Efficient solution of the electric and magnetic current combined-field integral equation with the multilevel fast multipole algorithm and block-diagonal preconditioning

Authors


Abstract

[1] We consider the efficient solution of electromagnetics problems involving dielectric and composite dielectric-metallic structures, formulated with the electric and magnetic current combined-field integral equation (JMCFIE). Dense matrix equations obtained from the discretization of JMCFIE with Rao-Wilton-Glisson functions are solved iteratively, where the matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm. JMCFIE usually provides well conditioned matrix equations that are easy to solve iteratively. However, iteration counts and the efficiency of solutions depend on the contrast, i.e., the relative variation of electromagnetic parameters across dielectric interfaces. Owing to the numerical imbalance of off-diagonal matrix partitions, solutions of JMCFIE become difficult with increasing contrast. We present a four-partition block-diagonal preconditioner (4PBDP), which provides efficient solutions of JMCFIE by reducing the number of iterations significantly. 4PBDP is useful, especially when the contrast increases, and the standard block-diagonal preconditioner fails to provide a rapid convergence.

1. Introduction

[2] For the solution of electromagnetics problems involving three-dimensional dielectric objects, the electric and magnetic current combined-field integral equation (JMCFIE) [Ylä-Oijala and Taskinen, 2005a, 2005b] is a preferable formulation in terms of accuracy and efficiency. In numerical solutions employing Rao-Wilton-Glisson (RWG) functions [Rao et al., 1982] on triangles, JMCFIE is more accurate than the normal (N) formulations, such as the combined normal formulation (CNF) [Ylä-Oijala et al., 2005b] and the modified normal Müller formulation (MNMF) [Ylä-Oijala and Taskinen, 2005b]. In addition, iterative solutions of problems involving large and complicated objects are more efficient with JMCFIE, which requires fewer iterations than MNMF and CNF [Ergül and Gürel, 2007, 2009]. For a given discretization with the RWG functions, the tangential (T) formulations, such as the combined tangential formulation (CTF) [Ylä-Oijala et al., 2005b] and the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation [Poggio and Miller, 1973; Chang and Harrington, 1977; Wu and Tsai, 1977], may provide more accurate results than JMCFIE. On the other hand, matrix equations obtained with the T formulations are difficult to solve iteratively [Ylä-Oijala et al., 2005b, 2008; Ergül and Gürel, 2007, 2009]. In fact, improving the accuracy of JMCFIE solutions to the levels of the T formulations by refining the discretization can be more efficient than using the T formulations with coarse discretizations. Moreover, JMCFIE becomes essential for large problems, which might not easily be solved with the T formulations.

[3] JMCFIE can easily be applied to electromagnetics problems involving multiple dielectric regions or composite structures with coexisting metallic and dielectric parts [Ylä-Oijala and Taskinen, 2005a, 2005b; Ylä-Oijala, 2008]. In general, equivalent problems, which are defined for all nonmetallic regions, are discretized with oriented basis and testing functions. Then, the related unknowns on the boundaries and the corresponding equations are combined to form a single matrix equation to solve. This procedure is detailed by Ylä-Oijala et al. [2005a, 2005b] in the context of a PMCHWT formulation, and is extended to JMCFIE in the work of Ylä-Oijala and Taskinen [2005a, 2005b]. As discussed by Ylä-Oijala and Taskinen [2005a, 2005b], JMCFIE is appropriate for complicated structures involving multiple dielectric and metallic regions.

[4] Electromagnetics problems involving large metallic, dielectric, and composite objects can be solved iteratively, where the required matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm (MLFMA) [Song et al., 1997; Sheng et al., 1998; Chew et al., 2001; Donepudi et al., 2003]. Recently, MLFMA is used to solve electromagnetics problems involving homogeneous dielectric objects formulated with JMCFIE [Ergül and Gürel, 2007, 2009]. In this study, we extend the MLFMA solution of JMCFIE to those problems involving multiple dielectric and composite dielectric-metallic structures. We mainly focus on the efficiency of the solutions and investigate the number of iterations for increasingly large objects. We show that iterative solutions of JMCFIE become difficult as the contrast increases, i.e., when electromagnetic parameters change significantly across dielectric interfaces. For efficient solutions of JMCFIE, we present a four-partition block-diagonal preconditioner (4PBDP), which reduces the iteration counts significantly. This preconditioner, which was originally developed by Ergül and Gürel [2009] for homogeneous dielectric objects, is particularly useful when a standard two-partition block-diagonal preconditioner (2PBDP) fails to provide a rapid convergence. In this paper, we present 4PBDP to accelerate the solution of more complicated problems involving multiple dielectric and metallic regions.

[5] The rest of the paper is organized as follows. Section 2 presents the matrix equations obtained with the JMCFIE formulation of electromagnetics problems involving multiple dielectric and metallic regions. MLFMA solutions are considered in section 3, where we provide the specific details of our implementation. Block-diagonal preconditioning is discussed in section 4, followed by numerical examples in section 5, and our concluding remarks in section 6. Time-harmonic electromagnetic fields with eiωt time dependence are assumed throughout the paper.

2. Solutions of Electromagnetics Problems With JMCFIE

[6] We consider the general case involving U regions, namely, D0, D1,…, DU−1, and D0 is a region extending to infinity. Each region Du for u = 0, 1,…, (U − 1) is either metallic with perfect conductivity or lossless dielectric with constant electromagnetic parameters, i.e., permittivity εu and permeability μu. We assume that there is no junction where three or more regions intersect and each region Du has a nonzero volume bounded by a closed surface Su. Then,

equation image

where Suv = Svu is the interface between the regions Du and Dv. We note that JMCFIE and MLFMA are also applicable to composite problems involving junctions and lossy dielectric regions [Sheng et al., 1998; Ylä-Oijala and Taskinen, 2005a, 2005b].

[7] Applying the equivalence principle and using the boundary conditions for the tangential electric and magnetic fields on surfaces, equivalent electric and magnetic currents are defined as

equation image
equation image

where equation image(r) is the unit normal vector. For an interface Suv for u < v, we choose equation image directed into the region Du. When Suv is perfectly conducting, the tangential electric field and the magnetic current M(r) on the surface are zero.

2.1. Discretization

[8] For numerical solutions, surface currents are expanded in a series of RWG functions, i.e.,

equation image
equation image

where bn(r) for n = 1, 2,…, N represents the nth basis function with a spatial support of An, while anJ and anM are the unknown coefficients. Since we assume that there is no junction, each RWG function is located on the interface of two regions, such as Du and Dv. For u < v, Du and Dv are called the “first” and “second” regions, respectively, of the RWG function. In addition, RWG functions are indexed by first considering the nonmetallic surfaces, which involve NDN basis functions. On these surfaces, which separate two dielectric regions, both the electric and magnetic currents are expanded in a series of the same set of RWG functions bn(r) for n = 1, 2,…, ND. The remaining (NND) RWG functions, if any, are defined on metallic surfaces to expand the electric current. Using a Galerkin scheme for the discretization, we employ the same set of RWG functions as the testing functions, i.e., there are N RWG functions to test the boundary conditions.

[9] In general, discretizations of JMCFIE lead to (N + ND) × (N + ND) dense matrix equations in the form of

equation image

where

equation image
equation image

are column vectors involving the coefficients for the expansion of the electric and magnetic currents. Matrix elements and the elements of the right-hand side (RHS) vector are derived in the next sections.

2.2. Matrix Elements

[10] Consider the interaction between a basis function bn(r) and a testing function tm(r), and let a dielectric region Du be common for the two functions. Then, the corresponding matrix element in the diagonal partition equation image(11) in (6) can be written as

equation image

where we use the “equation image” notation to indicate the “cumulative addition operation,” since the value in (9) may not be the only contribution to Zmn(11). Specifically, if tm(r) and bn(r) are on the same nonmetallic surface, both regions of these functions are common, and the corresponding matrix element Zmn(11) involves two sets of contributions, i.e., interactions of the basis and testing functions through the two regions.

[11] In (9), the integro-differential operators equation imageu and equation imageu for region Du are applied on the basis function, i.e.,

equation image
equation image

where PV indicates the principal value of the integral, ku = ωequation image is the wave number, and

equation image

denotes the homogeneous-space Green's function. Using a Galerkin scheme, both equation imageu and equation imageu operators are well tested in the diagonal partitions of JMCFIE [Ylä-Oijala and Taskinen, 2005a, 2005b; Ylä-Oijala et al., 2005b]. In (9), the signs γm = ±1 and γn = ±1 are determined by the orientation of the basis and testing functions. If the common region Du is the “first” region for the basis (testing) function, then γn = +1 (γm = +1); otherwise, γn = −1 (γm = −1).

[12] When the basis function bn(r) is not on a metallic surface, i.e., nND, there exists a matrix element Zmn(12) in (6). A contribution to this element due to the interaction of the basis and testing functions through the common region Du can be written as

equation image

where ηu is the impedance of the region. As opposed to the diagonal partitions, equation imageu and equation imageu operators are weakly tested in (13) using a Galerkin scheme. When the testing function tm(r) is not on a metallic surface, i.e., mND, there exists a matrix element Zmn(21) with a contribution as

equation image

Finally, when both basis and testing functions are not on metallic surfaces, there exists a matrix element Zmn(22), which is equal to the corresponding element of equation image(11), i.e.,

equation image

for mND and nND. If a structure does not involve any metallic surfaces, the diagonal partitions of the matrix equations obtained with JMCFIE are identical. This is a desirable property in terms of conditioning and iterative convergence [Ylä-Oijala et al., 2005b].

2.3. RHS Vector

[13] The RHS vector in (6) is obtained by testing the incident electromagnetic fields. In general, each nonmetallic region may host some external sources that produce incident electric and magnetic fields, i.e., Euinc(r) and Huinc(r). Consider an RWG function tm(r) on the surface of a nonmetallic region Du. The incident fields in Du are tested by tm(r) and added to the related element in the upper partition of the RHS vector, i.e.,

equation image

where γm = ±1 is determined by the orientation of tm(r). In addition, when the testing function tm(r) is not on a metallic surface, i.e., m < ND, there exists a corresponding element in the second partition of the RHS vector in (6). Contribution to this element due to the incident fields in region Du can be written as

equation image

3. MLFMA Solutions of Electromagnetics Problems Formulated With JMCFIE

[14] Matrix equations obtained with JMCFIE can be solved iteratively by employing a Krylov subspace algorithm, where the required matrix-vector multiplications are performed efficiently with MLFMA in &#55349;&#56490;(N logN) time using &#55349;&#56490;(N logN) memory [Song et al., 1997]. A multilevel tree structure with &#55349;&#56490;(logN) levels is constructed by placing the object in a cubic box and recursively dividing the computational domain into subdomains (clusters). Then, interactions of the basis and testing functions that are far from each other can be calculated approximately and efficiently in a group-by-group manner. In general, each matrix-vector multiplication performed by MLFMA involves three main stages called aggregation, translation, and disaggregation. These stages, which are performed on the multilevel tree structure, can be summarized as follows:

[15] 1. The first stage is aggregation. Radiated fields of clusters are calculated from the bottom of the tree structure to the highest level. In the lowest level, the radiated field of a cluster is obtained by combining the radiation patterns of the basis functions inside the cluster. In the upper levels, the radiated field of a cluster is the combination of the radiated fields of its subclusters.

[16] 2. The second stage is translation. Radiated fields computed during the aggregation stage are translated into incoming fields. For each cluster at any level, there are &#55349;&#56490;(1) clusters to translate the radiated field to.

[17] 3. The third stage is disaggregation. Total incoming fields at cluster centers are calculated from the top of the tree structure to the lowest level. The total incoming field for a cluster is obtained by combining incoming fields due to translations and the incoming field to the center of its parent cluster, if it exists. In the lowest level, incoming fields are received by testing functions.

[18] MLFMA is investigated extensively in various references [e.g., Song et al., 1997; Sheng et al., 1998; Chew et al., 2001; Donepudi et al., 2003]. In this paper, we provide only the specific details of our implementation.

[19] 1. For a general problem involving U regions, MLFMA must be applied for each nonmetallic region separately [Donepudi et al., 2003; Luo and Lu, 2007; Fostier and Olyslager, 2008]. This is because the Green's function depends on the electromagnetic parameters of the region, i.e., εu and μu. In addition, radiated and incoming fields of clusters are defined and sampled on the unit sphere, and the number of samples depends on the cluster size as measured by the wavelength [Koc et al., 1999]. Hence, the sampling rate and the resulting tree structure also depend on the electromagnetic parameters of the region.

[20] 2. For each region Du, we perform four matrix-vector multiplications with the four partitions of the system matrix, i.e.,

equation image
equation image

where the coefficients xJ and xM are provided by the iterative algorithm. We note that a set of aggregation, translation, and disaggregation stages is performed once for a multiplication with a partition, although each partition involves some combination of the integro-differential operators. This is possible, since the radiated and incoming fields do not depend on the type of the integro-differential operator [Ergül and Gürel, 2009]. Only the receiving patterns of the testing functions depend on the operator and the testing type, i.e., equation image · equation image, equation image × equation image, equation image · equation image, and equation image × equation image.

[21] 3. At the beginning of an aggregation stage, radiation patterns of the RWG functions are multiplied with the coefficients provided by the iterative algorithm and combined to obtain the radiated fields of the clusters in the lowest level. In the aggregation stage performed for a region Du, only the RWG functions located on the surface of the region (Su) are considered. Besides, for the partitions equation imageu(12) and equation imageu(22), basis functions located on metallic surfaces are omitted.

[22] 4. At the end of a disaggregation stage, incoming fields are received by the testing functions. Similar to the aggregation stage, a disaggregation stage performed for a region Du involves only the RWG functions located on the surface of the region. In addition, testing functions located on metallic surfaces do not receive incoming fields for the partitions equation imageu(21) and equation imageu(22).

[23] 5. The signs γm and γn in (9), (13), and (14), are introduced when the radiation patterns of the basis functions are combined or when the incoming fields are multiplied with the receiving patterns of the testing functions in the lowest level.

4. Block-Diagonal Preconditioning of JMCFIE

[24] MLFMA provides the solution of large problems by reducing the complexity of the matrix-vector multiplications required by the iterative solvers from &#55349;&#56490;(N2) to &#55349;&#56490;(N log N). For efficient solutions, however, the number of iterations should also be small, in addition to fast matrix-vector multiplications. In general, JMCFIE is a second-kind integral equation, and its Galerkin discretization involves well tested identity operators, which lead to well conditioned matrix equations [Ylä-Oijala and Taskinen, 2005a, 2005b; Ylä-Oijala et al., 2005b]. Matrix equations obtained with JMCFIE are easy to solve iteratively, especially when the contrasts between the neighboring dielectric regions are low, i.e., permittivity and permeability do not change significantly across dielectric interfaces. However, iterative solutions of JMCFIE become difficult as the contrast increases [Ylä-Oijala et al., 2005b; Ylä-Oijala, 2008; Ergül and Gürel, 2009], and effective preconditioners are required to reduce the number of iterations, especially when the problem size is large.

4.1. Effect of the Contrast in JMCFIE

[25] When a problem does not involve any metallic surface, the diagonal partitions of JMCFIE are identical. For composite structures with metallic surfaces, however, these partitions are not identical, and they have different sizes, which may deteriorate the numerical balance of the matrix equations. On the other hand, iterative solutions of JMCFIE become difficult with the increasing contrast, even in the case of nonmetallic objects. The main reason is the existence of off-diagonal partitions, which are numerically sensitive to the contrast. In general, off-diagonal partitions of JMCFIE are significantly unbalanced due to multiplications with ηu−1 and ηu in (13) and (14), respectively. Although this may not be critical for low contrasts, the off-diagonal partition equation image(21) dominates the overall matrix, as the contrast of the object increases. As a result, the overall matrix equation becomes significantly unbalanced and difficult to solve iteratively.

[26] For a further analysis, we consider a special case involving a dielectric object in homogeneous space. Matrix elements in this case are derived explicitly in Appendix A. In general, the numerical significance of the off-diagonal equation image(21) grows rapidly with the increasing contrast. Our investigations also show that a combined operator (η0equation image0η1equation image1) presents a major contribution in equation image(21) for relatively high contrasts. The related term can be written as

equation image

Using a Taylor series expansion for the exponential in the Green's function,

equation image

and

equation image

where R = (rr′) = equation imageR. We note that 1/R and 1/R2 singularities in (21) and (22) exist when μ0μ1 and equation image0equation image1, respectively. In addition, numerical values of the expressions in (21) and (22), thus the contribution of (η0equation image0η1equation image1) in equation image(21), grow rapidly with the increasing contrast. Finally, the resulting matrix equation becomes significantly unbalanced, due to large elements in equation image(21).

4.2. Preconditioning

[27] In general, the matrix equation in (6) can be preconditioned as

equation image

where equation image is a (N + ND) × (N + ND) preconditioner matrix. In MLFMA, there are &#55349;&#56490;(N) near-field interactions, which are calculated directly and are available for constructing preconditioners. These interactions are between the basis and testing functions that are located in the same cluster or in two touching clusters in the lowest level of the tree structure. During solutions with MLFMA, we reorder the RWG functions according to their positions in the multilevel tree. Let C be the number of clusters in the lowest level and N(c) represent the number of RWG functions in cluster c = 1, 2,…, C. Then, the RWG functions in cluster c are indexed from NT(c) + 1 to NT(c) + N(c), where

equation image

This way, the system matrix in (6) has a block structure, where each block represents the interaction of a pair of lowest-level clusters. In the sparse near-field matrix, only the blocks corresponding to the self interactions of the clusters or the interactions of two touching clusters involve nonzero elements.

[28] The block-diagonal preconditioner (BDP), which is based on using the self interactions of the lowest-level clusters, is commonly used to accelerate MLFMA solutions of electromagnetics problems involving perfectly conducting objects [Song et al., 1997; Chew et al., 2001]. The preconditioner matrix, which has a block-diagonal structure, can be inverted and used efficiently with &#55349;&#56490;(N) complexity. Although BDP is successful in reducing the iteration counts for second-kind integral equations [Song et al., 1997], such as the combined-field integral equation (CFIE), it may not accelerate the iterative solutions of first-kind integral equations [Gürel and Ergül, 2006], such as the electric field integral equation (EFIE). In fact, EFIE solutions are usually decelerated with BDP [Gürel and Ergül, 2003], and BDP is rarely useful for EFIE [Ubeda et al., 2006; Ergül et al., 2007].

4.3. Two-Partition Block-Diagonal Preconditioner

[29] A direct extension of BDP for dielectric problems, which we call 2PBDP, involves the self interactions of the lowest-level clusters in the diagonal partitions, i.e.,

equation image

where

equation image
equation image

are block-diagonal matrices. Then, a preconditioned matrix equation can be written as

equation image

where

equation image
equation image

are also block-diagonal matrices. As presented in section 5, 2PBDP accelerates iterative solutions of problems involving dielectric regions with relatively low contrasts. As the contrast increases, however, 2PBDP, which does not employ the large elements in equation image(21), can be insufficient to accelerate the iterations. For those problems, better preconditioners are required to reduce the iteration counts and to increase the efficiency of the solutions.

4.4. Four-Partition Block-Diagonal Preconditioner

[30] To improve the iterative solutions of JMCFIE, we propose 4PBDP, which is based on using the diagonal blocks, i.e., self interactions of the lowest-level clusters, in all four partitions of the matrix equations. This way, some of the large elements in equation image(21) are considered in constructing an effective preconditioner. The resulting preconditioner matrices are in the form of

equation image

where equation imageN×N(11) and equation imageequation image are block-diagonal matrices as in 2PBDP. The partitions

equation image
equation image

are also block matrices involving the self interactions of the clusters. However, these partitions are square and block-diagonal only when a problem does not involve metallic surfaces (ND = N); otherwise, they are rectangular matrices. In addition, the blocks in equation imageequation image and equation imageequation image are not necessarily square, and some of them can be rectangular, depending on the object and the clustering scheme in MLFMA.

[31] Using 4PBDP, a preconditioned matrix equation can be written as

equation image

where

equation image
equation image
equation image
equation image

and

equation image

is the Schur complement of equation imageN×N(11) [Gürel and Chew, 1990]. Matrix operations in (35)–(39), i.e., matrix-matrix multiplications, the inversion of equation imageN×N(11), and the inversion of equation imageequation image can be performed efficiently in &#55349;&#56490;(N) time using &#55349;&#56490;(N) memory. Our numerical experiments show that the extra cost of 4PBDP with respect to 2PBDP is always negligible, considering the overall cost of the solutions with MLFMA. Nevertheless, as demonstrated in the next section, 4PBDP can significantly improve the efficiency of solutions by reducing iteration counts, and it is especially useful when the acceleration provided by 2PBDP is not sufficient.

5. Results

[32] In this section, we present iterative solutions for various scattering problems and investigate their iteration counts when the solutions are accelerated with 2PBDP and 4PBDP, in addition to the no-preconditioner (NP) case. Scatterers are illuminated by a plane wave propagating in the −x direction with the electric field polarized in the y direction. Surfaces are discretized with about λ0/10 mesh size, where λ0 is the wavelength in medium D0 (free space) that extends to infinity. Iterative solutions are performed using the biconjugate-gradient-stabilized (BiCGStab) algorithm [Van der Vorst, 1992], which is known to provide rapid solutions for second-kind integral equations. In all solutions, matrix-vector multiplications are accelerated via MLFMA, and the relative residual error for the iterative convergence is set to 10−3.

[33] First, we consider scattering problems involving a dielectric sphere. Figure 1a presents the number of BiCGStab iterations when the relative permittivity of the sphere is 2.0 and the radius of the sphere (a) changes from λ0 to 7.5λ0. Discretizations of problems lead to 4142 and 412,998 unknowns, respectively, for radii 0.75λ0 and 7.5λ0. As depicted in Figure 1a, 2PBDP accelerates the iterative convergence substantially, compared to the NP case. Using 4PBDP further reduces the iteration counts, but the improvement with respect to 4PBDP is considerable only when a = 6λ0. Figure 1b presents the iteration counts with respect to the number of unknowns when the relative permittivity of the sphere is 4.0. In general, iterative solutions become difficult with increasing contrast. For a radius of 6λ0, convergence is not achieved in 1000 iterations without preconditioning. Furthermore, unlike the low-contrast (εr = 2.0) case, 2PBDP is unable to reduce the iteration counts when εr = 4.0. When we use 4PBDP, however, iterative solutions are accelerated significantly, and we obtain efficient solutions.

Figure 1.

Iteration counts for the solution of scattering problems involving a dielectric sphere with a relative permittivity of (a) 2.0 and (b) 4.0, when the radius of the sphere is in the range of 0.75λ0 to 7.5λ0, where λ0 is the wavelength in free space.

[34] Figure 2 depicts iteration counts for the solution of scattering problems involving a spherical object with multiple dielectric regions. A dielectric sphere of radius a is coated with a dielectric shell of radius 2a, where a changes from 0.5λ0 to 2.5λ0. Discretizations of problems lead to 13,176 and 316,032 unknowns, respectively, when a = 0.5λ0 and a = 2.5λ0. Figure 2a presents iteration counts with respect to the number of unknowns when the relative permittivities of the core and the shell are 4.0 and 2.0, respectively. In this case, 2PBDP reduces iteration counts substantially in comparison to the NP case, while 4PBDP does not provide a significant improvement over 2PBDP. On the other hand, when the permittivity of the shell and the core are exchanged, we obtain the iteration counts depicted in Figure 2b, where 4PBDP presents a superior performance in comparison to 2PBDP. Owing to the relatively high contrast between the shell and free space, solutions of JMCFIE become difficult without preconditioning. For example, when a = 1.67λ0, convergence cannot be achieved in 1000 iterations. 2PBDP accelerates the convergence for large problems, but the improvement is not sufficient. Using 4PBDP, the number of iterations is less than 100 for all solutions in Figure 2.

Figure 2.

Iteration counts for the solution of scattering problems involving a dielectric sphere of radius a coated with a dielectric shell of radius 2a, where a changes from 0.5λ0 to 2.5λ0. (a) Low-contrast case when the relative permittivities of the core and shell are 4.0 and 2.0, respectively. (b) High-contrast case when the relative permittivities of the core and shell are 2.0 and 4.0, respectively.

[35] Next, we consider iterative solutions of scattering problems involving a spherical composite object. In this case, a metallic sphere of radius a is coated with a dielectric shell of radius 2a, where a changes from 0.5λ0 to 2.5λ0. Figure 3 presents iteration counts with respect to the number of iterations. Similar to the previous example, 2PBDP reduces the iteration counts significantly for the low-contrast case, i.e., when the relative permittivity of the shell is 2.0, as depicted in Figure 3a. In this case, 4PBDP provides some improvement over 2PBDP, as the problem size grows. When the relative permittivity of the shell is 4.0, however, 4PBDP accelerates the iterative solutions significantly, compared to 2PBDP. In fact, 2PBDP decelerates the solutions for large problems, and there is a large discrepancy between the performances of 2PBDP and 4PBDP. Using 4PBDP, the number of iterations is again less than 100 for all solutions in Figure 3.

Figure 3.

Iteration counts for the solution of scattering problems involving a perfectly conducting sphere of radius a coated with a dielectric shell of radius 2a, where a changes from 0.5λ0 to 2.5λ0. The relative permittivity of the shell is (a) 2.0 and (b) 4.0.

[36] We also consider electromagnetics problems involving dielectric and composite objects with sharp edges and corners. Figure 4 presents iteration counts for the solution of scattering problems involving a coated dielectric cube. The core and shell have edges of a and 2a, respectively, where a changes from 0.5λ0 to 2.5λ0. Faces of the object are parallel to the coordinate axes. Discretizations of problems lead to matrix equations with 9864 to 228,132 unknowns. Figure 4a depicts iteration counts as a function of the number of unknowns, when the relative permittivities of the core and shell are 4.0 and 2.0, respectively. The results are similar to those for the spherical object depicted in Figure 2a, i.e., 2PBDP accelerates the iterative solutions significantly, and 4PBDP further reduces the iteration counts slightly compared to 2PBDP. When the relative permittivities of the core and shell are exchanged, however, 4PBDP performs much better than 2PBDP, as depicted in Figure 4b. On the other hand, unlike the solutions of the spherical object in Figure 2b, 2PBDP is also effective for the high-contrast case in Figure 4b. This is probably due to the larger numbers of elements being used in constructing 2PBDP for the cubic object, i.e., the lowest-level clusters in the multilevel tree are more populated for the cubic object than for the spherical object. Nevertheless, 4PBDP is again preferable for all solutions in Figure 4.

Figure 4.

Iteration counts for the solution of scattering problems involving a dielectric cube coated with a dielectric shell. The core and shell have edges of a and 2a, respectively, where a changes from 0.5λ0 to 2.5λ0. (a) Low-contrast case when the relative permittivities of the core and shell are 4.0 and 2.0, respectively. (b) High-contrast case when the relative permittivities of the core and shell are 2.0 and 4.0, respectively.

[37] Figure 5 presents the solution of scattering problems involving a coated metallic cube. The sizes of the core and shell are the same as those in the coated dielectric cube. Similar to previous examples, 4PBDP provides the most efficient results, and it presents improved convergence in comparison to 2PBDP, when the contrast is relatively high.

Figure 5.

Iteration counts for the solution of scattering problems involving a perfectly conducting cube coated with a dielectric shell. The core and shell have edges of a and 2a, respectively, where a changes from 0.5λ0 to 2.5λ0. The relative permittivity of the shell is (a) 2.0 and (b) 4.0.

[38] In Figure 6, we present the solution of scattering problems involving a spherical object with three dielectric regions. A dielectric sphere of radius 0.3a is coated with two dielectric shells of radii 0.5a and a, where a changes from 2λ0 to 2.6λ0. Relative permittivities of the core, inner shell, and outer shell are 1.44, 1.96, and 4.0, respectively. Only one discretization involving 215,304 unknowns is used for the entire frequency range. As depicted in Figure 6, solutions are performed efficiently with maximum 101 iterations using 4PBDP. Figure 6 also presents the normalized radar cross section (RCS/πa2) in the backscattering direction as a function of a in terms of the wavelength. We observe that computational values are in agreement with analytical values obtained by Mie-series solutions.

Figure 6.

Iteration counts for the solution of scattering problems involving a dielectric sphere of radius 0.3a coated with two dielectric shells of radii 0.5a and a, where a changes from 2λ0 to 2.6λ0. Relative permittivities of the core, inner shell, and outer shell are 1.44, 1.96, and 4.0, respectively. Normalized RCS (RCS/πa2) of the structure in the backscattering direction is also plotted as a function of a in terms of wavelength.

[39] Finally, to demonstrate the effectiveness of an implementation involving MLFMA, JMCFIE, and 4PBDP, we present the solution of large scattering problems discretized with 1,264,128 unknowns. A sphere of radius 5λ0 is placed inside another sphere of radius 10λ0. We consider four different cases: (1) Relative permittivities of the core and shell are 4.0 and 2.0, respectively. (2) Relative permittivities of the core and shell are 2.0 and 4.0, respectively. (3) The core is metallic and the relative permittivity of the shell is 2.0. (4) The core is metallic and the relative permittivity of the shell is 4.0.

[40] Scattering problems are solved via 6-level MLFMA, and the iteration counts are 101, 283, 75, and 187, respectively, for 10−3 residual error. Figures 7 and 8 present the normalized bistatic RCS (RCS/λ02) values on the x-z plane, where 0° and 180° correspond to the forward scattering and backscattering directions, respectively. We observe that computational and analytical results agree perfectly.

Figure 7.

Normalized bistatic RCS (RCS/λ02) of a structure involving spheres of radii 5λ0 and 10λ0, when (a) relative permittivities of the core and shell are 4.0 and 2.0, respectively, and (b) relative permittivities of the core and shell are 2.0 and 4.0, respectively.

Figure 8.

Normalized bistatic RCS (RCS/λ02) of a structure involving spheres of radii 5λ0 and 10λ0, when (a) the core is metallic and the relative permittivity of the shell is 2.0, and (b) the core is metallic and the relative permittivity of the shell is 4.0.

6. Concluding Remarks

[41] In this paper, we present an efficient solution of JMCFIE using MLFMA and block-diagonal preconditioners. We provide the details of an MLFMA implementation for the solution of electromagnetics problems involving multiple dielectric and metallic regions. In general, JMCFIE is a preferable formulation that provides well conditioned matrix equations that are easy to solve iteratively. However, iterative solutions of JMCFIE can be difficult for problems involving dielectric regions with relatively high contrasts. This is mostly due to the numerical imbalance of the off-diagonal partitions of the matrix equations obtained with JMCFIE. To accelerate the iterative solutions, we present 4PBDP, which is an efficient preconditioner based on using the diagonal blocks in all four partitions of the matrix equations. We show that 4PBDP reduces the iteration counts significantly and performs better than the standard 2PBDP, which can be insufficient to improve the iterative convergence as the contrast increases.

Appendix A

[42] In this Appendix A, we present the matrix equations for the special cases depicted in Figure A1, i.e., a dielectric object and a coated dielectric or metallic object in homogeneous space. We assume that the incident fields exist only in region D0, which extends to infinity.

Figure A1.

Electromagnetics problems involving (a) a single dielectric object and (b) a coated dielectric or metallic object located in homogeneous space.

[43] In the case of a dielectric object as depicted in Figure A1a, there are two regions, namely, D0 and D1. Since there is no metallic surface, ND = N, and the size of the matrix equation obtained from the discretization of JMCFIE is 2N × 2N. All RWG functions are located on the same surface, and each matrix element involves two sets of contributions due to the interactions of the basis and testing functions through the regions D0 and D1. Using (9), (13), and (14), we obtain

equation image
equation image
equation image

for m, n = 1, 2,…, N. In addition, by testing the incident electric and magnetic fields, elements of the RHS vector can be calculated as

equation image
equation image

for m = 1, 2,…, N.

[44] In the case of a coated dielectric object, there are three nonmetallic regions, namely, D0, D1, and D2, while D1 is between D0 and D2. Since there is no metallic surface, ND = N, and the size of the resulting matrix equation is again 2N × 2N. Let the first N01 RWG functions and the remaining N12 = (NN01) RWG functions be defined on surfaces S01 and S12, respectively. Each partition in (6) can be divided into four subpartitions, i.e.,

equation image

for a = 1, 2 and b = 1, 2. In (A6), equation image(ab,11) and equation image(ab,22) represent N01 × N01 and N12 × N12 matrices containing the interactions of the RWG functions located on S01 and S12, respectively. Calculations of these interactions are similar to those in (A1)–(A3). For m, n = 1, 2,…, N01,

equation image
equation image
equation image

For m, n = (N01 + 1), (N01 + 2),…, (N01 + N12), and (m′, n′) = (mN01, nN01),

equation image
equation image
equation image

On the other hand, the off-diagonal subpartitions, i.e., equation image(ab,12) and equation image(ab,21), involve the interactions of the basis and testing functions that are located on different surfaces. These basis and testing functions interact only through the region D1. For m = 1, 2,…, N01, n = (N01 + 1), (N01 + 2),…, (N01 + N12), and n′ = nN01,

equation image
equation image
equation image

For m = (N01 + 1), (N01 + 2),…, (N01 + N12), n = 1, 2,…, N01, and m′ = mN01,

equation image
equation image
equation image

Finally, incident electromagnetic fields are tested only by the RWG functions located on the surface S01, and the elements of the RHS vector for m = 1, 2,…, N01 are calculated as in (A4) and (A5).

[45] Next, we consider a coated metallic object in homogeneous space. In this case, there are two nonmetallic regions, i.e., D0 and D1, and the size of the matrix equation is (N + ND) × (N + ND), where ND < N. Let the first ND RWG functions be located on the surface S01 and the remaining N12 = (NND) RWG functions be located on the metallic surface S12. Then, the expressions for the matrix elements in the diagonal partition equation image(11) are the same as the expressions in (A7), (A10), (A13), and (A16), which are derived for the coated dielectric object. On the other hand, the off-diagonal partition equation image(12) becomes a rectangular matrix with two subpartitions, i.e.,

equation image

where the matrix elements Zmn(12,11) and Zmn(12,21) for m, n = 1, 2,…, ND and m′ = 1, 2,…, N12 are calculated as in (A8) and (A17). The off-diagonal partition equation image(21) also becomes a rectangular matrix, i.e.,

equation image

where the subpartitions Zmn(21,11) and Zmn(21,12) for m, n = 1, 2,…, ND, and n′ = 1, 2,…, N12 are calculated as in (A9) and (A15). The diagonal partition equation image(22) is an ND × ND matrix with elements Zmn(22) = Zmn(11) for m, n = 1, 2,…, ND. Finally, the elements of the RHS vector are calculated as in (A4) and (A5) for m = 1, 2,…, ND.

Acknowledgments

[46] This work was supported by the Turkish Academy of Sciences in the framework of the Young Scientist Award Program (LG/TUBA-GEBIP/2002-1-12), by the Scientific and Technical Research Council of Turkey (TUBITAK) under research grants 105E172 and 107E136, and by contracts from ASELSAN and SSM.

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