Discretization of Volume Integral Equation Formulations ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 11, NOVEMBER 2012

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Discretization of Volume Integral Equation Formulations for Extremely Anisotropic Materials Johannes Markkanen, Pasi Ylä-Oijala, and Ari Sihvola, Fellow, IEEE

Abstract—A stable volume integral equation formulation and its discretization for extremely anisotropic materials is presented. The volume integral equations are written in terms of the volume equivalent currents. The equivalent currents are expanded with piecewise constant basis functions, and the Galerkin’s scheme is applied for testing the equations. Numerical results show that the behavior of the formulation is more stable than the behaviors of the more conventional volume integral equation formulations based on fluxes or fields, when the scatterer is extremely anisotropic. Finally, the developed method is applied to analyze a highly anisotropic material interface which approximates the ideal DB boundary. Index Terms—Anisotropy, DB boundary, electromagnetic scattering, volume integral equation.

I. INTRODUCTION

E

LECTROMAGNETIC problems involving extremely anisotropic materials are of great interest in the field of electromagnetic metamaterials. The development of new electromagnetic materials has created an urgent need for accurate electromagnetic solvers for analyzing scattering properties of materials with complex anisotropy. In many cases, homogenizations of metamaterials lead to uniaxially anisotropic media with a strong contrast between an axial and transverse directions. Media with such strong anisotropy are considered here as extremely anisotropic media. An interface of extremely anisotropic material with vanishing axial permittivity and permeability acts as so-called DB boundary [1]. Due to the vanishing normal components of the fields, the power cannot propagate along the surface. Such a surface is known as an electromagnetic soft surface, and has many micro- and millimeter wave applications [2]. Another interesting property of the object with the DB boundary is that the backscattering by the object vanishes if the object is symmetric with respect to the incident field [3]. These properties can be advantageous in many electromagnetic engineering applications. Especially, the DB boundary has shown some interest in cloaking of the objects from electromagnetic waves. In [4], Yaghjian and Maci showed that the DB boundary conditions

Manuscript received October 14, 2011; revised March 30, 2012; accepted May 29, 2012. Date of publication July 10, 2012; date of current version October 26, 2012. This work was supported by the Graduate School of Electronics, Telecommunication, and Automation (GETA) and the Academy of Finland. The authors are with the Department of Radio Science and Engineering, Aalto University School of Electrical Engineering, FI-00076 Espoo, Finland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2012.2207675

are valid at the inner surface of spherical cloak. Later the results was generalized for arbitrarily shaped cloaks [5]. Another interesting extreme material is a material whose axial parameters tend to infinity. This material works as wave guiding medium as described in [6]. A quarter wavelength layer of the wave guiding medium, known as a quarter wave transformer, can be used to transform the PEC boundary to the PMC boundary and vice versa. Also, in [7] it was shown that the transformation between the DB and D B is possible with a quarter wave transformer. Analyzing the scattering behavior of arbitrarily shaped three-dimensional anisotropic objects with extreme parameters calls for numerical schemes since the exact solution can only be found for simple bodies such as spheres [8]–[10]. One possibility is to use volume integral equation (VIE) methods in which the anisotropic scatterer is replaced with the equivalent polarization currents, and the integral equations are derived for the total fields [11]. In the past, the volume integral equation methods have been developed for an anisotropic medium by several authors [12]–[15]. However, these methods have been applied only for objects with relatively low anisotropy. Applying volume integral equation methods to extremely anisotropic materials can be detrimental to the condition of the system, and thus might lead to the erroneous solution, or prevent a usage of the fast methods such as multilevel fast multipole algorithm (MLFMA). In this paper, we investigate properties of the numerical methods based on the volume integral equations in the case of the highly anisotropic materials. We consider three different types of formulations: the most widely used “DB-formulation” and , the in which the unknowns are the flux densities and as the unknowns, “EH-formulation” with the fields and the “JM-formulation” with which the volume equivalent are the unknowns. currents and These three integral equation formulations are equivalent with respect to the existence and uniqueness of the solution if both the permittivity and permeability functions are coercive and bounded, and are invertible [16]. However, in the case of the extreme anisotropies these conditions might not be fulfilled, and the numerical solutions of these formulations might be different. From the numerical point of view, the main difference in these formulations is the choice of the basis and testing functions. Basis functions are used for representing the unknowns; hence, they should satisfy the continuity conditions of the unknowns. Especially, it is crucial not to enforce any extra continuities. In other words, basis functions should span a proper vector space. Normal components of the flux densities must be continuous

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across the material interfaces, and therefore, the DB-formulation is usually discretized with the divergence conforming basis functions, known as the SWG functions or 3-D RWG functions [17]. In [18] the anisotropic EH-formulation was discretized with curl conforming functions [19] which keep the tangential continuity. The EH-formulation has also been discretized with the piecewise constant and linear functions in order to obtain a non-conformal method [20], [21]. In this case, the continuity of the fields are not directly enforced, but the solution will approximatively satisfy these conditions. The equivalent volume currents have no continuity across the material interfaces, and in the JM-formulation it is essential that the basis functions do not enforce any continuities across material interfaces. The equivalent currents are usually expanded with the piecewise constant functions, and the point-matching/Nyström techniques are applied to test the equations [22], [23]. The choice of the testing functions is also important in order to obtain an efficient numerical method. In [16] and [24], it is shown that the testing functions should span the dual space of the range of the integral operator to guarantee the convergence of the solution. In many previous numerical schemes this aspect has not been taken into consideration when the equations are discretized. Especially, when the differentiability of the testing functions is used explicitly it is not clear that the solution is restricted into the proper function space [16]. II. FORMULATIONS In this section, we introduce the volume integral equation formulations that we have used in the calculations. The total time harmonic electric and magnetic fields with the time factor can be expressed in an anisotropic, linear and inhomogeneous region bounded by volume via the volume equivalence principle as [25, (Chapter 11.2.1)]

A. DB-Formulation First we recall the most widely used volume integral equation formulation with the flux densities and as unknowns. By representing the equivalent volume currents (2) in terms of and , one obtains so called DB-formulation

(4) with contrast dyadic functions defined as

(5) and denote the inverses of the permitwhere dyadics tivity and permeability dyadics and . B. EH-Formulation The volume integral equations for the electric and magnetic fields can be obtained from (1). Since the total fields should be expanded with curl conforming functions, we want to transform the operator to the operator which contains curls rather than divergences to avoid extra boundary integrals. By using the fact that [18] (6) the EH-formulation can be written as [26]

(7) where the contrast dyadics are

(1) Here, and are the incident electric and magnetic fields, and is the wavenumber in homogeneous background medium with the constants and . The equivalent polarization currents are defined as

(2) where and are the relative permittivity and permeability dyadics, respectively, and is the identity dyadic. The volume integral operator in (1) can be expressed as (3) where

is the Green function of the background.

(8) C. JM-Formulation We can also derive formulations with the equivalent currents as the unknowns. Since the equivalent currents have no continuities, they should be expanded with the basis functions which do not enforce any continuities. We can write the JM-formulation as

(9) We note here that it would be possible to write the JM-formulation which contains the rather than the operator. In this paper we only consider the latter JM-formulation with the double curl operator since both formulations

MARKKANEN et al.: DISCRETIZATION OF VOLUME INTEGRAL EQUATION FORMULATIONS FOR EXTREMELY ANISOTROPIC MATERIALS

give practically identical results, and the latter one is easier to implement.

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and

III. DISCRETIZATION In this section, we consider the discretizations of the DB-, EH-, and JM-formulations. Let us divide the object with linear tetrahedral elements, and define the basis and the testing functions on the tetrahedral mesh. We have used Galerkin’s method to test the equations with the inner product defined as (10) where is the volume of the object. As pointed out in [16] and [24] to guarantee the convergence in the norm of the solution, the testing functions should span the dual space of the range of the integral operator. Therefore, we need to look at the mapping properties of the equations

(13) B. Discretization of EH-Formulation The unknown electric and magnetic fields have continuous tangential components on the element boundaries; therefore, the fields should be expanded with the curl conforming basis functions . Analogously to the case of the DB-formulation, we can discretize the operators of the EH-formulation with the curl conforming basis and testing functions as

(14)

DB-formulation EH-formulation

and

JM-formulation where and

is a function space with square integrable functions, (15) Now the surface integrals vanish on the interior faces because the tangential component of is continuous. (11)

Since and are dual to each other and is dual to itself, it is easy to see that the equations are tested in the dual space of the range when the Galerkin’s testing is applied. A. Discretization of DB-Formulation The DB-formulation is discretized with the Galerkin’s method using the lowest mixed order divergence conforming SWG basis and testing functions [17] which span the space. The hyper-singularity of the kernel in (4) is reduced by moving one derivative into the testing function by integrating by parts. Since the testing function is divergence conforming, surface integrals cancel out on each element boundaries except on the outer surface of the object. By moving the remaining derivative to the Green’s function, the discretized operator can be expressed as

C. Discretization of JM-Formulation The equivalent volume currents have no continuities, hence it is essential that basis functions do not enforce any continuity across the element interfaces. We use piecewise constant basis and testing functions to expand the unknowns (three functions in each tetrahedra). The singularity can be reduced by moving one derivative into the testing function as

(16)

where denotes the transpose of . The other term can be discretized as (17) Here it is important to note that the surface integrals in (16) do not vanish on the inner faces because the testing functions do not have any continuities. IV. NUMERICAL EXAMPLES

(12)

In this section, we analyze properties of the formulations with various numerical examples. We consider plane-wave scattering by different objects. To obtain as general analysis as possible, uniaxially anisotropic spheres and non-smooth cubes

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Fig. 1. Computed forward scattering cross sections of a small sphere of size as functions of the radial component of the permittivity and permeability dyadics.

Fig. 2. Iteration counts for the problem in Fig. 1. The resulting matrix equations . are solved by the GMRES solver with the stopping criterion

of different sizes are treated with varying axial permittivity and permeability. A. Radially Uniaxial Sphere Our first example is a radially uniaxial sphere with the permittivity defined in the spherical coordinate system as

(18) and similarly the permeability

dyadic is Fig. 3. Forward scattering cross sections of a sphere of size as functions of the radial component of the permittivity and permeability dyadics.

(19) where and are constants. Let us first consider scattering by a small sphere of radius . Fig. 1 shows calculated forward scattering cross sections of a sphere of size as functions of parameter while . The calculations are done by using the DB-, EH-, and JM-formulations. The sphere is meshed with 2793 tetrahedral elements giving 11636 for the DB-, 7246 for the EH-, and 16 758 unknowns for the JM-formulation. The reference result is obtained by analytical Mie solution for anisotropic spheres [8], [9]. The systems are solved iteratively with the GMRES solver, and the iteration counts are presented in Fig. 2. Next, we will investigate the properties of the formulations in the case of electrically larger sphere. In Fig. 3 the computed forward scattering cross sections are presented. Here the size parameter of the sphere is . The discretization is the same as in the previous case. The number of iterations required for the error to converge to are plotted in Fig. 4. Clearly, these examples show that the JM-formulation is the most stable one, and the agreement with the Mie solution is good with a large range of parameter . The DB-formulation has problems as the permittivity and permeability approach zero, but with higher values of , the solution is stable and accurate. The EH-formulation, on the other hand, gives stable solution with a small and the solution is unstable and inaccu-

Fig. 4. Iteration counts for the problem in Fig. 3. The resulting matrix equations . are solved by the GMRES solver with the stopping criterion

rate when is large in amplitude. Also, from numerical results in Figs. 1–4, it appears that these phenomena are independent of the sphere size. Another observation is that the solution of the JM-formulation is more accurate when is large in amplitude compared to the case where is close to zero. This is because the fields decay exponentially inside the material where the normal component of and are zero. Therefore, in order to model the fields accurately, the mesh should be denser near the surface.


Fig. 5. Forward scattering cross sections of a dielectric cube of size as functions of the component of the permittivity dyadic while .

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Fig. 7. Condition numbers of the systems as functions of frequencies. The radial component of the relative permittivity of the cube is either 1000 or 0.001, . and

the radial component of the permittivity is either or , and the . In the dielectric case the permeability . Fig. 7 shows the condition numbers of the matrices obtained by the JM-, EH-, and DB-formulations. As can be seen the JM-formulation is the most stable one at all frequencies. Another interesting observation is that the condition number of the DB-formulation increases at the high frequency region. C. Discussion

Fig. 6. Iteration counts for the problem in Fig. 5. The GMRES solver is used . with the stopping criterion

B. Anisotropic Dielectric Cube Our next example is an anisotropic dielectric cube with the permittivity dyadic in Cartesian coordinate defined as

(20) and the permeability . The cube is illuminated by a linearly polarized plane-wave propagating along the -axis. The frequency is chosen such that the size parameter of the cube is , where is the edge length of the cube. The cube is discretized with 2433 tetrahedral elements giving 5153, 3289, and 7299 unknowns for the DB-, EH-, and JM-formulation, respectively. Also, in the case of the cube, which has sharp wedges and corners, and the anisotropy is homogeneous, the behaviors of the formulations are very similar as in the case of the smooth sphere as can be seen from Figs. 5 and 6 where the forward scattering cross sections and the iteration counts are presented, respectively. Next we investigate the condition of the system as a function of frequency. The scatterer is a dielectric cube with edge length , and the center of the cube is at the origin. The permittivity is defined in the spherical coordinate system such that

According to our findings the JM-formulation seems to be stable at a wide range of permittivities and permeabilities, but both the EH- and DB-formulations suffer from the stability problems at extreme cases. As discussed in [16] and [24] the well-posedness of the formulations depends on the permittivity and permeability functions. They should be coersive and bounded: (21) for some , , and any . Obviously, at the limits and these conditions are not valid, and therefore, the JM-, DB-, and EH-formulations are not equivalent in the sense of the existence and the uniqueness of the solution. The JM-, and EH-formulations are equivalent when , and the JM- and DB-formulations are equivalent when . In the former case in the DB-formulation the contribution from the incident field vanishes, and in the latter case the incident field does not contribute at all in the EH-formulation. The above analysis is also valid for the permeability . From the physical point of view the constitutive relations and , and the finite energy assumption, point out the following about the flux densities and the fields at extreme cases:

(22) Therefore, in the case of the DB-formulation the unknown flux densities vanish when the permittivity and permeability go to zero, meaning that the flux densities inside the scatterer are

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independent of the incident field. Also, we can observe that the electric field and the magnetic field must go to zero when the permittivity and permeability tend to infinity because the energy must be bounded. Hence, in the EH-formulation the unknowns vanish for . When the contribution from the incident field vanishes the problem is to solve a matrix equation of type . Clearly, the trivial solution is physically correct; however, the existence of null-space allows nontrivial solutions. As discussed in [16] the discretization procedure used in the EH- and DB-formulations might give rise to a larger solution space in which case the null-space might exist, and the problem becomes ill-posed. Finally, we note that the equivalent volume currents and are linear combinations of the fluxes and fields, and thus the JM-formulation is well-defined in the case where the permittivity or permeability go to either zero or infinity. With the incident field in the JM-formulation goes to zero but the resulting system matrix is diagonal, and therefore, the problem is well-posed.

Fig. 8. Difference in total scattering cross section between the ideal DB sphere and its realization either with isotropic or radially uniaxial material.

V. APPLICATION TO THE DB BOUNDARY As an application of extreme anisotropy we study a material realization of the DB boundary condition [1], [27]. By the term “realization” we mean an interface of two media with given constitutive parameters that mimics the ideal boundary condition. Another step of the realization would be to find a metamaterial structure that effectively (when homogenized) gives rise to the desired medium parameters. The latter step is discussed in [28] and [29]. The DB boundary condition requires that the normal components of the electric and magnetic flux densities vanish on the boundary surface. Since the normal components of the flux densities must be continuous across material interfaces (interface conditions), it is clear that the material in which the normal components of the permittivity and permeability dyadics vanish should approximate the ideal DB boundary [1]. Obviously, the above conditions are also satisfied in the case of the simple isotropic material with , and hence we can have either an isotropic or uniaxially anisotropic realization for the DB boundary. First, we investigate how close to zero the permittivity and permeability have to be to approximate the ideal DB boundary. Fig. 8 shows the relative difference in the total scattering cross section of a sphere defined as follows: (23) where and are the total scattering cross sections of the realization and the ideal DB boundary, respectively. The total scattering cross sections are obtained by the Mie solution for either uniaxially anisotropic spheres [8] or ideal DB spheres [3]. Clearly, the isotropic material with small permittivity and permeability yields a better approximation to the ideal DB boundary than the uniaxial material. However, with decreasing the far field scattering of both materials approaches to that of the ideal DB boundary. So far, we have only considered spherical realizations of the DB boundary. To obtain more general analysis, we need to

Fig. 9. Cross section of a cube that approximates the ideal DB cube. The cube is built from six blocks in order to define the material parameters in which the . components normal to the interface vanish. The permeability is

Fig. 10. The -, -, and -components of the real part of the electric field near the cube on the electric field plane ( -plane). The solid curves denote the ideal DB cube, and stars and circles designate the isotropic and anisotropic approximations of the DB cube, respectively. The backscattering direction is at .

consider non-smooth scatterers with sharp corners and wedges since the DB wedges create singularities [30]. However, in the presence of wedges it is not clear how the anisotropy should be defined to approximate the DB boundary. One could start form a sphere, and use a proper transformation to obtain arbitrarily shaped object. Here, we use simpler approach, and construct a cube from six blocks in which the permittivities and permeabilities are defined. (Fig. 9.) The cube with edge length is illuminated by a -polarized plane-wave with frequency . Fig. 10 shows the -, -, and -components of the real part of the electric field calculated in the electric field plane ( -plane) at a distance from the center of the cube. The computations are


done for the ideal DB cube by using the surface integral equations (SIE) formulation for the DB boundary [31] (solid lines). For the isotropic (stars) and anisotropic (circles) realizations, the volume integral equation method with JM-formulation is used. The thicknesses of the blocks are 0.1 m, and the object is meshed with 4736 tetrahedral elements. We can observe that in the case of the cube, the isotropic material with exactly zero permittivity and permeability gives identical near field pattern to that of the ideal DB cube. The near field pattern of the anisotropic cube is almost identical as the ideal DB cube; however, we can see a small difference. Mainly, this error is due to the numerical approximation, and can be reduced by increasing the discretization density. Also, we note here that the far fields patterns are almost identical. VI. CONCLUSION The volume integral equation formulation based on the equivalent polarization currents (JM-formulation) for extremely anisotropic materials is presented, and compared to the more conventional formulations with either the fields (EH-formulation) or the flux densities (DB-formulation) as the unknowns. Our examples show that the JM-formulation is accurate and stable in terms of the conditioning of the system in a wide range of uniaxially anisotropic medium parameters. The EH-formulation suffers breakdown when some component of the permittivity or permeability dyadic is large in amplitude, and the DB-formulation does not work when the material parameters are close to zero. These observations agree with the mathematical theory presented in [16]. Applying the JM-formulation, and working fully in the solution is restricted to the proper function space, and thus, the problem is well-posed. Using discretization presented in this paper for the DB- and EH-formulations the actual solution space can be larger, and might lead to the ill-posed system. At extreme anisotropies the EH-, and DB-formulations become very sensitive to the small defects in discretization, and therefore, the iterative solutions do not converge. Finally we have applied the JM-formulation to analyze the properties of the material realizations of the DB boundary on which the flux densities normal to the boundary vanish. It is demonstrated that either the isotropic material with or the uniaxially anisotropic material with vanishing normal components of the permittivity and permeability approximate the ideal DB boundary. ACKNOWLEDGMENT The authors would like to thank M.Sc. T. Rimpiläinen and Dr. H. Wallén of the Department of Radio Science and Engineering, Aalto University, for providing the Mie scattering calculations. REFERENCES [1] I. V. Lindell and A. Sihvola, “Electromagnetic boundary condition and its realization with anisotropic material,” Phys. Rev. E, vol. 79, no. 2, p. 026604, 2009. [2] P. S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electron. Lett., vol. 24, pp. 168–170, Feb. 1988. [3] I. V. Lindell, A. Sihvola, P. Ylä-Oijala, and H. Wallén, “Zero backscattering from self-dual objects of finite size,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2725–2731, Sep. 2009.

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Johannes Markkanen received the M.Sc. (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), Helsinki, Finland, in 2009. He is currently working towards the D.Sc. (Tech.) degree in the Department of Radio Science and Engineering, Aalto University, Espoo, Finland. In the spring of 2012, he was a Visiting Researcher in the Department of Electrical and Computer Engineering, University of Kentucky. His research interests include computational methods in electromagnetics, and modeling of complicated electromagnetic materials and surfaces.

Pasi Ylä-Oijala received the M.Sc. and Ph.D. degrees in applied mathematics from the University of Helsinki, Helsinki, Finland, in 1992 and 1999, respectively. He is currently working as a Senior Researcher with Aalto University, School of Electrical Engineering, Department of Radio Science and Engineering, Espoo, Finland. His field of interest focuses on the development of efficient and stable integral equation techniques and solvers in computational electromagnetics, and analysis of complex electromagnetic phenomena.

Ari Sihvola (F’06) received the degree of Doctor of Technology from the Helsinki University of Technology, Helsinki, Finland (presently Aalto University), in 1987. Besides working for TKK, Aalto, and the Academy of Finland, he was Visiting Engineer in the Research Laboratory of Electronics of the Massachusetts Institute of Technology, Cambridge, from 198 to 1986. In 1990–1991, he worked as a Visiting Scientist at the Pennsylvania State University, State College. In 1996, he was Visiting Scientist at Lund University, Sweden. He was Visiting Professor at the Electromagnetics and Acoustics Laboratory of the Swiss Federal Institute of Technology, Lausanne (academic year 2000–2001), and in the University of Paris 11, in Orsay (June 2008). His research interests include waves and fields in electromagnetics, modeling of complex materials, remote sensing, and radar applications. He is currently a Professor in the Department of Radio Science and Engineering, Aalto University, Espoo, Finland. From 2005 to 2010, he served as Academy Professor of the Academy of Finland.