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The volume integral equation method is applied in electromagnetic scattering from arbitrarily shaped three-dimensional inhomogeneous objects. The properties of the volume electric and magnetic field integral equations (VEFIE and VMFIE) are investigated. Numerical experiments show that if the Galerkin's method with the lowest mixed-order basis functions is used to discretize the equations the accuracy of the VMFIE can be significantly poorer than the accuracy of the VEFIE, in particular, for high-contrast objects at high frequencies. The accuracy of the VMFIE can be essentially improved with full first order (linear) basis functions. The linear basis functions are found to be useful also when a single volume integral equation is used to model a general scatterer where both permittivity and permeability differ from the background.

The traditional thin dielectric sheet approximation (TDS) method models the electromagnetic scattering by thin dielectric sheet using surface integral equation. In this approximation, only the tangential current within the dielectric sheet is considered, and the vertical component is ignored. This article proposes a method that uses the volume integral equation solution to predict the vertical volume current and hence to improve the overall solution accuracy. The formulation is based on the hybrid surface (SIE) and volume integration equation (VIE) in which, the thin dielectric is modeled with SIE, and the thin layers are modeled by VIE. It is shown that the inclusion of the vertical current component does not increase memory and solution time. Numerical examples will be provided to demonstrate the effectiveness of this approach. © Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1978–1982, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23518

The hybrid volume and surface integral equation approach is applied to solve electromagnetic scattering and radiation problems involving conducting and/or dielectric objects. To flexibly and accurately model the complex structures and reduce the number of unknowns, mixed mesh scheme is developed to discretize the object. In this scheme, the triangles and quadrangles are used to discretize the conducting part of the object, and the tetrahedrons, hexahedrons, prisms and pyramids are used to model the dielectric volumes of the scatterer. Numerical results showed the solution accuracies from the mixed element meshes are of the same level compared with the single element meshes, but uses much less number of unknowns. This leads to flexibility for mesh generating and reduces the use of computing resources.

In practical applications, frequency-selective surfaces (FSSs) are finite, and sometimes even curved. In this paper, we present a hybrid volume-surface integral-equation approach to analyze the transmission and reflection characteristics of finite and curved FFS structures. The hybrid integral equations are established using the surface- and volume-equivalent principles. This approach has two advantages. One is the capability of modeling arbitrarily shaped FSS structures in detail, the other one allows us to easily apply the multilevel fast multiple algorithm to speed up the solution process. The scattering characteristics and frequency responses of several FSSs are analyzed. The simulation results show that for a finite-sized FSS, reducing the radius of curvature causes amplitude variation, frequency shift, and bandwidth change in the reflection and transmission responses. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 107–112, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20738

A high-order locally corrected Nystro/spl uml/m (LCN) method employing the mixed-order basis functions proposed by C$80al/spl iota/s$80kan and Peterson is presented for the electromagnetic scattering by targets composed of both dielectric and conducting bodies. An integral operator based on a combined field formulation for conducting surfaces and a Mu/spl uml/ller formulation for dielectric surfaces is used. It is found that for general scattering objects, mixed-order basis functions accelerate the convergence of the LCN solution, can eliminate spurious charges, and can significantly reduce the condition number of the impedance matrix.

This paper reviews the recent progress of fast integral equation solvers at the Center for Computational Electromagnetics and Electromagnetics laboratory, University of Illinois at Urbana-Champaign. We will demonstrate the ability to solve a variety of electromagnetic problems for complex structures and low-frequency structures as well as large scale scattering problems with over 10 million unknowns.

The volume integral equation (VIE) combined with the multilevel fast multipole algorithm (MLFMA) is applied to analyze antenna radiation in the presence of dielectric radomes. In solving the VIE, the radome is modeled by small volume cells of tetrahedron or hexahedron shape so that three-dimensional (3-D) complex radomes can be modeled accurately. When the induced volume current is determined by the method of moments, the total radiation is calculated by adding the induced current radiation to that of the antenna in the absence of the radome. The application of MLFMA reduced the computational complexity to a lower order and hence electrically large-sized radomes are analyzed. Numerical results are compared with the analytical solution for spherical shells with dipole array as excitations. The radiation patterns of dipole arrays in the presence of ogive, cone, and hemisphere radomes are also presented.

We consider the preconditioned iterative solution of large dense linear systems, where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering. For some scattering structures this matrix can be poorly conditioned. The main purpose of this study is to evaluate the efficiency of a class of incomplete LU (ILU) factorization preconditioners for solving this type of matrices. We solve the electromagnetic wave equations using the BiCG method with an ILU preconditioner in the context of a multilevel fast multipole algorithm (MLFMA). The novelty of this work is that the ILU preconditioner as constructed using the near part block diagonal submatrices generated from the MLFMA. Experimental results show that the ILU preconditioner reduces the number of BiCG iterations substantially, compared to the block diagonal preconditioner. The preconditioned iteration scheme also maintains the computational complexity of the MLFMA, and consequently reduces the total CPU time.

This paper discusses a special imaging reconstruction problem in which the object to be reconstructed resides in a homogeneous media but consists of a known part and an unknown part. The known part is combined with the homogeneous background media to form an inhomogeneous media. The dielectric permittivity profiles are reconstructed for the unknown part using the Born iteration (BI) method. Numerical examples demonstrate that this modified BI algorithm has faster convergence and higher accuracy than the regular BI method. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 91–95, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10684

Three-dimensional indoor radio-wave propagation is accurately modeled by solving the volume electric-field integral equation (VEFIE). The integral equation is discretized by the method of moments (MoM), in which the building walls, ceilings, floors, as well as any internal obstacles are represented by a collection of small volumetric cells. This discretization is capable of modeling a 3-D room structure to fine details. The solution to the MoM matrix equation is facilitated by the multilevel fast multipole algorithm (MLFMA), which significantly reduces memory requirements and CPU time. Numerical examples are shown to demonstrate the accuracy of the solution and the capability of modeling dielectric structures of nearly realistic sizes. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 29: 168–175, 2001.

Compared to the TM case, the inverse scattering problem for the TE incident field is more complicated due to its stronger nonlinearity. This work provides an effective method for the reconstruction of two-dimensional (2-D) inhomogeneous dielectric objects from TE scattering data. The algorithm applies the distorted Born iterative method to the integral equation of strong permittivity fluctuation to reconstruct scatterers with high-permittivity contrast. Numerical simulations are performed and the results show that the distorted Born iterative method (DBIM) for strong permittivity fluctuation (SPF-DBIM) converges faster and can obtain better reconstructions for objects with larger dimensions and higher contrasts in comparison with ordinary DBIM. A frequency hopping technique is also applied to further increase the contrast.

A combined solution of an electromagnetic (EM)-wave equation and head transfer equation is presented to analyze the microwave rewarming process of cryopreserved tissues. The solution process starts with an initial temperature of the tissue. The EM-field distribution inside the tissue is determined first by solving hybrid surface-volume integral equations. This solution provides a thermal source term for the heat-transfer equation. A finite-difference scheme is then applied to solve the heat-transfer equation, which determines the temperature distribution inside the tissue for the next time step. Since the tissue's electrical characteristics (/spl epsiv/ and /spl sigma/) are functions of temperature, their values are then updated based on the new temperature distribution. The iteration continues until a termination condition is satisfied. This combined iterative solution of wave equation and heat-transfer equation allows one to model the complex rewarming process. Numerical results are presented to demonstrate the application of the combined analysis approach.

An accurate and efficient technique called the thin-stratified medium fast-multipole algorithm (TSM-FMA) is presented for solving integral equations pertinent to electromagnetic analysis of microstrip structures, which consists of the full-wave analysis method and the application of the multilevel fast multipole algorithm (MLFMA) to thin stratified structures. In this approach, a new form of the electric-field spatial-domain Green's function is developed in a symmetrical form which simplifies the discretization of the integral equation using the method of moments (MoM). The patch may be of arbitrary shape since their equivalent electric currents are modeled with subdomain triangular patch basis functions. TSM-FMA is introduced to speed up the matrix-vector multiplication which constitutes the major computational cost in the application of the conjugate gradient (CG) method. TSM-FMA reduces the central processing unit (CPU) time per iteration to O(N log N) for sparse structures and to O(N) for dense structures, from O(N/sup 3/) for the Gaussian elimination method and O(N/sup 2/) per iteration for the CG method. The memory requirement for TSM-FMA also scales as O(N log N) for sparse structures and as O(N) for dense structures. Therefore, this approach is suitable for solving large-scale problems on a small computer.

This paper studies, in detail, a variety of formulations for the hybrid finite-element and boundary-integral (FE-BI) method for three-dimensional (3-D) electromagnetic scattering by inhomogeneous objects. It is shown that the efficiency and accuracy of the FE-BI method depends highly on the formulation and discretization of the boundary-integral equation (BIE) used. A simple analysis of the matrix condition number identifies the efficiency of the different FE-BI formulations and an analysis of weighting functions shows that the traditional FE-BI formulations cannot produce accurate solutions. A new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance. Finally, the multilevel fast multipole algorithm (MLFMA) is employed to significantly reduce the memory requirement and computational complexity of the proposed FE-BI method.

We present an accurate method of moments (MoM) solution of the combined field integral equation (CFIE) using the multilevel fast multipole algorithm (MLFMA) for scattering by large, three-dimensional (3-D), arbitrarily shaped, homogeneous objects. We first investigate several different MoM formulations of the CFIE and propose a new formulation, which is both accurate and free of interior resonances. We then employ the MLFMA to significantly reduce the memory requirement and computational complexity of the MoM solution. Numerical results are presented to demonstrate the accuracy and capability of the proposed method. The method can be extended in a straightforward manner to scatterers composed of different homogeneous dielectric and conducting objects.

Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements.

The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are reviewed. The number of modes required, block-diagonal preconditioner, near singularity extraction, and the choice of initial guesses are discussed to apply the MLFMA to calculating electromagnetic scattering by large complex objects. Using these techniques, we can solve the problem of electromagnetic scattering by large complex three-dimensional (3-D) objects such as an aircraft (VFY218) on a small computer.

The generalized network formulation is applied in combination with the method of moments to calculate the electromagnetic scattering from conducting objects containing near-resonant open-ended cavities. The presence of the cavity increases the iteration number in conjugate gradient iterations due to the near-resonant modes and the multiple wave bounces inside the cavity. The equivalence principle is used to separate the cavity region from the rest of the object, allowing an independent solution of the cavity problem by a direct inversion algorithm using a connection scheme. The solution of the cavity is then represented by a generalized admittance matrix. Numerical results for two-dimensional (2-D) composite targets show that the iteration number can be reduced significantly for objects containing long cavities. This algorithm can also be applied to three-dimensional problems.

In the solution of an integral equation using the conjugate gradient (CG) method, the most expensive part is the matrix-vector multiplication, requiring O(N2) floating-point operations. The fast multipole method (FMM) reduced the operation to O(N15). In this article we apply a multilevel algorithm to this problem and show that the complexity of a matrix-vector multiplication is proportional to N (log(N))2. © 1994 John Wiley & Sons, Inc.

The volume integral equation method is used to analyze electromagnetic scattering from arbitrarily shaped three-dimensional homogeneous objects whose permittivity and permeability differ from those of background medium. We consider three different types of formulations, the DB-formulation with flux densities as unknowns, the EH-formulation with fields as unknowns, and EE- and HH-formulations that contain only single unknown, either electric or magnetic field. The properties of the formulations are investigated with numerical examples. Numerical results show that at least fully linear curl conforming basis functions must be used when applying the single unknown volume intergal equation for general scatterers.

A three-dimensional (3-D) version of the nested equivalent principle algorithm (NEPAL) is presented. In 3-D, a scatterer is first decomposed into N subscatterers. Then, spherical wave functions are used to represent the scattered field of the subscatterers. Subscatterers are divided into different levels of groups in a nested manner. For example, each group consists of eight subgroups, and each subgroup contains eight sub-subgroups, and so on. For each subgroup, the scattering solution is first solved and the number of subscatterers of the subgroup is then reduced by replacing the interior subscatterers with boundary subscatterers using Huygens' equivalence principle. As a result, when the subgroups are combined to form a higher level group, the group will have a smaller number of subscatterers. This process is repeated for each level, and in the last level, the number of subscatterers is proportional to that of boundary size of the scatterers. This algorithm has a computational complexity of O(N/sup 2/) in three dimensions for all excitations and has the advantage of solving large scattering problems for multiple excitations. This is in contrast to Gaussian elimination which has a computational complexity of O(N/sup 3/). >

The Volume-Surface Integral Equation (VSIE) has been successfully used in computational electromagnetics for complicated structures, especially for coated objects. Since the VSIE only requires a mesh on the object, it can be used to solve scattering problems of electrically large complicated objects with high accuracy when combined with the Fast Multipole Method (FMM). The VSIE method is characterized by the use of different integral equations on perfect electric conductor (PEC) surfaces and in penetrable objects. The equations are discretized with the extended Galerkin projection method. Usually, scattering problems involving PEC surfaces are solved using Surface Integral Equations (SIE), where the RWG basis functions are commonly used to represent the unknown surface current density. Scattering problems involving dielectric objects can be solved using either the SIE (piecewise homogeneous object only) or the Volume Integral Equation (VIE). When it comes to the selection of a basis for the VIE, both the divergence-conforming and curl-conforming basis functions can be used with no particular advantage provided by one over the other [1]. Divergence-conforming basis functions are usually used with electric flux density as unknowns, because the normal continuity of electric flux density can be enforced. Curl-conforming basis functions are used with electric field as unknowns since the tangential continuity of electric field across the mesh element is automatically enforced. Research on coated objects using VSIE with RWG basis on surface and divergence-conforming basis functions in dielectric volume can be found in [6].

A coupled surface-volume integral equation approach is presented fur the calculation of electromagnetic scattering from conducting objects coated with materials. Free-space Green's function is used in the formulation of both integral equations. In the method of moments (MoM) solution to the integral equations, the target is discretized using triangular patches for conducting surfaces and tetrahedral cells for dielectric volume. General roof-top basis functions are used to expand the surface and volume currents, respectively. This approach is applicable to inhomogeneous material coating, and, because of the use of free-space Green's function, it can be easily accelerated using fast solvers such as the multilevel fast multipole algorithm.

This paper presented two formulations of the electric field volume integral equation, and discussed the use of two different types of volume basis functions in the representation of the volume current. In conclusion, for the two types of volume basis functions, the solution accuracies are of the same order as long as the numbers of basis functions are comparable. The numbers of iterations of the C-type basis function are generally smaller than that of the D-type basis functions if the initial is a zero vector. For multiple incident angles, the D-type basis case needs slightly less number of iterations when the initial solution is predicted from the solution of the previous incident angle.

The electromagnetic scattering by dielectric material sheet can be modeled in several different ways. In the integral equation approach to solve such a problem, there are surface equivalence method and volume equivalence method. When the thickness of a material sheet is much smaller than a wavelength, the thin dielectric sheet approximation (TDS) is often used. In this approximation, a volume equivalent problem is transferred to a surface integral equation for the material sheet, and hence number of unknowns is reduced. A major source of error in TDS is the lack of the vertical electric field component that is perpendicular to the sheet. This paper proposes a modification to TDS by adding the contribution of the vertical electrical field without increasing the computational cost. Numerical simulation results show that the accuracy is improved when this modification is employed.

The mixed mesh discretization scheme has been applied to solve hybrid integral equations for electromagnetic scattering and radiation. The mixed mesh scheme makes it easier to model complex objects without significantly increasing the number of unknowns. The program from this mixed scheme algorithm has been validated using exact solutions for spherical objects. When accelerated by MLFMA, this algorithm has the potential to solve many realistic problems as demonstrated by the examples shown in this paper

In this paper, a preconditioner in the implementation of the hybrid volume and surface integral equations (VSIE) with MLFMA solver is described. The algorithm is applied to the calculation of radiation pattern of a microstrip antenna residing in a large open-ended cavity. The numerical results show that the method can substantially reduce the iteration number with CG method. Using this preconditioner and MLFMA, the hybrid VSIE can be applied into the radiation analysis of many large scale problems with microstrip antennas.

An iteration algorithm is presented to obtain the electromagnetic field distribution and the temperature pattern for arbitrary shaped samples when they are in free space and inside cavities. The iteration involves solving the electromagnetic wave equation and the heat transfer equation alternatively. The electromagnetic solver models the cavity with quadrangle patches and the dielectric tissue by hexahedrons. Hence it can be used to simulate realistic microwave re-warming and heating processes, and provides an accurate and effective tool for virtual experiments, through which we can analyze the heating pattern and the heating rate as functions of sample's size, shape, and electric and heat conduction properties. The similar simulation can be performed using the FDTD method. Our preliminary comparison shows that the integral equation method has higher efficiency than the FDTD method for this example. These simulations are essential to determine what kind of cavity and what control process can lead to a desired heating pattern and re-warming history

We investigate preconditioned iterative solutions of large dense complex valued matrices arising from discretizing the hybrid integral equations of electromagnetic scattering. Due to the specific data pattern, we propose to construct two-level (TL) preconditioners to treat the conducting and dielectric parts more efficiently. The main purpose of this study is to show that the TL preconditioning is effective with the MLFMA and can reduce the number of Krylov iterations substantially by using subdivided domain of the near part matrix for each level. Our experimental results indicate that the TL preconditioning maintains the computational complexity of the MLFMA, but converges much faster, thus effectively reduces the overall simulation time for the coated material objects.

The algorithms based on the iterative solution of the integral equations and accelerated by fast solvers to provide efficient and accurate ways to calculate the scattering by large and complex objects. To further increase the efficiency in producing multi-angle and multi-frequency scattering data, attentions are focused on (a) developing precondition techniques to reduce the number of iterations for a converged solution and (b) developing advanced post processing methods which uses the information of the available solution to predict as much scattering data as possible. In this paper, an approximate extrapolation method is introduced to rapidly fill the angular and frequency far-field points using the solutions at sparsely sampled points.

Summary form only given. The surface integral equation formulation is applicable to homogeneous material only. If there is a slight change of dielectric contrast within the volume, or there is a small portion of the material region that has inhomogeneous permittivity, the SIE will no longer be valid. It has been proposed that an additional volume integral equation be introduced to model the inhomogeneous part of the material region. The idea is to use the induced volume current for the region that has dielectric contrast which is different from the majority of the dielectric region. This is called the hybrid PMCHWT and VIE formulation. In this presentation, we apply this method to calculate the scattering by three-dimensional material scatterers. Our formulation uses the quadrilateral mesh for the surface modeling and hexahedron mesh for the volume modeling. Methods to treat the singular integrals for various integral operators are discussed, and validation numerical results will be shown.

A high-order method of moment solution for the electromagnetic scattering by complex objects with composite materials is presented. The generalized technique can analyze material scattering by homogeneous and inhomogeneous material and conducting objects through both surface and volume equivalent currents. The solution employs high-order basis functions and a quadrature point-based discretization. It is shown that the solution leads to high-order convergence for the electromagnetic scattering by heterogeneous scatterers. This is validated here through the study of canonical scattering structures.

The hybrid volume-surface integral equation (VSIE) approach has been applied to investigate the curved FSS with the dielectric substrates of finite extent. The scattering RCS of a curved structure and a planar finite periodic structure have been presented. An important advantage of the VSIE algorithm is that it can be used to simulate the finite and arbitrary curved structure. When the finite structure size increases, the multilevel fast multipole algorithm can be applied to reduce the computational complexity.

A uniform formulation is presented for using the multilevel fast multipole algorithm (MLFMA) to solve a set of simultaneous integral equations for electromagnetic wave scattering. The formulation is based on two operators that apply to the aggregation and disaggregation matrices. With these two operators, we can easily extend the MLFMA that is developed for PEC scatterers to include material coatings (bulk dielectrics as well as approximate boundary conditions). The introduction of the P and Q operators (two operators that modify the matrices, based on the integral operators of the integral equations) does not cause an increase in computation time and memory. A numerical example is presented to illustrate the application of the uniform formulation.

A fast, high-order, hybrid integral equation solver for electromagnetic scattering is presented. The solver is based on a locally corrected Nystrom solution of a hybrid volume integral equation (VIE) and surface integral equation (SIE) formulation. The solution can be accelerated by quadrature-sampled pre-corrected FFT or the multilevel fast multipole method (MLFMM), or a combination of both schemes. The methods are validated in this paper through the study of the electromagnetic scattering of canonical geometries made of a composite of conducting and dielectric materials.

The paper presents a thorough study of the application of the mixed-order basis proposed by F. Caliskan and A.F. Peterson (see IEEE Antennas and Wireless Propag. Lett., vol.2, p.72-3, 2003) for the locally corrected Nystrom (LCN) method when analyzing electromagnetic scattering by targets composed of both dielectric and conducting materials. The integral formulation is based on a superposition of the classical combined field integral equation (CFIE) operator (Peterson et al., 1998) for metallic surfaces and a Muller formulation for penetrable objects (Muller, C., 1969; Mautz, J.R. and Harrington, R.F., 1979). It is demonstrated that for general scattering objects, mixed-order basis functions accelerate the convergence of the LCN solution, eliminate spurious charges for singular geometries, and can significantly reduce the condition number of the impedance matrix.

The author presents a method that combines the hybrid integral equations with the distorted Born iterative method (DBIM) to solve the half space inversion problems. The main idea is to approximate scattering by a half space with the surface contour integral or approximate the scattered field from the half space with buried objects using hybrid integral equations (HIE). Then, implementing the modified distorted Born iterative method, one can reconstruct objects buried under lossy earth. This method can be applied if there are some other objects besides the unknown objects under the lossy earth or if the earth surface is not flat.

Solving integral equation by iterative methods for cavity scattering problems can encounter slow convergence due to several factors, such as internal resonance, near resonance and multiple wave bounces. A numerical study is presented on the solution of the TETH-type (short for t/spl circ//spl middot/E and t/spl circ//spl middot/H) combined field integral equation in the context of the near-resonance decoupling approach (NRDA), in which the TETH-type formulation is discretized by the method of moments with roof-top basis and testing functions over quadrilateral shaped meshes.

The hybrid volume-surface integral equation approach is implemented to calculate the radiation and scattering by large microstrip arrays on curved substrate and ground plane. The multilevel fast multipole algorithm is applied to reduce the memory and CPU requirements for solving the matrix equation. Numerical results for input impedances of single patch and the scattering pattern for an array are presented.

Dielectric materials are often involved in electromagnetic simulation for real-world problems. Previously, researchers have applied the fast multipole method and a hybrid volume-surface integral equation (VSIE) formulation to solve scattering problems involving both PEC and inhomogeneous dielectric objects. Although the above solver can solve scattering from electrically large objects with optimal computational complexity, it only involves surface and volume basis functions to model currents on PEC surfaces and in dielectric media. In order to solve problems involving additional types of scatterers and radiators, such as wire antennas, thin dielectric sheets, and impedance surfaces, we have enhanced the multilevel fast multipole code in Chao et al. (1998) with the impedance boundary condition (IBC) and the hybrid VSIE formulation. The paper discusses how a variety of basis functions are incorporated into the hybrid VSIE formulation and the multilevel fast multipole algorithm (MLFMA).

The hybrid volume-surface integral equation (VSIE) approach has been applied to calculate the input impedance of the printed structure with finite ground plane. The calculated results for a rectangular patch antenna have been compared with experiment and good agreement is observed. When the ground plane size increases, the input impedance converges to the result of infinite ground plane. An important advantage of the VSIE algorithm is that it can be used to simulate the printed structure of finite, arbitrary shaped ground plane.

A high-order solution of the electric field integral equation (EFIE) for penetrable inhomogeneous material volumes is presented. The solution is based on a high-order method of moment solution with a quadrature point-based discretization. It is shown that the solution leads to high-order convergence for the electromagnetic scattering by inhomogeneous dielectric scatterers. This is validated through the study of canonical scattering objects.

[1] The surface integral equation approach and the hybrid surface-volume integral equation approach are compared for solving the problem of electromagnetic scattering by conducting objects with dielectric coating. The surface integral equation is formulated by the PMCHW method, and it is efficient for modeling bulk material scattering. The hybrid surface-volume integral equation approach is better conditioned for the volume scattering problems. For a special class of problems in which the dielectric coating is electrically thin (one to two grid sizes), the numbers of unknowns needed by the two approaches are of the same order, and the surface-volume integral equation approach converges significantly faster than the pure surface integral equation approach. Several numerical examples are given, and the results showed good agreements with the above statement.

[1] A high-order method of moment solution with quadrature-point-based sampling is presented for the solution of the volume electric field integral equation for the scattering of inhomogeneous dielectric bodies. The proposed scheme efficiently allows for the material profile to be inhomogeneous within a curvilinear cell. It is demonstrated that the method leads to exponential convergence in both the radar cross section (RCS) and the volume current density. It is also demonstrated that the method can be more efficient than surface field integral equation formulations for thin-material scattering.

We propose a hybrid integral equation approach that combines the volume integral equation (VIE) and the surface integral equation to model the mixed dielectric and conducting structures. The volume integral equation is applied to the material region and the surface integral equation (SIE) is enforced over the conducting surface. This results in a very general model as all the volume and surface regions are modeled properly. The advantage of this approach is that in the coated object scattering problem, the coating material can be inhomogeneous, and in the printed circuit and microstrip antenna simulation problem the substrate can be of finite size. Another advantage of this approach is the simplicity of the Green's function in both the VIE and SIE. However the additional cost here is the increase of the number of unknowns since the volume that is occupied by the dielectric material is meshed. This results in a larger memory requirement and longer solution time in solving the MoM matrix equation. But this deficiency can be overcome by applying fast integral equation solvers such as the multilevel fast multipole algorithm. We first give the formulation of the problem using the method of moments (MoM), and then show numerical simulation results to demonstrate the validity of the proposed method.

The fast multipole method (FMM) is an efficient method in the iterative solution of the matrix equation that is associated with the integral equation of wave scattering. The key idea of the FMM is to divide the interactions between groups of scatterers into near-field interactions and non-near-field interactions, and perform the non-near-field interactions efficiently with the aid of the multipole expansion of the fields. The far-field approximation is introduced in FMM to calculate the wave interactions between groups that are separated by a very large distance. The advantage of this approach is that it automatically switches back to the original FMM for small problems. Under the far-field approximation, the translation operator is much simpler than that used in the FMM. Numerical results show speed up of the modified FMM over the original FMM for problems as small as several thousands.

We have implemented a numerical algorithm for solving the integral equations of electromagnetic scattering from conducting objects using the method of moments. This code is implemented and developed for the computation of near-field interactions in the multilevel fast multipole algorithm (MLFMA). The implementation utilized the flat-triangular patch for the surface description and the Rao-Wilton-Glisson (1982) basis as the current expansion function. The triangular patch basis was introduced by Rao et. al. for solving the electrical field integral equation (EFIE). Our implementation is focused on the consideration of reducing the condition number of the impedance matrix. Hence, the implementation includes the electrical field integral equation (EFIE), the magnetic field integral equation (MFIE), and the combined field integral equation (CFIE). In the computation of the matrix elements associated with the nearby elements, the near-singular integrals are treated properly with the aid of the formulas given previously. This allows the code to deal with objects that contain very thin structures. Moreover, we added a partial diagonal preconditioner in the code in case the iterative solver does not provide any preconditioners. This is very important in the Galerkin's method. We have tested the code for various ill-conditioning problems to validate the performance of the implementation.

The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) have been applied for solving integral equations of electromagnetic scattering iteratively. FMM reduces the complexity of a matrix-vector multiply from O(N/sup 2/) to O(N