- Email: [email protected]
Electromagnetic scattering from an arbitrarily shaped PEC object coated by spherical dielectric material using equivalence principle algorithm
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Electromagnetic scattering from an arbitrarily shaped PEC object coated by spherical dielectric material using equivalence principle algorithm Mohammad Alian, Homayoon Oraizi∗ School of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran
a r t i c l e
i n f o
Keywords: Electromagnetic scattering Dielectric coating Equivalence principle Spherical wave harmonics Mode-matching,
a b s t r a c t A new method is presented for the analysis of electromagnetic scattering from arbitrarily shaped PEC object coated by homogeneous spherical dielectric material. The analysis of the internal PEC objects is performed using the equivalence principle algorithm (EPA). First, the scattering operator of EPA is evaluated for the PEC object in terms of RWG basis functions where the background is filled by coating material. Then a translation procedure is introduced to translate scattering operator evaluated in terms of RWG basis functions to a new scattering operator in terms of spherical wave harmonics (SWHs). This new scattering operator relates SWH coefficients of the scattered wave by internal PEC to those of the incident wave. By invoking the boundary conditions on the surface of spherical coating using the mode-matching technique, the coefficients of the scattered wave outside the coating material are evaluated. Since the scattering operator is independent from the coating radius, the problem may be solved by only analysis of boundary conditions on the surface of spherical coating for various values of coating radii without needing to repeat the analysis of internal PEC object. Several numerical examples are investigated to evaluate validity and efficiency of the proposed method.
1. Introduction The problem of electromagnetic wave scattering from coated conductors has extensively been studied in the last decades. The significance of the problem backs to its widespread applications in optimizing radar cross-section (RCS), shielding effectiveness, radome performance, scattering cancellation cloaking and so on [1–4]. So far numerous analytical [5–8] and numerical approaches [9–11] have been presented for the study of scattering from such objects. While the analytical methods are accurate and efficient, their applications are restricted to a few canonical geometries. On the other hand, the numerical methods usually cover a wide range of geometries, but they suffer from computational restrictions, such as CPU time and memory requirements. Nonetheless, in some cases an optimum solution may be adopted by employing a hybrid approach that benefits from the aforementioned merits of both the analytical and numerical approaches. This paper addresses the problem of electromagnetic wave scattering from an arbitrarily shaped PEC object coated by a dielectric sphere. Since this problem involves a dielectric medium, the volume integral equation (VIE) method may be employed as a numerical solution [12–15]. But the VIE necessitates a huge number of meshes for the discretization of the volume currents inside the dielectric media, resulting in computational inefficiencies. Fortunately, in recent years the alternative approach of surface integral equation (SIE) has effectively been ∗
employed for the problems involving homogeneous dielectric media [16–21]. In comparison to VIE that uses volume meshes, SIE considers the surface meshes for the discretization of the surface currents on the surface of a homogeneous volume. However, SIE is exclusively applicable to the piecewise homogeneous structures, but its computational costs are very lower than those of VIE, making it a popular numerical approach. In the current problem, the spherical geometry of the coating encourages the application of analytical method for the analysis of boundary conditions that are more efficient than the numerical approaches. The analytical solution employs the mode-matching technique for the SWHs. Furthermore, the arbitrary internal PEC object is treated numerically. Accordingly, the proposed method is a hybrid approach, that is expected to be computationally more efficient than the pure numerical approaches. It is shown that by employing the scattering operator of equivalence principle algorithm [22–30], the numerical solution of the internal PEC object is linked to that of the analytical solution involving SWHs. Furthermore, this procedure makes it possible to satisfy the boundary conditions on the surface of spherical coating, independent of the scattering operator of PEC object. Accordingly, as long as the coating material properties and the internal PEC object are not changed, for various values of radii of spherical coating, the same scattering operator of PEC object may be utilized for the satisfaction of boundary conditions on the surface of coating. Using the scattering operator, the equivalent
Corresponding author. E-mail address: [email protected] (H. Oraizi).
https://doi.org/10.1016/j.enganabound.2019.11.006 Received 22 June 2019; Received in revised form 30 October 2019; Accepted 15 November 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
scattering electric and magnetic currents on an imaginary surface enclosing the PEC object are evaluated from those of the incident currents in an unbounded medium with the same properties of coating material. Similar to many previous works, the well-known Rao–Wilton–Glisson (RWG) basis functions [31] are used in this paper for the numerical analysis of PEC object as well as for the discretization of surface currents on the equivalence surface. In this work, the equivalence surface is considered to be spherical. It is shown that by virtue of orthogonality of SWHs, the spherical equivalence surface facilitates the translation of scattering operator evaluated in terms of RWG basis functions to a new scattering operator in terms of SWHs. This new operator named SWHs scattering operator evaluates the SWHs coefficients of scattered field from those of the incident fields. The rest of the paper is arranged as follows. Section 2 describes the theory of proposed method where the evaluation of SWHs scattering operator is discussed in Section 2.1. Section 2.2 considers the boundary conditions using the mode-matching technique on the surface of spherical coating for the expanded fields in terms of SWHs. For the verification of the proposed method, several numerical examples are presented in Section 3. Finally, the conclusions are given in Section 4. 2. Theory Fig. 1. An arbitrarily shaped PEC object coated by a dielectric sphere of radius a with incident and scattered waves in each regions.
Consider the configuration of Fig. 1 where a PEC object is coated by a spherical dielectric medium and is illuminated by an arbitrary incident wave. The scattered field outside the spherical coating may be evaluated by the satisfaction of the boundary conditions on the surface of coating. The spherical surface of coating encourages the application of SWHs for the analysis of the problem. In terms of SWHs, the scattered and incident waves may be cast into four types, namely those of outside as well as inside the coating. Here the electromagnetic fields inside the coating are required in the region between rmin and a, where rmin is the radius of the smallest imaginary sphere enclosing the PEC object and concentric with the spherical coating and a is the radius of coating as shown in Fig. 1. Since both the investigation regions of r > a and rmin < r < a are homogenous source-free regions, their electric and magnetic fields may be expanded in terms of SWHs [32–35]: 𝐄𝓁 (𝑟, 𝜃, 𝜙) = 𝐇𝓁 (𝑟, 𝜃, 𝜙) =
∞ ∑ 𝑛 ∑ 𝑛=0 𝑚=−𝑛
𝑖) 𝑖) 𝑎𝑛𝑚 𝐦(𝑛𝑚, (𝑟, 𝜃, 𝜙) + 𝑏𝑛𝑚 𝐧(𝑛𝑚, (𝑟, 𝜃, 𝜙) 𝓁 𝓁
∞ 𝑛 𝑗 ∑ ∑ 𝑖) 𝑏 𝐦(𝑖) (𝑟, 𝜃, 𝜙) + 𝑎𝑛𝑚 𝐧(𝑛𝑚, (𝑟, 𝜃, 𝜙) 𝓁 𝜂𝓁 𝑛=0 𝑚=−𝑛 𝑛𝑚 𝑛𝑚,𝓁
respectively that are not used in the current work. 𝑃𝑛𝑚 (⋅) is the associated Legendre function of the first kind. The implicit time dependency of exp (j𝜔t) is assumed throughout the paper. The coefficients of the SWH expansion of the incident electric and magnetic fields in Fig. 1 can be evaluated by the orthogonality relations of SWHs [32,34,35] as follows: ( ) −𝑗𝜂 2𝑛 + 1 (𝑛 − 𝑚)! (1) 𝐦∗nm (5) 𝑎inc ⋅ 𝐇inc (𝑟) 𝑑Ω ))2 ∮ nm = 4𝜋𝑛(𝑛 + 1) (𝑛 + 𝑚)! ( ( 1 ,1 4𝜋 𝑗𝑛 𝑘1 𝑟 𝑏inc nm =
𝑖) 𝐧(mn ,𝓁
jm (𝑖) ( ) 𝑚 𝑧 𝑘 𝑟 𝑃𝑛 (cos𝜃) exp (jm𝜙) 𝜽̂ sin 𝜃 𝑛 𝓁 ( ) 𝜕 𝑚 −𝑧(𝑛𝑖) 𝑘𝓁 𝑟 𝑃 (cos𝜃) exp (jm𝜙) 𝝓̂ 𝜕𝜃 𝑛
𝑛(𝑛 + 1) (𝑖) ( ) 𝑚 = 𝑧𝑛 𝑘𝓁 𝑟 𝑃𝑛 (co𝑠𝜃) exp (jm𝜙) ̂ 𝐫 𝑘𝓁 𝑟 1 𝜕 [ (𝑖) ( )] 𝜕 𝑚 + 𝑟𝑧 𝑘 𝑟 𝑃 (cos𝜃) exp (jm𝜙) 𝜽̂ 𝑘𝓁 𝑟 𝜕𝑟 𝑛 𝓁 𝜕𝜃 𝑛 jm 𝜕 [ (𝑖) ( )] 𝑚 + 𝑟𝑧 𝑘 𝑟 𝑃𝑛 (cos𝜃) exp (jm𝜙) 𝝓̂ 𝑘𝓁 𝑟 sin 𝜃 𝜕𝑟 𝑛 𝓁
4𝜋
( ) (1 ) ⋅ 𝐄inc 𝐦∗nm (𝑟) 𝑑Ω 1 ,1
(6)
So, referring to Fig. 1 there exist totally six type of unknown SWH coefficients for the scattered field inside region 1 together with the incident and scattered fields inside regions 2. In Section 2.1 it is shown that using scattering operator of EPA, a comprehensive relation may be established between the SWH coefficients of the scattered field and those of incident field for the internal PEC object. This relation provides two sets of equations from the six required sets to evaluate six aforementioned types of unknowns. In Section 2.2 using the continuity of the tangential components of the electric and magnetic fields on the surface of spherical dielectric coating, four remaining sets of the equations are arranged, making it possible to have a unique solution.
(1)
(2)
√ where 𝜂𝓁 = 𝜇𝓁 ∕𝜖𝓁 is the intrinsic impedance of medium 𝓁 (𝓁 = 1, 2) (𝑖) 𝑖) and 𝐦𝑛𝑚,𝓁 and 𝐧(𝑛𝑚, are defined as follows: 𝓁 𝑖) 𝐦(mn = ,𝓁
2𝑛 + 1 (𝑛 − 𝑚)! 1 4𝜋𝑛(𝑛 + 1) (𝑛 + 𝑚)! (𝑗 (𝑘 𝑟))2 ∮ 𝑛 1
(3) 2.1. SWHs Scattering operator for the PEC object in an unbounded background region The scattered field SWH coefficients inside the coating may be directly evaluated from those of the incident field using the scattering operator of EPA. Consider the configuration of Fig. 2 where an arbitrary PEC scatterer is placed inside a homogenous dielectric background of permittivity 𝜖 2 and permeability 𝜇 2 and enclosed by an imaginary spherical surface of radius R. The equivalent electric and magnetic incident currents Jinc and Minc , defined on the interior surface of equivalence sphere irradiate the PEC object. The scattered field from the PEC object creates the equivalence surface currents Jsca and Msca on the exterior of equivalence surface. So the scattering operator acts on the incident currents to evaluate the scattering ones as follows: [ 𝑠𝑐𝑎 ] [ 𝑖𝑛𝑐 ] 𝐉 𝐉 = (7) 𝑠𝑐𝑎 𝐌 𝐌𝑖𝑛𝑐
(4)
√ in which 𝑘𝓁 = 𝜔 𝜇𝓁 𝜖𝓁 is the wave number of medium 𝓁. The superscript i distinguishes the scattered waves from the incident ones as for 𝑖 = 1 and 𝑖 = 4 the function 𝑧(𝑛𝑖) (⋅) stands for the spherical Bessel function of the first kind jn ( · ) and spherical Hankel function of the second kind ℎ(2) 𝑛 (⋅), respectively. Note the superscript i follows the same notation used in some main references pertaining to the SWHs such as Hansen [35]. In these references the superscript 𝑖 = 2 and 𝑖 = 3 are dedicated for the Bessel function of second kind and Hankel function of the firs kind, 179
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
On the other hand it is assumed that these surface currents are known in terms of RWG basis functions as: ∑ 𝑖𝑛𝑐 𝐌𝑖𝑛𝑐 (𝐫) = 𝐟𝑖𝑅𝑊 𝐺 (𝐫)𝐼𝑖𝑀 (13) 𝑖
𝐉𝑖𝑛𝑐 (𝐫) =
∑ 𝑖
𝐟𝑖𝑅𝑊 𝐺 (𝐫)𝐼𝑖𝐽
𝑖𝑛𝑐
(14) 𝑖𝑛𝑐
𝑖𝑛𝑐
The unknown amplitudes 𝐼𝑖𝑀 and 𝐼𝑖𝐽 may be evaluated from the equality of (11) and (13) as well as (12) and (14) [24]: [ 𝑖𝑛𝑐 ] ( ) 𝐼𝑀 = [𝑈 ]−1 [𝐴][𝑐 𝑖𝑛𝑐 ] + [𝐵][𝑑 𝑖𝑛𝑐 ] (15) [ 𝑖𝑛𝑐 ] −𝑗 ( ) [𝑈 ]−1 [𝐵][𝑐 𝑖𝑛𝑐 ] + [𝐴][𝑑 𝑖𝑛𝑐 ] 𝐼𝐽 = 𝜂2
Fig. 2. PEC object of Fig. 1, enclosed by equivalence sphere of radius R where the background is filled by the same dielectric medium of spherical coating of Fig. 1.
(8) where 𝐧̂ is outward unit normal vector on the equivalence surface and the subscripts io and oi show the inside-out and outside-in propagations, respectively. The near field operators and are defined as follows:
(𝐗) = ∇ ×
∫𝑆
∫𝑆
𝐺(𝐫 , 𝐫 ′ )𝐗(𝐫 ′ )𝑑 𝑠′ −
𝐺(𝐫 , 𝐫 )𝐗(𝐫 )𝑑 𝑠
𝐺(𝐫 , 𝐫 ′ )
′
′
𝑗 ∇ 𝐺(𝐫 , 𝐫 ′ )∇′ ⋅ (𝐗(𝐫 ′ ))𝑑 𝑠′ 𝑘2 ∫𝑆
where: ⟨ ⟩ 𝑈𝑗𝑖 = 𝐟𝑗𝑅𝑊 𝐺 (𝐫) ⋅ 𝐟𝑖𝑅𝑊 𝐺 (𝐫)
(17)
⟨ )⟩ ( 𝐴𝑗,𝑛𝑚 = 𝐟𝑗𝑅𝑊 𝐺 (𝐫) ⋅ 𝐫̂ × 𝐦(1) 𝑛𝑚
(18)
⟨ ( )⟩ 𝐵𝑗,𝑛𝑚 = 𝐟𝑗𝑅𝑊 𝐺 (𝐫) ⋅ 𝐫̂ × 𝐧(1) 𝑛𝑚
(19)
The notation ⟨ X(r).Y(r) ⟩ for the vector functions X(r) and Y(r) is defined in (A.4) (refer to the appendix). Note that for the expansion of fields in terms of SWHs, a finite number of radial modes Nso is considered for the evaluation of SWHs scattering operator to be discussed latter. A similar approach may be adopted for translation from RWG basis functions to SWHs of the scattered field. Here the scattering surface currents on the exterior surface of equivalence sphere may be represented in terms of SWHs using equations similar to (11) and (12) with the exception that the superscript (1) is replaced by (4) to show the diverging harmonics pertaining to the scattered fields. Consider the scattering surface currents to be determined in terms of RWG basis functions by 𝑠𝑐𝑎 and equations similar to (13) and (14). The unknown amplitudes 𝑐𝑛𝑚 𝑠𝑐𝑎 may be evaluated using the orthogonality relations for SWHs as: 𝑑𝑛𝑚
The scattering operator may be evaluated in three regular steps of outside-in propagation, analysis of the PEC object for the evaluation of induced electric surface currents and finally inside-out propagation [23]. Using these regular steps, the scattering operator may be evaluated as: [[ ] [ ]] [ ] ] 𝐧̂ × 𝒦io 11 12 [ ]−1 [ ℒoi − 𝜂1 𝒦oi = [ ] [ ] = 𝐵ii 2 ̂ × ℒio −𝜂2 𝒏 22 21
(𝐗) = −𝑗 𝑘 2
[𝑐 𝑠𝑐𝑎 ] = [𝜏][𝐺][𝐼 𝑀
𝑠𝑐𝑎
]
(10) 𝐫 ′ |)∕4𝜋|𝐫
]
(21)
( ⟨ )⟩ 𝐺𝑖,𝑛𝑚 = 𝐟𝑖𝑅𝑊 𝐺 (𝐫) ⋅ 𝐫̂ × 𝐦∗(4) 𝑛𝑚
in which = exp(−𝑗 𝑘2 |𝐫 − − is background Green’s function. Numerical evaluation of the scattering operator, necessitates the discretization of the equivalent currents on the surface of equivalence sphere. Similar to many previous works, the discretization of the surface currents may be done using RWG basis functions. However, since our analysis is based on SWHs, translations from RWG basis function to SWHs and vice versa are necessary for the scattered and incident fields, respectively. These translations make it possible to evaluate the scattering operator for the SWHs so that the SWH coefficients of the scattered field can be evaluated from those of the incident field. Assume that the incidence field be expanded in terms of SWHs with 𝑖𝑛𝑐 and 𝑑 𝑖𝑛𝑐 . The incident magnetic and the known complex amplitudes 𝑐𝑛𝑚 𝑛𝑚 electric surface currents on the interior surface of the equivalence sphere with radius R can be readily evaluated in terms of SWHs from the electric and magnetic fields, respectively, as follows:
(22)
and the diagonal matrix [𝜏] is defined as: ⎡𝜏1,−1 ⎢0 ⎢ [𝜏] = ⎢0 ⎢⋮ ⎢ ⎣0
0 𝜏1,0 0 ⋮ 0
0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ ⋮ ⎥ 𝜏𝑁,𝑁 ⎦
(23)
2𝑛 + 1 (𝑛 − 𝑚)! 𝑛(𝑛 + 1) (𝑛 + 𝑚)!
(24)
0 0 𝜏1,1 ⋮ 0
… … … ⋱ …
with the following elements: 𝜏𝑛𝑚 =
−1 | |2 4𝜋𝑅2 |ℎ(2) (𝑘 𝑅)| | 𝑛 2 |
By substitution of the coefficients of RWG basis functions of the incident currents from (15) and (16) into (7) and as well as those of scattering currents from (7) into (20) and (21) a new scattering operator ̃ is evaluated that constructs a bridge between the scattered and incident field SWHs as: [ 𝑠𝑐𝑎 ] [ 𝑖𝑛𝑐 ] [ ][ ] [𝑐 ] [̃11 ] [̃12 ] [𝑐 𝑖𝑛𝑐 ] ̃ [𝑐 ] = = (25) [𝑑 𝑠𝑐𝑎 ] [𝑑 𝑖𝑛𝑐 ] [̃21 ] [̃22 ] [𝑑 𝑖𝑛𝑐 ]
(11)
𝑛=0 𝑚=−𝑛
𝐉𝑖𝑛𝑐 (𝐫) = −𝐧̂ × 𝐇𝑖𝑛𝑐 (𝑅, 𝜃, 𝜙) ∞ 𝑛 ) ( ) −𝑗 ∑ ∑ 𝑖𝑛𝑐 ( 𝑖𝑛𝑐 ̂ (1) = 𝑑 𝐫̂ × 𝐦(1) 𝑛𝑚 + 𝑐𝑛𝑚 𝐫 × 𝐧𝑛𝑚 𝜂2 𝑛=0 𝑚=−𝑛 𝑛𝑚
𝑠𝑐𝑎
where:
𝐫′|
𝐌𝑖𝑛𝑐 (𝐫) = 𝐧̂ × 𝐄𝑖𝑛𝑐 (𝑅, 𝜃, 𝜙) ∞ ∑ 𝑛 ∑ ( ) ( ) 𝑖𝑛𝑐 ̂ 𝑖𝑛𝑐 ̂ (1) 𝐫 × 𝐦(1) = 𝑐𝑛𝑚 𝑛𝑚 + 𝑑𝑛𝑚 𝐫 × 𝐧𝑛𝑚
(20)
(9) [𝑑 𝑠𝑐𝑎 ] = 𝑗𝜂2 [𝜏][𝐺][𝐼 𝐽
′
(16)
where: ( ) [̃11 ] = [𝜏][𝐺] 𝑗∕𝜂2 [21 ][𝑈 ]−1 [𝐵] − [22 ][𝑈 ]−1 [𝐴]
(12) 180
(26)
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
( ) [̃12 ] = [𝜏][𝐺] 𝑗∕𝜂2 [21 ][𝑈 ]−1 [𝐴] − [22 ][𝑈 ]−1 [𝐵]
In a similar manner, the inner product of both sides of (32) by 𝐫̂ × 𝐧∗(1) 𝑛𝑚,1 and integrating on the surface of coating results in:
(27)
𝑠𝑐𝑎 𝑖𝑛𝑐 𝑠𝑐𝑎 𝑖𝑛𝑐 ̃ (2) ̃ ̃ ℎ̃ (2) 𝑛 (𝑘1 𝑎)𝑏𝑛𝑚 − 𝑗𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 − ℎ𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 = −𝑗𝑛 (𝑘1 𝑎)𝑏𝑛𝑚
( ) [̃21 ] = −[𝜏][𝐺] [11 ][𝑈 ]−1 [𝐵] + 𝑗𝜂2 [12 ][𝑈 ]−1 [𝐴]
(28)
( ) [̃22 ] = −[𝜏][𝐺] [11 ][𝑈 ]−1 [𝐴] + 𝑗𝜂2 [12 ][𝑈 ]−1 [𝐵]
(29)
where for the spherical Bessel function 𝑧(𝑛𝑖) (𝑥), the function 𝑧̃ (𝑛𝑖) (𝑥) is defined as follows: 1 𝑑 ( (𝑖) ) 𝑛 + 1 (𝑖) ) 𝑧̃ (𝑛𝑖) (𝑥) = 𝑧 (𝑥) − 𝑧(𝑛𝑖+1 (𝑥) (36) 𝑥𝑧𝑛 (𝑥) = 𝑥 𝑑𝑥 𝑥 𝑛 Analogous to the evaluation of (34) and (35) from (32), the following equations may be set up from (33):
Obviously, since the scattering operator is evaluated for the PEC object inside an unbounded background region, it is independent of the radius of spherical coating. The radius of coating is taken into account in the analysis of the boundary conditions on the coating surface as discussed in Section 2.2. 2.2. Analysis of the boundary conditions on the surface of spherical coating using mode-matching technique
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
(31)
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
𝑁𝑚𝑚
−
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛 𝑁𝑚𝑚
=−
∑
( ) ( ) 𝑖𝑛𝑐 ̂ 𝑖𝑛𝑐 ̂ 𝑐𝑛𝑚 𝐫 × 𝐦(1) + 𝑑𝑛𝑚 𝐫 × 𝐧(1) 𝑛𝑚,2 𝑛𝑚,2 ( ) ( ) 𝑠𝑐𝑎 ̂ 𝑠𝑐𝑎 ̂ 𝑐𝑛𝑚 𝐫 × 𝐦(4) + 𝑑𝑛𝑚 𝐫 × 𝐧(4) 𝑛𝑚,2 𝑛𝑚,2
𝑛 ∑
𝑛=0 𝑚=−𝑛
𝑗∕𝜂1
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
−𝑗 ∕𝜂2 −𝑗 ∕𝜂2
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
𝑁𝑚𝑚
∑
𝑛 ∑
𝑛=0 𝑚=−𝑛
= −𝑗∕𝜂1
𝑁𝑚𝑚
∑
(32)
(
)
( ) 𝑖𝑛𝑐 ̂ + 𝑐𝑛𝑚 𝐫 × 𝐧(1) 𝑛𝑚,2
(
)
( ) 𝑠𝑐𝑎 ̂ + 𝑐𝑛𝑚 𝐫 × 𝐧(4) 𝑛𝑚,2
𝑖𝑛𝑐 ̂ 𝑑𝑛𝑚 𝐫 × 𝐦(1) 𝑛𝑚,2
𝑠𝑐𝑎 ̂ 𝐫 × 𝐦(4) 𝑑𝑛𝑚 𝑛𝑚,2
𝑛 ∑
𝑛=0 𝑚=−𝑛
(
)
(
(1) (1) 𝑖𝑛𝑐 ̂ ̂ 𝑏𝑖𝑛𝑐 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑎𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
[0]
−[𝐽2 ]
[0]
−[𝐻2 ]
̃1 ] [𝐻
[0]
−[𝐽̃2 ]
[0]
[0]
𝜂 − 𝜂1 [𝐽̃2 ]
[0]
𝜂 ̃2 ] − 𝜂1 [ 𝐻
2
[𝐻1 ]
[0]
[0]
−[̃11 ]
[0]
−[̃21 ]
𝜂 − 𝜂1 [ 𝐽 2 ] 2
̃2 ] −[𝐻
2
[0]
−[̃12 ]
[𝐼]
−[̃22 ]
[0]
𝑖𝑛𝑐 ⎤ ⎡[𝐽1 ][𝑎 ]⎤ ⎥ ⎢ ⎥ 𝑖𝑛𝑐 ⎥ ⎢[𝐽̃1 ][𝑏 ]⎥ ⎥ ⎢ ⎥ 𝑖𝑛𝑐 ⎥ ⎢[𝐽̃1 ][𝑎 ]⎥ = − ⎥ ⎢ ⎥ 𝑖𝑛𝑐 ⎥ ⎢[𝐽1 ][𝑏 ]⎥ ⎥ ⎢ ⎥ ⎥ ⎢ [0] ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ [0] ⎦
⎤ ⎥ ⎥ ⎥ [0] ⎥ ⎥ 𝜂 − 𝜂1 [𝐻2 ]⎥ 2 ⎥ [0] ⎥ ⎥ ⎦ [𝐼] [0]
(39)
⎡𝑧 ( 𝑖 ) ( 𝑘 𝑎 ) ⎢ 1 𝓁 = ⎢⋮ ⎢0 ⎣
… ⋱ …
⎤ 0 ⎥ ⋮ ⎥ 𝑧(𝑁𝑖) (𝑘𝓁 𝑎)⎥⎦
(40)
̃𝓁 where 𝑍𝓁(𝑖) represents J𝓁 for 𝑖 = 1 and H𝓁 for 𝑖 = 4. Similarly 𝐽̃𝓁 and 𝐻 are defined as:
( ) ( ) (4) (4) 𝑠𝑐𝑎 ̂ ̂ 𝑏𝑠𝑐𝑎 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑎𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
𝑁𝑚𝑚
(38)
[𝑍𝓁(𝑖) ]
( ) ( ) (1) (1) 𝑖𝑛𝑐 ̂ ̂ 𝑎𝑖𝑛𝑐 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑏𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
𝑁𝑚𝑚
𝑠𝑐𝑎 𝑖𝑛𝑐 𝑠𝑐𝑎 𝑖𝑛𝑐 ̃ (2) ̃ ̃ 𝜂2 ℎ̃ (2) 𝑛 (𝑘1 𝑎)𝑎𝑛𝑚 − 𝜂1 𝑗𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 − 𝜂1 ℎ𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 = −𝜂2 𝑗𝑛 (𝑘1 𝑎)𝑎𝑛𝑚
Eqs. (34), (35), (37) and (38) are four equations to determine the 𝑠𝑐𝑎 𝑖𝑛𝑐 𝑖𝑛𝑐 𝑠𝑐𝑎 𝑠𝑐𝑎 six unknowns (𝑎𝑠𝑐𝑎 𝑛𝑚 , 𝑏𝑛𝑚 , 𝑐𝑛𝑚 , 𝑑𝑛𝑚 , 𝑐𝑛𝑚 and 𝑑𝑛𝑚 ). We must enforce two more relationships among these six unknowns before a unique solution can be obtained. The remaining equations may be adopted using the scattering operator as (25). Thereupon, a system of six equations with six unknowns may be established as (39) where [I] is the identity matrix and for J𝓁 and H𝓁 :
( ) ( ) (4) (4) 𝑠𝑐𝑎 ̂ ̂ 𝑎𝑠𝑐𝑎 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑏𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
𝑁𝑚𝑚
−
(37)
⎡ [𝑎𝑠𝑐𝑎 ] ⎢ 𝑠𝑐𝑎 ⎢ [𝑏 ] ⎢ 𝑖𝑛𝑐 ⎢ [𝑐 ] ×⎢ 𝑖𝑛𝑐 ⎢ [𝑑 ] ⎢ 𝑠𝑐𝑎 ⎢ [𝑐 ] ⎢ 𝑠𝑐𝑎 ⎣[𝑑 ]
where superscripts inc and sca represent the incident and scattered fields, respectively. Also subscript 𝓁, denotes the region r > a for 𝓁 = 1 and rmin < r < a for 𝓁 = 2 (refer to Fig. 1). Substitutions of the electric and magnetic fields in (30) and (31) by their SWH expansions lead to the following equations: 𝑁𝑚𝑚
𝑠𝑐𝑎 𝑖𝑛𝑐 (2) 𝑠𝑐𝑎 𝑖𝑛𝑐 𝜂2 ℎ(2) 𝑛 (𝑘1 𝑎)𝑏𝑛𝑚 − 𝜂1 𝑗𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 − 𝜂1 ℎ𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 = −𝜂2 𝑗𝑛 (𝑘1 𝑎)𝑏𝑛𝑚
⎡[ 𝐻 1 ] ⎢ ⎢[0] ⎢ ̃ ⎢[ 𝐻 1 ] ⎢ ⎢[0] ⎢ ⎢[0] ⎢ ⎣[0]
As the coating surface has a canonical shape the mode-matching technique is implemented that is an analytical approach without needing to surface discretization making it more efficient than the numerical full wave techniques.The boundary conditions for the electric and magnetic fields on the surface of spherical coating (𝑟 = 𝑎) may be written as: ( ) (30) 𝐫̂ × 𝐄sca (𝑎, 𝜃, 𝜙) − 𝐄inc (𝑎, 𝜃, 𝜙) − 𝐄sca (𝑎, 𝜃, 𝜙) = −𝐫̂ × 𝐄inc (𝑎, 𝜃, 𝜙) 1 2 2 1 ) ( 𝐫̂ × 𝐇sca (𝑎, 𝜃, 𝜙) − 𝐇inc (𝑎, 𝜃, 𝜙) − 𝐇sca (𝑎, 𝜃, 𝜙) = −𝐫̂ × 𝐇inc (𝑎, 𝜃, 𝜙) 1 2 2 1
(35)
⎡𝑧̃ (𝑖) (𝑘 𝑎) ⎢ 1 𝓁 (𝑖) ̃ [ 𝑍 𝓁 ] = ⎢⋮ ⎢0 ⎣
… ⋱ …
⎤ 0 ⎥ ⋮ ⎥ (𝑖) 𝑧̃ 𝑁 (𝑘𝓁 𝑎)⎥⎦
(41)
By the evaluation of the unknown coefficients of the scattered field from (39), the scattered field may be readily evaluated from (1) and (2). 2.3. SWHs truncation criteria )
The numerical implementation of the proposed method necessitates a truncated number of radial modes for the expansion of electric and magnetic fields in terms of SWHs in (1) and (2). According to the literature, the infinite series in these equations may be truncated by N modes as [35–37]:
(33)
where the upper limits of exterior series Nmm are the required number of radial modes in the mode-matching technique for an acceptable accuracy in the expansion of the fields on the surface of coating. Obviously, Nmm is a function of coating radius and material to be discussed later. The inner product of both sides of (32) by 𝐫̂ × 𝐦∗(1) and integrating 𝑛𝑚,1 on the surface of coating (𝑟 = 𝑎), thanks to the orthogonality relations for SWHs (refer to the appendix), gives: 𝑠𝑐𝑎 𝑖𝑛𝑐 (2) 𝑠𝑐𝑎 𝑖𝑛𝑐 ℎ(2) 𝑛 (𝑘1 𝑎)𝑎𝑛𝑚 − 𝑗𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 − ℎ𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 = −𝑗𝑛 (𝑘1 𝑎)𝑎𝑛𝑚
𝑁 ≃ 𝑘2 𝑟0 + 10
(42)
where r0 is the radius of a sphere encompassing the region with acceptable approximation of the actual fields by SWHs. In Section 2.1 a truncated number of modes Nso was considered for the evaluation of SWHs scattering operator. So, Nso may be evaluated from (42) where r0
(34) 181
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Fig. 3. PEC sphere coated by spherical dielectric of radius a, 𝜖𝑟 = 4 and 𝜇𝑟 = 1.
is replaced by the equivalence sphere radius R. Similarly, the truncated number of modes for mode-matching Nmm in Section 2.2 may be evaluated from (42), replacing r0 by spherical coating radius a. Since Nmm is a function of coating radius and material, it is independent of the complexity of PEC structure. The geometry of the PEC scatterer only affects the scattering operator which consider Nso number of radial modes for the expansion of the fields. When the number of modes is determined, according to (1) and (2) it can be shown that the number of unknowns for each set of the unknown fields is 2((𝑁mm + 1)2 − 1). As there are three sets of unknown fields, the total number of unknowns of the problem is 6((𝑁mm + 1)2 − 1). Note that if the coating radius is changed only the number of modes for the satisfaction of boundary conditions using the mode-matching technique should be changed, where the same SWHs scattering operator evaluated by Nso radial modes may be employed independent of coating radius. 3. Numerical examples For the verification of the proposed method, several numerical examples are investigated in this section. The incident wave is considered to be an 𝑥−polarized plane wave propagating along +𝑧 direction at 3 GHz. Note that if the electrical dimensions of the PEC scatterer be very large, some accelerating algorithms such as ACA or MLFMA can be adopted in the evaluation of the scattering operator. However, in the current work no accelerating algorithm is used.
Fig. 4. Comparison of the normalized RCS of coated PEC sphere of radius 5 cm for various values of coating radius a evaluated from proposed method (solid blue line) and that of exact analytical solution (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.1. Scattering from coated PEC sphere shown in Fig. 5. The same procedure of the previous example for the evaluation of the scattered field is applied here as well. In this case the normalized RCS evaluated by the proposed method is compared with that of surface integral equations solved by method of moments (SIEMoM) versus 𝜃 at the same planes of the previous example as shown in Fig. 6 for various values of coating radii. Once again the verification of the proposed method is approved by the acceptable consistency between the results. Furthermore, for the evaluation of the efficiency of the proposed method, its computational costs are compared to those of SIE-MoM in Table 1 for the case of a = 20 cm. Note that the set-up time of proposed method is more than that of SIE-MoM, which is mainly spent for the computation of scattering operator. However, the scattering operator needs to be computed once for various values of coating radii of a PEC object. Accordingly, the proposed method by decomposing the analysis domain improves the computational costs in comparison to a monolithic method such as SIE-MoM, particularly where different coating radii are investigated. The other cases of dielectric coating radii exhibit similar excellent performance, but they are not reported here due to space limitation.
This example considers a PEC sphere of radius 5 cm coated by a spherical dielectric material of radius a with the relative permittivity 𝜖𝑟 = 4 and permeability 𝜇𝑟 = 1, as shown in Fig. 3. First the scattering operator of the PEC sphere is evaluated in terms of RWG basis functions using (8). Then the RWG scattering operator is translated into that of SWHs using (26)–(29). Finally (39) is solved for the unknown SWH coefficients of the scattered field where the incident field SWH coefficients are substituted from (5)–(6). The normalized RCS of the coated sphere evaluated from the proposed method is compared with that of exact analytical solution [38–40] for various values of the spherical coating radius a versus zenith angle 𝜃 at 𝜙 = 0◦ and 𝜙 = 90◦ planes as shown in Fig. 4. The acceptable consistency between the results verifies the proposed method. 3.2. Scattering from coated PEC cube In this case the coated PEC object is considered to be a cube with edge length of 10 cm coated by a spherical dielectric material of radius a with the relative permittivity 𝜖𝑟 = 2 and permeability 𝜇𝑟 = 1, as 182
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Table 1 A comparison between the computational costs. Problem
Numerical approach
Number of unknowns
Set-up time (s)
Solution time (s)
Section 3.2 (a = 20 cm)
Proposed method SIE-MoM Proposed method SIE-MoM Proposed method SIE-MoM
5040
609
645
349
12462 8208
176 1204
1563 1684
1381 925
20268 5394
408 594
4108 727
3182 399
12743
188
1624
1467
Section 3.3 (a = 20 cm) Section 3.4 (a = 20 cm)
Memory usage (MB)
Fig. 5. PEC cube coated by spherical dielectric of radius a, 𝜖𝑟 = 2 and 𝜇𝑟 = 1.
Fig. 7. PEC cylinder coated by spherical dielectric of radius a, 𝜖𝑟 = 2 and 𝜇𝑟 = 2.
3.3. Scattering from coated PEC cylinder In this example the electromagnetic scattering from a coated cylinder of radius 5 cm and length 8 cm is considered as shown in Fig. 7. Here both the relative permittivity and permeability of the coating material are different from those of the background free-space. Consider 𝜖𝑟 = 2 and 𝜇𝑟 = 2, similar comparisons to the previous examples for the normalized RCS evaluated by the proposed method and that of SIE-MoM are depicted in Fig. 8. The comparisons show good agreement between the results of two methods. Similar comparison to the previous example for the computational costs are reported in the same Table 1 for the case of a = 20 cm. These comparisons show that the proposed method is faster and needs lower memory in comparison to SIE-MoM. Furthermore, in the proposed method while the PEC object is the same, it is sufficient to evaluate the SWHs scattering operator once independent of spherical coating radius. Observe that the proposed method inherits advantages of equivalence principle based domain decomposition method, that make it possible to solve some parts of the problem independently, saving analysis time and reducing memory requirements.
3.4. Scattering from coated diamond shape PEC The final example considers the electromagnetic scattering from a coated diamond shape PEC as shown in Fig. 9. Consider 𝜖𝑟 = 1.5 and 𝜇𝑟 = 1.5, similar comparisons to the previous examples for the normalized RCS evaluated by the proposed method and that of SIE-MoM are depicted in Fig. 10. Also similar comparison to the previous example for the computational costs are reported in the same Table 1 for the case of a = 20 cm.
Fig. 6. Comparison of the normalized RCS of coated PEC cube with radius 7 cm and length of 10 cm for various values of coating radius a evaluated by the proposed method (solid blue line) and that of simulated with SIE-MoM (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 183
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Fig. 10. Comparison of the normalized RCS of coated diamond shape PEC shown in Fig. 9 for various values of coating radius a evaluated by the proposed method (solid blue line) and that of simulated with SIE-MoM (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Comparison of the normalized RCS of coated PEC cylinder with radius 5 cm and length of 8 cm for various values of coating radius a evaluated by the proposed method (solid blue line) and that of simulated with SIE-MoM (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. (a) Diamond shape PEC object geometry and dimensions. (b) PEC diamond coated by spherical dielectric of radius a, 𝜖𝑟 = 1.5 and 𝜇𝑟 = 1.5.
184
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
4. Conclusion
[6] Sun J, Li L-W. Dispersion of waves over a PEC cylinder coated with two-layer lossy dielectric materials. IEEE Trans Antenna Propag 2007;55(3):877–81. [7] Partal HP, Mautz JR, Arvas E. The exact analytical solution for large circular loops radiating around a dielectric coated conducting sphere. IEEE Trans Antenn Propag 2009;57(2):436–43. [8] Geng Y-L, Qiu C-W, Yuan N. Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer. IEEE Trans Antenn Propag 2009;57(2):572–6. [9] Li L, Wang H, Ping X, Yin X. Comparison of the MIE series method and FEM in electromagnetic analysis of spherical dielectric-coated media. Progress in Electromagnetic Research Symposium (PIERS). IEEE; 2016. 2646–2646 [10] Caorsi S, Pastorino M, Raffetto M. Electromagnetic scattering by a conducting strip with a multilayer elliptic dielectric coating. IEEE Trans Electromagn Compatib 1999;41(4):335–43. [11] Shi Y, Liang C-H. An efficient single-source integral equation solution to em scattering from a coated conductor. IEEE Antenn Wirel Propag Lett 2015;14:547–50. [12] Schaubert D, Wilton D, Glisson A. A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies. IEEE Trans Antenn Propag 1984;32(1):77–85. [13] Wu Q. Computation of characteristic modes for dielectric bodies using volume integral equation and interpolation. IEEE Antenn Wirel Propag Lett 2017;16:2963–6. [14] Cai Q-M, Zhang Z-P, Zhao Y-W, Huang W-F, Zheng Y-T, Nie Z-P, et al. Nonconformal discretization of electric current volume integral equation with higher order hierarchical vector basis functions. IEEE Trans Antenn Propag 2017;65(8):4155–69. [15] Zhang L-M, Sheng X-Q. A discontinuous Galerkin volume integral equation method for scattering from inhomogeneous objects. IEEE Trans Antenn Propag 2015;63(12):5661–7. [16] Poggio AJ, Miller EK. Integral equation solutions of three-dimensional scattering problems. MB Associates; 1970. [17] Wu T-K, Tsai LL. Scattering from arbitrarily-shaped lossy dielectric bodies of revolution. Radio Science 1977;12(5):709–18. [18] Chang Y, Harrington R. A surface formulation for characteristic modes of material bodies. IEEE Trans Antenn Propag 1977;25(6):789–95. [19] Ylä-Oijala P, Taskinen M, Järvenpää S. Analysis of surface integral equations in electromagnetic scattering and radiation problems. Eng Anal Bound Elem 2008;32(3):196–209. [20] Cools K, Andriulli FP, Michielssen E. A Calderón multiplicative preconditioner for the PMCHWT integral equation. IEEE Trans Antenn Propag 2011;59(12):4579–87. [21] Ylä-Oijala P, Taskinen M, Järvenpää S. Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods. Radio Sci 2005;40(6). [22] Li M-K, Chew WC, Jiang LJ. A domain decomposition scheme based on equivalence theorem. Microw Opt Technol Lett 2006;48(9):1853–7. [23] Li M-K, Chew WC. Wave-field interaction with complex structures using equivalence principle algorithm. IEEE Trans Antenn Propag 2007;55(1):130–8. [24] Alian M, Oraizi H. Electromagnetic multiple PEC object scattering using equivalence principle and addition theorem for spherical wave harmonics. IEEE Trans Antenn Propag 2018;66(11):6233–43. [25] Ylä-Oijala P, Lancellotti V, De Hon BP, Järvenpää S. Domain decomposition methods combining surface equivalence principle and macro basis functions. ACES, 25; 2010. [26] Fu X, Jiang LJ, He SQ, et al. Performance enhancement of equivalence principle algorithm. IEEE Antenn Wirel Propag Lett 2016;15:480–3. [27] Alian M, Oraizi H. Application of equivalence principle for em scattering from irregular array of arbitrarily orientated PEC scatterers using both translation and rotation addition theorems. IEEE Trans Antenn Propag 2019. [28] Su T, Du L, Chen R. Electromagnetic scattering for multiple PEC bodies of revolution using equivalence principle algorithm. IEEE Trans Antenn Propag 2014;62(5):2736–44. [29] Shao H, Hu J, Chew WC. Single-source equivalence principle algorithm for the analysis of complex structures. IEEE Antenn Wirel Propag Lett 2014;13:1255–8. [30] He Z, Chen R-S, Wei E. An efficient marching-on-in-degree solution of transient multiscale em scattering problems. IEEE Trans Antenn Propag 2016;64(7):3039–46. [31] Rao S, Wilton D, Glisson A. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antenn Propag 1982;30(3):409–18. [32] Stratton JA. Electromagnetic theory. John Wiley & Sons; 2007. [33] Hald J. Spherical near-field antenna measurements, 26. Iet; 2008. [34] Liang C, Lo Y. Scattering by two spheres. Radio Sci 1967;2(12):1481–95. [35] Hansen JE. Spherical near-field antenna measurements, 26. Iet; 1988. [36] Jensen F, Frandsen A. On the number of modes in spherical wave expansions. AMTA Symposium, October; 2004. [37] MacPhie RH, Wu K-L. A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers. IEEE Trans Antenn Propag 2003;51(10):2801–5. [38] Pena O, Pal U. Scattering of electromagnetic radiation by a multilayered sphere. Comput Phys Commun 2009;180(11):2348–54. [39] Yang W. Improved recursive algorithm for light scattering by a multilayered sphere. Appl Opt 2003;42(9):1710–20. [40] Ericsson A., Sjöberg D., Larsson C., Martin T.. Scattering from a multilayered sphereapplications to electromagnetic absorbers on double curved surfaces. Technical Report LUTEDX/(TEAT-7249)/1-32/(2017)2017; 7249.
Electromagnetic scattering from PEC objects coated by spherical dielectric materials was investigated using a new method based on the equivalence principle. According to the spherical geometry of coating, the electric and magnetic fields are expanded by SWHs. The analysis of the PEC object was first considered by employing the scattering operator of equivalence principle algorithms using RWG basis functions. Then the translation of scattering operator from RWG basis functions to SWHs was discussed in detail. It was shown that using SWHs scattering operator for the PEC object, the SWH coefficients of the scattered field may be evaluated from those of the incident field independent of the coating radius. Accordingly, if the coating radius is changed it is possible to solve the problem using the same scattering operator of the PEC object only by repeating the analysis of the boundary conditions on the surface of coating. The verification of the proposed method was investigated through several numerical examples for various values of coating radius. The SWH expansions used in this work are extracted for a homogenous isotropic medium. Thereupon, the proposed method is not applicable for anisotropic coating material at least using the reported SWH expansions. However, in the case of lossy coating medium the equations may be corrected by considering complex permittivity or permeability that can be studied in a future work. Appendix. Orthogonality relations of transverse components of SWHs The orthogonality relations for SWHs may be found in many previous works [32,35]. From these orthogonality relations the following ones can be deduced for their transverse components as follows: ⟨( )( )⟩ ′ 𝑖) 𝐫̂ × 𝐦∗( (𝑟, 𝜃, 𝜙) . 𝐫̂ × 𝐦(𝑛𝑖′ 𝑚) ′ ,𝓁 ′ (𝑟, 𝜃, 𝜙) 𝑛𝑚,𝓁 = 4𝜋 ⟨(
′ 𝑛(𝑛 + 1) (𝑛 + 𝑚)! ∗(𝑖) 𝑧 (𝑘𝓁 𝑟)𝑧(𝑛𝑖 ) (𝑘𝓁 ′ 𝑟)𝛿𝑛,𝑛′ 𝛿𝑚,𝑚′ 2𝑛 + 1 (𝑛 − 𝑚)! 𝑛
(A.1)
)( )⟩ ′ 𝑖) (𝑟, 𝜃, 𝜙) . 𝐫̂ × 𝐧(𝑛𝑖′ 𝑚) ′ ,𝓁 ′ (𝑟, 𝜃, 𝜙) 𝐫̂ × 𝐧∗( 𝑛𝑚,𝓁
′ 𝑛(𝑛 + 1) (𝑛 + 𝑚)! ∗(𝑖) 𝑧̃ (𝑘𝓁 𝑟)𝑧̃ (𝑛𝑖 ) (𝑘𝓁 ′ 𝑟)𝛿𝑛,𝑛′ 𝛿𝑚,𝑚′ 2𝑛 + 1 (𝑛 − 𝑚)! 𝑛 ⟨( )( )⟩ ′ 𝑖) 𝐫̂ × 𝐦∗( (𝑟, 𝜃, 𝜙) . 𝐫̂ × 𝐧(𝑛𝑖′ 𝑚) ′ ,𝓁 ′ (𝑟, 𝜃, 𝜙) = 0 𝑛𝑚,𝓁
= 4𝜋
(A.2) (A.3)
where 𝛿 p,q is Kronecker delta function, 𝑧̃ (𝑛𝑖) (𝑥) is defined in (36) and for the vector functions X(r) and Y(r): ⟨𝐗(𝐫 ).𝐘(𝐫 )⟩ =
∫𝑆
𝐗(𝐫 ).𝐘(𝐫 )𝑑 𝑠
(A.4)
where r represents the position vector in the spherical coordinates system. References [1] Xu W, Duan B, Li P, Qiu Y. Study on the electromagnetic performance of inhomogeneous radomes for airborne applications–Part I: characteristics of phase distortion and boresight error. IEEE Trans Antenn Propag 2017;65(6):3162–74. [2] Valagiannopoulos C, Alitalo P, Tretyakov S. Dielectric-coated PEC cylinders which do not scatter electromagnetic waves. In: Proceedings of the 2012 International conference on electromagnetics in advanced applications (ICEAA). IEEE; 2012. p. 90–1. [3] Pay C, Ozgun O. A radar cross section reduction method using the concept of coordinate transformation and isotropic dielectric layers. In: Proceedings of the 2018 eighteenth mediterranean microwave symposium (MMS). IEEE; 2018. p. 101–3. [4] Guo L, Chen Y, Yang S. Scattering decomposition and control for fully dielectric-coated PEC bodies using characteristic modes. IEEE Antenn Wirel Propag Lett 2018;17:118–21. [5] Yamane T, Nishikata A, Shimizu Y. Resonance suppression of a spherical electromagnetic shielding enclosure by using conductive dielectrics. IEEE Trans Electromagn Compatib 2000;42(4):441–8.
185
Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Electromagnetic scattering from an arbitrarily shaped PEC object coated by spherical dielectric material using equivalence principle algorithm Mohammad Alian, Homayoon Oraizi∗ School of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran
a r t i c l e
i n f o
Keywords: Electromagnetic scattering Dielectric coating Equivalence principle Spherical wave harmonics Mode-matching,
a b s t r a c t A new method is presented for the analysis of electromagnetic scattering from arbitrarily shaped PEC object coated by homogeneous spherical dielectric material. The analysis of the internal PEC objects is performed using the equivalence principle algorithm (EPA). First, the scattering operator of EPA is evaluated for the PEC object in terms of RWG basis functions where the background is filled by coating material. Then a translation procedure is introduced to translate scattering operator evaluated in terms of RWG basis functions to a new scattering operator in terms of spherical wave harmonics (SWHs). This new scattering operator relates SWH coefficients of the scattered wave by internal PEC to those of the incident wave. By invoking the boundary conditions on the surface of spherical coating using the mode-matching technique, the coefficients of the scattered wave outside the coating material are evaluated. Since the scattering operator is independent from the coating radius, the problem may be solved by only analysis of boundary conditions on the surface of spherical coating for various values of coating radii without needing to repeat the analysis of internal PEC object. Several numerical examples are investigated to evaluate validity and efficiency of the proposed method.
1. Introduction The problem of electromagnetic wave scattering from coated conductors has extensively been studied in the last decades. The significance of the problem backs to its widespread applications in optimizing radar cross-section (RCS), shielding effectiveness, radome performance, scattering cancellation cloaking and so on [1–4]. So far numerous analytical [5–8] and numerical approaches [9–11] have been presented for the study of scattering from such objects. While the analytical methods are accurate and efficient, their applications are restricted to a few canonical geometries. On the other hand, the numerical methods usually cover a wide range of geometries, but they suffer from computational restrictions, such as CPU time and memory requirements. Nonetheless, in some cases an optimum solution may be adopted by employing a hybrid approach that benefits from the aforementioned merits of both the analytical and numerical approaches. This paper addresses the problem of electromagnetic wave scattering from an arbitrarily shaped PEC object coated by a dielectric sphere. Since this problem involves a dielectric medium, the volume integral equation (VIE) method may be employed as a numerical solution [12–15]. But the VIE necessitates a huge number of meshes for the discretization of the volume currents inside the dielectric media, resulting in computational inefficiencies. Fortunately, in recent years the alternative approach of surface integral equation (SIE) has effectively been ∗
employed for the problems involving homogeneous dielectric media [16–21]. In comparison to VIE that uses volume meshes, SIE considers the surface meshes for the discretization of the surface currents on the surface of a homogeneous volume. However, SIE is exclusively applicable to the piecewise homogeneous structures, but its computational costs are very lower than those of VIE, making it a popular numerical approach. In the current problem, the spherical geometry of the coating encourages the application of analytical method for the analysis of boundary conditions that are more efficient than the numerical approaches. The analytical solution employs the mode-matching technique for the SWHs. Furthermore, the arbitrary internal PEC object is treated numerically. Accordingly, the proposed method is a hybrid approach, that is expected to be computationally more efficient than the pure numerical approaches. It is shown that by employing the scattering operator of equivalence principle algorithm [22–30], the numerical solution of the internal PEC object is linked to that of the analytical solution involving SWHs. Furthermore, this procedure makes it possible to satisfy the boundary conditions on the surface of spherical coating, independent of the scattering operator of PEC object. Accordingly, as long as the coating material properties and the internal PEC object are not changed, for various values of radii of spherical coating, the same scattering operator of PEC object may be utilized for the satisfaction of boundary conditions on the surface of coating. Using the scattering operator, the equivalent
Corresponding author. E-mail address: [email protected] (H. Oraizi).
https://doi.org/10.1016/j.enganabound.2019.11.006 Received 22 June 2019; Received in revised form 30 October 2019; Accepted 15 November 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
scattering electric and magnetic currents on an imaginary surface enclosing the PEC object are evaluated from those of the incident currents in an unbounded medium with the same properties of coating material. Similar to many previous works, the well-known Rao–Wilton–Glisson (RWG) basis functions [31] are used in this paper for the numerical analysis of PEC object as well as for the discretization of surface currents on the equivalence surface. In this work, the equivalence surface is considered to be spherical. It is shown that by virtue of orthogonality of SWHs, the spherical equivalence surface facilitates the translation of scattering operator evaluated in terms of RWG basis functions to a new scattering operator in terms of SWHs. This new operator named SWHs scattering operator evaluates the SWHs coefficients of scattered field from those of the incident fields. The rest of the paper is arranged as follows. Section 2 describes the theory of proposed method where the evaluation of SWHs scattering operator is discussed in Section 2.1. Section 2.2 considers the boundary conditions using the mode-matching technique on the surface of spherical coating for the expanded fields in terms of SWHs. For the verification of the proposed method, several numerical examples are presented in Section 3. Finally, the conclusions are given in Section 4. 2. Theory Fig. 1. An arbitrarily shaped PEC object coated by a dielectric sphere of radius a with incident and scattered waves in each regions.
Consider the configuration of Fig. 1 where a PEC object is coated by a spherical dielectric medium and is illuminated by an arbitrary incident wave. The scattered field outside the spherical coating may be evaluated by the satisfaction of the boundary conditions on the surface of coating. The spherical surface of coating encourages the application of SWHs for the analysis of the problem. In terms of SWHs, the scattered and incident waves may be cast into four types, namely those of outside as well as inside the coating. Here the electromagnetic fields inside the coating are required in the region between rmin and a, where rmin is the radius of the smallest imaginary sphere enclosing the PEC object and concentric with the spherical coating and a is the radius of coating as shown in Fig. 1. Since both the investigation regions of r > a and rmin < r < a are homogenous source-free regions, their electric and magnetic fields may be expanded in terms of SWHs [32–35]: 𝐄𝓁 (𝑟, 𝜃, 𝜙) = 𝐇𝓁 (𝑟, 𝜃, 𝜙) =
∞ ∑ 𝑛 ∑ 𝑛=0 𝑚=−𝑛
𝑖) 𝑖) 𝑎𝑛𝑚 𝐦(𝑛𝑚, (𝑟, 𝜃, 𝜙) + 𝑏𝑛𝑚 𝐧(𝑛𝑚, (𝑟, 𝜃, 𝜙) 𝓁 𝓁
∞ 𝑛 𝑗 ∑ ∑ 𝑖) 𝑏 𝐦(𝑖) (𝑟, 𝜃, 𝜙) + 𝑎𝑛𝑚 𝐧(𝑛𝑚, (𝑟, 𝜃, 𝜙) 𝓁 𝜂𝓁 𝑛=0 𝑚=−𝑛 𝑛𝑚 𝑛𝑚,𝓁
respectively that are not used in the current work. 𝑃𝑛𝑚 (⋅) is the associated Legendre function of the first kind. The implicit time dependency of exp (j𝜔t) is assumed throughout the paper. The coefficients of the SWH expansion of the incident electric and magnetic fields in Fig. 1 can be evaluated by the orthogonality relations of SWHs [32,34,35] as follows: ( ) −𝑗𝜂 2𝑛 + 1 (𝑛 − 𝑚)! (1) 𝐦∗nm (5) 𝑎inc ⋅ 𝐇inc (𝑟) 𝑑Ω ))2 ∮ nm = 4𝜋𝑛(𝑛 + 1) (𝑛 + 𝑚)! ( ( 1 ,1 4𝜋 𝑗𝑛 𝑘1 𝑟 𝑏inc nm =
𝑖) 𝐧(mn ,𝓁
jm (𝑖) ( ) 𝑚 𝑧 𝑘 𝑟 𝑃𝑛 (cos𝜃) exp (jm𝜙) 𝜽̂ sin 𝜃 𝑛 𝓁 ( ) 𝜕 𝑚 −𝑧(𝑛𝑖) 𝑘𝓁 𝑟 𝑃 (cos𝜃) exp (jm𝜙) 𝝓̂ 𝜕𝜃 𝑛
𝑛(𝑛 + 1) (𝑖) ( ) 𝑚 = 𝑧𝑛 𝑘𝓁 𝑟 𝑃𝑛 (co𝑠𝜃) exp (jm𝜙) ̂ 𝐫 𝑘𝓁 𝑟 1 𝜕 [ (𝑖) ( )] 𝜕 𝑚 + 𝑟𝑧 𝑘 𝑟 𝑃 (cos𝜃) exp (jm𝜙) 𝜽̂ 𝑘𝓁 𝑟 𝜕𝑟 𝑛 𝓁 𝜕𝜃 𝑛 jm 𝜕 [ (𝑖) ( )] 𝑚 + 𝑟𝑧 𝑘 𝑟 𝑃𝑛 (cos𝜃) exp (jm𝜙) 𝝓̂ 𝑘𝓁 𝑟 sin 𝜃 𝜕𝑟 𝑛 𝓁
4𝜋
( ) (1 ) ⋅ 𝐄inc 𝐦∗nm (𝑟) 𝑑Ω 1 ,1
(6)
So, referring to Fig. 1 there exist totally six type of unknown SWH coefficients for the scattered field inside region 1 together with the incident and scattered fields inside regions 2. In Section 2.1 it is shown that using scattering operator of EPA, a comprehensive relation may be established between the SWH coefficients of the scattered field and those of incident field for the internal PEC object. This relation provides two sets of equations from the six required sets to evaluate six aforementioned types of unknowns. In Section 2.2 using the continuity of the tangential components of the electric and magnetic fields on the surface of spherical dielectric coating, four remaining sets of the equations are arranged, making it possible to have a unique solution.
(1)
(2)
√ where 𝜂𝓁 = 𝜇𝓁 ∕𝜖𝓁 is the intrinsic impedance of medium 𝓁 (𝓁 = 1, 2) (𝑖) 𝑖) and 𝐦𝑛𝑚,𝓁 and 𝐧(𝑛𝑚, are defined as follows: 𝓁 𝑖) 𝐦(mn = ,𝓁
2𝑛 + 1 (𝑛 − 𝑚)! 1 4𝜋𝑛(𝑛 + 1) (𝑛 + 𝑚)! (𝑗 (𝑘 𝑟))2 ∮ 𝑛 1
(3) 2.1. SWHs Scattering operator for the PEC object in an unbounded background region The scattered field SWH coefficients inside the coating may be directly evaluated from those of the incident field using the scattering operator of EPA. Consider the configuration of Fig. 2 where an arbitrary PEC scatterer is placed inside a homogenous dielectric background of permittivity 𝜖 2 and permeability 𝜇 2 and enclosed by an imaginary spherical surface of radius R. The equivalent electric and magnetic incident currents Jinc and Minc , defined on the interior surface of equivalence sphere irradiate the PEC object. The scattered field from the PEC object creates the equivalence surface currents Jsca and Msca on the exterior of equivalence surface. So the scattering operator acts on the incident currents to evaluate the scattering ones as follows: [ 𝑠𝑐𝑎 ] [ 𝑖𝑛𝑐 ] 𝐉 𝐉 = (7) 𝑠𝑐𝑎 𝐌 𝐌𝑖𝑛𝑐
(4)
√ in which 𝑘𝓁 = 𝜔 𝜇𝓁 𝜖𝓁 is the wave number of medium 𝓁. The superscript i distinguishes the scattered waves from the incident ones as for 𝑖 = 1 and 𝑖 = 4 the function 𝑧(𝑛𝑖) (⋅) stands for the spherical Bessel function of the first kind jn ( · ) and spherical Hankel function of the second kind ℎ(2) 𝑛 (⋅), respectively. Note the superscript i follows the same notation used in some main references pertaining to the SWHs such as Hansen [35]. In these references the superscript 𝑖 = 2 and 𝑖 = 3 are dedicated for the Bessel function of second kind and Hankel function of the firs kind, 179
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
On the other hand it is assumed that these surface currents are known in terms of RWG basis functions as: ∑ 𝑖𝑛𝑐 𝐌𝑖𝑛𝑐 (𝐫) = 𝐟𝑖𝑅𝑊 𝐺 (𝐫)𝐼𝑖𝑀 (13) 𝑖
𝐉𝑖𝑛𝑐 (𝐫) =
∑ 𝑖
𝐟𝑖𝑅𝑊 𝐺 (𝐫)𝐼𝑖𝐽
𝑖𝑛𝑐
(14) 𝑖𝑛𝑐
𝑖𝑛𝑐
The unknown amplitudes 𝐼𝑖𝑀 and 𝐼𝑖𝐽 may be evaluated from the equality of (11) and (13) as well as (12) and (14) [24]: [ 𝑖𝑛𝑐 ] ( ) 𝐼𝑀 = [𝑈 ]−1 [𝐴][𝑐 𝑖𝑛𝑐 ] + [𝐵][𝑑 𝑖𝑛𝑐 ] (15) [ 𝑖𝑛𝑐 ] −𝑗 ( ) [𝑈 ]−1 [𝐵][𝑐 𝑖𝑛𝑐 ] + [𝐴][𝑑 𝑖𝑛𝑐 ] 𝐼𝐽 = 𝜂2
Fig. 2. PEC object of Fig. 1, enclosed by equivalence sphere of radius R where the background is filled by the same dielectric medium of spherical coating of Fig. 1.
(8) where 𝐧̂ is outward unit normal vector on the equivalence surface and the subscripts io and oi show the inside-out and outside-in propagations, respectively. The near field operators and are defined as follows:
(𝐗) = ∇ ×
∫𝑆
∫𝑆
𝐺(𝐫 , 𝐫 ′ )𝐗(𝐫 ′ )𝑑 𝑠′ −
𝐺(𝐫 , 𝐫 )𝐗(𝐫 )𝑑 𝑠
𝐺(𝐫 , 𝐫 ′ )
′
′
𝑗 ∇ 𝐺(𝐫 , 𝐫 ′ )∇′ ⋅ (𝐗(𝐫 ′ ))𝑑 𝑠′ 𝑘2 ∫𝑆
where: ⟨ ⟩ 𝑈𝑗𝑖 = 𝐟𝑗𝑅𝑊 𝐺 (𝐫) ⋅ 𝐟𝑖𝑅𝑊 𝐺 (𝐫)
(17)
⟨ )⟩ ( 𝐴𝑗,𝑛𝑚 = 𝐟𝑗𝑅𝑊 𝐺 (𝐫) ⋅ 𝐫̂ × 𝐦(1) 𝑛𝑚
(18)
⟨ ( )⟩ 𝐵𝑗,𝑛𝑚 = 𝐟𝑗𝑅𝑊 𝐺 (𝐫) ⋅ 𝐫̂ × 𝐧(1) 𝑛𝑚
(19)
The notation ⟨ X(r).Y(r) ⟩ for the vector functions X(r) and Y(r) is defined in (A.4) (refer to the appendix). Note that for the expansion of fields in terms of SWHs, a finite number of radial modes Nso is considered for the evaluation of SWHs scattering operator to be discussed latter. A similar approach may be adopted for translation from RWG basis functions to SWHs of the scattered field. Here the scattering surface currents on the exterior surface of equivalence sphere may be represented in terms of SWHs using equations similar to (11) and (12) with the exception that the superscript (1) is replaced by (4) to show the diverging harmonics pertaining to the scattered fields. Consider the scattering surface currents to be determined in terms of RWG basis functions by 𝑠𝑐𝑎 and equations similar to (13) and (14). The unknown amplitudes 𝑐𝑛𝑚 𝑠𝑐𝑎 may be evaluated using the orthogonality relations for SWHs as: 𝑑𝑛𝑚
The scattering operator may be evaluated in three regular steps of outside-in propagation, analysis of the PEC object for the evaluation of induced electric surface currents and finally inside-out propagation [23]. Using these regular steps, the scattering operator may be evaluated as: [[ ] [ ]] [ ] ] 𝐧̂ × 𝒦io 11 12 [ ]−1 [ ℒoi − 𝜂1 𝒦oi = [ ] [ ] = 𝐵ii 2 ̂ × ℒio −𝜂2 𝒏 22 21
(𝐗) = −𝑗 𝑘 2
[𝑐 𝑠𝑐𝑎 ] = [𝜏][𝐺][𝐼 𝑀
𝑠𝑐𝑎
]
(10) 𝐫 ′ |)∕4𝜋|𝐫
]
(21)
( ⟨ )⟩ 𝐺𝑖,𝑛𝑚 = 𝐟𝑖𝑅𝑊 𝐺 (𝐫) ⋅ 𝐫̂ × 𝐦∗(4) 𝑛𝑚
in which = exp(−𝑗 𝑘2 |𝐫 − − is background Green’s function. Numerical evaluation of the scattering operator, necessitates the discretization of the equivalent currents on the surface of equivalence sphere. Similar to many previous works, the discretization of the surface currents may be done using RWG basis functions. However, since our analysis is based on SWHs, translations from RWG basis function to SWHs and vice versa are necessary for the scattered and incident fields, respectively. These translations make it possible to evaluate the scattering operator for the SWHs so that the SWH coefficients of the scattered field can be evaluated from those of the incident field. Assume that the incidence field be expanded in terms of SWHs with 𝑖𝑛𝑐 and 𝑑 𝑖𝑛𝑐 . The incident magnetic and the known complex amplitudes 𝑐𝑛𝑚 𝑛𝑚 electric surface currents on the interior surface of the equivalence sphere with radius R can be readily evaluated in terms of SWHs from the electric and magnetic fields, respectively, as follows:
(22)
and the diagonal matrix [𝜏] is defined as: ⎡𝜏1,−1 ⎢0 ⎢ [𝜏] = ⎢0 ⎢⋮ ⎢ ⎣0
0 𝜏1,0 0 ⋮ 0
0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ ⋮ ⎥ 𝜏𝑁,𝑁 ⎦
(23)
2𝑛 + 1 (𝑛 − 𝑚)! 𝑛(𝑛 + 1) (𝑛 + 𝑚)!
(24)
0 0 𝜏1,1 ⋮ 0
… … … ⋱ …
with the following elements: 𝜏𝑛𝑚 =
−1 | |2 4𝜋𝑅2 |ℎ(2) (𝑘 𝑅)| | 𝑛 2 |
By substitution of the coefficients of RWG basis functions of the incident currents from (15) and (16) into (7) and as well as those of scattering currents from (7) into (20) and (21) a new scattering operator ̃ is evaluated that constructs a bridge between the scattered and incident field SWHs as: [ 𝑠𝑐𝑎 ] [ 𝑖𝑛𝑐 ] [ ][ ] [𝑐 ] [̃11 ] [̃12 ] [𝑐 𝑖𝑛𝑐 ] ̃ [𝑐 ] = = (25) [𝑑 𝑠𝑐𝑎 ] [𝑑 𝑖𝑛𝑐 ] [̃21 ] [̃22 ] [𝑑 𝑖𝑛𝑐 ]
(11)
𝑛=0 𝑚=−𝑛
𝐉𝑖𝑛𝑐 (𝐫) = −𝐧̂ × 𝐇𝑖𝑛𝑐 (𝑅, 𝜃, 𝜙) ∞ 𝑛 ) ( ) −𝑗 ∑ ∑ 𝑖𝑛𝑐 ( 𝑖𝑛𝑐 ̂ (1) = 𝑑 𝐫̂ × 𝐦(1) 𝑛𝑚 + 𝑐𝑛𝑚 𝐫 × 𝐧𝑛𝑚 𝜂2 𝑛=0 𝑚=−𝑛 𝑛𝑚
𝑠𝑐𝑎
where:
𝐫′|
𝐌𝑖𝑛𝑐 (𝐫) = 𝐧̂ × 𝐄𝑖𝑛𝑐 (𝑅, 𝜃, 𝜙) ∞ ∑ 𝑛 ∑ ( ) ( ) 𝑖𝑛𝑐 ̂ 𝑖𝑛𝑐 ̂ (1) 𝐫 × 𝐦(1) = 𝑐𝑛𝑚 𝑛𝑚 + 𝑑𝑛𝑚 𝐫 × 𝐧𝑛𝑚
(20)
(9) [𝑑 𝑠𝑐𝑎 ] = 𝑗𝜂2 [𝜏][𝐺][𝐼 𝐽
′
(16)
where: ( ) [̃11 ] = [𝜏][𝐺] 𝑗∕𝜂2 [21 ][𝑈 ]−1 [𝐵] − [22 ][𝑈 ]−1 [𝐴]
(12) 180
(26)
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
( ) [̃12 ] = [𝜏][𝐺] 𝑗∕𝜂2 [21 ][𝑈 ]−1 [𝐴] − [22 ][𝑈 ]−1 [𝐵]
In a similar manner, the inner product of both sides of (32) by 𝐫̂ × 𝐧∗(1) 𝑛𝑚,1 and integrating on the surface of coating results in:
(27)
𝑠𝑐𝑎 𝑖𝑛𝑐 𝑠𝑐𝑎 𝑖𝑛𝑐 ̃ (2) ̃ ̃ ℎ̃ (2) 𝑛 (𝑘1 𝑎)𝑏𝑛𝑚 − 𝑗𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 − ℎ𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 = −𝑗𝑛 (𝑘1 𝑎)𝑏𝑛𝑚
( ) [̃21 ] = −[𝜏][𝐺] [11 ][𝑈 ]−1 [𝐵] + 𝑗𝜂2 [12 ][𝑈 ]−1 [𝐴]
(28)
( ) [̃22 ] = −[𝜏][𝐺] [11 ][𝑈 ]−1 [𝐴] + 𝑗𝜂2 [12 ][𝑈 ]−1 [𝐵]
(29)
where for the spherical Bessel function 𝑧(𝑛𝑖) (𝑥), the function 𝑧̃ (𝑛𝑖) (𝑥) is defined as follows: 1 𝑑 ( (𝑖) ) 𝑛 + 1 (𝑖) ) 𝑧̃ (𝑛𝑖) (𝑥) = 𝑧 (𝑥) − 𝑧(𝑛𝑖+1 (𝑥) (36) 𝑥𝑧𝑛 (𝑥) = 𝑥 𝑑𝑥 𝑥 𝑛 Analogous to the evaluation of (34) and (35) from (32), the following equations may be set up from (33):
Obviously, since the scattering operator is evaluated for the PEC object inside an unbounded background region, it is independent of the radius of spherical coating. The radius of coating is taken into account in the analysis of the boundary conditions on the coating surface as discussed in Section 2.2. 2.2. Analysis of the boundary conditions on the surface of spherical coating using mode-matching technique
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
(31)
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
𝑁𝑚𝑚
−
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛 𝑁𝑚𝑚
=−
∑
( ) ( ) 𝑖𝑛𝑐 ̂ 𝑖𝑛𝑐 ̂ 𝑐𝑛𝑚 𝐫 × 𝐦(1) + 𝑑𝑛𝑚 𝐫 × 𝐧(1) 𝑛𝑚,2 𝑛𝑚,2 ( ) ( ) 𝑠𝑐𝑎 ̂ 𝑠𝑐𝑎 ̂ 𝑐𝑛𝑚 𝐫 × 𝐦(4) + 𝑑𝑛𝑚 𝐫 × 𝐧(4) 𝑛𝑚,2 𝑛𝑚,2
𝑛 ∑
𝑛=0 𝑚=−𝑛
𝑗∕𝜂1
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
−𝑗 ∕𝜂2 −𝑗 ∕𝜂2
𝑛 ∑ ∑ 𝑛=0 𝑚=−𝑛
𝑁𝑚𝑚
∑
𝑛 ∑
𝑛=0 𝑚=−𝑛
= −𝑗∕𝜂1
𝑁𝑚𝑚
∑
(32)
(
)
( ) 𝑖𝑛𝑐 ̂ + 𝑐𝑛𝑚 𝐫 × 𝐧(1) 𝑛𝑚,2
(
)
( ) 𝑠𝑐𝑎 ̂ + 𝑐𝑛𝑚 𝐫 × 𝐧(4) 𝑛𝑚,2
𝑖𝑛𝑐 ̂ 𝑑𝑛𝑚 𝐫 × 𝐦(1) 𝑛𝑚,2
𝑠𝑐𝑎 ̂ 𝐫 × 𝐦(4) 𝑑𝑛𝑚 𝑛𝑚,2
𝑛 ∑
𝑛=0 𝑚=−𝑛
(
)
(
(1) (1) 𝑖𝑛𝑐 ̂ ̂ 𝑏𝑖𝑛𝑐 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑎𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
[0]
−[𝐽2 ]
[0]
−[𝐻2 ]
̃1 ] [𝐻
[0]
−[𝐽̃2 ]
[0]
[0]
𝜂 − 𝜂1 [𝐽̃2 ]
[0]
𝜂 ̃2 ] − 𝜂1 [ 𝐻
2
[𝐻1 ]
[0]
[0]
−[̃11 ]
[0]
−[̃21 ]
𝜂 − 𝜂1 [ 𝐽 2 ] 2
̃2 ] −[𝐻
2
[0]
−[̃12 ]
[𝐼]
−[̃22 ]
[0]
𝑖𝑛𝑐 ⎤ ⎡[𝐽1 ][𝑎 ]⎤ ⎥ ⎢ ⎥ 𝑖𝑛𝑐 ⎥ ⎢[𝐽̃1 ][𝑏 ]⎥ ⎥ ⎢ ⎥ 𝑖𝑛𝑐 ⎥ ⎢[𝐽̃1 ][𝑎 ]⎥ = − ⎥ ⎢ ⎥ 𝑖𝑛𝑐 ⎥ ⎢[𝐽1 ][𝑏 ]⎥ ⎥ ⎢ ⎥ ⎥ ⎢ [0] ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ [0] ⎦
⎤ ⎥ ⎥ ⎥ [0] ⎥ ⎥ 𝜂 − 𝜂1 [𝐻2 ]⎥ 2 ⎥ [0] ⎥ ⎥ ⎦ [𝐼] [0]
(39)
⎡𝑧 ( 𝑖 ) ( 𝑘 𝑎 ) ⎢ 1 𝓁 = ⎢⋮ ⎢0 ⎣
… ⋱ …
⎤ 0 ⎥ ⋮ ⎥ 𝑧(𝑁𝑖) (𝑘𝓁 𝑎)⎥⎦
(40)
̃𝓁 where 𝑍𝓁(𝑖) represents J𝓁 for 𝑖 = 1 and H𝓁 for 𝑖 = 4. Similarly 𝐽̃𝓁 and 𝐻 are defined as:
( ) ( ) (4) (4) 𝑠𝑐𝑎 ̂ ̂ 𝑏𝑠𝑐𝑎 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑎𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
𝑁𝑚𝑚
(38)
[𝑍𝓁(𝑖) ]
( ) ( ) (1) (1) 𝑖𝑛𝑐 ̂ ̂ 𝑎𝑖𝑛𝑐 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑏𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
𝑁𝑚𝑚
𝑠𝑐𝑎 𝑖𝑛𝑐 𝑠𝑐𝑎 𝑖𝑛𝑐 ̃ (2) ̃ ̃ 𝜂2 ℎ̃ (2) 𝑛 (𝑘1 𝑎)𝑎𝑛𝑚 − 𝜂1 𝑗𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 − 𝜂1 ℎ𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 = −𝜂2 𝑗𝑛 (𝑘1 𝑎)𝑎𝑛𝑚
Eqs. (34), (35), (37) and (38) are four equations to determine the 𝑠𝑐𝑎 𝑖𝑛𝑐 𝑖𝑛𝑐 𝑠𝑐𝑎 𝑠𝑐𝑎 six unknowns (𝑎𝑠𝑐𝑎 𝑛𝑚 , 𝑏𝑛𝑚 , 𝑐𝑛𝑚 , 𝑑𝑛𝑚 , 𝑐𝑛𝑚 and 𝑑𝑛𝑚 ). We must enforce two more relationships among these six unknowns before a unique solution can be obtained. The remaining equations may be adopted using the scattering operator as (25). Thereupon, a system of six equations with six unknowns may be established as (39) where [I] is the identity matrix and for J𝓁 and H𝓁 :
( ) ( ) (4) (4) 𝑠𝑐𝑎 ̂ ̂ 𝑎𝑠𝑐𝑎 𝑛𝑚 𝐫 × 𝐦𝑛𝑚,1 + 𝑏𝑛𝑚 𝐫 × 𝐧𝑛𝑚,1
𝑁𝑚𝑚
−
(37)
⎡ [𝑎𝑠𝑐𝑎 ] ⎢ 𝑠𝑐𝑎 ⎢ [𝑏 ] ⎢ 𝑖𝑛𝑐 ⎢ [𝑐 ] ×⎢ 𝑖𝑛𝑐 ⎢ [𝑑 ] ⎢ 𝑠𝑐𝑎 ⎢ [𝑐 ] ⎢ 𝑠𝑐𝑎 ⎣[𝑑 ]
where superscripts inc and sca represent the incident and scattered fields, respectively. Also subscript 𝓁, denotes the region r > a for 𝓁 = 1 and rmin < r < a for 𝓁 = 2 (refer to Fig. 1). Substitutions of the electric and magnetic fields in (30) and (31) by their SWH expansions lead to the following equations: 𝑁𝑚𝑚
𝑠𝑐𝑎 𝑖𝑛𝑐 (2) 𝑠𝑐𝑎 𝑖𝑛𝑐 𝜂2 ℎ(2) 𝑛 (𝑘1 𝑎)𝑏𝑛𝑚 − 𝜂1 𝑗𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 − 𝜂1 ℎ𝑛 (𝑘2 𝑎)𝑑𝑛𝑚 = −𝜂2 𝑗𝑛 (𝑘1 𝑎)𝑏𝑛𝑚
⎡[ 𝐻 1 ] ⎢ ⎢[0] ⎢ ̃ ⎢[ 𝐻 1 ] ⎢ ⎢[0] ⎢ ⎢[0] ⎢ ⎣[0]
As the coating surface has a canonical shape the mode-matching technique is implemented that is an analytical approach without needing to surface discretization making it more efficient than the numerical full wave techniques.The boundary conditions for the electric and magnetic fields on the surface of spherical coating (𝑟 = 𝑎) may be written as: ( ) (30) 𝐫̂ × 𝐄sca (𝑎, 𝜃, 𝜙) − 𝐄inc (𝑎, 𝜃, 𝜙) − 𝐄sca (𝑎, 𝜃, 𝜙) = −𝐫̂ × 𝐄inc (𝑎, 𝜃, 𝜙) 1 2 2 1 ) ( 𝐫̂ × 𝐇sca (𝑎, 𝜃, 𝜙) − 𝐇inc (𝑎, 𝜃, 𝜙) − 𝐇sca (𝑎, 𝜃, 𝜙) = −𝐫̂ × 𝐇inc (𝑎, 𝜃, 𝜙) 1 2 2 1
(35)
⎡𝑧̃ (𝑖) (𝑘 𝑎) ⎢ 1 𝓁 (𝑖) ̃ [ 𝑍 𝓁 ] = ⎢⋮ ⎢0 ⎣
… ⋱ …
⎤ 0 ⎥ ⋮ ⎥ (𝑖) 𝑧̃ 𝑁 (𝑘𝓁 𝑎)⎥⎦
(41)
By the evaluation of the unknown coefficients of the scattered field from (39), the scattered field may be readily evaluated from (1) and (2). 2.3. SWHs truncation criteria )
The numerical implementation of the proposed method necessitates a truncated number of radial modes for the expansion of electric and magnetic fields in terms of SWHs in (1) and (2). According to the literature, the infinite series in these equations may be truncated by N modes as [35–37]:
(33)
where the upper limits of exterior series Nmm are the required number of radial modes in the mode-matching technique for an acceptable accuracy in the expansion of the fields on the surface of coating. Obviously, Nmm is a function of coating radius and material to be discussed later. The inner product of both sides of (32) by 𝐫̂ × 𝐦∗(1) and integrating 𝑛𝑚,1 on the surface of coating (𝑟 = 𝑎), thanks to the orthogonality relations for SWHs (refer to the appendix), gives: 𝑠𝑐𝑎 𝑖𝑛𝑐 (2) 𝑠𝑐𝑎 𝑖𝑛𝑐 ℎ(2) 𝑛 (𝑘1 𝑎)𝑎𝑛𝑚 − 𝑗𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 − ℎ𝑛 (𝑘2 𝑎)𝑐𝑛𝑚 = −𝑗𝑛 (𝑘1 𝑎)𝑎𝑛𝑚
𝑁 ≃ 𝑘2 𝑟0 + 10
(42)
where r0 is the radius of a sphere encompassing the region with acceptable approximation of the actual fields by SWHs. In Section 2.1 a truncated number of modes Nso was considered for the evaluation of SWHs scattering operator. So, Nso may be evaluated from (42) where r0
(34) 181
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Fig. 3. PEC sphere coated by spherical dielectric of radius a, 𝜖𝑟 = 4 and 𝜇𝑟 = 1.
is replaced by the equivalence sphere radius R. Similarly, the truncated number of modes for mode-matching Nmm in Section 2.2 may be evaluated from (42), replacing r0 by spherical coating radius a. Since Nmm is a function of coating radius and material, it is independent of the complexity of PEC structure. The geometry of the PEC scatterer only affects the scattering operator which consider Nso number of radial modes for the expansion of the fields. When the number of modes is determined, according to (1) and (2) it can be shown that the number of unknowns for each set of the unknown fields is 2((𝑁mm + 1)2 − 1). As there are three sets of unknown fields, the total number of unknowns of the problem is 6((𝑁mm + 1)2 − 1). Note that if the coating radius is changed only the number of modes for the satisfaction of boundary conditions using the mode-matching technique should be changed, where the same SWHs scattering operator evaluated by Nso radial modes may be employed independent of coating radius. 3. Numerical examples For the verification of the proposed method, several numerical examples are investigated in this section. The incident wave is considered to be an 𝑥−polarized plane wave propagating along +𝑧 direction at 3 GHz. Note that if the electrical dimensions of the PEC scatterer be very large, some accelerating algorithms such as ACA or MLFMA can be adopted in the evaluation of the scattering operator. However, in the current work no accelerating algorithm is used.
Fig. 4. Comparison of the normalized RCS of coated PEC sphere of radius 5 cm for various values of coating radius a evaluated from proposed method (solid blue line) and that of exact analytical solution (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.1. Scattering from coated PEC sphere shown in Fig. 5. The same procedure of the previous example for the evaluation of the scattered field is applied here as well. In this case the normalized RCS evaluated by the proposed method is compared with that of surface integral equations solved by method of moments (SIEMoM) versus 𝜃 at the same planes of the previous example as shown in Fig. 6 for various values of coating radii. Once again the verification of the proposed method is approved by the acceptable consistency between the results. Furthermore, for the evaluation of the efficiency of the proposed method, its computational costs are compared to those of SIE-MoM in Table 1 for the case of a = 20 cm. Note that the set-up time of proposed method is more than that of SIE-MoM, which is mainly spent for the computation of scattering operator. However, the scattering operator needs to be computed once for various values of coating radii of a PEC object. Accordingly, the proposed method by decomposing the analysis domain improves the computational costs in comparison to a monolithic method such as SIE-MoM, particularly where different coating radii are investigated. The other cases of dielectric coating radii exhibit similar excellent performance, but they are not reported here due to space limitation.
This example considers a PEC sphere of radius 5 cm coated by a spherical dielectric material of radius a with the relative permittivity 𝜖𝑟 = 4 and permeability 𝜇𝑟 = 1, as shown in Fig. 3. First the scattering operator of the PEC sphere is evaluated in terms of RWG basis functions using (8). Then the RWG scattering operator is translated into that of SWHs using (26)–(29). Finally (39) is solved for the unknown SWH coefficients of the scattered field where the incident field SWH coefficients are substituted from (5)–(6). The normalized RCS of the coated sphere evaluated from the proposed method is compared with that of exact analytical solution [38–40] for various values of the spherical coating radius a versus zenith angle 𝜃 at 𝜙 = 0◦ and 𝜙 = 90◦ planes as shown in Fig. 4. The acceptable consistency between the results verifies the proposed method. 3.2. Scattering from coated PEC cube In this case the coated PEC object is considered to be a cube with edge length of 10 cm coated by a spherical dielectric material of radius a with the relative permittivity 𝜖𝑟 = 2 and permeability 𝜇𝑟 = 1, as 182
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Table 1 A comparison between the computational costs. Problem
Numerical approach
Number of unknowns
Set-up time (s)
Solution time (s)
Section 3.2 (a = 20 cm)
Proposed method SIE-MoM Proposed method SIE-MoM Proposed method SIE-MoM
5040
609
645
349
12462 8208
176 1204
1563 1684
1381 925
20268 5394
408 594
4108 727
3182 399
12743
188
1624
1467
Section 3.3 (a = 20 cm) Section 3.4 (a = 20 cm)
Memory usage (MB)
Fig. 5. PEC cube coated by spherical dielectric of radius a, 𝜖𝑟 = 2 and 𝜇𝑟 = 1.
Fig. 7. PEC cylinder coated by spherical dielectric of radius a, 𝜖𝑟 = 2 and 𝜇𝑟 = 2.
3.3. Scattering from coated PEC cylinder In this example the electromagnetic scattering from a coated cylinder of radius 5 cm and length 8 cm is considered as shown in Fig. 7. Here both the relative permittivity and permeability of the coating material are different from those of the background free-space. Consider 𝜖𝑟 = 2 and 𝜇𝑟 = 2, similar comparisons to the previous examples for the normalized RCS evaluated by the proposed method and that of SIE-MoM are depicted in Fig. 8. The comparisons show good agreement between the results of two methods. Similar comparison to the previous example for the computational costs are reported in the same Table 1 for the case of a = 20 cm. These comparisons show that the proposed method is faster and needs lower memory in comparison to SIE-MoM. Furthermore, in the proposed method while the PEC object is the same, it is sufficient to evaluate the SWHs scattering operator once independent of spherical coating radius. Observe that the proposed method inherits advantages of equivalence principle based domain decomposition method, that make it possible to solve some parts of the problem independently, saving analysis time and reducing memory requirements.
3.4. Scattering from coated diamond shape PEC The final example considers the electromagnetic scattering from a coated diamond shape PEC as shown in Fig. 9. Consider 𝜖𝑟 = 1.5 and 𝜇𝑟 = 1.5, similar comparisons to the previous examples for the normalized RCS evaluated by the proposed method and that of SIE-MoM are depicted in Fig. 10. Also similar comparison to the previous example for the computational costs are reported in the same Table 1 for the case of a = 20 cm.
Fig. 6. Comparison of the normalized RCS of coated PEC cube with radius 7 cm and length of 10 cm for various values of coating radius a evaluated by the proposed method (solid blue line) and that of simulated with SIE-MoM (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 183
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
Fig. 10. Comparison of the normalized RCS of coated diamond shape PEC shown in Fig. 9 for various values of coating radius a evaluated by the proposed method (solid blue line) and that of simulated with SIE-MoM (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Comparison of the normalized RCS of coated PEC cylinder with radius 5 cm and length of 8 cm for various values of coating radius a evaluated by the proposed method (solid blue line) and that of simulated with SIE-MoM (dotted red line) at 𝜙 = 0◦ and 𝜙 = 90◦ planes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. (a) Diamond shape PEC object geometry and dimensions. (b) PEC diamond coated by spherical dielectric of radius a, 𝜖𝑟 = 1.5 and 𝜇𝑟 = 1.5.
184
M. Alian and H. Oraizi
Engineering Analysis with Boundary Elements 111 (2020) 178–185
4. Conclusion
[6] Sun J, Li L-W. Dispersion of waves over a PEC cylinder coated with two-layer lossy dielectric materials. IEEE Trans Antenna Propag 2007;55(3):877–81. [7] Partal HP, Mautz JR, Arvas E. The exact analytical solution for large circular loops radiating around a dielectric coated conducting sphere. IEEE Trans Antenn Propag 2009;57(2):436–43. [8] Geng Y-L, Qiu C-W, Yuan N. Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer. IEEE Trans Antenn Propag 2009;57(2):572–6. [9] Li L, Wang H, Ping X, Yin X. Comparison of the MIE series method and FEM in electromagnetic analysis of spherical dielectric-coated media. Progress in Electromagnetic Research Symposium (PIERS). IEEE; 2016. 2646–2646 [10] Caorsi S, Pastorino M, Raffetto M. Electromagnetic scattering by a conducting strip with a multilayer elliptic dielectric coating. IEEE Trans Electromagn Compatib 1999;41(4):335–43. [11] Shi Y, Liang C-H. An efficient single-source integral equation solution to em scattering from a coated conductor. IEEE Antenn Wirel Propag Lett 2015;14:547–50. [12] Schaubert D, Wilton D, Glisson A. A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies. IEEE Trans Antenn Propag 1984;32(1):77–85. [13] Wu Q. Computation of characteristic modes for dielectric bodies using volume integral equation and interpolation. IEEE Antenn Wirel Propag Lett 2017;16:2963–6. [14] Cai Q-M, Zhang Z-P, Zhao Y-W, Huang W-F, Zheng Y-T, Nie Z-P, et al. Nonconformal discretization of electric current volume integral equation with higher order hierarchical vector basis functions. IEEE Trans Antenn Propag 2017;65(8):4155–69. [15] Zhang L-M, Sheng X-Q. A discontinuous Galerkin volume integral equation method for scattering from inhomogeneous objects. IEEE Trans Antenn Propag 2015;63(12):5661–7. [16] Poggio AJ, Miller EK. Integral equation solutions of three-dimensional scattering problems. MB Associates; 1970. [17] Wu T-K, Tsai LL. Scattering from arbitrarily-shaped lossy dielectric bodies of revolution. Radio Science 1977;12(5):709–18. [18] Chang Y, Harrington R. A surface formulation for characteristic modes of material bodies. IEEE Trans Antenn Propag 1977;25(6):789–95. [19] Ylä-Oijala P, Taskinen M, Järvenpää S. Analysis of surface integral equations in electromagnetic scattering and radiation problems. Eng Anal Bound Elem 2008;32(3):196–209. [20] Cools K, Andriulli FP, Michielssen E. A Calderón multiplicative preconditioner for the PMCHWT integral equation. IEEE Trans Antenn Propag 2011;59(12):4579–87. [21] Ylä-Oijala P, Taskinen M, Järvenpää S. Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods. Radio Sci 2005;40(6). [22] Li M-K, Chew WC, Jiang LJ. A domain decomposition scheme based on equivalence theorem. Microw Opt Technol Lett 2006;48(9):1853–7. [23] Li M-K, Chew WC. Wave-field interaction with complex structures using equivalence principle algorithm. IEEE Trans Antenn Propag 2007;55(1):130–8. [24] Alian M, Oraizi H. Electromagnetic multiple PEC object scattering using equivalence principle and addition theorem for spherical wave harmonics. IEEE Trans Antenn Propag 2018;66(11):6233–43. [25] Ylä-Oijala P, Lancellotti V, De Hon BP, Järvenpää S. Domain decomposition methods combining surface equivalence principle and macro basis functions. ACES, 25; 2010. [26] Fu X, Jiang LJ, He SQ, et al. Performance enhancement of equivalence principle algorithm. IEEE Antenn Wirel Propag Lett 2016;15:480–3. [27] Alian M, Oraizi H. Application of equivalence principle for em scattering from irregular array of arbitrarily orientated PEC scatterers using both translation and rotation addition theorems. IEEE Trans Antenn Propag 2019. [28] Su T, Du L, Chen R. Electromagnetic scattering for multiple PEC bodies of revolution using equivalence principle algorithm. IEEE Trans Antenn Propag 2014;62(5):2736–44. [29] Shao H, Hu J, Chew WC. Single-source equivalence principle algorithm for the analysis of complex structures. IEEE Antenn Wirel Propag Lett 2014;13:1255–8. [30] He Z, Chen R-S, Wei E. An efficient marching-on-in-degree solution of transient multiscale em scattering problems. IEEE Trans Antenn Propag 2016;64(7):3039–46. [31] Rao S, Wilton D, Glisson A. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antenn Propag 1982;30(3):409–18. [32] Stratton JA. Electromagnetic theory. John Wiley & Sons; 2007. [33] Hald J. Spherical near-field antenna measurements, 26. Iet; 2008. [34] Liang C, Lo Y. Scattering by two spheres. Radio Sci 1967;2(12):1481–95. [35] Hansen JE. Spherical near-field antenna measurements, 26. Iet; 1988. [36] Jensen F, Frandsen A. On the number of modes in spherical wave expansions. AMTA Symposium, October; 2004. [37] MacPhie RH, Wu K-L. A plane wave expansion of spherical wave functions for modal analysis of guided wave structures and scatterers. IEEE Trans Antenn Propag 2003;51(10):2801–5. [38] Pena O, Pal U. Scattering of electromagnetic radiation by a multilayered sphere. Comput Phys Commun 2009;180(11):2348–54. [39] Yang W. Improved recursive algorithm for light scattering by a multilayered sphere. Appl Opt 2003;42(9):1710–20. [40] Ericsson A., Sjöberg D., Larsson C., Martin T.. Scattering from a multilayered sphereapplications to electromagnetic absorbers on double curved surfaces. Technical Report LUTEDX/(TEAT-7249)/1-32/(2017)2017; 7249.
Electromagnetic scattering from PEC objects coated by spherical dielectric materials was investigated using a new method based on the equivalence principle. According to the spherical geometry of coating, the electric and magnetic fields are expanded by SWHs. The analysis of the PEC object was first considered by employing the scattering operator of equivalence principle algorithms using RWG basis functions. Then the translation of scattering operator from RWG basis functions to SWHs was discussed in detail. It was shown that using SWHs scattering operator for the PEC object, the SWH coefficients of the scattered field may be evaluated from those of the incident field independent of the coating radius. Accordingly, if the coating radius is changed it is possible to solve the problem using the same scattering operator of the PEC object only by repeating the analysis of the boundary conditions on the surface of coating. The verification of the proposed method was investigated through several numerical examples for various values of coating radius. The SWH expansions used in this work are extracted for a homogenous isotropic medium. Thereupon, the proposed method is not applicable for anisotropic coating material at least using the reported SWH expansions. However, in the case of lossy coating medium the equations may be corrected by considering complex permittivity or permeability that can be studied in a future work. Appendix. Orthogonality relations of transverse components of SWHs The orthogonality relations for SWHs may be found in many previous works [32,35]. From these orthogonality relations the following ones can be deduced for their transverse components as follows: ⟨( )( )⟩ ′ 𝑖) 𝐫̂ × 𝐦∗( (𝑟, 𝜃, 𝜙) . 𝐫̂ × 𝐦(𝑛𝑖′ 𝑚) ′ ,𝓁 ′ (𝑟, 𝜃, 𝜙) 𝑛𝑚,𝓁 = 4𝜋 ⟨(
′ 𝑛(𝑛 + 1) (𝑛 + 𝑚)! ∗(𝑖) 𝑧 (𝑘𝓁 𝑟)𝑧(𝑛𝑖 ) (𝑘𝓁 ′ 𝑟)𝛿𝑛,𝑛′ 𝛿𝑚,𝑚′ 2𝑛 + 1 (𝑛 − 𝑚)! 𝑛
(A.1)
)( )⟩ ′ 𝑖) (𝑟, 𝜃, 𝜙) . 𝐫̂ × 𝐧(𝑛𝑖′ 𝑚) ′ ,𝓁 ′ (𝑟, 𝜃, 𝜙) 𝐫̂ × 𝐧∗( 𝑛𝑚,𝓁
′ 𝑛(𝑛 + 1) (𝑛 + 𝑚)! ∗(𝑖) 𝑧̃ (𝑘𝓁 𝑟)𝑧̃ (𝑛𝑖 ) (𝑘𝓁 ′ 𝑟)𝛿𝑛,𝑛′ 𝛿𝑚,𝑚′ 2𝑛 + 1 (𝑛 − 𝑚)! 𝑛 ⟨( )( )⟩ ′ 𝑖) 𝐫̂ × 𝐦∗( (𝑟, 𝜃, 𝜙) . 𝐫̂ × 𝐧(𝑛𝑖′ 𝑚) ′ ,𝓁 ′ (𝑟, 𝜃, 𝜙) = 0 𝑛𝑚,𝓁
= 4𝜋
(A.2) (A.3)
where 𝛿 p,q is Kronecker delta function, 𝑧̃ (𝑛𝑖) (𝑥) is defined in (36) and for the vector functions X(r) and Y(r): ⟨𝐗(𝐫 ).𝐘(𝐫 )⟩ =
∫𝑆
𝐗(𝐫 ).𝐘(𝐫 )𝑑 𝑠
(A.4)
where r represents the position vector in the spherical coordinates system. References [1] Xu W, Duan B, Li P, Qiu Y. Study on the electromagnetic performance of inhomogeneous radomes for airborne applications–Part I: characteristics of phase distortion and boresight error. IEEE Trans Antenn Propag 2017;65(6):3162–74. [2] Valagiannopoulos C, Alitalo P, Tretyakov S. Dielectric-coated PEC cylinders which do not scatter electromagnetic waves. In: Proceedings of the 2012 International conference on electromagnetics in advanced applications (ICEAA). IEEE; 2012. p. 90–1. [3] Pay C, Ozgun O. A radar cross section reduction method using the concept of coordinate transformation and isotropic dielectric layers. In: Proceedings of the 2018 eighteenth mediterranean microwave symposium (MMS). IEEE; 2018. p. 101–3. [4] Guo L, Chen Y, Yang S. Scattering decomposition and control for fully dielectric-coated PEC bodies using characteristic modes. IEEE Antenn Wirel Propag Lett 2018;17:118–21. [5] Yamane T, Nishikata A, Shimizu Y. Resonance suppression of a spherical electromagnetic shielding enclosure by using conductive dielectrics. IEEE Trans Electromagn Compatib 2000;42(4):441–8.
185