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Keywords:

  • surface integral equation;
  • method of moments;
  • electromagnetic scattering;
  • dielectric object;
  • composite object;
  • iterative solution

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[1] In this paper, formulation of the surface integral equations for solving electromagnetic scattering by dielectric and composite metallic and dielectric objects with iterative methods is studied. Four types of formulations are considered: T formulations, N formulations, the combined field integral equation formulation, and the Müller formulation. By studying properties of the integral equations and their testing in the Galerkin method, “optimal” forms for each formulation type are derived. Numerical examples demonstrate that the developed new formulations lead to clear improvements in the convergence rates when the matrix equations are solved iteratively with the generalized minimal residual method. Both the Rao-Wilton-Glisson and Trintinalia-Ling (TL) basis functions are used in expanding the unknown electric and magnetic surface current densities. In particular, the first-order TL basis functions are required in the N formulations to maintain the solution accuracy when the surfaces include sharp edges.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[2] The surface integral equation method is one of the most popular numerical methods in the electromagnetic scattering analysis of metallic, homogeneous dielectric and composite metallic and dielectric objects [see, e.g., Kolundzija and Djordjevic, 2002]. Traditional surface integral equation methods are, however, restricted to rather small-scale problems, because discretization of an integral equation results in a system matrix which is fully populated and expensive to store and solve. In order to consider complex large-scale problems including a high number of unknowns, advanced methods like the fast multipole method [Rokhlin, 1990; Chew et al., 2001] and the adaptive integral method [Bleszynski et al., 1996], are required. These fast integral equation methods are based on the iterative solution of the matrix equation. Consequently, for the fast methods to be “fast” it is important to have sufficiently rapidly converging iterative solutions. However, many of the traditional surface integral equation formulations lead to ill-conditioned matrix equations and slowly converging iterative solutions [Zhao and Chew, 2000]. Conditioning of the system matrix can be improved with a better choice of an integral equation formulation [Wilton and Wheeler, 1991; Lloyd et al., 2004] and in some cases the number of iterations may be further decreased, for example, with preconditioning [Kas and Yip, 1987] or with a better choice of an iterative solver [Kolundzija and Sarkar, 2001].

[3] In the case of metallic objects the fundamental integral equations are the electric field integral equation (EFIE), the magnetic field integral equation (MFIE) and the combined field integral equation (CFIE) [Mautz and Harrington, 1978]. Iterative solution of these formulations have been studied [e.g., Wilton and Wheeler, 1991; Song and Chew, 1995; Pocock and Walker, 1997; Makarov and Vedantham, 2002].

[4] For homogeneous dielectric objects there are infinitely many possibilities to derive a set of coupled integral equation formulations [Harrington, 1989]. The most popular formulations are the PMCHWT (Poggio-Miller-Chang-Harrington-Wu-Tsai) formulation [Mautz and Harrington, 1979] and the Müller formulation [Müller, 1969]. In recent years iterative solution of these formulations have attained a lot attention in the literature. Yeung [1999] reported that the PMCHWT formulation leads to slowly converging iterative solutions. Li et al. [2002] and Lloyd et al. [2004] showed that convergence of the PMCHWT formulation can be improved with an appropriate choice of the coupling coefficients. Lloyd et al. [2004] and Zhu and Gedney [2004] observed that the Müller formulation leads to faster converging iterative solutions than the PMCHWT formulation.

[5] These integral equation formulations can be rather straightforwardly extended for piecewise dielectric and composite metallic and dielectric objects [see, e.g., Kishk and Shafai, 1986]. However, in the case where three or more subregions meet at a common edge, and form a junction, a special care is required [Kolundzija, 1999; Ylä-Oijala et al., 2005]. Lu and Luo [2003] apply coupled EFIE-PMCHWT formulation for scattering by metallic objects with dielectric coating and observed that iterative solution converges very slowly. Zhu and Gedney [2004] apply CFIE-Müller and CFIE-PMCHWT formulations for a similar problem and found that the CFIE-Müller formulation gives a better conditioned matrix equation.

[6] In addition to the conditioning of the system matrix and the convergence of an iterative solver an important issue associated with the formulation of integral equations is the accuracy of the solution. For example, it is well known that in the case of nonsmooth metallic objects the MFIE formulation with the low-order basis functions, like the Rao-Wilton-Glisson (RWG) [Rao et al., 1982] functions, leads to poorer accuracy than the EFIE formulation. Recently, Ergül and Gürel [2004] observed that the accuracy of the MFIE formulation can be significantly improved by applying the Trintinalia-Ling (TL) [Trintinalia and Ling, 2001] first-order basis functions. In addition, Zhu and Gedney [2004] reported that the accuracy of the Müller formulation decreases compared to the PMCHWT formulation when the permittivity of the object is increased.

[7] In this paper we study formulation of the surface integral equations and their properties in solving electromagnetic scattering by homogeneous dielectric and piecewise dielectric objects as well as by closed metallic and composite objects with iterative methods. Four types of formulations are considered: T formulations, N formulations, combination of the T and N formulations, that is, the CFIE formulation, and the Müller formulation. After having studied the properties of the integral equations and their testing in the Galerkin method, we derive “optimal” form for each formulation type. The developed new formulations lead to clear improvements in the conditioning of the system matrices and in the convergence rates when the matrix equations are solved iteratively with the generalized minimal residual (GMRES) method. Both the RWG and TL basis functions are used in expanding the unknown electric and magnetic surface current densities. The TL basis functions are required in the N formulations to maintain the solution accuracy in the case of metallic objects with sharp edges and nonsmooth dielectric objects with a high electric permittivity.

2. Electric and Magnetic Field Integral Equations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[8] Consider time-harmonic electromagnetic scattering (time factor is eiωt) by a composite object made of metallic and homogeneous dielectric regions. Let Dl, l = 1,…, L, denote the regions with constant, possibly complex, electromagnetic parameters ɛl and μl. The surface of the domain Dl is denoted by Sl and nl denotes the inner unit normal of Sl pointing into domain Dl. Let Elinc and Hlinc denote the incident fields in Dl due to the sources in Dl and vanishing if there are no sources in Dl.

[9] The electric and magnetic field integral equations, EFIEl and MFIEl, of the domains Dl, l = 1,…, L, can be formulated in two alternative ways [Kolundzija and Djordjevic, 2002]. We define the T equations, T-EFIEl and T-MFIEl, as

  • equation image
  • equation image

where tan denotes tangential component. The other set of equations, the N equations, N-EFIEl and N-MFIEl, are

  • equation image
  • equation image

Here Jl = nl × Hl and Ml = −nl × El are the equivalent electric and magnetic surface current densities on Sl and El and Hl are the total fields in Dl. The integral operators Els and Hls are defined on the surface Sl as follows [Kolundzija and Djordjevic, 2002]

  • equation image
  • equation image

where ηl = equation image and the operators equation imagel and equation imagel are

  • equation image
  • equation image

Here Gl is the homogeneous space Green's function of domain Dl and kl = ωequation image.

3. Galerkin Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[10] Equations (1)–(4) are solved with the method of moments [Harrington, 1968] using Galerkin testing procedure. Let us next suppose that the surfaces are divided into some simple elements. The unknown surface current densities, Jl and Ml, are expanded as linear combinations of the given divergence conforming basis functions fn defined on these elements as

  • equation image

Note that on the metallic surfaces Ml vanishes and thus, in the presence of metallic objects the number of the magnetic unknowns βn is lower than the number of the electric unknowns αn.

3.1. Criteria for an Optimal and Stable Method

[11] Since in the Galerkin method the basis and testing functions are equal, for an application of the Galerkin method the basis functions should span both the domain and range of an integral operator [Sarkar et al., 1985]. This requirement is, in particular, satisfied if the domain and range spaces coincide. Here we, however, do not require that the domain and range spaces should be mathematically equal, rather we say that functions which have similar physical properties (e.g., the same boundary conditions) and have equal dimension belong to the same function space.

[12] In the Galerkin method it is essential to test the equations so that the resulting matrix becomes diagonally dominant, that is, largest elements appear on the diagonal [Kolundzija and Djordjevic, 2002] and that all unknowns become well tested [Sheng et al., 1998]. Let fm be a testing function and let fn be a basis function. An integral operator equation image is said to be well tested if the associated matrix equation with elements

  • equation image

is well conditioned and has a low condition number.

[13] To summarize, in this paper we seek a Galerkin solution of equations (1)–(4) which satisfies the following two criteria:

[14] 1. Applicability: For the Galerkin method to be applicable the range and domain space of an integral operator should be (physically) equal.

[15] 2. Stability: For the Galerkin method to be stable the resulting system matrix should be diagonally dominant and all unknowns should be well tested.

[16] It is not necessary that all integral operators of an integral equation system are well tested and usually suffices to test well at least one integral operator per each unknown. In addition, although both the above criteria are satisfied, the solution of the Galerkin method is not necessarily accurate and we have to consider the accuracy of the solution separately. Here the important factor is the choice of the basis functions.

3.2. Mapping Properties

[17] Let us first study the mapping properties of the integral operators in a source free medium. Intuitively, by using the definition of the surface currents, we find that the operators in equations (1)–(4) have the following mapping properties:

  • equation image

In other words, operators (Els)tan and (Hls)tan map the currents onto the tangential field components, n × M and −n × J, respectively, and operators nl × Els and nl × Els map the currents onto −M and J, respectively. Since the spaces of the surface current densities and the tangential field components are not physically identical, the first criterion of section 3.1 can be satisfied only by operators n × Els and n × Hls.

[18] Next we (approximately) modify the operators (Els)tan and (Hls)tan by using the impedance boundary conditions

  • equation image

so that also they are mappings onto the currents

  • equation image

An integral operator is said to be in the J or M space if it maps the currents J and M onto J or M, respectively. From (11) and (13) we find that equations 1/η T-EFIE and N-MFIE include operators which are in the J space. Hence we say that these equations are in the J space. For the same reason, equations ηT-MFIE and -N-EFIE belong to the M space. By combining the equations which are in the same space, we get the following two combined field integral equations (CFIE) in region Dl

  • equation image
  • equation image

where al, bl, cl and dl, l = 1,…, L, are appropriate complex coefficients. Equation (14) is called an electric current CFIE (JCFIE) and (15) is called a magnetic current CFIE (MCFIE) [Ylä-Oijala and Taskinen, 2005a].

[19] If we now consider equations (14) and (15) together as a matrix valued operator, where the coefficients al, bl, cl and dl have values 0 or 1, we see that this operator is a mapping from the joint space (J, M) onto itself, and, thus satisfies the first criterion of section 3.1.

3.3. Testing Procedure

[20] Consider next testing of equations (1)–(4) with the Galerkin method. Using the results of Sheng et al. [1998] and Ylä-Oijala and Taskinen [2005a], we find that the following matrix elements become well tested in the Galerkin method,

  • equation image

and the following elements are poorly tested

  • equation image

By studying integral equations (1)–(4) and results (equation (16)) we find that J is well tested in T-EFIE, N-MFIE and in JCFIE, and M is well tested in T-MFIE, N-EFIE and in MCFIE. Thus a stable formulation should include either T-EFIE or N-MFIE, or both, for J and either T-MFIE or N-EFIE, or both, for M.

[21] There is, however, a significant difference in the T and N equations. In the T-EFIE (T-MFIE) J(M) is well tested with respect to the operator equation imagetan and in the N-MFIE (N-EFIE) J (M) is well tested with respect to the operator n × equation image + equation image, where equation image denotes the identity operator. In the case of metallic objects operators equation imagetan and n × equation image + equation image correspond to the operators of the EFIE and MFIE formulations. Hence these operators are here called as the EFIE and MFIE operators, respectively. This also motivates us call the T equations as “EFIE-type” equations and the N equations as “MFIE-type” equations. Since JCFIE (MCFIE) is a combination of T-EFIE and N-MFIE (N-EFIE and T-MFIE), in JCFIE (MCFIE) J (M) is well tested with respect to both the EFIE and MFIE operators.

[22] The fact that an operator is well tested usually leads to a diagonally dominant matrix equation. However, in the case of coupled integral equations, (e.g., as in (29)), it is essential to choose the coupling coefficients so that any significant cancellation on the diagonal operators does not appear. This issue is more carefully considered in section 7.

[23] Finally, we like to note that some authors [e.g., Sheng et al., 1998; Jung et al., 2002, 2004] consider testing with both fm and n × fm. Because testing with n × fm does not correspond to the Galerkin method, it is not considered here. This kind of coupled testing results in so-called TENETHNH formulation in which equations in the J and M spaces are combined. As shown by Ylä-Oijala and Taskinen [2005a], TENETHNH formulation leads to very slowly converging iterative solutions.

4. Integral Equation Formulations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[24] In this section we derive integral equation formulations which, when solved with the Galerkin method, satisfy the two criteria given in section 3.1.

4.1. PEC and PMC Objects

[25] Consider first closed PEC and PMC objects. Since on the metallic surfaces M vanishes, the equations in the M space can be omitted and the EFIE, MFIE and CFIE formulations satisfying the two criteria of section 3.1 are obtained from (14) with the coefficients (cl = 0 = dl)

  • equation image
  • equation image
  • equation image

These equations corresponds to the usual EFIE, MFIE and CFIE formulations for metallic surfaces.

[26] Analogously on PMC surfaces J vanishes and stable EFIE, MFIE and CFIE-type formulations are obtained from (15) with (al = 0 = bl)

  • equation image
  • equation image
  • equation image

4.2. Homogeneous Dielectric Objects

[27] Let us next consider a homogeneous dielectric object. In this case we have two domains and L = 2. Let D1 denote the exterior and let D2 denote the object. Because of the continuity of the tangential field components on the surface of the object, the surface currents satisfy

  • equation image

Thus we have only two independent unknowns, one electric current and one magnetic current. Since discretization of (1)–(4) gives eight equations for each testing function, in order to have the same number of equations and unknowns the number of equations have to be decreased. First, we combine the equations similarly as in (14) and (15) separately in domains D1 and D2. Next we combine equations (14) and (15) of the domains so that again only equations which are in the same space are combined. More precisely, we consider the following combined formulation

  • equation image

Here we suppose that (1)–(4) are first converted into matrix equations with the Galerkin method using oriented basis and testing functions (see Figure 1 with M = 2 for the oriented basis functions on an interface of two domains) separately in D1 and D2 and thereafter the discretized equations are combined as in (25).

image

Figure 1. Dielectric and composite metal-dielectric junctions with M dielectric regions. Metallic region (PEC) is denoted by shading, and arrows indicate positive direction of the oriented basis functions associated with a junction edge ej.

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[28] Depending on the choice of the coefficients al, bl, cl and dl, l = 1, 2, we can derive several integral equation formulations with different properties. For example, the choice

  • equation image

with l = 1, 2, gives the well-known PMCHWT (Poggio-Miller-Chang-Harrington-Wu-Tsai) formulation [Mautz and Harrington, 1979]. This formulation is here denoted by T-PMCHWT since it includes the T equations only. As can be seen from (11), equations of the T-PMCHWT formulation are in the spaces of tangential field components n × M and −n × J, respectively.

[29] In order to have a T formulation in which the equations are in the J and M spaces, and satisfy the two criteria of section 3.1, we consider the following combined T formulation (CTF)

  • equation image

An N formulation, called as a combined N formulation (CNF), satisfying the two criteria of section 3.1 is obtained as

  • equation image

Analogously a CFIE formulation, denoted by JMCFIE [Ylä-Oijala and Taskinen, 2005a], satisfying the two criteria given in section 3.1 is obtained from (25) with

  • equation image

Another possible formulation for dielectric objects is the Müller formulation [Müller, 1969]. Since the Galerkin method for the T-Müller formulation leads to an instable solution [Ylä-Oijala and Taskinen, 2005b] (see also section 7), here we consider only the N-Müller formulation. This formulation is obtained from (25) with the coefficients

  • equation image

for l = 1, 2. The problem with the N-Müller formulation is that the equations are neither in the J nor M space. Rather these equations are in the spaces of n × B and n × equation image. To approximately map the equations onto the J and M spaces, we consider the following modified N-Müller (mN-Müller) formulation: a1 = 0 = a2, d1 = 0 = d2 and

  • equation image

We note that all formulations considered in this section remove the problem of internal resonances.

4.3. Piecewise Dielectric and Composite Objects

[30] Next the formulations are generalized for piecewise dielectric and composite metallic and dielectric objects.

[31] Let us shortly consider the integral equation formulations at a junction of M > 2 dielectric regions. For more details of the definition of the basis and testing functions and integral equation formulations at the junctions we refer to Ylä-Oijala et al. [2005] [see also Kolundzija, 1999]. A junction and oriented basis functions are illustrated in Figure 1.

[32] Because of the boundary conditions of the fields, the unknown coefficients of the oriented basis functions of the same order and type associated with the same junction edge are not independent, but they are all the same. Thus, at a junction of M dielectric regions, these unknowns can be combined into a single unknown and there are only two independent unknowns for each basis function, one for J and another for M. In order to have the same number of equations and unknowns, also at the junctions, the discretized integral equations associated with a junction edge ej are combined as follows

  • equation image
  • equation image

Here M denotes the number of testing functions, defined in the subregions, and associated with edge ej. It is important to note that, similarly as in the previous section, (1)–(4) are first discretized individually in each subregion and thereafter the discretized equations are combined as in (32) and (33).

[33] The CTF, CNF, JMCFIE and mN-Müller formulations at the dielectric junctions are now obtained from (32) and (33) with the coefficients

  • equation image
  • equation image
  • equation image

and

  • equation image

respectively, where l = 1, …, M.

[34] If the object consists of both dielectric and metallic regions, we apply 1/ηlT-EFIE, N-MFIE, JCFIE or N-MFIE formulation on the metallic surfaces and the CTF, CNF, JMCFIE or mN-Müller formulation on the dielectric surfaces and junctions. At a metal-dielectric junction M vanishes [Kolundzija, 1999] and the equations in the M space can be omitted. The formulations on a metal-dielectric junction read

  • equation image
  • equation image
  • equation image
  • equation image

Here M is the number of testing functions defined in subregions and associated with a composite metal-dielectric junction edge ej between two metallic surfaces (see Figure 1) and bl is the coupling coefficient of the mN-Müller formulation defined in (37).

5. On Numerical Implementation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[35] Let us next shortly discuss on the numerical implementation of the system matrix elements of the integral equation formulations. In the T formulations we have two symmetric matrix blocks (one for the operator equation imagetan and another one for the operator equation imagetan); hence the computational cost in each subregion is of order 1/2 + 1/2 = 1. In the N formulations the matrix blocks are nonsymmetric and the computational cost in each subregion is of order 1 + 1 = 2. Because JMCFIE includes both the matrix elements of the T and N formulations, the computational cost of the JMCFIE formulation is of order 3. Thus, when iterative solvers are applied in the JMCFIE, three times more calculations are needed at each iteration step than in the T formulations.

[36] The T formulations have an additional benefit, namely in these formulations the singularity of the hypersingular integral operator equation image can be decreased by moving the gradient from the Green's function into a testing function. In the N formulations moving the gradient of the equation image operator into a testing function results in a line integral over the boundaries of the support of the testing function [Medgyesi-Mitschang et al., 1994; Ylä-Oijala and Taskinen, 2003].

[37] In the numerical examples we suppose that the surfaces are divided into planar triangular patches. The surface currents are expanded with both the Rao-Wilton-Glisson (RWG) [Rao et al., 1982] and the Trintinalia-Ling (TL) [Trintinalia and Ling, 2001] basis functions. The system matrix elements of the T and N formulations with both the RWG and TL basis functions, including the possible line integrals, are calculated with the singularity subtraction technique presented by Ylä-Oijala and Taskinen [2003] and Järvenpää et al. [2005].

6. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[38] In this section accuracy of the developed integral equation formulations and convergence rates when the matrix equations are solved iteratively with the generalized minimal residual (GMRES) method [Saad and Schultz, 1986] (without restarts) are studied in several example cases. In all considered cases the incident wave is a y polarized plane wave propagating along the −z axis, the frequency is 100 MHz and the objects are in vacuum. The iteration is stopped after 2000 steps.

6.1. Nonsmooth Closed Metallic Object

[39] Let us begin by considering a nonsmooth metallic object, that is, a metallic cube. The side length of the cube is λ, where λ is the wavelength in vacuum and the faces of the cube are parallel to the coordinate axes.

[40] The top two plots in Figure 2 show the monostatic RCS as a function of the number of RWG and TL unknowns. The two vertical lines indicate how many elements per wavelength the mesh includes. At the line on the left hand side there are five elements per wavelength and at the line on the right hand side there are ten elements per wavelength. Note that to the left of the λ/5 line the discretization with the RWG functions is probably too coarse to give accurate results. The results clearly show that the solution of the MFIE converges very slowly if the RWG basis functions are used. With the TL basis functions the converge is essentially faster and the accuracy of the MFIE formulation is almost the same as can be obtained with the EFIE and CFIE formulations and RWG functions. Note that when TL basis functions are applied there are two unknowns per edge instead of one and thus the TL basis functions double the number of unknowns compared to the RWG basis functions if the same mesh is used.

image

Figure 2. (top) Monostatic RCS and (bottom) number of GMRES iterations for a metallic cube as a function of (left) RWG and (right) TL unknowns.

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[41] Next we study iterative solution of the matrix equations based on the EFIE, MFIE and CFIE formulations. The number of iterations required to obtain relative residual error of 10−6 with the GMRES method is displayed in the bottom two plots of Figure 2 as a function of the number of the RWG and TL unknowns. The important result here is that the convergence rates do not significantly change when the RWG basis functions are replaced with the TL basis functions. Here the frequency is chosen so that it does not correspond to the interior resonance of the object. At the resonance frequencies, or close to them, EFIE and MFIE become singular and CFIE is the preferred choice [Song and Chew, 1995].

6.2. Smooth Dielectric Object

[42] Next we consider a smooth dielectric object, that is, a homogeneous dielectric sphere. The radius of the sphere r = 0.5 m, that is, at 100 MHz k0r ≈ 1, where k0 is the wave number in vacuum.

[43] First we checked the accuracy of the solution by studying bistatic RCS. We found that all formulations give practically the same accuracy for four dielectric spheres with ɛr = 1.1, 4, 10 and ɛr = 36 + 0.3i. Then we study convergence of the GMRES method. The first columns of Tables 1 and 2 show the number of iterations required to obtain relative residual error of 10−6 with the GMRES method for two dielectric spheres with ɛr = 4 and ɛr = 36 + 0.3i. The results show that in both cases the T-PMCHWT formulation gives the slowest convergence rate and the fastest rate is obtained with the mN-Müller formulation. In addition, we find that for a low-contrast sphere CNF converges very rapidly, but for a high-contrast sphere the convergence rate of the CNF significantly decreases.

Table 1. Number of GMRES Iterations for a Dielectric Sphere With k0r = 1, ɛr = 4, and 2940 RWG Unknowns
FormulationWithout PreconditioningDiagonalElement
T-PMCHWT1329175150
CTF136124117
CNF211816
N-Müller771414
mN-Müller141312
JMCFIE454338
Table 2. Number of GMRES Iterations for a Dielectric Sphere With k0r = 1, ɛr = 36 + 0.3i, and 2940 RWG Unknowns
FormulationWithout PreconditioningDiagonalElement
T-PMCHWT1441346340
CTF137126161
CNF126113159
N-Müller2174049
mN-Müller443746
JMCFIE9583133

[44] Next we consider the effect of a simple preconditioner on the convergence rates. The number of iterations with a diagonal preconditioner and an incomplete LU (ILU) preconditioner assembled using the nearby elements of the system matrices are listed in the second and third columns of Tables 1 and 2. In the latter case, the ILU algorithm is applied using those matrix elements in which the supports of the basis and testing functions overlap. The results show that preconditioning has a clear effect only on the formulations (T-PMCHWT and N-Müller) in which the equations are not in the current spaces. It is interesting to see that the ILU preconditioning works better than diagonal preconditioning only in the case of a low-contrast sphere. Our other studies have shown that the diagonal preconditioner is even less effective in the case of more complex objects [Ylä-Oijala and Taskinen, 2005a]. Tables 1 and 2 show also that the N formulations are more sensitive to the increase of the permittivity than the T formulations. In addition, we see that the mN-Müller formulation without preconditioning and the N-Müller formulation with a preconditioner lead to roughly the same convergence rates, but without preconditioning mN-Müller gives clearly higher convergence rate. Note that the N-Müller formulation applied in this paper is not identical with the N-Müller formulation developed earlier by Ylä-Oijala and Taskinen [2005b] because in these formulations the equations (N-EFIE and N-MFIE) appear in reverse order.

[45] A usual way to predict the convergence rate is to study the condition number of the system matrix and typically high condition number correlates with poor convergence. Table 3 shows the condition numbers (cond) and the spectral condition numbers, denoted by econd and defined as a ratio of the maximum and minimum absolute value of the eigenvalues for two dielectric spheres with ɛr = 4 and ɛr = 36 + 0.3i.

Table 3. Condition (Cond) and Spectral Condition Numbers (Econd) for a Dielectric Sphere With k0r = 1, ɛr = 4, 36 + 0.3i, and 2940 RWG Unknowns
Formulationɛr = 4ɛr = 36 + 0.3i
CondEcondCondEcond
T-PMCHWT2.10 × 1072.10 × 1072.28 × 1062.28 × 106
CTF3.48 × 1042.89 × 1029.12 × 1048.72 × 101
CNF4.62 × 1063.58 × 1003.11 × 1062.82 × 101
N-Müller2.17 × 1052.17 × 1051.24 × 1051.24 × 105
mN-Müller6.90 × 1043.92 × 1001.02 × 1051.94 × 101
JMCFIE3.17 × 1051.51 × 1013.50 × 1051.87 × 101

[46] By studying the results of Tables 1–3, we find that all formulations have a rather high condition number, but their spectral condition numbers and convergence rates are clearly different. This indicates that the condition number is not always a good indicator the convergence rate of the iterative solution. Rather one should study the spectral condition number and a low econd is a better indicator for a fast convergence rate. However, a low econd does not necessarily mean high convergence rate, because our other studies have shown that two formulations may have almost identical spectral condition numbers, but their convergence rates can be clearly different [Ylä-Oijala and Taskinen, 2005a]. For example, the CNF and a similar formulation with negative b coefficients in (28) have the same condition and spectral condition numbers, but the latter one leads to a clearly slower convergence speed. The reason is that, although the spectral condition numbers are identical, the distribution of the eigenvalues are not. This also shows that it is important to derive the equations so that they are mappings from (J, M) onto (J, M), not onto (−J, M) or (J, −M), for example.

6.3. Nonsmooth Dielectric Object

[47] Next we study the accuracy and convergence rates of the formulations in the case a nonsmooth dielectric object. We consider a homogeneous dielectric cube with the side length l = 1.0 m.

[48] Figure 3 displays the monostatic RCS as a function of the number of RWG unknowns. The results show that if the permittivity is high the solutions of the N formulations converge slower than the solutions of the T formulations. Since this phenomenon seems to be similar as has been observed previously in the case of nonsmooth metallic objects with the MFIE and EFIE formulations by Ergül and Gürel [2004], next we study the effect of using the TL basis functions on the solution accuracy. The results in Figure 4 indicate that in the case of dielectric objects with sharp edges and high permittivity the accuracy of the MFIE-type N formulations can be improved by increasing the order of the basis functions. Similarly as in Figure 2 the two vertical lines show when the maximum edge length of the discretization is roughly λ2/5 (on the left) and λ2/10 (on the right), where λ2 is the wavelength inside the cube.

image

Figure 3. Monostatic RCS for four dielectric cubes with ɛr = 1.1, 4, 10, and 36 + 0.3i as a function of the number of RWG unknowns.

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image

Figure 4. Monostatic RCS for four dielectric cubes with ɛr = 1.1, 4, 10, and 36 + 0.3i as a function of the number of TL unknowns.

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[49] Then we study iterative solution of the integral equation formulations using both the RWG and TL basis functions. Figures 5 and 6 display the number of iterations as a function of the number of unknowns. Similarly as in the case of a metallic cube, convergence rates do not significantly change as the RWG functions are replaced with the TL functions.

image

Figure 5. Number of GMRES iterations for four dielectric cubes with ɛr = 1.1, 4, 10, and 36 + 0.3i as a function of the number of RWG unknowns.

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image

Figure 6. Number of GMRES iterations for four dielectric cubes with ɛr = 1.1, 4, 10, and 36 + 0.3i as a function of the number of TL unknowns.

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6.4. Piecewise Dielectric Object

[50] Next we consider a piecewise homogeneous dielectric object with a junction. The object is an inhomogeneous dielectric sphere made of two homogeneous dielectric hemispheres. The radius of the sphere r = 0.5 m. We consider two cases, a low-contrast sphere, ɛr1 = 4 and ɛr2 = 8, and a high-contrast sphere, ɛr1 = 4 and ɛr2 = 36 + 0.3i, where ɛr1 and ɛr2 are the permittivities of the hemispheres. The interface of the two hemispheres is in the x, y plane (parallel to the propagation direction of the incident wave) so that the hemisphere with ɛr1 is on the top, that is, on the side of excitation.

[51] Figure 7 displays the monostatic RCS and the number of iterations as a function of the number of unknowns. In this case only the RWG functions are applied. Again the T formulations and the JMCFIE formulation give better accuracy than the N formulations. When the number of unknowns is increased we find that JMCFIE formulation gives the fastest convergence rate. Similarly as in Figures 3 and 4 the two vertical lines show when there are 5 or 10 elements per wavelength inside the hemisphere with the highest permittivity.

image

Figure 7. (top) Monostatic RCS and (bottom) number of GMRES iterations for two inhomogeneous dielectric spheres with (left) ɛr1 = 4, ɛr2 = 8 and (right) ɛr1 = 4, ɛr2 = 36 + 0.3i as a function of the number of RWG unknowns.

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6.5. Composite Objects

[52] Next we consider two composite metallic and dielectric objects. First we consider a metallic sphere with a dielectric coating. The radius of the metallic sphere r = 0.5 m and the thickness of the coating is r/10, that is, 0.05 m. We consider two cases. In the first case the relative permittivity of the coating is 4 and in the second case ɛr = 36 + 0.3i. Figure 8 displays the monostatic RCS and the number of GMRES iterations as a function of the number of the RWG unknowns. The Mie series solution is denoted by a solid horizontal line. In this case (a composite object with smooth surfaces and no junctions), the results are rather similar as in the case of a homogeneous dielectric sphere and all formulations give similar accuracy.

image

Figure 8. (top) Monostatic RCS and (bottom) number of GMRES iterations for two metallic spheres with a dielectric coating with (left) ɛr = 4 and (right) ɛr = 36 + 0.3i as a function of the number of RWG unknowns.

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[53] Consider next a similar inhomogeneous sphere as in section 6.4. Now the second hemisphere at the bottom is PEC; that is, the structure includes a metal-dielectric junction. Monostatic RCS and the number of GMRES iterations as a function of RWG unknowns are displayed in Figure 9. Similarly as in section 6.4 also in this case performance of the N formulations decreases.

image

Figure 9. (top) Monostatic RCS and (bottom) number of GMRES iterations for two composite spheres with (left) ɛr = 4 and (right) ɛr = 36 + 0.3i as a function of the number of RWG unknowns.

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7. Further Analysis and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[54] The results of the previous section can be summarized as follows:

[55] 1. In the case of metallic, homogeneous dielectric and composite objects with low permittivities and with no junctions, the N formulations give clearly faster converging iterative solutions than the T formulations. In all these cases, also with a high electric permittivity, the fastest convergence rate is obtained with the N-MFIE/mN-Müller formulation.

[56] 2. In the case of objects with junctions convergence rates of the N formulations decrease and in many cases the fastest convergence rate is obtained with the JMCFIE formulation.

[57] 3. In the case of nonsmooth objects (metallic or dielectric with high permittivity contrast or objects with junctions) the accuracy of the N formulations is poorer than the accuracy of the T formulations when the RWG basis functions are applied.

[58] 4. The accuracy of the N formulations can be improved by using the TL basis functions.

[59] In this section we more carefully analyze reasons for the above results.

7.1. Homogeneous Dielectric Objects

[60] Let us first consider the developed integral equation formulations in the case of a homogeneous dielectric object. Let us denote n = n1 = −n2. The integral operators associated with the T formulations (al ≠ 0, bl = 0, cl = 0, dl ≠ 0) can be written in the matrix form as follows

  • equation image

The corresponding operators of the N formulations (al = 0, bl ≠ 0, cl ≠ 0, dl = 0) read

  • equation image

The equations in (42) and (43) are arranged so that the operators on the diagonal are well tested and the poorly tested operators appear on the off-diagonal blocks. As a consequence, the matrix equations will be diagonally dominant. In each of the CTF, CNF and JMCFIE formulations the matrix blocks on the diagonal are identical. Thus, also in this sense, the developed CTF, CNF and JMCFIE formulations are “optimal” EFIE-, MFIE- and CFIE-type formulations. Note that this is not generally true for the mN-Müller formulation.

[61] From (42) we find that a T formulation with a1/k1 = −a2/k2 and d1/k1 = −d2/k2 leads to a diagonally less dominant matrix equation, because in that case the main singularity of the equation imagetan operator cancels. T-Müller (with coefficients a1 = ɛ1η1, a2 = −ɛ2η2, d1 = μ11 and d2 = −μ22) is an example of an unstable T formulation [Ylä-Oijala and Taskinen, 2005b]. Similar result holds also for the N formulations. A N formulation with b1 = −b2 and c1 = −c2 leads to an integral equation of the first kind (N-PMCHWT formulation) which is very ill conditioned [Lloyd et al., 2004] because the identity operator cancels and any of the operators is not well tested. In the mN-Müller (and N-Müller) the main singularity of the n × equation image operator cancels and in the CNF the main singularity of the n × equation image operator cancels. In the mN-Müller formulation the cancellation appears on the off-diagonal blocks and hence even a more diagonally dominant matrix equation is obtained. In the CNF this cancellation, however, takes place on the diagonal and most probably is the source of numerical problems observed with the CNF in the case of objects with high permittivity contrast. In that case also the off-diagonal blocks of the CNF become very unbalanced. However, for the low-contrast objects the CNF works well and in the case ɛ1 = ɛ2 and μ1 = μ2, we have

  • equation image
  • equation image
  • equation image

and the solution of the CNF for the problem is J = n × Hinc and M = −n × Einc, that is, the tangential components of the incident field.

[62] The worst convergence rate is always obtained with the T-PMCHWT formulation. The main problems of the PMCHWT formulation are that it is an integral equation of the first kind and that the matrix elements have clearly different magnitudes. The convergence of the T-PMCHWT formulation can be improved with the proper weighting coefficients as in the CTF.

[63] The N formulations, excluding N-PMCHWT, lead to integral equations of the second kind with a well tested identity operator. The T formulations, excluding T-PMCHWT, lead also to integral equations of the second kind, but the identity operator is not well tested. Since the N formulations usually lead to faster converging iterative solutions than the T formulations, the identity operator, and how it is tested, plays a crucial role in deriving integral equation formulations with good spectral properties and rapidly converging iterative solutions. Note that each of the CNF, JMCFIE and mN-Müller formulations are derived so that the identity operators of the coupled system have an equal coefficient. This property, which is a consequence of defining the equations so that they are mappings from the surface currents onto themselves, seems to be also important for formulations with good iterative properties.

[64] There are two reasons why the mN-Müller works so well with homogeneous dielectric objects. First, because of the cancellation of the main singularity of the n × equation image operator, mN-Müller (and N-Müller) removes the low-frequency breakdown and leads to a well conditioned matrix equation on a wide frequency range [Ylä-Oijala and Taskinen, 2005b]. Second, the equations are scaled so that they are (approximately) in the current spaces. We remind that in the N-Müller formulation this is not the case.

7.2. Inhomogeneous and Nonsmooth Objects

[65] The results of section 6 indicate that in the case of nonsmooth and inhomogeneous objects the N formulations do not work as well as they do in the case of smooth homogeneous objects. There are (at least) two reasons for that. First, the currents have singularities at the edges and junctions and in that case the RWG functions are not sufficiently smooth functions for expanding the unknowns and testing the equations in the MFIE-type N formulations [see also Ergül and Gürel, 2004]. Secondly the integral operators of the combined formulations have different performance at the junctions. As mentioned in the previous section, the main singularity of the n × equation image operator cancels in the CNF and the main singularity of the n × equation image operator cancels in the mN-Müller formulation. In the case of objects with junctions, however, these cancellations are not complete, and as a consequence, the mN-Müller formulation, for example, does not remove the low-frequency breakdown.

[66] Moreover, in the case of piecewise dielectric and composite objects the operators on the diagonal of the CTF, CNF and JMCFIE formulations are not generally identical. This also implies that the developed formulations are not as optimal as they are in the case of homogeneous objects.

7.3. Other Frequencies

[67] The calculations of the previous section were performed at a fixed frequency. We finish this section by considering the same geometries as in sections 6.2, 6.4 and 6.5, by varying the frequency so that k0r varies from 0.25 to 4.25. Here r is the radius of the sphere and k0 is the wavelength in vacuum. In the case of homogeneous dielectric objects (for both sphere and cube) all formulations give similar accuracy as in the case k0r = 1 in section 6 if the number of unknowns is chosen so that the discretization is dense enough. In addition, in all formulations the number of iterations increases smoothly for increased frequency, as shown in the top left plot of Figure 10.

image

Figure 10. Number of GMRES iterations for (top left) a homogeneous dielectric sphere (ɛr = 4), (top right) an inhomogeneous dielectric sphere (ɛr = 4, 8), (bottom left) a dielectrically coated metallic sphere (ɛr = 4), and (bottom right) a composite sphere (ɛr = 4) as a function of the size parameter k0r, where k0 is the wavelength in vacuum and r is the radius of the sphere. The numbers of the RWG unknowns are 4860, 4646, 4410, and 5077, respectively.

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[68] In the case of piecewise dielectric and composite objects the situation, however, is different. Also in this case all formulations give similar accuracy as in the case k0r = 1 (better accuracy is obtained at lower frequencies, because the number of unknowns was fixed), but the convergence rates of the N formulations essentially decrease after some threshold. This is clearly shown in the left-hand plots of Figure 10. This threshold seems to depend on the discretization and mesh density. Similar phenomenon can be observed also in Figures 7 and 9, where the number of iterations of the CNF and mN-Müller formulations increase more rapidly than that of the other formulations as the number of unknowns is increased. The N formulations show strange behavior also in the case of a PEC sphere with a dielectric coating at k0r = 0.25 (bottom right plot of Figure 10), although in that case the surfaces are smooth. Note also that in most cases the T-PMCHWT formulation does not converge in 2000 iterations, and hence the results for the T-PMCHWT formulation are not shown in all cases. In Figure 10, the two vertical lines show for which size parameter k0r the maximum edge length of the mesh is roughly λ/10 (on the left) and λ/5 (on the right), where λ is the wavelength of the region with the highest permittivity.

8. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information

[69] Surface integral equation formulations and their properties in the electromagnetic scattering analysis of metallic, dielectric, piecewise dielectric and composite objects are studied. By investigating the properties of the integral operators and their testing in the Galerkin method, new integral equation formulations are proposed. In these formulations the equations are derived so that the integral operators are mappings from the surface currents onto themselves. The numerical examples illustrate that the system matrices of the new formulations have good spectral properties and lead to significant improvements in the convergence rates when the matrix equations are solved iteratively by the GMRES method (without preconditioning).

[70] The numerical results indicate that using the integral equations of the second kind with a well tested identity operator, that is, the MFIE-type N formulations, including the N-Müller formulations, instead of the integral equations of the first kind, or integral equations of the second kind with a poorly tested identity operator, that is, the EFIE-type T formulations, better converging iterative solutions can be obtained. However, the RWG functions are not sufficiently smooth functions for expanding the unknowns and testing the equations in the N formulations when the surfaces include sharp edges. The accuracy of the N formulations can be improved by increasing the order of the basis functions. The first-order TL basis functions are found to lead to both accurate results and rapidly converging iterative solutions.

[71] In the case of homogeneous dielectric objects, the developed modified form of the N-Müller formulation of Ylä-Oijala and Taskinen [2005b] (mN-Müller) has been found to give clearly the most efficient formulation for iterative solvers. The mN-Müller formulation also removes the problem of the low-frequency breakdown. For general piecewise dielectric and composite objects with junctions, the performance of the N formulations decrease and the most robust formulation is the JMCFIE formulation developed earlier by Ylä-Oijala and Taskinen [2005a].

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Electric and Magnetic Field Integral Equations
  5. 3. Galerkin Method
  6. 4. Integral Equation Formulations
  7. 5. On Numerical Implementation
  8. 6. Numerical Results
  9. 7. Further Analysis and Discussion
  10. 8. Conclusions
  11. Acknowledgments
  12. References
  13. Supporting Information
FilenameFormatSizeDescription
rds5162-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5162-sup-0002-t02.txtplain text document0KTab-delimited Table 2.
rds5162-sup-0003-t03.txtplain text document1KTab-delimited Table 3.

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