Nonconformal Discretization of Electric Current Volume ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 8, AUGUST 2017

4155

Nonconformal Discretization of Electric Current Volume Integral Equation With Higher Order Hierarchical Vector Basis Functions Qiang-Ming Cai, Zhi-Peng Zhang, Yan-Wen Zhao, Wei-Feng Huang, Yu-Teng Zheng, Zai-Ping Nie, Fellow, IEEE, and Qing Huo Liu, Fellow, IEEE

Abstract— A nonconformal discretization of the electric current volume integral equation (JVIE) is presented for the electromagnetic scattering analysis of inhomogeneous dielectric objects. This JVIE is based on higher order geometrical and current modeling, where curved tetrahedral elements and higher order hierarchical vector (HOHV) basis functions are adopted. In order to implement the nonconformal discretization, the half face-based HOHV basis function is introduced to address the nonconformal volumetric elements. Compared to the conventional low-order conformal/nonconformal VIE, our nonconformal HOHV-based JVIE has the advantage of improving the computational efficiency and accuracy. Compared to the previous VIE discretized by other higher order functions, our nonconformal JVIE is based on the higher order geometric modeling and the HOHV bases. In order to improve the efficiency, the basis expansion and recombination technique is introduced to significantly accelerate the matrix filling. Moreover, the flexibility of basis order selection is further enhanced by the mixed order schemes. Numerical results are given to demonstrate its accuracy, efficiency, and flexibility. Index Terms— Electric current volume integral equation (JVIE), higher order hierarchical vector (HOHV) basis functions, nonconformal discretization.

I. I NTRODUCTION OLUME integral equation (VIE) has been widely employed for the numerical modeling of electromagnetic scattering from inhomogeneous dielectric objects, which is of great significance in many engineering applications [1], [2], e.g., wireless communication and microwave imaging. In these applications, the modeled dielectric targets are commonly of large scale, multiscale, or high contrast [3], [4]. For the traditional low-order VIE [5], the resulting computational cost may be huge [6]. In order to improve its efficiency, many fast

V

Manuscript received May 21, 2016; revised April 10, 2017; accepted May 20, 2017. Date of publication May 31, 2017; date of current version August 2, 2017. This work was supported in part by the Natural Science Foundation of China under Grant 61371050 and Grant 61231001 and in part by the National Ministry Foundation of China under Grant 9140A03010613DZ02030. (Corresponding author: Yan-Wen Zhao.) Q.-M. Cai, Z.-P. Zhang, Y.-W. Zhao, Y.-T. Zheng, and Z.-P. Nie are with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). W.-F. Huang and Q. H. Liu are with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2017.2710211

algorithms or techniques have been developed over the past decades [7]–[13]. One effective technique is the higher order approach, which involves the higher order basis function and the higher order geometric modeling [14]–[16]. It has been widely recognized that the higher order approach can enhance the efficiency and accuracy of VIE substantially [17]–[24]. However, for targets of multiscale or high contrast, the merits of the higher order VIE may be considerably damaged by using the traditional conformal discretization. This is mainly because, with the existence of small geometric structures or high-contrast subdomains, conformal discretization usually produces many extra mesh elements, thus yielding many unnecessary unknowns. In addition, for the extremely multiscaled targets, conformal discretization will make the meshing process very tedious and time consuming. Based on the above, it is valuable to integrate nonconformal discretization into the higher order VIE, and researchers have paid much attention to this topic [25], [26]. In [25], a higher order VIE based on the Nyström method is implemented over flat tetrahedrons, rather than the curved ones. In [26], a higher order VIE based on the curvilinear tetrahedrons is proposed with the Lagrange interpolation polynomials and the point matching. Given the merits of the higher order hierarchical vector (HOHV) basis functions [15]–[24], [27]–[29], this paper aims to extend nonconformal discretization into VIE and improve its flexibility and efficiency to a higher level. To the best of our knowledge, this HOHV-based VIE with nonconformal discretization has never been reported. In this paper, the adopted VIE formulation is the so-called electric current VIE, i.e., J-formulation VIE (JVIE) in terms of equivalent polarization currents J. The main reason for choosing JVIE is that for the simulation of inhomogeneous objects, JVIE is more robust than the conventional D-formulation VIE (DVIE) and E-formulation VIE (EVIE) [10], [30]–[34], where detailed investigations can be found. M. C. van Beurden and S. J. L. van Eijndhoven [31] and [32], based on their theoretical analysis, suggested the use of Galerkin JVIE, whose convergence in norm can be guaranteed if basis and testing functions space are the space of square integrable functions L 2 . This is because when the JVIE maps L 2 to L 2 , it is well posed from L 2 to L 2 , and can be discretized with the basis

0018-926X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

4156


and testing functions belonging to L 2 , and with the L 2 inner product [31], [32]. The convergence properties of JVIE have been further demonstrated by numerical cases of extremely anisotropic materials [34], bianisotropic materials [30], multilayered materials [35], and highly inhomogeneous materials [10]. For JVIE, the equivalent volume current is physically noncontinuous across the material interfaces, and hence “it is essential that basis functions do not enforce any continuity across material interfaces” [34]. This was further demonstrated by the utilization of the square integrable functions, e.g., the piecewise constant functions [34], the piecewise linear functions [36], the half rooftop basis function [35], and the half Schaubert–Wilton–Glisson (SWG) basis functions [37]. In [38]–[41], the half SWG and the half rooftop basis functions were extended into the nonconformal JVIE. Based on these studies, half HOHV functions are defined to discretize JVIE over the nonconformal mesh. It should be pointed out that: (1) in comparison to the loworder nonconformal VIE schemes [38]–[41], our HOHV-based nonconformal JVIE has much better accuracy and efficiency, and much less computational cost and (2) in contrast to the traditional higher order VIE, its flexibility can be further enhanced by adjusting both the order of HOHV bases and the size of the nonconformal mesh elements. In addition, a novel technique, which we refer to as the basis expansion and recombination (BER) technique, is proposed to reduce the matrix-filling time. The rest of this paper is organized as follows. Section II elaborates the implementation of the HOHV-based JVIE and its nonconformal discretization. Section III presents two techniques i.e., BER and the mixed order schemes for HOHV-based JVIE. Section IV gives numerical results to demonstrate the accuracy, efficiency, and flexibility of the nonconformal higher order JVIE. Section V is the conclusion. II. N ONCONFORMAL JVIE W ITH HOHV F UNCTIONS There are three parts in this section. The first part gives the basic formulation of JVIE, where the equivalent electric current J is chosen for discretization. The second part explains the definition of HOHV basis functions over higher order curvilinear tetrahedrons, especially the definition of half HOHV functions. The third part expounds the implementation of the nonconformal JVIE based on these HOHV bases. A. Electric Current Volume Integral Equation

ε(r) − εb . (2) ε(r) With this definition, the relation between J(r) and the total field E(r) can be expressed as κ(r) =

J(r) = j ωκ(r)ε(r)E(r).

(1)

where Einc (r) is the incident electric field with angular frequency ω, J(r) is the equivalent volume current, and κ(r) is

(3)

The vector potential A(r) and scalar potential (r) are A(r) = μb J(r )G(r, r ) dv (4) V ηb ∇ · J(r )G(r, r ) dv (r) = − j kb V + nc (r ) · (J1 (r ) − J2 (r ))G(r, r ) ds Sdc

(5) where kb and ηb are the wavenumber and the intrinsic impedance of the background medium, respectively. G(r, r ) = e− j kb R /4π R is Green’s function in the homogeneous medium, where R = |r − r | is the distance between the field point r and the source point r . Sdc is the interface where the discontinuity of ε(r ) lies. As a unit vector normal to Sdc , nc is directed from medium 2 [with current J2 (r)] to medium 1 [with current J1 (r)]. B. HOHV Basis Functions for JVIE Since the J(r) do not necessarily satisfy any continuity conditions, it is essential that basis functions do not enforce any continuity across the element interfaces. Therefore, some appropriate basis functions in L 2 can be used to discretize the JVIE [31], [32]. In this paper, in order to completely expand J(r), three types of HOHV basis functions, i.e., the full face-based, the volumebased, and the half face-based functions, will be utilized. They are defined over second-order curved tetrahedrons. For each tetrahedral element, a unique mapping between the local curvilinear coordinate system (ξ1 , ξ2 , ξ3 , ξ4 ) and the physical space coordinate (x, y, z) can be established by the secondorder Lagrange interpolation polynomials [14], [16], [23]. In the following, we first discuss the full face-based basis f function Fi,k1 k2 and the volume-based basis function Fvi,l , whose definitions are the same as the traditional case. Assof ciated with the local face i (ξi = 0), Fi,k1 k2 and Fvi,l are expressed by [18] and [23] f

Consider an arbitrarily shaped 3-D object with the complex permittivity ε(r) and permeability μ(r) = μb , which is embedded in the homogeneous isotropic background of εb and μb . According to the volume equivalence theorem [5], JVIE can be formulated as J(r) + j ωA(r) + ∇(r) = Einc (r) j ωε(r)κ(r)

the contrast ratio of the permittivity, that is

f

Fi,k1 k2 = Pi,k1 k2 (ξ1 , ξ2 , ξ3 , ξ4 )i (r) v (ξ1 , ξ2 , ξ3 , ξ4 )i (r) Fvi,l = Pi,l

(6) (7)

where i (r) is the commonly used zeroth-order curved f v are the SWG functions, i.e., CSWG [18]. Pi,k1 k2 and Pi,l scalar hierarchical face-based and volume-based polynomials, respectively. Their fourth-order (k1 + k2 = l = 4) expressions are listed in Table I. Their expressions of other three lower orders can be found in [18] and [23]. Based on the definition of Fvi,l , it is evident that Fvi,l vanishes on the tetrahedral faces. In JVIE, Fvi,l is employed to

CAI et al.: JVIE WITH HOHV BASIS FUNCTIONS

4157

TABLE I FACE - AND V OLUME -BASED H IERARCHICAL S CALAR P OLYNOMIALS OF THE F OURTH -O RDER A SSOCIATED W ITH FACE 1 (ξ1 = 0) FOR T ETRAHEDRON [18]

enhance the current approximation throughout the dielectric region V. Since its definition is restricted to the inside of a tetrahedron, Fvi,l can be utilized to represent the volume current inside any tetrahedron no matter whether the mesh is conformal or not. f As for Fi,k1 k2 , it is defined on the common face shared by a pair of conformal tetrahedrons of the same material parameters, i.e., similar to the definition of SWG functions [5]. f This suggests that Fi,k1 k2 can only be used to represent the equivalent current associated with the shared face of two conformal tetrahedrons in the same materials domain. This is because only under such situation, the continuous f normal component of J(r) can be well represented by Fi,k1 k2 . Otherwise, the half HOHV basis function Fhi,k1 k2 should be defined and employed. Before the definition of Fhi,k1 k2 , we first consider two scenarios: (1) two neighboring tetrahedrons are nonconformal, or their common face are not completely overlapping and (2) two neighboring tetrahedrons have different material f parameters. For Scenario 1, it is evident that Fi,k1 k2 cannot be defined on the common face of the corresponding tetrahedron pair. For Scenario 2, the normal component of J(r) across the common face is physically not continuous, which contradicts f with the normal continuity of Fi,k1 k2 . Thus, it can be conf cluded that for either scenario, Fi,k1 k2 should be replaced by more reasonable basis functions, e.g., the half HOHV basis function Fhi,k1 k2 . f Different with Fi,k1 k2 , the definition domain of Fhi,k1 k2 is merely a single tetrahedral element, rather than a tetrahedron pair. In its supporting tetrahedron, Fhi,k1 k2 has the same f expression as Fi,k1 k2 . By regarding Fhi,k1 k2 as one “half” of f Fi,k1 k2 [18], [23], its expression directly follows from (6), that is: f

Fhi,k1 k2 = Pi,k1 k2 (ξ1 , ξ2 , ξ3 , ξ4 )i (r).

(8)

Based on the above, it is evident that Fhi,k1 k2 is readily applicable to both scenarios. Since Fhi,k1 k2 is defined over the volume, rather than on the surface, their application for

both scenarios will only generate surface charge densities, instead of the line charge densities [37], [38]. This suggests that no singular field will be produced by applying Fhi,k1 k2 to Scenarios 1 and 2. f Here, we give some remarks on Fvi,l , Fi,k1 k2 , and Fhi,k1 k2 . For f Fvi,l and Fi,k1 k2 , their definition and application are the same as the case of conformal discretization. When the nonconformal discretization is addressed, the definition of Fhi,k1 k2 can solve the difficulties yielded by nonconformal meshes, and Fhi,k1 k2 plays a key role in the implementation of nonconformal JVIE. f Specially, Fvi,l /Fi,k1 k2 belongs to the space L 2 but does not f span L 2 , because Fvi,l /Fi,k1 k2 spans the space H (div)(H (div) ∈ {f| f ∈ L 2 ∇ · f ∈ L 2 }) which is the subspace of L 2 [18], [32]. The proper basis functions for completely expanding J(r) are often demanded to span L 2 . Thus, in the implementation of Galerkin JVIE, Fhi,k1 k2 is needed to extend the space H (div) to the proper space L 2 . Since Fhi,k1 k2 is defined only over a f single element and is obtained by splitting Fi,k1 k2 , Fhi,k1 k2 also belongs to the space L 2 . In order to clearer illustrate three types of HOHV basis functions defined over tetrahedral elements, i.e., the full face-based f basis function Fi,k1 k2 , the volume-based basis function Fvi,l , and the half face-based basis function Fhi,k1 k2 , a dielectric object with two homogeneous subdomains meshed with tetrahedrons in free space is shown in Fig. 1. C. Nonconformal Discretization of HOHV-Based JVIE In previous studies, the nonconformal discretization of the low-order JVIE has been implemented [38]–[41], but it requires much more computational cost than the HOHV-based nonconformal JVIE. In this section, the nonconformal JVIE is numerically solved by the Galerkin method to guarantee the convergence in the norm of the solution. In this JVIE, first we consider the mapping property, namely mapping from the space L 2 to the space H (curl) (H (curl) ∈ {f | f ∈ L 2 ∇ × f ∈ L 2 }). In [31] and [32], it is suggested that testing functions should span the dual space of the range of the integral

4158


the element of [V] is Vm =

1 j kb ηb

Vm

fm (r) · Einc (r)dv

(11)

and the element of [Z] is fm (r) · fn (r) 1 dv + Zmn = − 2 fm (r) · An (r)dv kb Vm κ(r)εrn (r) Vm 1 + 2 ∇ · fm (r)n (r)dv kb Vm 1 f (r) · nm (r)n (r)ds (12) − 2 k b Sm m Fig. 1. Dielectric object with two homogeneous subdomains, and the illustration of three types of HOHV basis functions defined over tetrahedral f elements, i.e., the full face-based basis function Fi,k k , the volume-based 1 2

basis function Fvi,l , and the half face-based basis function Fhi,k k . 1 2

operator to guarantee the convergence in the norm of the solution. Therefore, our higher order JVIE should be paired with the testing functions belonging to H (div) to form a proper reaction integral. However, in order to implement the nonconformal JVIE, the testing function space is enlarged into the proper space L 2 . The convergence in the norm of the solution often can be guaranteed by Galerkin’s testing. This has been demonstrated by the numerical examples reported in this paper or in [35] and [38]–[41]. The reason is that the dual space of the range of our nonconformal JVIE is a subspace of L 2 . Moreover, as pointed out in [31] and [32], the discretization schemes can be improved by using less smooth testing functions, e.g., the square integral functions L 2 . Therefore, it is feasible to extend the testing functions space to the proper space L 2 for this nonconformal JVIE. First, we expound the nonconformal discretization of JVIE based on the HOHV basis functions defined above. With f Fi,k1 k2 , Fhi,k1 k2 , and Fvi,l , J(r) can be expanded as f

J(r) =

f

K N n=1 k1 ,k2 =0

h

f αn,ik1 k2 Fi,k1 k2 +

n=1 k1 ,k2 =0 v

+

h

K N

βn,ik1 k2 Fhi,k1 k2

v

3 N K

γn,il Fvi,l

(9)

n=1 i=1 l=1

where N f and N h are the total number of tetrahedral faces f associated with Fi,k1 k2 and Fhi,k1 k2 , respectively, and N v is the total number of tetrahedrons. The subscript i denotes the local number for a tetrahedral face. K f , K h , and K v , respectively, f represent the global order number of Fi,k1 k2 , Fhi,k1 k2 , and Fvi,l corresponding to the i th face. αn,ik1 k2 , βn,ik1 k2 , and γn,il are the unknown coefficients. Next, substituting (9) into (1) and applying the Galerkin testing, we obtain the matrix equation system [Z][I] = [V]

(10)

where I is the vector of unknown coefficients, Z is the impedance matrix, and V is the excitation vector. With the testing function fm (r) and the basis function fn (r), which are, respectively, defined on the mth and nth tetrahedrons,

where nm is the outward unit vector normal to Sm of the mth element. The divergence theorem fm ·∇n = ∇ ·(n fm )− n ∇ · fm has been applied in the derivation of Zmn . According to (4) and (5), An and n are given by fn (r )G(r, r ) dv (13) An (r) = Vn n (r) = − Vn

∇ · fn (r )G(r, r ) dv

+

Sdc

nc (r ) · fn (r )G(r, r ) ds

. (14)

In (13) and (14), only weakly singular integrals are involved, and can be easily addressed by existing singularity-handling methods, e.g., Duffy’s transform [22], [42]. In comparison with those works on nonconformal VIE with higher order or low-order basis functions, where the hypersingular integrals are contained [25], [26], [43], [44], the above integrals are much easier to calculate. III. T WO T ECHNIQUES FOR HOHV-BASED VIE In order to improve the efficiency and flexibility of VIE, two techniques will be recommended in this section. One is the novel BER technique for reducing the computational complexity of matrix filling. The other is the mixed order schemes for basis functions, which can reduce the degrees of freedom (DoFs) and enhance the flexibility of choosing basis orders. A. Basis Expansion and Recombination For the higher order method, it is well known that the memory requirement can be reduced significantly as the basis order increases. However, the computational efficiency may considerably deteriorate at the same time. For example, as reported in [23], the matrix-filling time still soars when the third-order bases are employed, even if the technique proposed in [45] has been adopted. In order to improve the efficiency of matrix filling, a BER technique for VIE is proposed here. This technique is an extension of our previous work on the HOHV-based surface integral equation (SIE) [46]. Without loss of generality, we explain BER by an integral based on (13). The adopted basis function is β γ

fn (r ) = ξiα ξ j ξk l (r )

(15)


4159

where the zeroth-order CSWG for local face l can be written as 2 l (r ) = r +ξ1 r˜l,1 +ξ2 r˜l,2 +ξ3 r˜l,3 +˜rl,4 (16) ξ1 , ξ2 , ξ3 where (ξ1 , ξ2 , ξ3 ) is the Jacobian factor of tetrahedrons, and the position vector r is determined by the coordinates (ξ1 , ξ2 , ξ3 , ξ4 ) in curved tetrahedrons [14], [16], [23]. r˜l, j ( j = 1, 2, 3, and 4) is the constant vectors associated with each tetrahedron. For example, when l = 1, r˜l, j are given by r˜1,1 = (4r8 − r4 − 3r1 )/2 r˜1,2 = (4r8 − r4 + r2 − 4r5 )/2 r˜1,3 = (4r8 − r4 + r3 − 4r7 )/2 r˜1,4 = (−4r8 + r1 + r4 )/2 (17) where ri (i = 1 ∼ 10) are the nodes of the curved tetrahedron. Thus, from (13) and (15), it suffices to consider the following integral: P= fn (r )G(r, r )dv =

Vn 1 1−ξ1

0

1−ξ1 −ξ2

0 0 1 1−ξ1 1−ξ1 −ξ2

β γ

ξiα ξ j ξk G(r, r )l (r )dξ3 dξ2 dξ1 β γ

=2 ξiα ξ j ξk G(r, r ) 0 0 0 · r + ξ1 r˜l,1 + ξ2 r˜l,2 + ξ3 r˜l,3 + r˜l,4 dξ3 dξ2 dξ1

(18)

which can be rewritten as P=

N

ak (˜rl,1 , r˜l,2 , r˜l,3 , r˜l,4 ) Pk

(19)

k=1

where ak is the constant vector only related to r˜l, j , and N is the number of the subterms Pk , e.g., N = 22 when fn (r ) = f Fi,01 (r ). The subterm Pk is given by 1 1−ξ 1−ξ −ξ 1 1 2 Ik ξ1 , ξ2 , ξ3 G(r, r )dξ3 dξ2 dξ1 Pk = 0

0

0

(20) where Ik are fundamental terms only related to (ξ1 , ξ2 , ξ3 , ξ4 ), e.g., ξ1 , ξ12 , ξ1 ξ2 , and ξ1 ξ22 ξ33 . For the fixed basis and testing tetrahedrons, Pk can be first precomputed and stored. When computing Zmn , its value can be obtained by the linear combination of Pk . It is obvious that the matrix-filling time will be remarkably reduced by employing BER, and this reduction will be much more prominent when the number of integral points or the order of HOHV bases increases. It should be noted that a similar idea of BER has been previously adopted in [21] and [47]. According to the scheme in [21] and [47], the integral (18) should be decomposed into subintegrals involving each nodal vector ri (i = 1 ∼ 10). In this paper, the integral (18) is only decomposed into subintegrals involving a vector and four scalars by using r˜l, j in (17), implying that the BER scheme adopted here makes the decomposition of original integral more concise, while comparable efficiency is also achieved. The concision of BER facilitates its numerical implementation substantially, especially for the cases of increasing basis orders. The proposed BER is also different from the scheme in [45] and has

Fig. 2. RMS error of the E-plane bistatic RCS with respect to the number of unknowns, which is obtained by the HOHV-based JVIE (with and without MOI) for a dielectric sphere having the relative permittivity 2.0 and radius 0.55λ0 . (a) RMS error in the RCS (bistatic) versus the number of unknowns obtained by the HOHV-based JVIE. (b) RMS error in the RCS (bistatic) versus the number of unknowns obtained by the first (or new) type of mixed order schemes, i.e., MOI.

higher efficiency for filling matrix. This is because the scheme utilized in [45] is only proposed to eliminate those repeated computations associated with ((r∂w × n) × r∂w ) · ∇G(r, r ) and (r∂w × r∂w ) [45], but the computation of ((r∂w × n) × r∂w ) · ∇G(r, r ) and (r∂w × r∂w ) can be further improved by the BER or the scheme in [21] and [47]. B. Mixed Order Schemes In [23], given the difference of Fi,k1 k2 and Fvi,l , a new mixed order scheme for choosing basis orders was proposed. For a f single tetrahedron, let K f denote the order of Fi,k1 k2 , and K v denote the order of Fvi,l . In traditional higher order VIE, K f = K v must be ensured for each tetrahedron. However, via theoretical analysis and numerical experiments, it is found that for many targets, K f can be unequal to K v [23], while the same accuracy is achieved. In [23], this new mixed order scheme is titled as MOI(K v = a, K f = b). One example is illustrated in Fig. 2(b). In order to further demonstrate the advantage of the first (or new) type of mixed order schemes, i.e., MOI and clarify relevant numerical comparisons, another two mixed order schemes, i.e., MOII and MOIII, will be defined. In this paper, the simulated target may be divided into f several subdomains. If the orders of Fi,k1 k2 and Fvi,l are always f

4160


the same in any subdomain, but they can be different among different subdomains, we refer to this kind of mixed order scheme as MOII. One example is illustrated in Fig. 8. As defined above, MOII requires K f = K v in each subdomain. If in at least one subdomain K f = K v , whereas K f = K v in the other subdomains, we refer to this kind of mixed order scheme as MOIII. One example is shown in Fig. 8. In previous works on the HOHV-based VIE, only MOII is available for choosing the basis orders. With the aid of MOI and MOIII, it is evident that the flexibility of basis order selection is further enhanced. This enhancement should be more important when nonconformal meshes are adopted for simulation. In addition to the flexibility enhancement, as demonstrated in [23] and the following numerical experiments, these three mixed order schemes can effectively remove those “wasted” DoFs. Note that in all three schemes, the orders f of Fi,k1 k2 and Fhi,k1 k2 are always the same. IV. N UMERICAL R ESULTS AND D ISCUSSION In this section, we validate the accuracy and efficiency of our conformal HOHV-based JVIE first by targets with conformal meshes in Section III-A, and then by targets with nonconformal meshes in Section III-B, whose capability is further demonstrated in Section III-C. All simulations are performed on a computer with 4 GHz Intel(R) Core(TM) I7-4790K CPU. Unless otherwise mentioned, the excitation for all examples is a plane wave Einc = xe− j kb z with the frequency of 300 MHz. The matrix equation (10) is solved by the conjugate gradient method with iterative tolerance 0.001, and the normalization basis function (NBF) technique reported in [21] and [23] is utilized to improve its convergence. A. Validation of Conformal Discretization 1) Homogeneous Sphere: The first example is a homogeneous dielectric sphere with a radius of 0.55λ0 and a relative permittivity of εr = 2.0. The accuracy is measured by the root-mean-square (rms) error of the bistatic radar cross section (RCS), which is defined as [48] NS 1

|σref,i − σJVIE,i |2 (21) RMS error(dB) = NS i=1

where N S is the number of sampling points or bistatic observation angles, and σref,i and σJVIE,i denote the reference and the computed RCS results, respectively. Both σref,i and σJVIE,i are in decibels. The reference results in this paper are obtained either by Mie’s series or by the commercial software FEKO. In the FEKO simulations, only the method of moments SIE solver with Rao-Wilton-Glisson (RWG) bases [49] is utilized, √ and the adopted average mesh size is 0.05λ D (λ D = λ0 / εr ) to ensure the accuracy. In this case, by computing the rms error of the bistatic RCS with respect to the number of unknowns, the accuracy and convergence for different basis orders are obtained and plotted in Fig. 2(a). It is found that better accuracy is achieved for higher basis orders, and the associated accuracy improvement is pronounced. Moreover, the solution error decreases quickly

Fig. 3. Bistatic RCS obtained by using the HOHV-based JVIE solution from a dielectric sphere with the relative permittivity εr = 3.0 − 0.3 j and radius 0.55λ0 .

with the increase in the number of unknowns in higher order solutions, thus confirming the higher order convergence of the HOHV-based JVIE. However, it is also found that the curve of the fourth order decreases more slowly than the curve of the third-order bases to some extent. It is not an exception. Similar phenomenon was observed in higher order VIE [23], [26] and SIE [16], [48]. This indicates that a dramatic improvement in accuracy may not be achieved when the order of the bases grows (e.g., the fourth-order or higher order bases with coarse mesh sizes) in comparison with lower order bases. This is because problems associated with ill-conditioning rapidly increase in higher order JVIE when the polynomial orders of HOHV bases increase to 4. This increasingly illconditioning is often an open problem and such an observation has been reported in previous publications on higher order basis functions [19]–[23]. In our higher order JVIE, here are two potential reasons. One is the increasingly ill-conditioned nature of the nonconformal JVIE linear system. The second one probably comes from the fact that accurate integration of the reaction integrals becomes difficult as the order increases, and the singular and nearly singular integrals play an important role in the higher order method. It should be noted that, no special treatment has been applied to remedy the near singularity in this paper. This may result in the accuracy loss of the numerical integral when higher order bases, e.g., fourthorder bases are used. However, this problem can be further alleviated by adopting a proper method, e.g., the singularity subtraction technique [47], [51], to calculate (near-) singular integrals involving higher order bases over curved tetrahedral elements. Similar performance is also observed for the case of the first type of mixed order schemes, i.e., MOI, as shown in Fig. 2(b). Note that for both cases, the reference results are the Mie series, and the Gaussian elimination solver and reasonable Gaussian integral points are adopted to ensure high accuracy. With the same radius but different permittivity εr = 3.0 − 0.3 j , this sphere is simulated by JVIE with different meshes and different orders of the HOHV basis. Simulation details are listed in Table II, and the scattering results of soln. no. 1, 6, and 10 are shown in Fig. 3, where the JVIE curves agree well with the exact Mie’s series.


4161

TABLE II S TATISTICS OF THE S IMULATIONS OF A H OMOGENEOUS S PHERE W ITH R ELATIVE P ERMITTIVITY εr = 3.0 − 0.3 j AND R ADIUS 0.55λ0

According to Table II, it is obvious that for a given mesh size, the higher accuracy is obtained as the basis order increases, by comparing soln. no. 3 and 8, 4 and 9, and 6 and 10. By conducting similar comparisons, the accuracy influence from the average mesh size and the number of Gaussian integral points becomes very clear. Note that the points N O and N I in Table II represent the out and inner integral points of the volume–volume integral in (12), respectively. As for the efficiency, in comparison with the lower order basis function, higher order basis functions can achieve higher accuracy with fewer unknowns, which can be concluded by comparing soln. no. 1, 4, 6, 8, and 9. As shown by soln. no. 1 and 6, the number of unknowns, the memory, and the matrix-filling time are, respectively, reduced by 16.1, 70.7, and 5.1 times. Moreover, the prominent efficiency gain resulting from BER and NBF is also demonstrated. Based on Table II, it should be easy to achieve a good tradeoff between accuracy, memory, and computational time for most situations. 2) Continuously Inhomogeneous Sphere Shell: The second example is a spherical shell with a continuous radial inhomogeneity [50], which is not suitable for the SIE simulation [1], [6]. Its relative permittivity is ⎧ (r − a)2 ⎨ 2.0 + 7.0, a < r < b (22) εr (r) = (b − a)2 ⎩ 1, other where a = 0.3λ0 and b = 0.5λ0 are, respectively, the inner and outer radii of this shell. For this example, two discretization sizes are adopted. The larger one is 0.15 m, which results in 831 curved tetrahedrons, whereas the smaller one is 0.051 m, generating 24 280 tetrahedrons. Besides, it is worth mentioning that in our JVIE simulation, the relative permittivity varies continuously within the tetrahedral elements, instead of assuming a constant value for the entire tetrahedral element. For the coarser mesh, applying MOI (1st, 2nd) yields 13 923 unknowns. Benefiting from the BER and NBF, 3456 s computational time and 322 iterations are taken, respectively. For the other mesh, the zeroth-order basis function is utilized

Fig. 4. Bistatic RCS of the spherical shell with radial inhomogeneous profile.

and 50 406 unknowns are yielded, which results in 10 066 s computational time and 147 iterations. As shown in Fig. 4, good agreement between the JVIE results and the reference results [50] is also obtained, which proves the accuracy of the proposed HOHV-based JVIE for the simulation of challenging inhomogeneous objects. B. Validation of Nonconformal Discretization 1) Cube: The first nonconformal case is a lossy homogeneous cube whose edge length is 0.85λ0 and dielectric constant is εr = 2.35 − 0.15 j . This cube is cut into two halves of the same size, and they are meshed separately with the same average mesh size. Four typical nonconformal meshes are plotted in Fig. 5. The HOHV-based JVIE is used to simulate this cube with different meshes and different orders of the HOHV basis. Simulation details are listed in Table III, and the scattering results of soln. no. 1, 4, and 8 are shown in Fig. 6, where the JVIE curves overlap with the reference results obtained by the commercial software FEKO. Note that very small mesh size, i.e., 0.05λ D , is used in FEKO to guarantee the good accuracy.

4162


TABLE III S TATISTICS OF THE S IMULATIONS OF A H OMOGENEOUS C UBE W ITH R ELATIVE P ERMITTIVITY εr = 2.35 − 0.15 j AND S IZE 0.85λ0

Fig. 6. Bistatic RCS obtained by using the HOHV-based JVIE solution from a dielectric cube whose edge length is 0.85λ0 and the relative permittivity is εr = 2.35 − 0.15 j. Fig. 5. Four nonconformal discretization of a cube whose edge length is 0.85λ0 and the relative permittivity is εr = 2.35 − 0.15 j. (a) 0.1λ D . (b) 0.4λ D . (c) 0.65λ D . (d) 0.85λ D .

According to Table III, it can be observed from soln. no. 1, 3, and 4 that the memory and computational time are significantly reduced by using the higher order basis function, whereas the solution accuracy remains the same level. As shown by soln. no. from 5 to 8 in Table III, reduction of the unknown number can be obtained by using MOI, while the higher accuracy is achieved. For example, from soln. no. 1 and 7, it is evident that the number of unknowns, the memory, and the matrix-filling time are reduced by 11.6, 48.8, and 11.2 times, respectively. Moreover, the flexibility of basis order selection is further enhanced by MOI, which can avoid “wasting” unknowns and make the HOHV-based nonconformal JVIE much more universal. We also observe that the iteration steps of the nonconformal (or conformal) JVIE increase with the growing basis order, as shown by soln. no. 1–4 of Table III (or by soln. no. 1 and 3–6 of Table II). This is not out of our expectation. Such an observation has been reported in previous publications on higher order basis functions [19]–[23]. For example,

as listed in [21, Table 2], the condition number of impedance matrix rapidly increases as the basis order increases. It is worth mentioning that the orthogonality and conditioning properties of the HOHV bases can be further improved as demonstrated in [17], [20], [22], [23], and [28], which often yields a faster convergence for iterative solvers. In this paper, the NBF technique [22], [23] is used to reduce the condition number. As summarized in Tables II and III, the number of iterations steps is considerably reduced, and thus the overall computational time is reduced. 2) Ogive: The second case is a dielectric ogive with the relative permittivity εr = 4.0 − 0.166 j . As shown in Fig. 7(a), this ogive is rationally symmetric. In the z = 0 plane, the generating curve is y = h/2 + r 2 − x 2 − r, {x = −L/2, L/2} (23) where r = (L 2 + h 2 )/4h, L = 0.6 m is the length of the major axis, and h = 0.3 m is the diameter at x = 0. When the traditional low-order VIE is used, the fine discretization is required for the two ogive ends. This yields a


4163

Fig. 7. Ogive. (a) Nonconformal discretization with curved geometrical modeling. (b) Nonconformal discretization with planar geometrical modeling. (c) Conformal discretization with curved geometrical modeling.

large number of unknowns, and the associated computational cost is very high. For example, with the mesh size 0.1λ D √ (λ D = λ0 / εr ), 68 384 tetrahedrons will be generated, and the number of associated SWG bases is 139 284. As illustrated in Fig. 7(a), the ogive is separately meshed with different mesh sizes, i.e., nonconformal discretization. The average mesh size of the two ends is 0.075λ D so that 3497 tetrahedrons are generated for them. As for the middle part, its average mesh size is 0.35λ D , leading into 278 tetrahedrons. The second type of mixed order schemes, i.e., MOII, is utilized for this target, where the second-order basis is utilized for the middle part and the zeroth-order basis for two ends. The resulting unknown number is 14 882. With help of BER and NBF, 5651 s and 887 iteration steps are demanded for the simulation. In order to further reduce the computational time, the third type of mixed order schemes, i.e., MOIII, is also utilized. Different from MOII, MOIII employs the second-order facebased basis but the first-order volume-based basis for the middle part, which results in 12 380 unknowns. The associated computational time is 4907 s with 226 iteration steps. As shown in Fig. 8, scattering results of both MOII and MOIII are compared with the reference results from FEKO, and they agree well with each other. Note that the excitation is a plane wave Einc = xe− j kb z with the frequency of 750 MHz. This is denoted as the original scenario, and in the following another two scenarios will be considered. The first scenario is to investigate the solution with the degraded geometrical accuracy, namely when the ogvie is discretized by the low-order (i.e., planar) elements, as shown in Fig. 7(b). With the same average edge length as the mesh adopted for the original scenario, there result 4072 planar tetrahedrons. This scenario results in higher computational cost, i.e., 12 935 unknowns, 5107 s and 260 iteration steps. Bistatic RCS results are calculated and compared with the FEKO results, as shown in Fig. 8, where relatively poor agreement is observed. This suggests that in the higher order JVIE,

Fig. 8. Simulation results of a dielectric ogive with the relative permittivity εr = 4.0 − 0.166 j, which are obtained by the HOHV-based JVIE solution, the HOHV-based DVIE method [23], and FEKO-SIE. (a) E-plane bistatic RCS. (b) H-plane bistatic RCS.

higher order geometrical modeling is indispensable to ensure accuracy, which is also demonstrated in [16], [21], and [23]. The second scenario is to show the flexibility and advantage of the proposed nonconformal JVIE in comparison with the conventional higher order VIE, e.g., the DVIE in [23], with the conformal discretization. With the same average edge length as the mesh adopted for the original scenario, the DVIE in [23] requires 4135 curved tetrahedral elements when conformal discretization is adopted. The associated mesh is shown in Fig. 7(c). Employing the same MOIII scheme as that of the original scenario produces 18 972 unknowns. As shown in Fig. 8, the bistatic RCS results obtained by the higher order DVIE [23] agree very well with the FEKO results. For the E-plane case, the rms errors of the original and second scenarios are, respectively, 0.124 and 0.195 dB, which imply the same accuracy level. However, higher computational time (i.e., 17 974 s) with 426 iteration steps is required by the higher order DVIE in [23], which demonstrates that our nonconformal higher order JVIE has the advantage of reducing computational cost. C. Application To further show the application of our nonconformal JVIE, we study the electromagnetic problems of some complex

4164


Fig. 9. (a) Dipole array of eight elements covered by a three-layer hemispherical dielectric radome. (b) Nonconformal discretization of the dielectric radome.

or practical objects, i.e., a radome, a combinational object, a room, and a dielectric lens antenna. 1) Hemispherical Radome: The first case is a three-layer hemispherical dielectric radome, as shown in Fig. 9(a). The thicknesses and the relative permittivities of the outer and the inner layers are τ1 = τ3 = 3 mm and εr1 = εr3 = 2 − 0.005 j , respectively, and the counterparts of the middle layer are τ2 = 8.8 mm and εr2 = 1.5 − 0.035 j , respectively. The diameter D of the radome is 100 mm. A linear array with eight small dipoles, each of which is fed with equal excitation amplitude, 0.25π progressive phase, and 0.01 m element spacing, is placed along the x-axis and centered in the radome. The currents on the dipoles flow along the x-direction. The operating frequency is 3 GHz. In this simulation, the nonconformal mesh is adopted and shown in Fig. 9(b). The average mesh size of the outer and inner layers is 0.22λ D1, and that of the middle layer is 0.4λ D2 . (λ Di is the dielectric wavelength.) The total number of tetrahedrons is 1480, where only 108 large curved elements are used to model the middle layer. The adopted MOIII is that the first-order face-based but the zeroth-order volumebased bases are utilized for the first and third layers, and the second-order face-based but the first-order volume-based bases are utilized for the middle layer. The resulting unknown number is 11 652, and the required computational cost is 7464 s with 663 iteration steps. Note that when this radome is discretized by the traditional conformal mesh size, i.e., 0.1λ D , there result 22 293 tetrahedrons and 46 697 unknowns for the traditional zeroth-order VIE. It is evident that the higher order nonconformal JVIE is much superior to the traditional loworder one. As given in Fig. 10, the normalized radiation pattern of this dipole array is compared with the reference results of FEKO, and very good agreement is observed. Moreover, the radiation pattern without the hemispherical radome is also plotted. It is evident that the presence of the radome has a tremendous influence on the array’s radiation pattern. 2) Combinational Object: The second case is a combinational object consisting of four small cones, a large cone, and a cube. As shown in Fig. 11(a), this cube is split by a plane. The discretization size of the four small cones, the large cone, the upper part, and the bottom part of the cube is, respectively, √ 0.02λ D1, 0.2λ D2 , 0.8λ D3 and 0.5λ D3 , where λ Di = λ0 / εri (i = 1, 2, 3). As depicted in Fig. 11(b), the volume

Fig. 10. Normalized far-field radiation of an eight-dipole array with and without a hemispherical radome.

Fig. 11. (a) Combinational object consisting of four small cones, one larger cone, and a cube. (b) Nonconformal discretization of the combinational object.

discretization is nonconformal and the element number of these four parts is 5421, 74, 100, and 107, respectively, where the maximum size of the tetrahedral element is about 30 times of the minimum one. The adopted MOIII is that the zeroth-order bases are utilized for the small cones, the first-order face-based but the zerothorder volume-based bases for the large cone, the third-order bases for the upper part of the cube, and the second-order facebased but the first-order volume-based bases for the bottom part of the cube. The simulation of the higher order JVIE generates 15 982 unknowns, and requires 8050 s and 156 iterations to converge to the relative residue 0.001. As shown in Fig. 12, the calculated bistatic RCS results agree very well with the reference results obtained by FEKO with the conformal discretization. In order to further show the flexibility of our nonconformal JVIE method, the object is simulated by the higher order DVIE [23], where the conformal discretization is required. With the same average edge length and the same basis order as that of our JVIE, 73 545 unknowns are generated, which are based on 6732 curved tetrahedral elements. As shown in Fig. 12, the bistatic RCS results obtained by the DVIE [23] agree with the FEKO results. For the E-plane case, the rms errors of our JVIE and this DVIE [23] are, respectively,


4165

Fig. 12. Comparison of the HOHV-based JVIE results with FEKO solution for the E- and H-planes bistatic RCS for the combinational object in Fig. 11(a).

0.141 and 0.134 dB, which imply the same accuracy level. However, this higher order DVIE [23] produces higher computational cost. Note that the unknown number of higher order DVIE may be reduced by the hp-refinement technique [21], but this technique should not be as effective as the adoption of nonconformal mesh. Moreover, this technique also increases the difficulty of mesh generation to some extent. Besides, if the conformal discretization with the average edge length 0.1λ Di is used, 28 942 tetrahedrons will be yielded, and the conventional low-order VIE will produce 60 767 unknowns. Therefore, through this example, it can be demonstrated that the proposed HOHV-based nonconformal JVIE is very suitable for the scattering simulation of multiscale dielectric structures. 3) Room Containing Materials of High Contrast: The third case is a room including both multiscale and high-permittivity contrast materials, as shown in Fig. 13(a). The sizes of the room, the door, and the window are 3.6 m × 3.6 m × 3.0 m, 1.0 m × 2.0 m, and 0.6 m × 0.6 m, respectively. The thickness of the wall, the door, and the window are 0.3, 0.06, and 0.02 m, respectively. The door and the window reside in the middle of the wall. At the center of the floor, there is a dielectric desk, including the square desktop of 1.2 m × 1.2 m × 0.2 m and four cylindrical legs of the radius 0.05 m and the height 0.5 m. Other geometrical parameters shown in Fig. 13(a) are h 1 = 1.5 m, h 2 = 0.5 m, w1 = 1.5 m, and w2 = 1.3 m. The simulation domain is subdivided into five parts whose relative permittivity εr values are listed in Table IV. As shown in Fig. 13(b) and (c), nonconformal meshes are used to discretize the model. Their average edge length and the resultant number of elements for each subpart are given in Table IV. It can be observed from Table IV that although the window, the door, and the desk’s leg are very small compared with the wall, about 64% of the total elements are used to discretize these three parts. In this case, different orders of the HOHV bases are used for different parts, which are summarized in Table IV. The simulation frequency is 100 MHz. There are 60 321 unknowns,

Fig. 13. room.

(a) Room model. (b) and (c) Nonconformal discretization of the

TABLE IV N ONCONFORMAL D ISCRETIZATION AND THE A SSOCIATED BASIS O RDERS FOR THE D IELECTRIC R OOM S HOWN IN F IG . 13(a)

and the computational time is 22 578 s with 665 iteration steps. As shown in Fig. 14, the bistatic RCS results of the E- and H-planes obtained by the higher order JVIE agree well with the FEKO results. Scattering of the room without the window, the door, and the desk is also computed. It can be observed that those fine structures indeed have an important influence. This also demonstrates that the proposed JVIE can successfully capture the influence. Besides, it should be noted that the conventional zeroth-order DVIE [5] with the traditional conformal discretization may result in 110 730 elements and about 237 801 unknowns, and the zeroth-order EVIE [43], [44] with the conventional noncoformal mesh may result in 76 196 elements and about 228 588 unknowns. 4) Dielectric Lens Antenna: The last case is a dielectric lens antenna of the diameter D = 512.83 mm and the thickness t = 50 mm, which is made of the polytetrafluoroethylene material (εr = 2.25). As shown in Fig. 15(a), the hyperbolical surface of the lens can be obtained by rotating the curve AOB around the z-axis. The antenna is excited by a line array with five small dipoles. The distance F between the line array and

4166


Fig. 14. Comparison of the HOHV-based JVIE results with FEKO solution for the E- and H-planes bistatic RCS for the room shown in Fig. 13(a). Fig. 16. Comparison of the proposed JVIE results with FEKO solution for the normalized far-field radiation E- and H-planes for the lens antenna shown in Fig. 15.

Fig. 15. (a) Geometry of the lens antenna with matching slots. (b) Nonconformal discretization.

the bottom of the lens antenna is 1211.2 mm. In the dipole array, each dipole is fed with the equal excitation amplitude, 0.25π progressive phase, and 0.01 m element spacing, and is placed along the x-axis. The currents on the dipoles flow along the x-direction. The operating frequency is 3 GHz. For reducing the effects of internal reflections inside the lens, the dielectric matching slots are used. The slot depth is h 1 = √ λ D /(4 εm ), and the width is h 2 = 1.5h 1 , where λ D is the √ wavelength in the lens and εm = εr is the uniform dielectric constant of the sheet. In order to reduce the number of unknowns, the nonconformal discretization is adopted for the multiscale antenna, as illustrated in Fig. 15(b). Based on different mesh sizes for different antenna parts, different orders of the HOHV bases are used. In this simulation, the discretization size of the fine matching layers is 0.2λ D so that 5311 tetrahedrons are generated, and the associated basis function is the firstorder face-based but the zeroth-order volume-based bases. The average edge length of the dielectric lens is 0.65λ D so that 766 tetrahedrons are generated, and the associated basis function is of the third-order face-based but the second-order volume-based bases. Using the HOHV bases, the proposed nonconformal JVIE with the MOIII scheme results in 68 471 unknowns and takes 18 496 s. As shown in Fig. 16, the normalized radiated far fields obtained by the higher order JVIE are compared

with the reference results from FEKO, and good agreement is evident. Note that this model is not easy to simulate by conventional low-order SIE, which requires 131 754 unknowns and 41 264 triangular patches with the average mesh size 0.1λ D , because there are too many subdomains, whereas it is suitable for our method in this paper. Moreover, the radiation of the dielectric lens antenna filled with materials of continuously varying permittivity is also considered here. The continuous profile of the permittivity is given as z (1.25 − 1.0 j ), {z = 0, t + h 1 }. (24) εr (z) = 2.25 + t + h1 By keeping other simulation parameters the same, the associated radiated fields obtained by the nonconformal higher order JVIE are also presented in Fig. 16. It is within the expectation that the inhomogeneous material has an influence on the radiation pattern of the antenna. It should be pointed out that for this continuously inhomogeneous case, the proposed JVIE, rather than SIE, is very suitable. Note also that when the conformal mesh with the average edge length 0.1λ D is utilized into the dielectric lens antenna, 194 260 tetrahedrons will be yielded, and the conventional loworder VIE will produce 406 923 unknowns. V. C ONCLUSION In this paper, an HOHV-based nonconformal JVIE method is presented to simulate the scattering from dielectric structures. Different from previous studies on VIE, our nonconformal HOHV-based JVIE utilizes curved nonconformal tetrahedral meshes for geometric modeling, and the associated HOHV basis functions for current modeling. To the best of our knowledge, such a nonconformal JVIE method with HOHV bases is first reported here for the scattering by dielectric objects. In order to improve the efficiency of the proposed JVIE, the efficient BER technique is introduced to significantly


reduce the matrix-filling time. The NBF technique [22], [23] is utilized to improve the conditioning of impedance matrices so that the efficiency is further enhanced. Moreover, the mixed order schemes associated with HOHV bases, which were recently reported for VIE [23], have been extended into this paper to improve its flexibility. This extension is more important and valuable when nonconformal meshes are adopted. Therefore, with the help of BER, NBF, and the mixed order schemes, our nonconformal HOHV-based JVIE is very accurate, efficient, and flexible. This is demonstrated by the numerical results of several typical targets discretized by conformal or nonconformal meshes. Further, the higher order JVIE can be combined with fast algorithms like multilevel fast multipole algorithm (MLFMA) [11], [48] for applications in real life where the targets are generally large scale and multiscale. Note that, though the advantages of higher order bases and MLFMA cannot be directly combined, several efficient techniques [6], [11], [52]–[55] have provided possible solutions to combine MLFMA with higher order basis functions. For example, the block diagonal preconditioner [6], [53] and the sparse approximate inverse [54], [55] are always helpful for the iterative solution of higher order MLFMA. The adaptive grouping approach and the sparse matrix storage technique [11], [52], [53] are utilized for MLFMA implementation higher order bases for large groups. Note also that, though increasing basis order can be beneficial in terms of memory compared with lower orders, ill-conditioning always increases in higher order JVIE when the order grows, especially going beyond the fourth order. Thus, we caution against recommending a relatively high polynomial expansion order (e.g., the basis order is greater than 3 or 4) for its developed fast solver version, when there is no effective use of precondition techniques. More details need further investigating. It is worth mentioning that for some special structures, when the HOHV basis functions are adopted, choosing different basis orders for different regions may also be able to simulate them with a smaller number of unknowns. However, this scheme is evidently not as universal as the nonconformal scheme, and it does not alleviate the difficulty of mesh generation. Moreover, when both schemes are combined together, the resulting nonconformal HOHV-based JVIE becomes more flexible and powerful. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which have significantly improved the quality of this paper. R EFERENCES [1] R. F. Harrington, Field Computation by Moment Methods. Piscataway, NJ, USA: IEEE Press, 1993. [2] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York, NY, USA: IEEE Press, 1995. [3] J. P. Kottmann and O. J. F. Martin, “Accurate solution of the volume integral equation for high-permittivity scatterers,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1719–1726, Nov. 2000. [4] J. Markkanen, C.-C. Lu, X. Cao, and P. Ylä-Oijala, “Analysis of volume integral equation formulations for scattering by high-contrast penetrable objects,” IEEE Trans. Antennas Propag., vol. 60, no. 5, pp. 2367–2374, May 2012.

4167

[5] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. AP-32, no. 1, pp. 77–85, Jan. 1984. [6] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Eds., Fast and Efficient Algorithms in Computational Electromagnetics. Norwood, MA, USA: Artech House, 2001. [7] Z. Q. Zhang and Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 1, pp. 102–105, 2002. [8] C. C. Lu, “A fast algorithm based on volume integral equation for analysis of arbitrarily shaped dielectric radomes,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 606–612, Mar. 2003. [9] K. Sertel and J. L. Volakis, “Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1686–1692, Jul. 2004. [10] A. G. Polimeridis, J. F. Villena, L. Daniel, and J. K. White, “Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects,” J. Comput. Phys., vol. 227, no. 14, pp. 7052–7068, 2014. [11] O. Borries, P. Meincke, E. Jorgensen, and P. C. Hansen, “Multilevel fast multipole method for higher-order discretizations,” IEEE Trans. Antennas Propag., vol. 62, no. 9, pp. 3119–3129, Sep. 2014. [12] Z. R. Yu, W. J. Zhang, and Q. H. Liu, “The mixed-order BCGSFFT method for the scattering of three-dimensional inhomoge-neous anisotropic magnetodielectric objects,” IEEE Trans. Antennas Propag., vol. 63, no. 12, pp. 5709–5717, Dec. 2015. [13] D. Dault and B. Shanker, “A mixed potential MLFMA for higher order moment methods with application to the generalized method of moments,” IEEE Trans. Antennas Propag., vol. 64, no. 2, pp. 650–662, Feb. 2016. [14] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 329–342, Mar. 1997. [15] J. P. Webb, “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, Aug. 1999. [16] B. M. Notaros, “Higher order frequency-domain computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2251–2276, Aug. 2008. [17] M. M. Botha, “Fully hierarchical divergence-conforming basis functions on tetrahedral cells with applications,” Int. J. Numer. Methods Eng., vol. 71, no. 2, pp. 127–148, 2007. [18] R. D. Graglia and A. F. Peterson, “Hierarchical and conforming Nédélec elements for surface and volumetric cells,” IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 5215–5227, Nov. 2012. [19] O. S. Kim, P. Meincke, O. Breinbjerg, and E. Jørgensen, “Method of moments solution of volume integral equations using higher-order hierarchical Legendre basis functions,” Radio Sci., vol. 39, no. 5, pp. 1–7, Oct. 2004. [20] O. S. Kim, P. Meincke, O. Breinbjerg, and E. Jørgensen, “Solution of volume-surface integral equations using higher-order hierarchical Legendre basis functions,” Radio Sci., vol. 42, no. 4, pp. 1–8, Aug. 2007. [21] E. Chobanyan, M. M. Ilić, and B. M. Notaroš, “Double-higherorder large-domain volume/surface integral equation method for analysis of composite wire-plate-dielectric antennas and scatterers,” IEEE Trans. Antennas Propag., vol. 61, no. 12, pp. 6051–6063, Dec. 2013. [22] M. M. Kostic and B. M. Kolundžija, “Maximally orthogonalized higher order bases over generalized wires, quadrilaterals, and hexahedra,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3135–3148, Jun. 2013. [23] Q.-M. Cai, Y.-W. Zhao, Y.-T. Zheng, M.-M. Jia, Z. Zhao, and Z.-P. Nie, “Volume integral equation with higher order hierarchical basis functions for analysis of dielectric electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 63, no. 11, pp. 4964–4975, Dec. 2015. [24] E. Chobanyan, D. I. Olćan, M. M. Ilić, and B. M. Notaroš, “Volume integral equation-based diakoptic method for electromagnetic modeling,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 10, pp. 3097–3107, Aug. 2016. [25] M. S. Tong, Z. G. Qian, and W. C. Chew, “Nyström method solution of volume integral equations for electromagnetic scattering by 3D penetrable objects,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1645–1652, May 2010.

4168


[26] S. F. Tao and R. S. Chen, “A higher-order solution of volume integral equation for electromagnetic scattering from inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett., vol. 13, pp. 627–630, 2014. [27] J. Wang and J. P. Webb, “Hierarchical vector boundary elements and p-adaption for 3-D electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, Aug. 1997. [28] E. Jorgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2985–2995, Nov. 2004. [29] Z. P. Nie, W. M. Ma, Y. Ren, Y. W. Zhao, J. Hu, and Z. Q. Zhao, “A wideband electromagnetic scattering analysis using MLFMA with higher order hierarchical vector basis functions,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3169–3178, Oct. 2009. [30] P. Ylä-Oijala, J. Markkanen, and S. Järvenpää, “Current-based volume integral equation formulation for bianisotropic materials,” IEEE Trans. Antennas Propag., vol. 64, no. 8, pp. 3470–3477, Aug. 2016. [31] M. C. van Beurden and S. J. L. van Eijndhoven, “Gaps in present discretization schemes for domain integral equations,” presented at the Int. Conf. Electromagn. Adv. Appl. ICEAA, Torino, Italy, 2007. [32] M. C. van Beurden and S. J. L. van Eijndhoven, “Well-posedness of domain integral equations for a dielectric object in homogeneous background,” J. Eng. Math., vol. 62, no. 3, pp. 289–302, 2008. [33] M. M. Botha, “Solving the volume integral equations of electromagnetic scattering,” J. Comput. Phys., vol. 218, no. 1, pp. 141–158, 2006. [34] J. Markkanen, P. Ylä-Oijala, and A. Sihvola, “Discretization of integral, equation formulations for extremely anisotropic materials,” IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 5195–5202, Nov. 2012. [35] Y. Schols and G. A. E. Vandenbosch, “Separation of horizontal and vertical dependencies in a surface/volume integral equation approach to model quasi 3-D structures in multilayered media,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1086–1094, Apr. 2007. [36] J. Markkanen and P. Ylä-Oijala, “Discretization of electric current volume integral equation with piecewise linear basis functions,” IEEE Trans. Antennas Propag., vol. 62, no. 9, pp. 4877–4880, Sep. 2013. [37] L.-M. Zhang and X.-Q. Sheng, “Solving volume electric current integral equation with full- and half-SWG functions,” IEEE Antennas Wireless Propag. Lett., vol. 14, pp. 682–685, 2015. [38] X. Zheng et al., “Volumetric method of moments and conceptual multilevel building blocks for nanotopologies,” IEEE Photon. J., vol. 4, no. 1, pp. 267–282, Feb. 2012. [39] C. C. Lu, X. D. Cao, and A. Frotanpour, “Method of moments for partially structured mesh,” in Proc. 31st Int. Rev. Prog. Appl. Comput. Electromagn., Mar. 2015, pp. 1215–1226. [40] Q.-M. Cai, Y.-W. Zhao, L. Gu, Z.-P. Nie, and Q. H. Liu, “Electromagnetic scattering by inhomogeneous dielectric and magnetic scatterers using VIE with a normalization basis function (NBF) technique,” in Proc. Int. IEEE AP-S Symp., Jul. 2016, pp. 1609–1610. [41] Y. L. Hu, J. Li, D. Z. Ding, and R. S. Chen, “Analysis of transient EM scattering from penetrable objects by time domain nonconformal VIE,” IEEE Trans. Antennas Propag., vol. 64, no. 11, pp. 360–365, Jan. 2016. [42] M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a singularity,” SIAM J. Numer. Anal., vol. 19, no. 6, pp. 1260–1262, Dec. 1982. [43] N. A. Ozdemir and J.-F. Lee, “A nonconformal volume integral equation for electromagnetic scattering from penetrable objects,” IEEE Trans. Magn., vol. 43, no. 4, pp. 1369–1372, Apr. 2007. [44] L. M. Zhang and X. Q. Sheng, “A discontinuous Galerkin volume integral, equation method for scattering from inhomogeneous objects,” IEEE Trans. Antennas Propag., vol. 63, no. 12, pp. 5661–5667, Dec. 2015. [45] X. Mu, H.-X. Zhou, K. Chen, and W. Hong, “Higher order method of moments with a parallel out-of-core LU solver on GPU/CPU platform,” IEEE Trans. Antennas Propag., vol. 62, no. 11, pp. 5634–5646, Nov. 2014. [46] Z.-P. Zhang, Y.-W. Zhao, Q.-M. Cai, Y.-T. Zheng, L. Gu, and Z.-P. Nie, “An efficient matrix fill-in scheme for surface integral equation with higher order hierarchical vector basis functions,” in Proc. Int. 5th IEEE APCAP Symp., Jul. 2016, pp. 177–178. [47] B. M. Notaroš and B. D. Popović, “Optimized entire-domain moment method analysis of 3D dielectric scatterers,” Int. J. Numer. Modell. Electron. Netw. Devices Fields, vol. 10, no. 3, pp. 177–192, 1997. [48] G. Kang, J. Song, W. C. Chew, K. C. Donepudi, and J.-M. Jin, “A novel grid-robust higher order vector basis function for the method of moments,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 908–915, Jun. 2001.

[49] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [50] S. D. Gedney and C. C. Lu, “High-order solution for the electromagnetic scattering by inhomogeneous dielectric bodies,” Radio Sci., vol. 38, no. 1, art. no. 1015, 2003. [51] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra,” Int. J. Numer. Methods Eng., vol. 58, no. 8, pp. 1149–1165, 2003. [52] T. F. Eibert, “A diagonalized multilevel fast multipole method with spherical harmonics expansion of the k-space integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 814–817, Feb. 2005. [53] Y. L. Wang, S. Yan, and Z. P. Nie, “A point-adaptive grouping scheme of MLFMA for electrically large scattering simulation,” IEEE Trans. Antennas Propag., vol. 64, no. 12, pp. 5527–5530, Dec. 2016. [54] J. Lee, J. Zhang, and C. C. Lu, “Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in elecromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2277–2287, Sep. 2004. [55] X.-M. Pan, L. Cai, and X.-Q. Sheng, “An efficient high order multilevel fast multipole algorithm for electromagnetic scattering analysis,” Prog. Electromagn. Res., vol. 126, pp. 85–100, 2012.

Qiang-Ming Cai received the B.S. degree in physics from Sichuan Normal University, Chengdu, China, in 2011. He is currently pursuing the Ph.D. degree in electromagnetic field and microwave technology with the Department of Microwave Engineering, University of Electrical and Science Technology of China, Chengdu. From 2015 to 2016, he was a Visiting Scholar with the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA. His current research interests include numerical methods in electromagnetics, especially integral equation-based higher order moment of methods and fast algorithms, electromagnetic scattering, and radiation.

Zhi-Peng Zhang received the B.S. degree in electronic and information engineering from Anhui University, Anhui, Hefei, China, in 2013. He is currently pursuing the Ph.D. degree in electromagnetic field and microwave technology with the Department of Microwave Engineering from the University of Electrical and Science Technology of China, Chengdu, China. His current research interests include numerical methods in electromagnetics, especially integral equation-based higher order moment of methods and fast algorithms, electromagnetic scattering, and radiation.

Yan-Wen Zhao received the M.S. degree in geophysical exploration from the University of Chang Jiang, Sichuan, Chengdu, China, in 1994, and the Ph.D. degree in electrical engineering from the University of Electrical and Science Technology of China (UESTC), Chengdu. Since 2008, he has been a Professor and a Supervisor with UESTC. He has authored or co-authored over 100 journal and conference papers. His current research interests include computational electromagnetics (EMs), inhomogeneous media field and wave, EM scattering, and inverse scattering. Dr. Zhao is a Senior Member of the Chinese Institute of Electronics. He was a recipient of the National Science and Technology Progress Second Award and the Provincial Science and Technology Progress Award.


Wei-Feng Huang received the B.S. degree in electromagnetic field and radio technology and the M.S. degree in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2011 and 2014, respectively. He is currently pursuing the Ph.D. degree with Duke University, Durham, NC, USA. Since 2014, he has been a Research Assistant with the Department of Electrical and Computer Engineering, Duke University. His current research interests include high-frequency methods, integral equation methods, the finite-element method, low-frequency simulations, electromagnetic scattering in layered media, and well-logging applications.

Yu-Teng Zheng received the B.S. degree in electromagnetics (EMs) and microwave technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2010, where he is currently pursuing the Ph.D. degree in electromagnetic field and microwave technology. From 2015 to 2016, he was a Visiting Scholar with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, USA. His current research interests include numerical methods in computational EMs, especially integral equation-based methods and fast algorithms.

4169

Zai-Ping Nie (SM’96–F’13) was born in Xi’an, China, in 1946. He received the B.S. degree in radio engineering and the M.S. degree in electromagnetic (EM) field and microwave technology from the Chengdu Institute of Radio Engineering [now University of Electronic Science and Technology of China (UESTC)], Chengdu, China, in 1968 and 1981, respectively. From 1987 to 1989, he was a Visiting Scholar at the Electromagnetics Laboratory, University of Illinois at Urbana–Champaign, Champaign, IL, USA. He is a currently a Professor with the Department of Microwave Engineering, UESTC. He has authored more than 300 journal papers. His current research interests include antenna theory and techniques, fields and waves in inhomogeneous media, computational EMs, EM scattering and inverse scattering, new techniques for antenna in mobile communications, and transient EM theory and applications. Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, China, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, Urbana, IL, USA. His research interests include computational electromagnetics and acoustics, inverse problems, and their application in nanophotonics, geophysics, biomedical imaging, and electronic packaging. He has published over 400 papers in refereed journals and 500 papers in conference proceedings. He was with the Electromagnetics Laboratory at the University of Illinois at UrbanaChampaign as a Research Assistant from 1986 to 1988, and as a Postdoctoral Research Associate from 1989 to 1990. He was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT from 1990 to 1995. From 1996 to 1999 he was an Associate Professor with New Mexico State University. Since 1999 he has been with Duke University where he is now a Professor of Electrical and Computer Engineering. Dr. Liu is a Fellow of the Acoustical Society of America, the Electromagnetics Academy, and the Optical Society of America. Currently he serves as the founding Editor-in-Chief of the new IEEE J OURNAL ON M ULTISCALE AND M ULTIPHYSICS C OMPUTATIONAL T ECHNIQUES , the Deputy Editor in Chief of Progress in Electromagnetics Research, an Associate Editor for IEEE T RANSACTIONS ON G EOSCIENCE AND R EMOTE S ENSING, and an Editor of Journal of Computational Acoustics. He received the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, the 1997 CAREER Award from the National Science Foundation, and the ACES Technical Achievement Award in 2017. He served as an IEEE Antennas and Propagation Society Distinguished Lecturer from 2014 to 2016.