARISONS OF COUPLED VSIE AND NONCOUPLED VSIE FORMULATIONS

Coupled volume-surface integral equation (CVSIE) and non-coupled volume-surface integral equation (NCVSIE) are compared in this paper. For scattering problems, results with similar accuracy are given. For the radiation problems, different impedance behaviors are arrived using the two approaches, by comparing the numerical results to the results from measurement or reference papers. The CVSIE approach has better performances in the input impedance calculation.

Electromagnetic Scattering by Arbitrarily Shaped PEC Targets Coated with Anisotropic Media Using Equivalent Dipole-Moment Method

The equivalent dipole-moment method (EDM) is extended and applied in the analysis of electromagnetic (EM) scattering by arbitrarily shaped perfect electric conductor (PEC) targets coated with electric anisotropic media in this paper. The scattering targets are discretized into tetrahedral volume elements in the material region and into triangle patches on the conducting surface, where the volume-surface integral equation (VSIE) is set up. Then the method of moments (MoM) is employed to solve the VSIE. In the impedance matrix, the near field interaction elements are computed by the conventional MoM while the far field interaction elements are modeled by the EDM. The proposed approach is sufficiently versatile in handling arbitrarily shaped objects coated with general electric anisotropic media and is easily constructed through a simple procedure. Numerical results are given to demonstrate the accuracy and efficiency of this method.