Electromagnetic scattering from composite objects using a mixtureof exact and impedance-boundary conditions

Different surface integral equations for characterizing the electromagnetic scattering from a surface impedance object partially coated with dielectric materials are presented. The impedance boundary condition (IBC) is applied on the impedance surface and the exact boundary condition is applied on the dielectric surface. The resulting integral equations are solved for bodies of revolution using the method of moments. The numerical results are compared with the exact solution for a sphere. Other geometries are considered, and their results are verified by comparing results of the numerical solutions which were obtained using different formulations. The internal resonance problem is examined. It is found that the combined field integral equation (CFIE) can be used at any frequency and with any surface impedance     

Hybrid solutions for large-impedance coated bodies of revolution

Electromagnetic scattering solutions are developed for coated perfectly conducting bodies of revolution (BOR) that satisfy the impedance boundary condition. The integral equation arising from the impedance (Leontovich) boundary condition is solved by use of the method of moments (MM) technique along with an Ansatz for the surface currents that is derived from physical optics (PO) and the Fock theory that is modified for imperfectly conducting surfaces. The MM solution is expressed in terms of two integral (Galerkin) operators. The form of the Galerkin expansion used results in a symmetric MM system matrix. The hybrid solution is specialized for BOR's although the approach is applicable to a broader class of scatterers as well. The results are compared with the Mie solution for penetrable spherical scatterers, which satisfy the impedance boundary condition, and with recently published MM solutions for nonspherical scatterers.     

Scattering by superquadric dielectric-coated cylinders using higherorder impedance boundary conditions

A general method for deriving higher order impedance boundary conditions is described. It is based on solving an appropriate canonical problem exactly in the spectral domain. After approximating the spectral impedance terms as a ratio of polynomials in the transform variable, elementary properties of the Fourier transform are used to obtain the corresponding boundary condition in the spatial domain. The method is applicable to multilayer coatings with arbitrary constitutive relations. Higher-order boundary conditions which neglect the effects of curvature are derived for a dielectric coating using the method. The boundary condition equation and the magnetic field integral equation are solved simultaneously using the method of moments, yielding the bistatic and monostatic radar cross section for dielectric-coated superquadric cylinders. The method is also applicable to a combined field integral equation (CFIE) solution, which can be used to eliminate the internal resonance problem associated with either the electric field integral equation (EFIE) or magnetic field integral equation (MFIE)     

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.     

Integral equation formulations for imperfectly conducting scatterers

Integral equation formulations are presented for characterizing the electromagnetic (EM) scattering interaction for nonmetallic surfaced bodies. Three different boundary conditions are considered for the surfaces: namely, the impedance (Leontovich), the resistive sheet, and its dual, the magnetically conducting sheet boundary. The integral equation formulations presented for a general geometry are specialized for bodies of revolution and solved with the method of moments (MM). The current expansion functions, which are chosen, result in a symmetric system of equations. This system is expressed in terms of two Galerkin matrix operators that have special properties. The solutions of the integral equation for the impedance boundary at internal resonances of the associated perfectly conducting scatterer are examined. The results are compared with the Mie solution for impedance-coated spheres and with the MM solutions of the electric, magnetic, and combined field formulations for impedance-coated bodies.     

Exact surface impedance/admittance boundary conditions for complexgeometries: theory and applications

A methodology useful to derive exact and higher order surface impedance/admittance boundary conditions (HOI/ABC's) for complex geometries is presented. It is shown that exact surface boundary conditions are always expressed through dyadic integral operators involving the tangential magnetic and electric fields all over the surface of the body. Quasi-local surface boundary conditions that include curvature effects are shown to be computable through an asymptotic approximation of the integral operators. Finally, an example of a surface admittance boundary condition useful to analyze a structure exhibiting discontinuities along its surface boundary is presented. Practical examples to demonstrate the feasibility of the proposed methodology, as well as the accuracy of the resulting surface boundary conditions are also presented     

Scattering analysis of a large body with deep cavities

A numerical scheme is presented for simulating electromagnetic scattering from a large and arbitrarily shaped body, coated with inhomogeneous composite materials, with large and deep cavities. This numerical scheme employs the higher order vector finite-element method (FEM) to discretize the fields inside the cavities and coatings and the higher order boundary integral (BI) method to terminate the FEM computational domain. A highly efficient special solver is designed to eliminate the unknowns inside the cavities, which yields a computed relation (CR) matrix over the cavity's aperture between the tangential electric and magnetic fields. This CR matrix is then combined with the finite element-boundary integral (FE-BI) matrix equation to form a complete linear system for the discrete fields everywhere in the computational domain. The resulting system is solved iteratively using a novel preconditioner derived by replacing the BI with a corresponding absorbing boundary condition (ABC).     

Comparison of different integral equation formulations for bodiesof revolution with anisotropic surface impedance boundaryconditions

To obtain an accurate solution in the method of moments it is vital that an appropriate integral equation be used. In solving the problem of scattering from bodies of revolution with anisotropic surface impedance boundary conditions, different answers may result from seemingly minor differences in the integral equation formulations used. Different kinds of integral equations are compared with one another when they are applied to bodies of revolution     

High-order impedance boundary conditions for multilayer coated 3-Dobjects

The scattering problem by a multilayer coated three-dimensional (3-D) object where the coating is modeled by an impedance boundary condition (IBC) is considered. First, the exact boundary condition is obtained for an infinite planar coating with an arbitrary number of layers. Then, various approximations for the pseudodifferential operators involved in this exact condition are proposed. In the expressions of the resulting IBCs, all tangential derivatives of the fields of order higher than two are suppressed. These IBCs are compared, in terms of numerical efficiency, by computing either the reflection coefficients on an infinite planar metal-backed coating or the radar cross section (RCS) of a perfectly conducting coated sphere using the tangent plane approximation. In both cases, it is found that the highest order IBC models the coating with a good accuracy. Finally, some guidance is given on how this IBC may be numerically implemented in an integral equation or a finite-element formulation for an arbitrarily shaped object     

Surface integral formulation for calculating conductor anddielectric losses of various transmission structures

The power-loss method, along with a surface integral formulation, has been used to compute the attenuation constant in microstrip and coplanar structures. This method can be used for the analysis of both open and closed structures. Using the surface equivalence principle, the waveguide walls are replaced by equivalent electric surface currents and dielectric surfaces are replaced by equivalent electric and magnetic surface currents. Enforcing the appropriate boundary condition, and E-field integral equation (EFIE) is developed for these currents. Method of moments with pulse expansion and point matching testing procedure is used to transform the integral equation into a matrix one. The relationship between the propagation constant and frequency is found from the minimum eigenvalue of the moment matrix. The eigenvector pertaining to the minimum eigenvalue gives the unknown electric and magnetic surface currents     

A hybrid symmetric FEM/MOM formulation applied to scattering byinhomogeneous bodies of revolution

A new symmetric formulation of the hybrid finite element method (HFEM) is described which combines elements of the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) for the exterior region along with the finite element solution for the interior region. The formulation is applied to scattering by inhomogeneous bodies of revolution. To avoid spurious modes in the interior region a combination of vector and nodal based finite elements are used. Integral equations in the exterior region are used to enforce the Sommerfeld radiation condition by matching both the tangential electric and magnetic fields between interior and exterior regions. Results from this symmetric formulation as well as formulations based solely on the EFIE or MFIE are compared to exact series solutions and integral equation solutions for a number of examples. The behaviors of the symmetric, EFIE, and MFIE solutions are examined at potential resonant frequencies of the interior and exterior regions, demonstrating the advantage of this symmetric formulation     

A method for treating junctions between bodies of revolution andarbitrary surfaces

A method to treat junctions between bodies of revolution (BOR) and arbitrary surfaces is presented. The method is applicable to formulations employing the method of moments to solve surface integral equations for structures consisting of combined BOR and arbitrary surfaces. Harmonic entire domain basis functions are used on the bodies of revolution and triangular surface patches are used on the arbitrary surfaces. A junction is formed by creating a spatially coincident region between a body of revolution and an arbitrary surface. The geometry of the junction region is determined by the arrangement of the triangular surface patches used to form the junction. It determines the accuracy with which the junction currents are modeled. Numerical results are presented to demonstrate the accuracy of the junction models for both radiation pattern and input impedance predictions     

基于阻抗边界条件建模的涂敷目标电磁散射分析

针对飞行器上常用的涂敷吸波材料结构开展电磁散射数值建模和散射特性分析。利用涂敷结构表面电磁场的阻抗边界条件,建立表面电流和表面磁流的新型积分方程形式,并利用快速算法进行求解。数值结果表明:该型积分方程在不增加额外计算量和存储量的条件下,显著改善了迭代求解收敛性,为复杂涂敷结构的电磁散射分析提供了快速、可靠的技术途径。     

Scattering from multiple conducting and dielectric bodies ofarbitrary shape

A simple moment-method solution is presented for the problem of electromagnetic scattering from structures consisting of multiple perfectly conducting and dielectric bodies of arbitrary shape. The system is excited by a plane wave. The surface equivalence principle is used to replace the bodies by equivalent electric and magnetic surface currents, radiating into an unbounded medium. A set of coupled integral equations, involving the surface currents, is obtained by enforcing the boundary conditions on the tangential components of the total electric and magnetic fields. The method of moments is used to solve the integral equations. The surfaces of the bodies are approximated by planar triangular patches, and linearly varying vector functions are used for both expansion and testing functions. Some of the limitations of the method are briefly discussed. Results for the scattering cross sections are presented. The computed results are in very good agreement with the exact solutions and with published data     

Scattering by Conducting Bodies Coated With Bi-Isotropic Materials

Electromagnetic scattering by arbitrarily shaped conducting bodies coated with general bi-isotropic materials is formulated in terms of the surface integral equation method. In order to facilitate the implementation of the surface equivalence principle, a field decomposition scheme is utilized to split a bi-isotropic media into two equivalent isotropic media. By enforcing the boundary condition on the interfaces of the body, a set of coupled integral equations is finally obtained for the unknown surface currents and then numerically solved using the moment methods combined with the vector triangular basis function. The fast multipole technique has been embedded into the algorithm to accelerate the solution process. The validity of theoretical formulations is verified by numerical results and their comparisons. The calculated results for bi-isotropically coated conducting spheres and oblate spheroids are compared with the exact solution and the existing data, and excellent agreements are observed.     

Scattering from narrow rectangular filled grooves

The solution of the integral equation for a small width rectangular groove is considered. It is shown that by retaining the dominant mode supported by the rectangular groove, the resulting quasi-static integral equations are comparable to those associated with the perfectly conducting narrow strip. They are, therefore, amenable to analytic solution yielding the exact field distribution or equivalent currents across the groove's aperture. The derived currents exhibit the same edge behavior as that associated with the currents of a perfectly conducting half-plane. The corresponding current behavior based on a (numerical) impedance simulation of the groove is quite different. However the resulting echowidths are comparable. Both transverse electric (TE) and transverse magnetic (TM) polarizations are treated     

Radar cross section of arbitrarily shaped bodies of revolution

The radar cross section (RCS) of an arbitrarily shaped, homogeneous dielectric body of revolution (BOR) is evaluated by the surface integral equation (SIE) formulation and the method of moments. Method accuracy is verified by the good agreement with the exact solutions for the RCS of a dielectric sphere. To demonstrate the advantages of this method, the RCS for a complex BOR model of human torso is computed with a nonaxially incident plane wave. Seven Fourier modes are considered in the computation. The SIE and approximate integral equation (AIE) formulations are next given for the RCS evaluation of a composite dielectric and conducting BOR. For the cases considered, both formulations give the same surface currents and RCS results. However, significant savings in computer storage and CPU time are realized for the AIE approach, since only one current (electric or magnetic) need be determined for RCS evaluation     

An application of the boundary element method to the magnetic fieldintegral equation

A boundary element method (BEM) for the solution of electromagnetic scattering problems using the magnetic field integral equation (MFIE) is discussed. The discretized form of the MFIE is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation. By considering several different types of boundary elements, it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution. It is shown that a higher-order approximation for the current could significantly improve the accuracy of the numerical solution. The superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower-order approximations. The BEM results are compared to the results obtained using the dielectric bodies of revolution (DBR) code     

Generalized formulations for electromagnetic scattering fromperfectly conducting and homogeneous material bodies-theory andnumerical solution

A generalized E-field formulation for three-dimensional scattering from perfectly conducting bodies and generalized coupled operator equations for three-dimensional scattering from material bodies are introduced. A fictitious electric current flowing on a mathematical surface enclosed inside the body is used to simulate the scattered field, and, in the material case, a fictitious electric current flowing on a mathematical surface enclosing the body is used to simulate the diffracted field inside the body. Application of the respective boundary conditions lead to operator equations to be solved for the unknown fictitious currents, which facilitates calculation of the fields in the various regions, using the magnetic vector potential integral. The existence and uniqueness of the solution are discussed. These alternative operator equations are solvable using the method of moments. The numerical solution is simple to execute, rapidly converging, and general in that bodies of smooth but otherwise arbitrary surface, both lossless and lossy, can be handled effectively. Comparison of the results with available analytic solutions demonstrates the accuracy of the moment procedure     

On combined source solutions for bodies with impedance boundary conditions

Numerical solutions to the impedance boundary condition (IBC) combined source integral equation (CSIE) for scattering from impedance spheres are presented. The CSIE formulation is a well-posed alternative to the IBC electric and magnetic field integral equations which can be contaminated by spurious resonant modes. Compared with the IBC combined field integral equation (CFIE), CSIE solutions have the same accuracy when the combined source coupling admittance is chosen to be the same value as the combined field coupling admittance. However, the CSIE formulation is better suited than the CFIE for creating a general purpose computer code capable of handling aperture radiation problems and/or a scatterer which has a spatially varying surface impedance.