meshfree solutions of volume integral equations for electromagnetic scattering by anisotropic...

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 9, SEPTEMBER 2012 4249

Meshfree Solutions of Volume Integral Equations forElectromagnetic Scattering by Anisotropic Objects

Mei Song Tong

Abstract—Volume integral equations (VIEs) are indispensablefor solving inhomogeneous or anisotropic electromagnetic (EM)problems by integral equation approach. The solution of VIEsstrongly relies on the discretization of volume integral domains,and tetrahedral elements in discretization are usually preferredfor arbitrary geometric shapes. Unlike discretizing a surfacedomain, discretizing a volume domain is very inconvenient inpractice, and special commercial software is needed in generaleven for a simple and regular geometry. To release the burden ofdescretizing volume domains, especially to remove the constraintof mesh conformity in the traditional method of moments (MoM),we propose a novel meshfree scheme for solving VIEs in this paper.The scheme is based on the transformation of volume integralsinto boundary integrals through the Green–Gauss theorem whenintegral kernels are regularized by excluding a small cylinderenclosing the observation node. The original integral domainrepresented by the object is also expanded to a cylindrical domaincircumscribing the object to facilitate the evaluation of boundaryintegrals. The singular integrals over the small cylinder are spe-cially handled with singularity subtraction techniques. Numericalexamples for EM scattering by inhomogeneous or anisotropicobjects are presented to illustrate the scheme, and good resultsare observed.

Index Terms—Anisotropic object, electromagnetic (EM) scat-tering, meshfree method, volume integral equation (VIE).

I. INTRODUCTION

N UMERICAL solutions for theoretical and practical prob-lems are based on computer technology and require a

domain discretization for governing equations or mesh descrip-tion for involved geometries [1]. Though mesh generation withthe help of special commercial software is not thought of as aproblem in general, it could be very tedious for some complexstructures in which multiscale components or multiple materialproperties exist and geometric discontinuities (or geometricsingularities) are popular. Multiscale property of structuresrequires nonuniform discretization, and it could be very hardto perfectly merge meshes in different scales [2]. Defectivemeshes could be easily produced near the junctions of com-ponents in such structures, and they are intolerable in somenumerical methods. Also, repeatedly remeshing is requiredand very tedious when structures have geometric deformationor moving boundaries [3]. Therefore, removing the need of

Manuscript received August 04, 2011; revised March 15, 2012; acceptedApril 16, 2012. Date of publication July 03, 2012; date of current version Au-gust 30, 2012.The author is with the School of Electronics and Information Engineering,

Tongji University, Shanghai 201804, China (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2012.2207052

meshing or remeshing and developing meshfree or meshlessmethods could be very attractive for solving such problems.Meshfree or meshless methods aim at the reduction of

meshing or remeshing costs in the numerical solution of prob-lems and have received an extensive attention in mechanicalengineering due to the practical need for solving the problemswith moving boundaries, such as extrusion molding process,growth of cracks, and propagation of interfaces between solidsand liquids [1]–[13]. Traditionally, meshfree methods employdiscrete nodes to replace meshes in the geometric descriptionof objects, and this can lead to a great reduction of costs ingeometric discretization because generating a set of nodeswithout connection is usually much easier than generatingmeshes. Meshfree methods are also desirable for solving elec-tromagnetic (EM) problems, though they have not been paidsufficient attention. For example, the widely used method ofmoments (MoM) with the Rao–Wilton–Glisson (RWG) basisfunction [14] for solving EM surface integral equations (SIEs)requires high-quality meshes, and remeshing could be fre-quently encountered for complex structures because of theexistence of unqualified (nonconformal) meshes. This is be-cause the RWG basis function is defined over a pair of triangleswith a common edge, and many triangles near a junction butbelonging to two different components in multiscale structuresmight not form a qualified (conformal) triangle pair [15]. Inaddition, the solution of inverse scattering problems for recon-structing unknown objects also requires remeshing involvedgeometries because their profiles are refined or changed iter-atively [16]. There have been some publications addressingthe meshfree methods for EM applications in recent years,but they mainly deal with differential equations for static orquasi-static problems and also only deal with the SIEs in theintegral equation approach [17]–[27].The integral equation method in EM includes the numerical

solutions for SIEs or volume integral equations (VIEs). ThoughSIEs are preferred whenever available, the VIEs are indispens-able for inhomogeneous/anisotropic structures or inverse scat-tering problems [15]. Solving VIEs requires the volumetric el-ements in discretization, and tetrahedral elements are usuallypreferred to match arbitrary geometric shapes [28]. However,volumetric discretization is much more inconvenient than sur-face discretization in general, and special commercial softwarefor mesh generation is definitely needed even for very simplegeometries [29]. Also, the MoM for solving the VIEs tradi-tionally uses the Schaubert–Wilton–Glisson (SWG) basis func-tion to represent unknown currents [30]. The SWG basis func-tion is defined over a pair of tetrahedral elements, and the de-fective or nonconformal meshes could be easily produced inthe discretization for complex structures [31]. Moreover, the

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4250 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 9, SEPTEMBER 2012

Fig. 1. Scattering by a 3-D inhomogeneous anisotropic object embedded in ahomogeneous background with a constant permittivity and constant perme-ability . A volumetric electric current and magnetic currentare induced inside the object.

SWG-based MoM cannot be implemented for solving inversescattering problems because the object profile or material inter-face is unknown. Therefore, it would be great if one can avoidthe use of volumetric elements in the numerical solutions.In this paper, we develop a novel meshfreemethod for solving

the VIEs in which there is no constraint of mesh conformity inthe geometric discretization. The method transforms the volumeintegral over a volume domain into a boundary (surface) inte-gral plus a one-dimensional (1-D) line integral so that the di-rect volume integration over the volume domain can be avoidedand the volumetric discretization is not necessary. To facili-tate the evaluation of boundary integrals, the original domainrepresented by the object is expanded to a cylinder circum-scribing the object, and the involved integrands are redefined.The above transformation is based on the Green–Gauss theoremand is only valid for nonsingular integrals [32]. Since the inte-gral kernels with the double gradient of the scalar Green’s func-tion in the VIEs are hypersingular, we need to exclude a smallcylinder including the observation node, and the transformationis only performed for the remaining part. For the singular in-tegrals over the small cylinder, we specially handle them byusing a singularity subtraction technique and deriving analyticsolutions for subtracted singular integrals under the Cauchy-principal-value (CPV) sense [33]. In addition, the moving leastsquare (MLS) approximation is used to represent the unknowncurrents [18], and a point-matching procedure is implementedto transform the VIEs into matrix equations. Numerical exam-ples for EM scattering by inhomogeneous or anisotropic objectsare presented to illustrate the scheme and good solutions areobtained.

II. VIES

VIEs are used to describe inhomogeneous or anisotropicproblems and can be derived from a vector wave equationby introducing the dyadic Green’s function. Consider theEM scattering by a three-dimensional (3-D) inhomogeneousanisotropic object embedded in a homogeneous backgroundwith a constant permittivity and constant permeability(see Fig. 1); the VIEs can be written as

(1)

(2)

where and are the incident electric and mag-netic field, respectively, while and are the total elec-tric and magnetic field inside the object, respectively. Also, theintegral kernel is the 3-D dyadic Green’s functionfor the homogeneous background medium with a wavenumber

and is defined as

(3)

where is the 3-D scalar Green’sfunction in which is the distance between an obser-vation point and a source point , and is the identity dyad.In addition

(4)

where is the relative permeability tensor,and is the relative permittivity tensor. Notethat the above equations are not computationally friendly andwe can recast them into the following form [15]:

(5)

(6)

where

(7)

(8)

are the two matrices related to the object’s permeability tensorand permittivity tensor, respectively, and

(9)

(10)

are the induced volumetric electric current and magnetic currentinside the object, respectively.If the object is inhomogeneous but isotropic with a permit-

tivity and a permeability , the corresponding VIEscan be written as [15]

(11)

TONG: MESHFREE SOLUTIONS OF VIEs FOR EM SCATTERING BY ANISOTROPIC OBJECTS 4251

Fig. 2. Descriptive geometry for describing the meshfree scheme. An objectis circumscribed by a cylinder , and an observation node is enclosed witha small cylinder . The cylinder has a bottom , a top , and a sidewall , while has a bottom , a top , and a side wall . There are somediscrete nodes inside the object, and they are chosen as observation nodes in thepoint-matching procedure. The volume integrals over are transformedinto the boundary integrals based on the Green–Gauss theorem.

(12)

where

(13)

are the induced volumetric electric and magnetic currents in-side the object, respectively. Furthermore, if we assume that theobject has the same permeability as the background, which isusually true, then and the above two equationscan be reduced to

(14)

(15)

and we only need to solve one of the two equations to obtainthe unknown current . We can treat either , thetotal electric field or electric flux density inside theobject as an unknown function to be solved in the above VIEs,but is chosen in the MoM with the SWG basis functionsince the flux density is normally continuous across materialinterfaces.

III. MESHFREE SOLUTIONS OF VIES

To solve the above VIEs with an efficient meshfree scheme,we first enclose the object using a cylinder as shown in Fig. 2and enforce the wall of the cylinder to shrink until touching theprofile of the object. We then choose some discrete nodes insidethe object without any connection each other. Although it is not

necessary to require that the nodes distribute equally, it is betterthat they distribute as uniformly as possible because the uni-form distribution can better express the unknown function withan interpolation function. Since no available commercial soft-ware can be used to automatically choose the discrete nodes fora given 3-D object currently, we select those nodes by taking thecentroids of tetrahedral elements after tetrahedrally discretizingthe object with IDEAS software. We believe that this is just atemporary measure and expect that mesh-generator companieswill develop related software for meshfree discretization of ge-ometries in the future. The VIEs are transformed into a matrixequation by performing a point-matching procedure based onthose chosen nodes. We choose each node as an observationpoint subsequently and evaluate the resultant volume integralsin matrix elements, which represent the field contribution fromthe volume current inside the object. The value of unknown cur-rent at an arbitrary point is represented with an interpolationfunction based on the values of the current at some neighboringnodes within its compact support, and this is known as movingleast square (MLS) approximation for the unknown function [9].The MLS approximation has never been applied to the VIEs,and it can be briefly described as follows. If we want to findthe value of an unknown function , such as a component of3-D unknown volumetric current vector, at an arbitrary point(for example, a quadrature point), we can approximate it with afirst-order polynomial function (linear approximation) in a localcoordinate system , i.e.,

(16)

where , and are the unknown coefficients to be de-termined. We can then obtain a set of equations by matching theunknown function on some neighboring nodes within a compactsupport of that point, i.e.,

(17)

where are the unknown function’s valueson those neighboring nodes and is the total number of neigh-boring nodes for that point. The neighboring nodes are the nodesenclosed in a sphere centered at that point, and the radius of thesphere is called the compact support. The compact support ischosen such that the number of enclosed neighboring nodes isclose to the number of unknown coefficients. Since the numberof the unknown coefficients is not equal to the number of theneighboring nodes in general, the set of equations can only besolved with the least square method (LSM) [12] by defining afunctional

(18)

which is the sum of weighted residue errors for the equations( is the th weight). Minimizing the functional or enforcingthe derivative of the functional with respect to the coefficientsto vanish, i.e.,

(19)


we can determine the unknown coefficients, i.e.,

(20)

where

(21)

are the coefficient vector and the vector of function values atthe neighboring nodes, respectively (the superscript denotesa transpose). The matrices and are related to the locationsof those neighboring nodes and the chosen weights and can begiven by

(22)

(23)

where

......

...... (24)

is a matrix formed by evaluating the monomial basis functionat those neighboring nodes, and

(25)

is a diagonal matrix whose diagonal elements are the weightsput on the residue errors of the equations. The weight for eachneighboring node is chosen in terms of the distance between thenode and the quadrature point, and one choice is

ifif

(26)

where is the distance between the th node and the quadraturepoint and is the compact support. With the solved unknowncoefficients, the unknown function within the compact supportcan be expressed as

(27)

where is called shape function. The aboveprocess is theMLS approximation for an unknown function, andwe can use it to represent the unknown current (or total electricfield) function within a compact support for EM equations, i.e.,

(28)

where is the value of unknown cur-rent function at the th neighboring node. The unknown currentvalues at all nodes are the unknowns to be solved in the matrixequation.

The matrix elements are obtained by evaluating the corre-sponding volume integrals over the object domain in the tra-ditional meshfree methods. The traditional meshfree methodsrequire background meshes in the evaluation, so they are notthought of as truly meshfree methods sometimes [5]. In thispaper, we remove the need of background meshes (volumetricelements) by transforming the volume integrals into boundaryintegrals, leading to a truly meshfree scheme. To evaluate thevolume integrals without using volumetric elements, we con-sider the following generalized case:

(29)

where represents an integral kernel in the VIEs. Sincethe integral kernel in the VIEs is singular, we choose a smallcylinder enclosing the observation node and exclude thissmall part in the volume integral so that the integrand within

is regular. For the integral over the small cylinder, wewill specially treat it by using the singularity subtraction tech-nique [33]. To evaluate the above integral more conveniently,we choose a circumscribing cylinder for the object as shown inFig. 2 and redefine the integrand as follows:

(30)

Obviously

(31)

where is the volume of the circumscribing cylinder. Theabove volume integral can be changed to a boundary integralby applying the Green–Gauss theorem [32]

(32)

where is the boundary or surface of a volume domain andis the th component of the unit normal vector

on the boundary. Also, is a position vector in the smoothfunction , and is its th coordinate. If we choose

(33)

where is an arbitrary constant, then we have

(34)

because

(35)

In the above, is the boundary or surface of the circumscribingcylinder , and is the boundary or surface of the smallcylinder . Also, in the system corresponds to


in the system. The boundary integral in (34) can bereduced to

(36)

because on the surface of the walls of the two cylinders.Furthermore, if we choose , then

(37)

because on . Therefore

(38)

The evaluation of the above three integrals is convenient. Wehave on and , and on . Also,is a constant on those surfaces, so we can conveniently findby discretizing those surfaces and evaluating the corresponding

with numerical integration. We take (14) as an ex-ample to illustrate the implementation of the above scheme forsolving EM integral equations. Equation (14) can be rewritteninto the following scalar equations after using the relationshipbetween and :

(39)

where is the location of th observation node and is thetotal number of nodes. Also

(40)

where andare the components of the dyadic Green’s

function at the observation node. According to (38), the volumeintegral in the above can be written as

(41)

where represents the heights of the three surfaces ,and , respectively, and the integration over will be han-dled individually through the singularity treatment technique inthe next section. Note that outside the

Fig. 3. Algorithmic outline for the meshfree method.

object , so the actual integral interval for variable can be re-duced. The integrand includes the three com-ponents of the unknown total electric field, and their values ata quadrature point are represented with their values at someneighboring nodes within a compact support as shown in (28),namely

(42)

where is the value of the unknown total elec-tric field ( component) at the th neighboring node located at. Each neighboring node has a global index and could ap-

pear in the compact supports of other quadrature points, so wehave to search and then add all integrations for a node with thesame global index to form the matrix element for that node. Todescribe the above process more clearly, we provide an algo-rithmic outline in Fig. 3.

IV. EVALUATION OF SINGULAR INTEGRALS

The integration over the small cylinder needs to be han-dled specially. The primary integral kernel in the VIEs is thedyadic Green’s function, which incudes a double gradient onthe scaler Green’s function, yielding a hypersingularity whenthe observation node is inside the small cylinder. The hypersin-gularity cannot be reduced to aweaker singular form because wedo not use any basis and testing functions in the scheme, whichare used to weaken the singularity in the MoM. The dyadicGreen’s function has nine components if expanded, but onlysix of them are independent due to its symmetry. After usingthe MLS approximation or polynomial interpolation represen-tation for the unknown function in the VIEs, the strongest sin-gular or hypersingular integrals come from the combination ofthe dyadic Green’s function with the constant term in the poly-nomial. The combination of the kernel with higher-order terms


weakens the degree of singularity and can be treated with exis-tent techniques, so we do not address them here. For the hyper-singular integrals, the kernels are one of the following compo-nents in the dyadic Green’s function (we omit the constant termfrom the polynomial):

(43)

where

(44)

and we have used the local coordinate system to de-fine them. The evaluation of hypersingular integrals requires asingularity subtraction process that can be illustrated as followsif we take the first component as an example:

(45)

The singular terms of the kernels can be distinguished interms of the series expansion of the scalar Green’s functionand have been extracted in the above. The first integral aboveis regular or bounded now and can be evaluated using numer-ical quadrature rules. The second integral only includes ormore weakly singular term in the integrand and can be evaluatedusing the Duffy’smethod [34].We do not address these two inte-grals because they can be handled with existing techniques andtheir accuracy can be controlled by using appropriate quadraturerules [35], [36]. We only treat the third integral, which includesa hypersingular kernel. If expanding other components of thedyadic Green’s function as well, we can find all the hypersin-gular integrals from the above singularity subtraction process,i.e.,

(46)

where represents the source point anddenotes the observation node in the local coordinate system

as shown in Fig. 4. The local coordinate system is setup over the bottom of the small cylinder, and its origin is locatedat the center of the bottom. The projection of the observationnode on the bottom coincides with the origin. In such a local

Fig. 4. Derivation of singular integrals over a small cylinder . A local coor-dinate system is set up over the bottom of the cylinder, and its origin islocated at the center of the bottom. The observation node is located at ,and its projection on the bottom coincides with the center of the bottom.

coordinate system, the above hypersingular integrals over thesmall cylinder, for instance, the first integral in (46), can bewritten as

(47)

The outer integral with respect to in is regular and is evalu-ated by a numerical quadrature rule (Gaussian quadrature rule).The inner surface integral could be regular, near-singular, or sin-gular, depending on the spatial relation between the observationnode and the source area , which is moving along the

axis. However, nomatter how close the observation nodeis to the source area , the surface integral is always inte-grable under the CPV sense, and we can find its closed-formsolution, which will be needed in the numerical integration inthe outer integral.From those volume integrals in (46), we can find the fol-

lowing surface integrals for which we need to derive the closed-form solutions:

(48)

The first three integrals in (46) can be obtained by appropri-ately combining the first three integrals with the last integral in(48), respectively. With the help of integral tables [37], we canfind that


(49)

It can be seen that some of integrals above vanish, and thisis due to the symmetry of the small cylinder in the azimuthaldirection. The nonvanishing integrals , and are in-dividually divergent when , i.e., the observation nodeapproaches the source area. However, when they are combinedappropriately to form the surface integrals with the same inte-grands as in the first three integrals of (46), the resulting expres-sions are convergent because the divergent terms cancel eachother. Thus, the solutions of the surface integrals correspondingto the volume integrals in (46) can be written as

(50)

where the superscript represents the surface integral. Theabove formulas are effective no matter how close the obser-vation point is to the source area, and they can be used toaccurately evaluate the near-interaction matrix elements. Whenthe observation point is inside the source area, we only need totake a limit of those formulas by letting and obtain thefollowing solutions for the first three integrals:

(51)

V. NUMERICAL EXAMPLES

To demonstrate the developed novel meshfree scheme forsolving VIEs, we present several numerical examples for EM

Fig. 5. Geometries of scatterers. (a) Dielectric sphere with two-layer dielec-tric coatings. (b) Homogeneous dielectric cube. (c) Plasma anisotropic sphericalshell.

scattering by different penetrable objects as shown in Fig. 5. Itis assumed that the incident wave is a plane wave with a fre-quency MHz and is propagating along direction infree space ( , and ) in all examples. Wecalculate the scattered near electric field or bistatic radar crosssection (RCS) observed along the principal cut ( and

to 180 ) for the scatterers with both vertical polariza-tion (VV) and horizontal polarization (HH). The solutions arecompared to the available analytical solutions in Mie series, ortraditional MoM solutions.In the first example, we consider the scattering by an inho-

mogeneous dielectric object, namely, a dielectric sphere withtwo-layer concentric dielectric coatings; see Fig. 5(a). The radiiof three interfaces are , and ,respectively, and the relative permittivity of each layer is

, respectively (the relative perme-ability is assumed except stated otherwise). We select5846 discrete nodes in total inside the object for describing thegeometry, and the node densities are different in different mate-rials. Specifically, there are 1542 nodes in the core, 1026 nodesin the inner coating, and 3278 nodes in the outer coating. Thesenodes are chosen by discretizing each material body indepen-dently without considering their emergence along the materialinterface, so they are nonconformal in the sense of SWG basisfunction. We calculate the scattered near electric field along theprincipal cut of observation surface, as plotted inFig. 6, and they are close to the corresponding analytical solu-tions. To see the convergence property of the scheme, we usedifferent numbers of nodes to represent the object, and thencompare the scattered near electric field with the correspondingexact solutions by calculating the root mean square (RMS) er-rors. Fig. 7 shows the RMS error versus the node density, andit is clear that the numerical error can be exponentially reducedwith the increase of node density.The second example illustrates the scattering by a homoge-

neous dielectric cube with a side length and a relativepermittivity ; see Fig. 5(b). We choose 2542 discretenodes inside the cube to represent the geometry. The bistaticRCS solutions are depicted in Fig. 8, and they are comparedto the corresponding MoM solutions (using SIEs) because of


Fig. 6. Solutions of scattered near electric field at for a dielectricsphere with two-layer dielectric coatings. The radii of three interfaces are

, and , respectively, and the relative permittivityof each layer is , and , respectively.

Fig. 7. Convergence property of the scheme. The RMS error of the scatterednear electric field for a dielectric sphere with two-layer dielectriccoatings can be exponentially reduced with the increase of node density.

lacking the analytic solutions. It can be seen that the solutionsfrom the two different approaches are in good agreement.The above examples include homogeneous or piecewise ho-

mogeneous objects that can be solved with SIEs more conve-niently, and the illustrations are mainly for verifying the accu-racy of solutions. In the third example, we consider the scat-tering by an inhomogeneous and anisotropic object that is ofinterest in many applications [38], and its solutions can only beobtained by solving the VIEs. The object is a plasma anisotropicspherical shell as sketched in Fig. 5(c), and the radii of the innerand outer interfaces are and , respec-tively (inside the inner interface is an empty space). The plasmamaterial is characterized by a gyrotropic-tensor permittivity(the permeability is an identity tensor) or a gyrotropic-tensor

Fig. 8. Bistatic RCS solutions for a homogeneous dielectric cube with a sidelength and a relative permittivity .

Fig. 9. Bistatic RCS solutions for a plasma anisotropic spherical shell. Theradii of the inner and outer interfaces are and ,respectively. The plasma material is characterized by a gyrotropic-tensor (G-T)permittivity or a gyrotropic-tensor (G-T) permeability .

permeability (the permittivity is an identity tensor). The gy-rotropic tensors are defined by

(52)

We choose 3462 discrete nodes in the geometric description, andFig. 9 shows the corresponding bistatic RCS solutions when thepermittivity is the gyrotropic tensor , or the permeabilityis the gyrotropic tensor . Although we do not have accu-rate results with which to compare, the solutions are quite closeto those provided in [39] or [40], which are obtained with verydifferent approaches.Note that the developed meshfree scheme does not worsen

the conditioning of system matrix in general because it is just


TABLE ICOMPARISON ON CONDITION NUMBERS (CNS) OF SYSTEM MATRICES FOR

THREE NUMERICAL EXAMPLES

a new way to evaluate the volume integrals in the matrix el-ements. The property of system matrix is mainly determinedby the method of transforming the integral equations into ma-trix equations, and the scheme uses the point-matching proce-dure that is widely used in mechanical engineering. Since theVIEs are the second kind of integral equations, the resultantsystem matrices are usually well conditioned due to the dom-inance of diagonal elements. Table I shows the condition num-bers of system matrices for the three numerical examples with acomparison to those of traditional SWG-based MoM, and theycan verify our estimation.

VI. CONCLUSION

Solving the VIEs is indispensable in the integral equation ap-proach if the problems involve inhomogeneous or anisotropicmedia. The numerical solutions of VIEs strongly rely on goodvolumetric discretization for the involved structures, and thisis usually tedious and expensive. In this paper, we propose anovel meshfree scheme for solving the VIEs, and the constraintof costly mesh conformity as required in the traditional MoMcan be removed. The scheme employs a point-matching pro-cedure to transform the VIEs into matrix equations, and theunknown current functions are represented with the MLS ap-proximation. The resultant volume integrals in matrix elementsare then changed into boundary (surface) integrals based on theGreen–Gauss theorem after the integral kernels are regularizedby excluding a small cylinder enclosing the observation node.To evaluate the produced boundary integrals more efficiently,we expand the original integral domain represented by the ob-ject into a cylindrical domain circumscribing the object by re-defining the integrands, so that the integrals over the wall ofthe cylinder vanish. The remaining integrals over the top andbottom of the circumscribing cylinder and small cylinder can beeasily evaluated. The singular integrals over the small cylinderare handled with a singularity subtraction technique in which theintegrals with subtracted hypersingular cores are derived analyt-ically under the CPV sense. Numerical examples for EM scat-tering by inhomogeneous or anisotropic objects are presented todemonstrate the scheme, and good performance can be observedby comparing the solutions to the available analytic solutions orMoM counterparts.

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Mei Song Tong received the Ph.D. degree in elec-trical engineering from Arizona State University,Tempe, in 2004.He is currently a Distinguished Professor and the

Head of the Department of Electronic Science andTechnology, School of Electronics and InformationEngineering, Tongji University, Shanghai, China.Before he joined Tongji University, he was a Re-search Scientist with the Center for ComputationalElectromagnetics and Electromagnetics Laboratory,Department of Electrical and Computer Engineering,

University of Illinois at Urbana–Champaign. He has published more than 90papers in refereed journals and conference proceedings, and coauthored a book.His research interests include computational electromagnetics, antenna theoryand design, simulation and design of RF/microwave circuits and devices,interconnect and packaging analysis, and inverse electromagnetic scatteringfor imaging.Prof. Tong is a Fellow of the Electromagnetics Academy, a Full Member

(Commission B) of the U.S. National Committee for the International Union ofRadio Science, and a member of the Applied Computational ElectromagneticsSociety and the Sigma Xi Honor Society. He is serving as a technical reviewerfor many international journals and conferences, and serving as an Associate Ed-itor or Guest Editor for several well-known international journals, including theIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE TRANSACTIONSON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, the In-ternational Journal of Numerical Modeling: Electronic Networks, Devices andFields, Progress in Electromagnetics Research, and the Journal of Electromag-netic Waves and Applications. He also served as an Editorial Board memberof Applications and Applied Mathematics, and as an Associate Editor or GuestEditor for Waves in Random and Complex Media. In addition, he has servedas a session organizer, session chair, and technical program committee memberfor many prestigious international conferences, including the annual IEEE In-ternational Symposium on Antennas and Propagation and USNC/URSI Na-tional Radio Science Meeting and the Progress in Electromagnetics ResearchSymposium.

meshfree solutions of volume integral equations for electromagnetic scattering by anisotropic...

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