An enhanced volume integral equation method and augmented equivalence principle algorithm for low frequency problems

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AN ENHANCED VOLUME INTEGRAL EQUATION METHOD AND AUGMENTED EQUIVALENCE PRINCIPLE ALGORITHM FOR LOW FREQUENCY PROBLEMS

BY LIN SUN

DISSERTATION

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Electrical and Computer Engineering in the Graduate College of the

University of Illinois at Urbana-Champaign, 2010

Urbana, Illinois Doctoral Committee:

Professor Weng Cho Chew, Chair Professor Jianming Jin

Professor Jennifer T. Bernhard Professor Martin D. F. Wong

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ABSTRACT

Two techniques based on integral equation methods are addressed. Firstly, a novel volume integral equation method is proposed to characterize the scattering properties of dielectric objects involving inhomogeneous and anisotropic permittivity and permeability. Two algorithms are available: con-ventional method of moments and reciprocity preserving method. Both of them are applied to both the permittivity and permeability terms. Curl-conforming edge elements are used to model the electric field distributions. Integration by parts is applied to deal with the singularities at the boundary introduced by the discontinuities of the material properties. Duffy’s method formulations are derived for all the surface and volume singular integrations. Moreover, the multilevel fast multipole algorithm (MLFMA) is utilized to accelerate the matrix vector product process for large problems. Representative numerical results are shown to be excellent.

Secondly, the present equivalence principle algorithm (EPA) is augmented by introducing charge densities as extra unknowns. This helps to separate the vector potential term and scalar potential term and avoid the imbalance at low frequencies. The current continuity constraint is enforced in both the scattering operator and translation operator. These further form a new augmented EPA equation system. With this technique, the low-frequency breakdown of EPA is removed. The augmented system serves not only as a stable low-frequency method, but also as a substitute over the whole frequency band. The new scheme is verified by numerical examples.

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ACKNOWLEDGMENTS

My heartful thanks goes to my advisor, Professor Weng Cho Chew, for his guidance throughout my PhD study. I feel very fortunate to have joined his research group. Not only has his understanding of a wide variety topics in electromagnetics enriched my mind, but also his deep insight into physics and mathematics has inspired my interests in different areas. Also, his patience, encouragement and caring to students impressed me a lot. I believe all of these will continue to affect my future career and life very positively.

I also thank Professor Jianming Jin, who has generously let me join his group meetings and given his time and guidance to my work during the time when Professor Chew was working in Hong Kong. I have benefited a lot from his profound knowledge in electromagnetics.

I would also like to thank my other doctoral committees, Professor Jennifer T. Bernhard and Profes-sor Martin Wong, for their valuable comments and suggestions on my work. I thank ProfesProfes-sor Andreas Cangellaris and Professor Jose Schutt-Aine in the Center for Computational Electromagnetics at UIUC for their advice during my PhD study. I would also like to thank my colleagues in the Center for Computational Electromagnetics at UIUC, for they have never tired of discussing things with me and answering my questions.

Also, I would like to thank Dr. Roberto Suaya and his group at Mentor Graphics, and Dr. Alina Deutch and her group at IBM T. J. Watson Center. Working with them during the summer internships equipped me with meaningful experience and polished my knowledge in electromagnetics.

Finally, I dedicate this thesis to my parents, brother and extended family. I thank them for their unconditional love and support during my life.

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LIST OF FIGURES

2.1 RCS of Mie series (solid line) and the proposed methods. The radius of the spherea= 0.15

λwith the parameters²r = 1.0,µr= 2.2. . . 14

2.2 RCS of the sphere with radius ofa= 0.15λand parameters²r = 1.0,µr= 2.2for different meshes. . . 15

2.3 Order of convergence for theµrterm of the proposed method with different mesh densities. . 16

2.4 RCS of the sphere with radius ofa= 0.2λand parameters²r= 2.2,µr = 1.0for different meshes. . . 17

2.5 RCS of Mie series (solid line) and the proposed methods. The radius of the spherea= 0.2 λwith the parameters²r = 2.2,µr= 1.0. . . 17

2.6 Order of convergence for the²rterm of the proposed method with different mesh densities. . . 18

2.7 RCS of Mie series (solid line) and the proposed method. The radius of the spherea= 0.1λ with the parameters²r= 1.5,µr = 2.2. . . 18

2.8 RCS of the proposed methods for uniaxial sphere with radius 1 m and ²r,xx = ²r,yy = 3, ²r,zz = 2, µr,xx =µr,yy =µr,zz = 1at frequency of 45 MHz in E plane and RCS of its duality case in H plane. . . 19

2.9 RCS of the proposed methods for gyrotropic spherical shell with thickness of 0.5 m and outer radius 1 m. ²r,xx =²r,yy = 5, ²r,yx =−²r,xy =i, ²r,zz = 7, µr =Iat frequency of 60 MHz in E plane and RCS of its duality case in H plane. . . 19

2.10 RCS of the MLFMA method for 0.0322 m thick spherical shell, 1 m outer radius shell with ²r= 2.2, µr = 1, at frequencyf = 0.6 GHz, in E-plane. . . 20

2.11 A spherical shell with outer radius of 2λand 58,490 unknowns. The GMRES reduced to the error to10−3with 18 iterations. . . 20

2.12 RCS of the MLFMA method for 0.0322 m thick spherical shell, 1 m outer radius shell with ²r= 2.2, µr = 1.5, at frequencyf = 0.6 GHz, in E-plane. . . 21

2.13 (a) The four sub-tetrahedra domain inx-y-zspace. (b) The mapping of one sub-tetrahedron fromx-y-zspace tou-v-wspace. . . 22

3.1 An illustration of multi-region problem in EPA . . . 27

3.2 Outside-in propagation in equivalence principle operator. . . 29

3.3 Inside-out propagation in equivalence principle operator. . . 32

3.4 An example of the interaction between two equivalence surfaces. . . 32

3.5 An example of the interpolation points and the rectangular mesh of an equivalence surface. . . 33

3.6 The mesh of a rectangular strip loop. A single delta-gap voltage source is assigned in the middle of the bottom side. Unit: 10µm. . . . 35

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3.7 Magnitude of admittance of three methods: the augmented EFIE, the RWG based EFIE, the

EPA for a PEC strip loop in free space. . . 35 4.1 A PEC sphere is wrapped with a Huygens’ box discretized with RWG bases and rectangular

patches respectively. . . 39 4.2 The comparison of tested field using outside-in operator and direct method whena= 0.1λ

by original EPA. . . 40 4.3 The comparison of tested field using outside-in operator and direct method when a =

0.0001λby original EPA. . . 41 4.4 Relative error of the tested field by the outside-in operator whena= 0.0001λby original

EPA . . . 42 4.5 Imaginary part of the tested field using outside-in operator and direct method whena =

0.0001λby the proposed method. . . 43 4.6 Relative error of the imaginary part of the tested field whena= 0.0001λby the proposed

method. . . 43 4.7 Electric field error convergence with mesh densities at low frequency by the proposed method. 44 4.8 Tested electric field inside the Huygens’ box by the proposed method. . . 44 4.9 The relative error of the tested electric field inside the Huygens’ box by the proposed method. . 45 4.10 An equivalence surface with an electric dipole outside. . . 45 4.11 Electric field inside the equivalence surface at the frequency of 0.136364 GHz by the

pro-posed method. . . 46 4.12 Relative error of the electric field inside the equivalence surface at the frequency of 0.136364

GHz by the proposed method. . . 46 4.13 Electric field inside the equivalence surface at the frequency of 1.36364×10−5_{GHz by the}

proposed method. . . 47 4.14 Relative error of the electric field inside the equivalence surface at the frequency of 1.36364×10−5

GHz by the proposed method. . . 47 4.15 Electric field inside the equivalence surface at the frequency of 1.36364×10−6GHz by the

proposed method. . . 48 4.16 Relative error of the electric field inside the equivalence surface at the frequency of 1.36364×10−6

GHz by the proposed method. . . 48 4.17 An equivalence surface with an electric dipole inside. . . 49 4.18 Electric field outside of the equivalence surface at the frequency of 0.1364 GHz by the

proposed method. . . 49 4.19 Relative error of the electric field outside of the equivalence surface at the frequency of

0.1364 GHz by the proposed method. . . 50 4.20 Electric field outside of the equivalence surface at the frequency of 1.364×10−4 GHz by

the proposed method. . . 50 4.21 Relative error of the electric field outside of the equivalence surface at the frequency of

1.364×10−4GHz by the proposed method. . . 51 4.22 Electric field outside of the equivalence surface at the frequency of 1.364×10−5 GHz by

the proposed method. . . 51 4.23 Relative error of the electric field outside of the equivalence surface at the frequency of

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4.24 Error convergence of electric field radiated from the equivalence currents with mesh density

at frequency of 1.364×10−5_{GHz by the proposed method.} _{. . . .} ₅₂

5.1 Two equivalence surfaces with an electric dipole in one of them. . . 56 5.2 Electric field inside the right equivalence surface at the frequency of 0.136364 GHz when

box distance is 0.1 m by using originalT operator. . . 57 5.3 Relative error of the electric field inside the right equivalence surface at the frequency of

0.136364 GHz when box distance is 0.1 m by using originalT operator. . . 57 5.4 Electric field inside the right equivalence surface at the frequency of 1.36364×10−4 GHz

when box distance is 0.1 m by using originalT operator. . . 58 5.5 Relative error of the electric field inside the right equivalence surface at the frequency of

1.36364×10−4GHz when box distance is 0.1 m by using originalT operator. . . 58 5.6 Electric field inside the right equivalence surface at the frequency of 1.36364×10−4 GHz

1.36364×10−4GHz when box distance is 0.5 m by using originalT operator. . . 59 5.8 Electric field inside the right equivalence surface at the frequency of 1.36364×10−4 GHz

1.36364×10−4GHz when box distance is 0.8 m by using originalT operator. . . 60 5.10 Electric field inside the right equivalence surface at the frequency of 1.36364×10−5 _GHz

1.36364×10−5_{GHz when box distance is 0.1 m by using original}_T _operator. _{. . . .} ₆₂

5.12 Electric field inside the right equivalence surface at the frequency of 1.36364×10−6 GHz

1.36364×10−6GHz when box distance is 0.1 m by using originalT operator. . . 63 5.14 Electric field inside the right equivalence surface at the frequency of 0.136364 GHz when

box distance is 0.1 m by using augmentedT˜ operator. . . 64 5.15 Relative error of the electric field inside the right equivalence surface at the frequency of

0.136364 GHz when box distance is 0.1 m by using augmentedT˜ operator. . . 64 5.16 Electric field inside the right equivalence surface at the frequency of 1.364×10−4 _GHz

when box distance is 0.1 m by using augmentedT˜ operator. . . 65 5.17 Relative error of the electric field inside the right equivalence surface at the frequency of

1.364×10−4_{GHz when box distance is 0.1 m by using augmented}_T˜ _operator. _{. . . .} ₆₆ 5.18 Electric field inside the right equivalence surface at the frequency of 1.364×10−5 GHz

1.364×10−5GHz when box distance is 0.1 m by using augmentedT˜ operator. . . 67 5.20 Electric field inside the right equivalence surface at the frequency of 1.364×10−6 GHz

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6.1 The RCS of a PEC sphere ata= 0.01λ. . . 73

6.4 Geometry configuration of a 2-turn spiral loop.. . . 74

6.5 Current distribution of a spiral loop at the frequency of 10 GHz . . . 75

6.6 Charge distribution of a spiral loop at the frequency of 10 GHz. . . 75

6.7 Two PEC spheres wrapped with Huygens’ boxes. . . 75

6.8 RCS of two PEC spheres excited by the plane wave ata= 0.001λ. . . 76

6.9 Plane wave scattering of a metallic strip using tap basis. . . 77

6.10 Scattered electric field along a straight line at 0.0003 GHz. . . 77

6.11 A small loop conductor enclosed by two Huygens’ boxes. . . 78

6.12 Magnitude of the input impedance of a loop conductor with two Huygens’ boxes at low frequencies.. . . 78

6.13 Magnitude of the input impedance of a loop conductor with two Huygens’ boxes at high frequencies.. . . 79

6.14 A parallel plate capacitor in EPA algorithm. . . 79

6.15 Magnitude of the input impedance of a parallel plate at low frequencies. . . 80

6.16 A microstrip line from a sample package. . . 80

6.17 A microstrip line divided into four Huygens’ boxes with three connected regions. . . 81

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CHAPTER 1

INTRODUCTION

1.1

Background

Computational electromagnetics (CEM) has been becoming an important mathematical modeling tool for modern electromagnetic engineering and for the analysis and design of myriad electromagnetic sys-tems. There are two main categories of computational electromagnetic methods: One is the differential-equation-based method, in which the electromagnetic equations in their partial differential equation (PDE) forms are solved directly. This includes the element method (FEM) [1, 2, 3] and the finite-difference method (FDM) [4]. The other is the integral-equation-based solvers, in which the PDE’s are converted into relevant integral equations using Green’s functions. The typical integral equation method is the method of moments [5, 6].

In this dissertation, two techniques based on the integral equation method will be discussed: a novel volume integral equation (VIE) for anisotropic and inhomogeneous media, and an augmented equivalence principle algorithm (AEPA) for low frequency problems.

1.2

Volume Integral Equation for Inhomogeneous and Anisotropic

Media

The scattering by penetrable scatterers has been a research topic of great interest. In the early days, approximate methods such as the geometric theory of diffraction were used [7]. The extended bound-ary condition method was also investigated as a possible approach to solve such problems [8]. More recently, numerical methods have been adopted to tackle them, such as the finite element method [1, 2, 9, 10], the finite difference time domain method [11], the generalized multipole method [12], as well as the method of moments [5].

When a scatterer is piecewise constant in its inhomogeneity, it can be solved with surface integral equations [13]. The volume integral equation(VIE) is useful when the scattering solution from a highly inhomogeneous medium is sought. Due to the numerical computational intensity needed, early work was done in two dimensions [14, 15].

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A 3D scatterer involving highly inhomogeneous media can also be solved with either the finite element method [1, 2, 10] or the volume integral equation [6, 16, 17, 18, 19]. The advantage of the volume integral equation is that the solution satisfies the radiation condition automatically.

The general formulation of the VIE has been presented in [6]. The use of that formulation for dielectric scatterers where the permittivity is inhomogeneous has been demonstrated in [19, 20]. When the permeability is inhomogeneous, the use of this formulation has not been demonstrated. Other formulations, however, have been used in the literature [21, 22].

In this work, a set of novel methods to solve the volume integral equation of electromagnetic scat-tering from arbitrarily shaped inhomogeneous objects with anisotropic permittivity and permeability tensors is proposed. This set of methods is based on the vector wave equation of the electric field for anisotropic, inhomogeneous media. Curl-conforming basis is used to model the electric field [1, 2]. Because there is only one set of unknowns for the electric field, the unknown number is reduced by half compared to the method with unknowns for both electric field and magnetic field. Curl-conforming basis is also used as the testing function to simplify the matrix representation of the equation. The scattered field mainly includes two parts: one is from the contribution of induced magnetic polarization current, the other is from the induced electric polarization current. We will present the matrix represen-tations for both of them. In addition, for each of these two terms, we propose a reciprocity preserving method to get much simpler integrations. Furthermore, to reduce the computation cost for the gener-ally large problem, the multilevel fast multipole algorithm (MLFMA) [23, 24, 25, 26, 27, 28, 29, 30] is applied.

1.3

Augmented Equivalence Principle Algorithm at Low Frequencies

The equivalence principle algorithm (EPA) is an integral equation based domain decomposition method for solving 3D electromagnetic wave equations based on the equivalence theorem or Huygens’ theorem [31]. By using the equivalence principle, Lu and Chew introduced a nested equivalence principle algo-rithm (NEPAL) for volume integral equation in both two and three dimensions [32, 33]. This algoalgo-rithm transforms the volume unknowns to surfaces unknowns on the boundary of the volume scatterer by using multipole expansions. It solves the matrix equation directly with computational complexity of

O(N1.5₎_{for two-dimensional problems and of}_O₍_N2₎_{for three-dimensional problems.}

Later, Li and Chew applied the equivalence principle to both the volume integral equation and the surface integral equation and named the new algorithm the equivalence principle algorithm or EPA [34, 35]. In EPA, regions with highcomplexity are enclosed by equivalence surfaces (ES). The scattered currents on them are calculated by equivalence principle operatorsS containing the information of the inside scatterers. This includes three steps: outside-in, current solver, and inside-out. The translation

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operatorT characterizes the interactions among the equivalence surfaces. In this way, the unknowns of the inside scatterers are transferred to the unknowns on the equivalence surface that encloses the elements in the final matrix equation. This scheme provides an efficient way to solve structures with repeated elements, such as array antennas, and also objects with fine details, such as circuit structures.

In the present EPA, a challenge arises when the frequency is very low. Using the present EPA scheme to solve low frequency problems will result in big errors. This is the consequence of the decoupling of electric and magnetic fields in Maxwell’s equations right around zero frequencies. This decoupling manifests as a natural Helmholtz decomposition in the current by separating itself into a solenoidal (divergence-free) component and a complementary irrotational (nonsolenoidal or curl-free) component. These two parts of the current are in different orders at low frequency. And the contribution of the solenoidal current to the scalar potential part of theLoperator is strictly equal to zero. Because of the finite machine precision, it cannot cancel out completely. Therefore, the error from the contribution of solenoidal current will spill over to that of irrotational current, which will introduce big errors to the EPA operators.

The low frequency problem is usually remedied by the loop-tree decomposition method [36, 37, 38, 39].However, for complicated structures, this method suffers from the need to search for loops. Since EPA is a domain decomposition method for multiscale problems, finding loops for the subdomains, especially for connected subdomains cases, becomes very involved. Besides, the loop tree bases are usually needed to switch back to the traditional bases when the frequency goes higher.

In recent years, researchers have proposed multiple current and charge equations for low frequency breakdown problems [40, 41, 42]. By introducing both the current and charge unknowns in the sur-face integral equations and enforcing the current continuity condition, these methods can improve the conditioning of the matrix equations at low frequencies. Moreover, these methods are not burdened by loop search. For example, [40] proposes a current and charge integral equation (CCIE) for scattering of both metallic and dielectric objects, while the way it adds in the current continuity condition will introduce the inaccuracy problem for certain problems. Later, a separated potential integral equation (SPIE) using the same idea was proposed in [41]. But it works for lossy structures only and still breaks down for lossless problems.

Recently, an augmented EFIE method was proposed to solve the low frequency problem of EFIE [42, 43]. This method introduces the charge density unknowns and enforces the current continuity equation as an additional equation. It is better formulated than the previous methods. However, this method has only been used for the surface integral equations based on the RWG bases. It has not been applied to more complicated equations or equations with high-order point sampling method.

This work proposes an augmented EPA formulation based on the high-order point sampling scheme. It introduces charge density unknowns on the equivalence surface (ES). The current continuity condi-tion for both the electric and magnetic currents is enforced as addicondi-tional equacondi-tions. Hence, both the

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scattering operatorSand the translation operatorT are reformulated and the low frequency breakdown of operators in EPA is removed. The new augmented scheme is stable over a wide frequency range.

1.4

Organization of the Thesis

The main contributions in this dissertation are listed as follows. Chapter 2 presents the novel volume integral equation method for anisotropic and inhomogeneous objects. The MOM scheme is proposed for both the magnetic polarization current and electric polarization current terms. To simplify the matrix representation, curl conforming basis is used to discretize both the electric field and magnetic field. In addition, a reciprocity preserving approach is proposed to get the symmetric matrix for reciprocal media. In order to solve large problems, the MLFMA scheme is combined with the proposed method. Finally numerical results that verify all the methods proposed are provided.

Chapters 3−6 focus on the augmented EPA scheme. Chapter 3 discusses the low frequency break-down issues of the present EPA scheme. Reasons for low frequency breakbreak-down for different operators in EPA are analyzed. Chapter 4 introduces a current charge separation method to overcome the low frequency breakdown of field projection operators, especially the outside-in and inside-out operator. Numerical examples show the validity of this method to suppress the field projection error at low fre-quencies. Chapter 5 proposes an augmented form for the translation operator. By including the charge density unknowns on the equivalence surface, the augmented translation operator is shown to be stable over a very wide frequency band. Then in Chapter 6, the augmented EPA is proposed. Numerical examples at low frequencies validate this method.

In the last chapter, we draw conclusions and discuss some possible research in the future to improve these methods.

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CHAPTER 2

A NOVEL VOLUME INTEGRAL EQUATION

METHOD FOR ELECTROMAGNETIC SCATTERING

PROBLEM

A novel method to solve the volume integral equation involving inhomogeneous and anisotropic per-mittivity and permeability dielectric objects is introduced. A curl-conforming edge element is used to model the electric field distributions. This simplifies the process of finding the matrix representation of the integral equations. Furthermore, a reciprocity preserving method to solve the volume integral equation is presented based on the reciprocity theorem. By introducing a delta function in the vol-ume integral equation, this method decomposes one complicated integral into several simple integrals, which simplifies the calculation of integration. MLFMA is utilized to accelerate the matrix vector product process for large problems. Duffy’s method is applied for all the surface and volume singular integrations. Representative numerical results are shown to be excellent.

2.1

Volume Integral Equation

In this section, the VIE is derived in a manner similar to that in [6]. The vector wave equation for a general inhomogeneous, anisotropic medium is given by

∇ ×µ−1

r (r)· ∇ ×E(r)−ω2²r(r)·µ0²0E(r) = iωµ0J(r) (2.1)

whereµ_r(r) and²r(r) are the relative permeability and permittivity of the media. After subtracting ∇ × ∇ ×E(r)−ω2_µ

0²0E(r)from both sides of the equation, the above can be rewritten as

∇ × ∇ ×E(r)−ω2_µ 0²0E(r) = iωµ0J(r) +∇ × £ I−µ−1 r (r) ¤ · ∇ ×E(r) −ω2_µ 0²0 £ I−²r(r) ¤ ·E(r) (2.2)

The dyadic Green’s function corresponding to the differential operator on the left-hand side can be derived, viz.,

∇ × ∇ ×G(r,r0₎₋_k2

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wherek0 =ω√µ0²0, and G(r,r0_{) =} µ I+∇∇ k2 0 ¶ g(r,r0₎ _(2.4)

G(r,r0₎_·_a _{can be thought of as the field due to a point source} _aδ₍_r₋_r0₎_{located at} _r ₌ _r0_{. By the}

principle of linear superposition, we treat the right-hand side of (2.2) as equivalent volume sources, and write down the solutionE(r)as

E(r) =iωµ0 Z V+ G(r,r0₎_·_J₍_r0₎_dr0 + Z V+ G(r,r0)· ∇0×£I−µ_r−1(r0)¤· ∇0 ×E(r0)dr0 −k2 0 Z V+ G(r,r0₎_·£_I₋_² r(r0) ¤ ·E(r0₎_dr0 _(2.5)

whereV+is a volume that is slightly larger than the support of the scattererV defined by the volume whereµ_ror²rdeparts fromI. The first term can be considered the field generated by the current source

Jin the absence of the scatterer. Hence, we call itEinc(r), a known field. Consequently, (2.5) becomes an integral equation forE(r), or

E(r) = Einc₍_r_{) +} Z V+ G(r,r0₎_{· ∇}0_×£_I₋_µ−1 r (r0) ¤ · ∇0_×_E₍_r0₎_dr0 −k2 0 Z V+ G(r,r0₎_·£_I₋_² r(r0) ¤ ·E(r0₎_dr0 _(2.6)

The second term on the right-hand side represents the scattered field due to induced magnetic polar-ization current from the inhomogeneous permeability, while the third term represents that due to the induced electric polarization current from the inhomogeneous permittivity. A similar derivation has been presented in [6] for isotropic media. This derivation cleanly separates the scattered field of the inhomogeneous permeability from that of the inhomogeneous permittivity.

The above is not suitable for computational electromagnetics. However, the equation can be made computationally friendly using integration by parts. The dyadic Green’s function in the second term on the right-hand side of (2.6) has a term that involves the double del operator. But it can be shown that

∇∇ Z V+ g(r,r0₎_{· ∇}0_×_E₍_r0₎_dr0 ₌_−∇ Z V+ ∇0_g₍_r,_r0₎_{· ∇}0_×_E₍_r0₎_dr0 =−∇ Z V+ ∇0 _{× ∇}0_g₍_r,_r0₎_·_E₍_r0₎_dr0 _{= 0} _(2.7)

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Consequently, (2.6) becomes E(r) = Einc(r) + Z V+ g(r,r0)∇0×£I−µ_r−1(r0)¤· ∇0×E(r0)dr0 −k2 0 Z V+ G(r,r0₎_·£_I₋_² r(r0) ¤ ·E(r0₎_dr0 _(2.8)

2.1.1

The First Method for

µ

_r

Term

Because the curl operator acts on the function to its right, there is a singularity in theµ_rterm in (2.8) on the surface of the object. To make it computationally friendly, we rewrite it in two ways, which means either move the curl operator out of the integral by integration by parts or move the curl operation to the Green’s function. Here, we take the first method; then the formulation including theµ_rterm only is E(r) = Einc₍_r_{) +}_{∇ ×}_M µ(E(r0)) (2.9) where Mµ(E(r0)) = Z V+ g(r,r0)£I−µ−_r1(r0)¤· ∇0×E(r0)dr0 (2.10)

2.1.2

The Second Method for

µ

_r

Term: The Reciprocity Preserving Approach

We present a reciprocity preserving approach [44] in this section. When a subspace is spanned by a set of non-orthogonal basisfn(r),n= 1, ..., N,we can expand a functionf(r)in that subspace by

f(r) = N

X

n=1

anfn(r) (2.11)

The above is a 3 vector, or a3×1matrix. Testing the above withf_mt(r),m = 1, . . . , N,we have hf_mt,fi=X

n

anhfmt,fni (2.12)

Usually, it is understood that the vector at the left of the bracket notation is a transpose vector, but we explicitly indicate it here for clarity. To solve foran, we can rewrite the above as

an=

X

m

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where bnm = h F−1 i nm = £ B¤_nm, £F¤_mn =hft m,fni= Z V ft m(r)·fn(r)dr (2.14) Using (2.13) in (2.11), we have f(r) =X nm fn(r)bnmhfmt,fi= Z dr0X nm fn(r)bnmfmt(r0)·f(r0) (2.15)

From (2.15), we get the expression for the approximateδˆfunction as ˆ δ(r,r0) =X nm fn(r)bnmfmt(r0) = f t (r)·B·f(r0) (2.16)

whereft(r) = [f1(r),f2(r), ...,fN(r)]. Here,ˆδ(r,r0)is approximating the delta functionδ(r−r0). Note thatftis a3×N matrix, whilef isN ×3.

By insertingδˆ(r0,r00)into theµ_rterm in (2.8), we get

E(r) =Einc(r) + Z V Z V dr0dr00g(r,r0)ˆδ(r0,r00)DµE(r00) (2.17) whereDµE(r00) = ∇00× £ I−µ−1 r (r00) ¤ · ∇00_×_E₍_r00_{). Let} E(r00) = X m fm(r00)em =f t (r00)·e (2.18)

Using (2.18) in (2.17), we then have

ft(r)·e=Einc₍_r_{) +} Z V Z V dr0_dr00_g₍_r,_r0_)ˆ_δ₍_r0_,_r00₎_D µf t (r00₎_·_e _(2.19)

Multiplying the above byf(r)and integrating, we have

hf,fti ·e=hf,Einc_i₊_h_f_{, g,}_δ,_ˆ _D µ,f

t

i ·e (2.20)

In the above bracket notation, each comma represents a one-fold integration. Hence, the first term on the right-hand side has one-fold integration, while the second term has four-fold integration. Substituting the definition ofδˆ=ft·B·f, we have

hf,fti ·e=hf,Einc_i₊_h_f_{, g,}_ft_{i ·}_B_{· h}_f_,_D µ,f

t

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The above is F·e=hf,Einci+g·B·Dµ·e (2.22) whereF=hf,fti,g=hf, g,fti,Dµ=hf,Dµ,f t i. Besides, £ Dµ ¤ ij = Z V fi(r)· ∇ × £ I−µ−1 r (r) ¤ · ∇ ×fj(r)dr = Z V ∇ ×fi(r)· £ I−µ−_r1(r)¤· ∇ ×fj(r)dr (2.23) We call this the reciprocity preserving approach because the matrix system solved corresponds to a symmetric matrix when the medium is reciprocal. Reciprocity is deeply related to symmetry of opera-tors [45]. Equation (2.22) can be rearranged to yield

¡ F·g−1_·_F₋_D µ ¢ ·e =F·g−1_{· h}_f_,_Einc_i_. (2.24) The matrix operator on the left side is a symmetric matrix becauseF,gandDµare symmetric matrices, while most methods of solving the VIE do not preserve this symmetry.

2.1.3

The First Method for

²

r

Term

The meaning ofG(r,r0)in the²r term of (2.8) still has to be properly defined since the dyadic Green’s function is plagued with singularities whose evaluation has to be taken with great care. However, the effect of the singularity can be mitigated if we define the action of the dyadic Green’s operator on

£ I−²r(r0) ¤ ·E(r0₎_{to mean} −k2 0 Z V dr0_G₍_r,_r0₎_·£_I₋_² r(r0) ¤ ·E(r0_{) =} ₋_k2 0 µ I+∇∇ k2 0 ¶ · Z V g(r,r0₎£_I₋_² r(r0) ¤ ·E(r0₎_dr0 = −k₀2A²(E(r0))− ∇G²(E(r0)) (2.25) where A²(E(r0)) = Z V g(r,r0)£I−²r(r0) ¤ ·E(r0)dr0 (2.26) G²(E(r0)) = Z V ∇g(r,r0₎_·£_I₋_² r(r0) ¤ ·E(r0₎_dr0 _(2.27)

The formulation above is preferable for anisotropic media. Duffy’s method is used to deal with the singular integration in the second term. For the isotropic media, the singularity of the second term

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above can be further reduced by applying integration by parts. Besides, due to the property of the first order edge element, the divergence of the basis function inside each tetrahedron disappears. In this case, the second term can be reformulated as a surface integral over each tetrahedron.

∇ Z V ∇0g(r,r0)·£I−²r(r0)¤·E(r0)dr0 =∇ Z S dS0· {g(r,r0)£I−²r(r0)¤·E(r0)} (2.28) Duffy’s method for the surface integral is used to calculate the singular term in this surface integration.

2.1.4

The Second Method for

²

r

Term

The dyadic Green’s function for an unbounded, homogeneous medium can also be written as

G(r,r0) = 1

k2 0

[∇ × ∇ ×Ig(r,r0)−Iδ(r,r0)] (2.29) By substituting it into the²r term of (2.8), we can easily get the other formulation for the²rterm,

²r(r)·E(r) = Einc(r)− ∇ ×

Z

V

©

∇g(r,r0)×£I−²r(r0)¤·E(r0)ªdr0 (2.30) where we have moved the delta function contribution from (2.29) in (2.30) to the left-hand side.

Like the first method, when the medium is isotropic, we can reduce the singularity further by inte-gration by parts. In this case, (2.30) can be reformulated as

−∇ × Z V © ∇g(r,r0)×£I−²r(r0)¤·E(r0)ªdr0 =−∇ × Z V g(r,r0₎_∇0_{× {}£_I₋_² r(r0) ¤ ·E(r0₎_}_dr0 _(2.31)

which is preferable for the isotropic case.

2.1.5

The Third Method for

²

r

Term: The Reciprocity Preserving Approach

The expression of G(r,r0) in the manner of (2.29) allows its matrix representation to be found in a simple manner using curl conforming basis. Using curl conforming basis, the matrix representation of hf,G,ftiis

hf(r),G(r,r0),ft(r0)i= 1

k2 0

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Using the above in Esca(r) = −k2₀ Z dr0G(r,r0)·[I−²r(r0)]·E(r0) (2.33) we have b=hf(r),Esca₍_r₎_i₌₋_k2 0hf(r),G(r,r0)·[I−²r(r0)]·E(r0)i (2.34) By lettingE(r) =ft(r)·eand insertingδˆ=ft·B·f, we have

b = −k2₀hf(r),G(r,r0),ft(r0)i ·B· hf(r00),[I−²r(r00)]·ft(r00)i ·e

= −k2

0g·B·D²·e (2.35)

Thus, the matrix representation of the²r term can be easily found in the same manner as that of µr term. And from (2.22) and (2.35), we see thatgandBare the same matrix. Therefore, we need only to calculate them once and this can significantly reduce the time of matrix filling. In this way, to get the general solution, the matrix-vector production is increased to three times instead of one.

Equation (2.8), due to the∇∇operator inside the integration operating ong(r−r0), has a singularity of1/|r−r0|3 when r → r0. Consequently, it has to be redefined in this case for it does not converge uniformly, specifically, when r is also in the source region occupied by J(r). Therefore, in all the methods above, we never move both the double ∇ operators completely to the Green’s function to render it ill defined. That is to say, the integrals in (2.10), (2.25), (2.30) are always well defined. Duffy’s methods for both surface and volume integrals are used to calculate these singular integrations.

2.2

Application of MLFMA

Like other MOM based methods, the VIE algorithm has the same dense-matrix bottleneck for large problems. This severely limits the capability of the VIE method in dealing with large objects since the dense matrix has a memory requirement ofO(N2)and computational complexity ofO(N3)to compute the matrix-vector products.

One solution to the problem above is to accelerate the matrix-vector products using the multilevel fast multipole algorithm (MLFMA). To implement MLFMA, the entire object is first enclosed in a large cube, which is divided into eight smaller cubes. Each subcube is then recursively subdivided into smaller cubes until the edge length of the finest cube is about a half wavelength or less. For two points in the same or nearby finest cubes, their interaction is calculated in a direct manner. However, when the two points are in different nonnearby cubes, their interaction is calculated by MLFMA. The level

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of cubes on which MLFMA is applied depends on the distance between the two points. The detailed description of MLFMA is given in [30].

To get the general matrix equation, Equation (2.6) can be discretized by first approximatingV by a sum of tetrahedra and then expandingE(r)as

E(r) = Ne

X

i=1

IiNi(r) (2.36)

whereNedenotes the total number of edges inV andNi(r)denotes the curl conforming basis function on thei-th edge: there are six edge bases in each tetrahedron and each edge is shared by more than one tetrahedron. UsingNi(r)as the testing function, we obtain the matrix equation as follows:

Ne X j=1 AijIj =bi, i= 1,2, ..., Ne (2.37) where Aij =Aiij +Aµij +A²ij (2.38) Ai ij =− Z Vi Ni(r)·Nj(r)dv (2.39) Aµ_ij = Z Vi Ni(r)· ∇ × Z Vj g(r,r0₎₍_I₋_µ−1 r (r0))· ∇0×Nj(r0)dv0dv (2.40) A²_ij =−k2₀ Z Vi Ni(r)· Z Vj G(r,r0)·(I−²r(r0))·Nj(r0)dv0dv (2.41) When the interactions are between the nonnearby groups,Ai_ij vanishes, and Aµ_ij andA²_ij are computed through the fast multiple expansion as

Aµ_ij = I d2_ˆ_kVµ f,im(ˆk)·Tmm0(ˆk·rˆ_mm0)Vµ s,jm0(ˆk) (2.42) A² ij = I d2ˆ_kV² f,im(ˆk)·Tmm0(ˆk·rˆ_mm0)V²_s,jm0(ˆk) (2.43)

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are the radiation and receiving patterns of theµ_rand²rterms, respectively, with definition of Vµ_f,im(ˆk) = Z V eik·rim_N i(rim)dv (2.44) Vµ_s,jm0(ˆk) = ik× Z V e−ik·rjm0₍_I₋_µ−1 r (rjm0))· ∇ ×Nj(r_jm0)dv0 (2.45) V² f,im(ˆk) = Z V eik·rim_N i(rim)dv (2.46) V²_s,jm0(ˆk) = −k2₀ Z V e−ik·rjm0₍_I₋ˆ_kˆ_k₎_·₍_I₋_² r(rjm0))·N_j(r_jm0)dv0 (2.47)

Here, Ni(rim) resides in a group m centered at rm, Nj(rjm0) resides in a group m0 centered atr0_m. rim=ri−rm,rjm0 =r_j −r0_m,r_mm0 =r_m−r0_m. T_mm0 takes the same form as in [30], which is

Tmm0(ˆk·rˆ_mm0) = k 2 0 (4π)2 L X l=0 il₍₂_l_{+ 1)}_h(1) l (krmm0)P_l(ˆr_mm0 ·ˆk) (2.48)

In the above, noticing that bothVµ_s,jm0(ˆk)andV_s,jm² 0(ˆk)have onlyθandφcomponents, then only theθˆ

andφˆcomponents ofVµ_f,im(ˆk)andV²_f,im(ˆk)are needed. Moreover, sinceVµ_f,im(ˆk)andV²_f,im(ˆk)have the same form,Vµ_s,jm0(ˆk)andV²_s,jm0(ˆk)can be summed together as one radiation pattern and share one

aggregation-translation-disaggregation process. Finally, we can rewrite the matrix-vector multiply as Ne X j=1 AijIj = X m0_∈_B_m X j∈G0 m AijIj + I d2_kV_ˆ µ,² f,im(ˆk)· X m0_∈_/_B_m Tmm0(ˆk·rˆ_mm0) X j∈G0 m (Vµ_s,jm0(ˆk) +V²_s,jm0(ˆk)) (2.49)

fori ∈ Gm, whereGm denotes all the elements in the m-th group, andBm denotes all nearby groups of them-th group. The first term is the contribution from nearby groups, and the second term is the far interaction calculated by FMM.

2.3

Numerical Results

2.3.1

Isotropic Spheres

To validate the algorithms proposed above, three examples of plane wave scattering of the isotropic sphere with radius of 1 m are introduced first. The sphere is placed in free space and the incident

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construct 5,053 edge unknowns. Different frequencies and parameters are tested. RCS results obtained are compared with those obtained by the Mie series code. The observation points are atθ = [0o,180o] andφ = 0o_.

For the first case, the radius of the sphere equals0.15λ. Here,λrefers to the free-space wavelength. The permittivity and permeability are chosen to be ²r = 1.0, µr = 2.2. The incident wave is φ -polarized. As can be seen in Figure 2.1, the results obtained from proposed methods in Sections 2.2.1 and 2.2.2 for isotropic media agree excellently with that of Mie series. Figure 2.2 shows the RCS plots with different mesh densities. It can be seen that the RCS converges to the analytical value as the mesh density increases. The RCS errors with respect to the number of unknowns associated with theµrterm are plotted in Figure 2.3. The results show that the decrease rate of the RCS errors for theµr term is about in the 0-th order.

0 20 40 60 80 100 120 140 160 180 −65 −60 −55 −50 −45 −40 −35 −30 −25 −20 −15 θ(Degrees) RCS(dB/ λ 2 ) Mie First Method Second Method

Figure 2.1: RCS of Mie series (solid line) and the proposed methods. The radius of the spherea= 0.15λwith the parameters²r = 1.0,µr= 2.2.

For the second case, the radius of the sphere is 0.2 λ. The constitutive parameters are ²r = 2.2,

µr = 1.0. The incident wave is θ-polarized. As shown in Figure 2.4, the results by methods for isotropic media proposed in Sections 2.2.3, 2.2.4 and 2.2.5 have a good agreement with that of Mie series. Figure 2.5 shows the RCS convergence with different mesh densities. Furthermore, Figure 2.6 studies the order of convergence for the²rterm with different number of discretizations. It shows that the²r term has the first-order convergence rate by the proposed method, which is higher than the µr term. In the proposed method, since linear elements are employed to expand the electric field, andµr term is associated with the curl operation of the electric field, by taking the curl operation, the order of basis element is reduced. However, for theµrterm, high accuracy can be achieved by employing more discretizations.

In the third case, the radius of the sphere is chosen to be 0.1λ, and²r= 1.5,µr= 2.2. The incident wave isφ-polarized. As shown in Figure 2.7, again the results from method in Sections 2.2.1 and 2.2.3

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0 20 40 60 80 100 120 140 160 180 −80 −70 −60 −50 −40 −30 −20 −10 θ(Degrees) RCS Error(dB/ λ 2 ) Mie

VIE with 364 meshes VIE with 784 meshes VIE with 1174 meshes VIE with 3601 meshes

Figure 2.2:RCS of the sphere with radius ofa= 0.15λand parameters²r= 1.0,µr = 2.2for different meshes. both agree well with that of Mie series.

2.3.2

Anisotropic Spherical Shell

In this part, methods proposed in Sections 2.2.1 and 2.2.3 for anisotropic media are verified by scatter-ing from uniaxial sphere and gyrotropic spherical shell.

For the first case, assume that a homogeneous, uniaxial anisotropic sphere of radius 1 m is cen-trally located in the free space. The permittivity and permeability tensors are characterized by the two matrices ²=²t(ˆxxˆ+ ˆyyˆ) +²zzˆzˆ=    ²t 0 0 0 ²t 0 0 0 ²z    µ=µt(ˆxxˆ+ ˆyyˆ) +µzzˆzˆ=    µt 0 0 0 µt 0 0 0 µz   

The incident electric field is a plane wave, polarized in the xˆ direction, and propagating in the +ˆz direction. The radius of the sphere is equal to 0.15λ. It is discretized into 3,601 tetrahedra with 5,053 edge unknowns. To demonstrate the validity of the methods for anisotropic media, we compare the RCS in the E plane (solid line) for a sphere with constitutive parameters²t= 3²0,²z = 2²0,µt=µz = 1

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0 500 1000 1500 2000 2500 3000 3500 4000 10−2 10−1 100 Number of Unknowns RCS Error(dB/ λ 2 )

Figure 2.3:Order of convergence for theµrterm of the proposed method with different mesh densities. 1²0. By duality, the RCS results for these two cases are the same. By the methods proposed in Section

2.2.1 and 2.2.3, one can see the RCSs for these two cases have an excellent agreement, which is shown in Figure 2.8.

The second example is a source-free plasma anisotropic spherical shell, which is illuminated by a plane wave electric field polarized inxˆand that propagates in+ˆzdirection. The electric dimensions of the outer and inner spherical surfaces are chosen ask0a2= 0.2πandk0a1= 0.1π. The permittivity and

permeability are gyrotropic tensors, characterized by Hermitian matrix

²=    ²1 −i²0 0 i²0 ²1 0 0 0 ²3    µ=    µ1 −iµ0 0 iµ0 µ1 0 0 0 µ3   

The meshes include 3,354 tetrahedra with 4,824 edge unknowns. In Figure 2.9, the solid line shows the RCS in the E plane scattering from the spherical shell with permittivity²1 = 5²0, and²3= 7²0, and

the permeability is an identity tensor. The dashed line shows the RCS for its duality case, which is in the H plane and the spherical shell has the constitutive parametersµ1 = 5 µ0 andµ3 = 7µ0, while the

permittivity is an identity tensor. As shown, the RCS results for these two cases both agree well with the result given by Reference [46] .

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0 20 40 60 80 100 120 140 160 180 −90 −80 −70 −60 −50 −40 −30 −20 θ(Degrees) RCS Error(dB/ λ 2 ) Mie

VIE with 364 meshes VIE with 784 meshes VIE with 1174 meshes VIE with 3601 meshes

Figure 2.4:RCS of the sphere with radius ofa= 0.2λand parameters²r = 2.2,µr= 1.0for different meshes.

0 20 40 60 80 100 120 140 160 180 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 θ(Degrees) RCS(dB/ λ 2 ) Mie First method Second method Third method

Figure 2.5: RCS of Mie series (solid line) and the proposed methods. The radius of the spherea= 0.2λwith the parameters²r = 2.2,µr= 1.0.

2.3.3

MLFMA

Using the MLFMA method described in Section 2.3, both the CPU memory and costs are reduced tremendously for large problems. The following two examples are to test the accuracy of the MLFMA scheme.

The first example calculated is the scattering of a spherical shell with outer radius of 1 m and thickness of 0.0322 m. The permittivity is ²r = 2.2, and the permeability is an identity. The shell is excited by anxˆ-polarized electric field wave which propagates in the −zˆdirection at frequency 0.6 GHz. The mesh consists of 35,094 tetrahedra with 58,490 edge unknowns. The observation points are atθ = [0o,180o]andφ = 0o. Figure 2.10 shows the MLFMA and Mie RCS results in E-plane. They agree well with each other. The matrix solving takes 18 steps and 1 min 23.70 s to converge to10−3by

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0 500 1000 1500 2000 2500 3000 3500 4000 10−1.6 10−1.5 10−1.4 10−1.3 10−1.2 10−1.1 Number of Unknowns RCS Error(dB/ λ 2 )

Figure 2.6:Order of convergence for the²rterm of the proposed method with different mesh densities.

0 20 40 60 80 100 120 140 160 180 −70 −65 −60 −55 −50 −45 −40 −35 −30 −25 −20 θ(Degrees) RCS(dB/ λ 2 ) VIE Mie

Figure 2.7: RCS of Mie series (solid line) and the proposed method. The radius of the spherea= 0.1λwith the parameters²r = 1.5,µr= 2.2.

GMRES method. Figure 2.11 shows the details of the convergence.

The second example is the same spherical shell as in the first case, while the constitutive parameters are ²r = 2.2, µr = 1.5. The spherical shell is excited by the plane wave at frequency 0.6 GHz and discretized by 86,295 tetrahedra with 143,825 edge unknowns. Figure 2.12 shows the RCS result by MLFMA and Mie. They have a good agreement. The matrix solving takes 20 steps and 3 min 13.27 s to converge to 10−3 by GMRES method. Both of these cases show the advantage of VIE in good convergence.

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0 20 40 60 80 100 120 140 160 180 −60 −55 −50 −45 −40 −35 −30 −25 −20 −15 −10 θ(Degrees) RCS(dB/ λ 2 ) Method in 2.1, E−plane Method in 2.3, H−plane

Figure 2.8: RCS of the proposed methods for uniaxial sphere with radius 1 m and²r,xx = ²r,yy = 3, ²r,zz = 2, µr,xx =µr,yy=µr,zz = 1at frequency of 45 MHz in E plane and RCS of its duality case in H plane.

0 20 40 60 80 100 120 140 160 180 −45 −40 −35 −30 −25 −20 θ(Degrees) RCS(dB/ λ 2 ) Method in 2.1.1,E−plane Method in 2.1.3,H−plane Result of Geng, Wu and Li

Figure 2.9:RCS of the proposed methods for gyrotropic spherical shell with thickness of 0.5 m and outer radius 1 m. ²r,xx =²r,yy = 5, ²r,yx =−²r,xy =i, ²r,zz = 7, µr=Iat frequency of 60 MHz in E plane and RCS of its duality case in H plane.

2.4

Duffy’s Method for Volume Integration

Because of the1/R and1/R3 term in the integrand, the integrals in (2.10), (2.25), (2.30) are singular if a testing pointris inside a source cell. A regular numerical scheme cannot obtain accurate results, while it can be shown that these terms are integrable. In this paper, Duffy’s transform method is used to do these integrations [30].

In what follows, we will consider the integration of

Iv(r) = Z V R R3 ·F(r 0₎_dr0_,_r0 _∈_V (2.50) The integral domainV in (2.50) is a tetrahedron andF(r0)is a regular function ofr0. First we divide the

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0 20 40 60 80 100 120 140 160 180 −40 −30 −20 −10 0 10 20 30 40 θ(Degrees) RCS(dB/ λ 2) Mie MLFMA

Figure 2.10: RCS of the MLFMA method for 0.0322 m thick spherical shell, 1 m outer radius shell with ²r = 2.2, µr = 1, at frequencyf = 0.6 GHz, in E-plane.

Figure 2.11:A spherical shell with outer radius of 2λand 58,490 unknowns. The GMRES reduced to the error to10−3with 18 iterations.

tetrahedron into four sub-tetrahedra. All of them have a common vertex ofr. As a result, the integral in (2.50) can be expressed as the sum of four integrals

Iv =I1+I2+I3+I4 (2.51)

Let us consider one of the four integrals

Ij = Z Vj R R3 ·F(r 0₎_dr0 _(2.52)

A transformation of(x, y, z)→(u, v, w)is performed such that

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0 20 40 60 80 100 120 140 160 180 −20 −10 0 10 20 30 40 θ(Degrees) RCS(dB/ λ 2 ) Mie MLFMA

Figure 2.12: RCS of the MLFMA method for 0.0322 m thick spherical shell, 1 m outer radius shell with ²r = 2.2, µr = 1.5, at frequencyf = 0.6 GHz, in E-plane.

y= (ya−y0)u+ (yb−ya)v+ (yc−yb)w+y0 (2.54)

z = (za−z0)u+ (zb−za)v+ (zc−zb)w+z0 (2.55)

The domain is shown in Figure 2.13 andxa, ya, za, xb, yb, zb, xc, yc, zcare the(x, y, z)coordinates of the three noncommon vertex of the four sub-tetrahedra. Since the transformation from the(x, y, z)system to(u, v, w)system is linear, the Jacobian is a constant.

√ gj =Aj = ¯ ¯ ¯ ¯ ¯ ¯ ¯ a11 a12 a13 a21 a22 a23 a31 a32 a33 ¯ ¯ ¯ ¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯ ¯ ¯ ¯ xa−x0 xb−xa xc−xb ya−y0 yb−ya yc−yb za−z0 zb−za zc−zb ¯ ¯ ¯ ¯ ¯ ¯ ¯ (2.56)

Using the transformation in (2.53), (2.54) and (2.55), the integral given in (2.52) can be expressed in the(u, v, w)system as Ij =√gj Z ₁ 0 Z _u 0 Z _v 0 R R3 ·F(r 0₎_dwdvdu (2.57) In the last step, two more new variablessandtare introduced such that v =tu, w =sv =stu, hence

dv=udt, dw=utds. Then,

Ij =√gj Z ₁ 0 Z ₁ 0 Z ₁ 0 R R3 ·F(r 0₎_u2_tdsdtdu _(2.58)

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(1,0,0) (1,1,0) (1,1,1) (0,0,0) u v w 1 2 3 4 V1 V2 V3 V4 (a) (b)

Figure 2.13: (a) The four sub-tetrahedra domain inx-y-zspace. (b) The mapping of one sub-tetrahedron from x-y-zspace tou-v-wspace.

where

R=−(a11u+a12tu+a13stu, a12u+a22tu+a23stu, a31u+a32tu+a33stu) =uR0 (2.59)

and

R0 =−(a11+a12t+a13st, a21+a22t+a23st, a31+a32t+a33st) (2.60)

Note thatR0 6= 0for all s ∈ [0,1], t ∈ [0,1], since for nontrivial cases, s+t 6= 0. Therefore, the

integral in (2.58) is finally transformed to a form that can be evaluated using a numerical method.

Ij =√gj Z ₁ 0 Z ₁ 0 Z ₁ 0 R0 R3 0 ·F(r0₎_tdsdtdu _(2.61)

By the same method, we can derive the numerical expression of another singular integral in (2.10), (2.25) and (2.30): Z V R R ·F(r 0₎_dr0 ₌ 4 X j=1 √ gj Z ₁ 0 Z ₁ 0 Z ₁ 0 R0 R0 ·F(r0)tdsdtdu (2.62)

For the singular integral over a surface triangle, the procedure is similar to the above. The idea is to partition the domain in(x, y, z)into three sub-triangles that share a common vertex(x0, y0, z0). Then

we use the two steps of transformation to transfer the integration in a triangle to a regular integration that can be calculated with numerical method. The transformations needed are

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y= (ya−y0)u+ (yb−ya)v+y0 :=a21u+a22v+y0 (2.64)

z = (za−z0)u+ (zb−za)v +z0 :=a31u+a32v+z0 (2.65)

and

v =tu (2.66)

where the parametersxa, xb, ya, yb, za, zb are thex, y, z coordinates of the two noncommon vertices of the three sub-triangles. After applying (2.63)-(2.66), we get

Z S R R ·F(r 0₎_dr0 ₌ 4 X j=1 gj Z ₁ 0 Z ₁ 0 R0 R0 ·F(r0₎_dtdu _(2.67)

where the Jacobiangj is a constant, which is

gj = p (g11∗g22−g122), g11= q (a2 11+a221+a231), g22= q (a2 12+a222+a232), g12= p (a11∗a12+a21∗a22+a31∗a32) (2.68) andR0 is given by R0 =−(a11+a12t, a21+a22t, a31+a32t) (2.69)

2.5

Conclusions

A set of VIE formulations for application to dielectric objects has been given. Compared to the previous methods, a general inhomogeneous anisotropic medium is considered. Curl conforming basis is used to represent the electric field. The resulting matrix representation of the integral equation is simple when edge elements are used for the basis and testing functions.

Furthermore, this work discusses the way to deal with discontinuities of the material properties at the boundary, or of the basis and testing functions at the boundary of these functions. The derivative of the step discontinuities in material properties or functions generates delta function singularities which introduce the surface integral terms.

In addition, the reciprocity preserving approach is proposed to simplify the calculation of matrix elements. And MLFMA combined with the proposed method is utilized to reduce the memory cost and matrix-vector computation.

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Finally, numerical examples of different cases show the validity of the proposed methods. We have validated the formulation for scattering from an isotropic medium by comparing with the Mie series solution. We verified the formulation for a general anisotropic medium by scattering of a uniaxial sphere and gyrotropic spherical shell. And scattering from a large spherical shell was studied to show the validity of the code for large problems.

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CHAPTER 3

LOW FREQUENCY BREAKDOWN OF THE

EQUIVALENCE PRINCIPLE ALGORITHM

The equivalence principle algorithm (EPA) serves as an attractive domain decomposition method to deal with multiscale problems [47, 35]. It provides an efficient electromagnetic solver for structures like random antenna arrays, periodic structures with defects, etc. However, like many of the method of moments (MoM) techniques, EPA has a low frequency breakdown problem when the frequency goes down to nearly zero, which severely limits the application of the present EPA scheme for mid-frequency problems to the small structures. As is known, in the present EPA scheme, the equivalence principle replaces currents distributed in a volume by equivalent current residing on the bounding surface. In-side the equivalence surface, EFIE serves as one of the current solvers to compute the currents on the conductor. When the object size is small compared with the wavelength, EPA loses its accuracy and be-comes invalid. In addition, the present EPA method also has the low frequency breakdown problem for field projection operators. Therefore, a study of the low frequency breakdown of the EPA is presented in this chapter and it provides a guideline for the application of EPA in the low frequency regime.

3.1

Introduction

EPA is based on the equivalence principle, also known as Huygens’ principle [6, 31, 32, 33], which shows that the fields inside or outside a closed surface can be determined by the tangential components of the fields on the surface. The electric and magnetic field can be written respectively as

E(r) = −∇ × Z s dS0_g₍_r,_r0₎_M s(r0)− 1 iω²∇ × ∇ × Z s dS0_g₍_r,_r0₎_J s(r0) =−KS EM(r,r0)Ms(r0)− LSEJ(r,r0)Js(r0) (3.1) H(r) =KS HJ(r,r0)Js(r0)− 1 η2L S HM(r,r0)Ms(r0) (3.2) whereM =−nˆ×E,J = ˆn×H, andg(r,r0)is the Green’s function in the embedding medium. The formnˆ(r)is the normal direction pointing towards the region whereris located.

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Equations (3.1) and (3.2) display the way of decomposing the whole solution domain into several subdomains using the equivalence currents on the surfaces of these subdomains. The scattered field is calculated via the equivalence surface instead of directly from the object. The scattering via an equivalence surface is described by the equivalence principle operator, which is defined as [35]

" Jsca s 1 ηMscas # = " −nˆ0 _{× K}S HJ −1 ηnˆ0× LSEJ # ·[Lpp]−1· h −LS EJ −ηKSEM i · " Jinc s 1 ηMincs # =S · " Jinc s 1 ηMincs # (3.3) In the above, the equivalence principle operatorSincludes three steps: The first step is the outside-in propagation, which calculates the outside-incident currents on the object from the current on the equivalence surface. The second step is the solving for the current on the object by MOM method. EFIE is used for conductors and VIE is used for dielectrics. The third step is called inside-out propagation. Once the current on the object is known, the scattered electric and magnetic currents on the equivalence surface can be computed. Therefore, the scattered field outside can be calculated from the scattered currents on the equivalence surface (ES), which are solved given the incident currents on the ES by Eq. (3.3).

Besides equivalence principle operator, the radiation from one equivalence surface to the other can be captured using the translation operator as

" J2 1 ηM2 # =Thh_· " J1 1 ηM1 # = " ˆ n× KS HJ −1ηnˆ× LSHM 1 ηnˆ× LSEJ nˆ× KSEM # · " J1 1 ηM1 # (3.4)

Here, 1_η is the normalization factor. In EPA, the translation operator describes the equivalence currents on one equivalence surface induced by the currents on another ES.

After both the equivalence principle operatorS and the translation operatorT are set up, the EPA scheme can be used to solve the multi-region problem. For a general multi-region problem, the nota-tions of the surfaces and regions are illustrated in Figure 3.1. In this figure, there are two equivalence surfaces and one PEC. The interactions between the electric and magnetic currents on ES1 and ES2

are defined byThh. The interactions between the electric current on PEC3 and the equivalent electric

and magnetic currents on ES1 and ES2 are defined byThpandTph. As is derived in [35], the electric

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Figure 3.1:An illustration of multi-region problem in EPA following matrix equation:

" Jsca 1 1 ηMsca1 # − S11· T12hh· " Jsca 2 1 ηMsca2 # − S11· T13hp·J3 =S11· " Jinc 1 1 ηMinc1 # −S22· T

An enhanced volume integral equation method and augmented equivalence principle algorithm for low frequency problems