Scattering Analysis of Electromagnetic Materials Using Fast Dipole Method Based on Volume Integral Equation

Abstract

The fast dipole method (FDM) is extended to analyze the scattering of dielectric and magnetic materials by solving the volume integral equation (VIE). The FDM is based on the equivalent dipole method (EDM) and can achieve the separation of the field dipole and source dipole, which reduces the complexity of interactions between two far groups (such as group 𝑖 and group 𝑗) from 𝑂(𝑁_𝑖𝑁_𝑗) to 𝑂(𝑁_𝑖+𝑁_𝑗), where 𝑁_𝑖 and 𝑁_𝑗 are the numbers of dipoles in group 𝑖 and group 𝑗, respectively. Targets including left-handed materials (LHMs), which are a kind of dielectric and magnetic materials, are calculated to demonstrate the merits of the FDM. Furthermore, in this study we find that the convergence may become much slower when the targets include LHMs compared with conventional electromagnetic materials. Numerical results about convergence characteristics are presented to show this property.

1. Introduction

The electromagnetic materials have gained wide attention and interest for a long time. The method of moments (MoM) based on the volume integral equation (VIE) [1] is one of the most popular numerical methods to analyze the scattering of the electromagnetic materials. However, the conventional MoM needs 𝑂(𝑁²) CPU time and memory requirement for an iterative solver. The complexity is very expensive for a personal computer with limited resources. Fortunately, many methods have been proposed in the past decades to deal with this problem, such as multilevel fast multipole algorithm (MLFMA) [2–4], the adaptive integral method (AIM) [5, 6], and the conjugate gradient-fast Fourier transform (CG-FFT) [7]. These methods improve the efficiency (computer storage and solution time) of the conventional MoM.

Recently, the equivalent dipole method (EDM) [8–10] is proposed to simplify the impedance matrix element filling procedure for the MoM, in which the Schaubert-Wilton-Glisson (SWG) [11] elements are equivalent to an infinitely small dipole. However, it cannot save memory and reduce iterative time. More recently, the fast dipole method (FDM) [12, 13] is developed to mitigate the problem of the EDM, which can efficiently calculate the interactions between far groups. It reduces the complexity of interactions between two far groups, such as group 𝑖 and group 𝑗, from 𝑂(𝑁_𝑖𝑁_𝑗) to 𝑂(𝑁_𝑖+𝑁_𝑗), where 𝑁_𝑖 and 𝑁_𝑗 are the numbers of the dipoles in groups 𝑖 and 𝑗, respectively.

In this work, we extended the FDM to analyze the scattering of dielectric and magnetic materials by solving the volume integral equation (VIE). These materials not only contain the conventional materials but also contain the left-handed materials (LHMs) and LHM-RHM (right-handed material) composite materials. We find that it usually converges slowly when solving the targets involving LHMs.

2. Volumetric Integral Equations for Electromagnetic Materials

Consider an arbitrarily shaped 3D scattering electromagnetic object illuminated by a plane wave incident fields (𝐄^𝑖,𝐇^𝑖), the permittivity and permeability of the object are characterized by 𝜀(𝐫) and 𝜇(𝐫). The background space surrounding the dielectric targets with region 𝑉 is free space with 𝜀₀=𝜇₀=1. According to the equivalence principle and boundary conditions, the total electromagnetic field (𝐄,𝐇) can be described as follows: 𝐄=𝐄^𝑖+𝐄^𝑠𝐉^𝑒_𝑣+𝐄^𝑠𝐉^𝑚_𝑣,𝐇=𝐇^𝑖+𝐇^𝑠𝐉^𝑚_𝑣+𝐇^𝑠𝐉^𝑒_𝑣, (1) where 𝐄^𝑠and𝐇^𝑠 denote the scattering electromagnetic fields. 𝐉^𝑒_𝑣(𝐫) and 𝐉^𝑚_𝑣(𝐫) denote equivalent volumetric electric and magnetic currents, respectively. 𝐄^𝑠(𝐉^𝑒_𝑣), 𝐄^𝑠(𝐉^𝑚_𝑣), 𝐇^𝑠(𝐉^𝑚_𝑣), and 𝐇^𝑠(𝐉^𝑒_𝑣) are the scattered electric field and magnetic field produced by the equivalent volume currents 𝐉^𝑒_𝑣 and 𝐉^𝑚_𝑣 and can be found in [10]. 𝐉^𝑒_𝑣(𝐫) and 𝐉^𝑚_𝑣(𝐫) can be expressed as 𝐉^𝑒_𝑣(𝐫)=𝑗𝜔𝜅^𝑒(𝐫)(𝐫),𝐉⋅𝐃_𝑚𝑣(𝐫)=𝑗𝜔𝜅^𝑚(𝐫)⋅𝐁(𝐫), (2) where 𝜅^𝑒(𝐫) and 𝜅^𝑚(𝐫) are the contrast ratios, and 𝜅^𝑒(𝐫)=1−𝜀_𝑟⁻¹(𝐫),𝜅^𝑚(𝐫)=1−𝜇_𝑟⁻¹(𝐫). (3) The region 𝑉 is discretized into small tetrahedral elements. Then, the unknown volume currents 𝐉^𝑒_𝑣 and 𝐉^𝑚_𝑣 can be expanded by a set of SWG basis functions [11] as 𝐉^𝑒_𝑣(𝐫)=^𝑁𝑣_𝑛=1𝐼^𝑒_𝑛𝜅^𝑒_𝑛𝐟_𝑛(𝐫),𝐉^𝑚_𝑣(𝐫)=𝜂^𝑁𝑣_𝑛=1𝐼^𝑚_𝑛𝜅^𝑚_𝑛𝐟_𝑛(𝐫), (4) where 𝐼^𝑒_𝑛 and 𝐼^𝑚_𝑛 are the unknown expansion coefficients for the electric and magnetic currents, respectively. 𝜅^𝑒_𝑛 and 𝜅^𝑚_𝑛 are the contrast ratios in the 𝑛th SWG basis function. 𝐟_𝑛 denotes the 𝑛th volume basis function defined in two adjoining tetrahedrons in the volume meshes associated with common face 𝑛. 𝑁_𝑣 is the number of faces of the tetrahedral meshes. 𝜂=𝜇₀/𝜀₀ is the impedance of the free space. Using the Galerkin’s method and testing the resultant integral equations with a set of testing functions, as a result, the integral equations are converted into a matrix equation, and can be written in the matrix form as 𝐙^𝐸𝐸𝐙^𝐸𝑀𝐙^𝑀𝐸𝐙^𝑀𝑀𝐈^𝐸𝐈^𝑀=𝐕^𝐸𝐕^𝑀. (5) The impedance elements of the submatrices 𝐙^𝐸𝐸, 𝐙^𝐸𝑀, 𝐙^𝑀𝐸, and 𝐙^𝑀𝑀 are represented as 𝑍^𝑒𝑒_𝑚𝑛=𝐟_𝑚,𝐄−𝐄^𝑠𝐉^𝑒_𝑣𝑛,𝑍^𝑒𝑚_𝑚𝑛=𝐟_𝑚,−𝐄^𝑠𝐉^𝑚_𝑣𝑛,𝑍^𝑚𝑒_𝑚𝑛=𝜂𝐟_𝑚,−𝐇^𝑠𝐉^𝑒_𝑣𝑛,𝑍^𝑚𝑚_𝑚𝑛=𝜂𝐟_𝑚,𝐇−𝐇^𝑠𝐉^𝑚_𝑣𝑛, (6) where 𝐉^𝑒_𝑣𝑛 and 𝐉^𝑚_𝑣𝑛 denote equivalent volume electric and magnetic currents corresponding to the 𝑛th SWG element. The elements in voltage vectors 𝐕^𝐸 and 𝐕^𝑀 are 𝑉^𝑒_𝑚=𝐟_𝑚,𝐄^𝑖,𝑉^𝑚_𝑚=𝜂𝐟_𝑚,𝐇^𝑖. (7)

3. Fast Dipole Method for Electromagnetic Materials

The FDM is an efficient way to solve the VIE, which is based on the EDM. In the EDM, each SWG element can be approximated as an infinitely small dipole with an equivalent moment. Referring to [10], the 𝑛th volume electric dipole moment 𝐦^𝑒_𝑛 and magnetic dipole moment 𝐦^𝑚_𝑛 can be represented as 𝐦^𝑒_𝑛=𝑎_𝑛𝜅_𝑛^𝑒−𝐫_𝑛^𝑐−−𝐫^𝑐_𝑛𝑠+𝑎_𝑛𝜅_𝑛^𝑒+𝐫^𝑐_𝑛𝑠−𝐫_𝑛^𝑐+,𝐦^𝑚_𝑛=𝑎_𝑛𝜅_𝑛^𝑚−𝐫_𝑛^𝑐−−𝐫^𝑐_𝑛𝑠+𝑎_𝑛𝜅_𝑛^𝑚+𝐫^𝑐_𝑛𝑠−𝐫_𝑛^𝑐+, (8) where 𝐫_𝑛^𝑐± is the position vector of the centroid of   𝑇^±_𝑛 and 𝐫^𝑐_𝑛𝑠 is the position vector of the centroid of the common face of the 𝑇^±_𝑛. 𝑎_𝑛 is the area of the nth common face associated with 𝑇^±_𝑛. 𝜅_𝑛^𝑒± and 𝜅_𝑛^𝑚± stand for the contrast ratio of  𝑇^±_𝑛. The impedance matrix elements in (6) can be calculated by the EDM: 𝑍^𝑢𝑢_𝑚𝑛=𝑗𝑘𝜂𝐦^′_𝑚⋅𝐆(𝐑)⋅𝐦^𝑢_𝑛𝑍,𝑢=𝑒,𝑚,_{𝑚𝑒𝑚𝑛}=𝑗𝑘𝜂𝐦^′_𝑚⋅𝐆(𝐑)×𝐦^𝑒_𝑛,𝑍^𝑒𝑚_𝑚𝑛=−𝑗𝑘𝜂𝐦^′_𝑚⋅𝐆(𝐑)×𝐦^𝑚_𝑛, (9) where 𝐆(𝐑)=𝑒^{−𝑗𝑘𝑅}4𝜋𝑅𝐈11++1𝑗𝑘𝑅𝑗𝑘𝑅²−𝐑𝐑31++3𝑗𝑘𝑅𝑗𝑘𝑅²,𝐆(𝐑)=𝑒^{−𝑗𝑘𝑅}14𝜋𝑅1+𝐦𝑗𝑘𝑅𝐑,_′𝑚=𝑎_𝑚𝐫_𝑚^𝑐−−𝐫_𝑚^𝑐+. (10)(11)(12) In (10)–(12), 𝐑=𝐫_𝑚−𝐫_𝑛, 𝑅=|𝐑|, and 𝐑=𝐑/𝑅. In addition, 𝐫_𝑛=(𝐫_𝑛^𝑐−+𝐫_𝑛^𝑐+)/2 and 𝐫_𝑚=(𝐫_𝑚^𝑐−+𝐫_𝑚^𝑐+)/2. 𝐫_𝑛^𝑐± (𝐫_𝑚^𝑐±) is the position vector of the centroid of 𝑇^±_𝑛 (𝑇^±_𝑚). According to [8–10], the EDM can be applied when the distance between the source and the field dipole is greater than 0.15𝜆_𝑔 (𝜆_𝑔 is the wavelength in dielectric). It should be noted that 𝜆_𝑔 is the maximum wavelength in dielectric when the medium is inhomogeneous.

In [12, 13], the FDM is developed to efficiently calculate the interactions between far groups. In order to describe how the FDM works for dielectric and magnetic materials, we consider two dipoles m and n, which belong to groups j and i, respectively and suppose the two groups are a far group pair. The distance between the two dipoles can be written as 𝐑=𝐫_𝑚−𝐫_𝑛=𝐫_𝑗𝑖+𝐫_𝑚𝑗+𝐫_𝑛𝑖, where 𝐫_𝑗𝑖=𝐫_𝑜𝑗−𝐫_𝑜𝑖, 𝐫_𝑚𝑗=𝐫_𝑚−𝐫_𝑜𝑗, and 𝐫_𝑛𝑖=𝐫_𝑛−𝐫_𝑜𝑖, 𝐫_𝑜𝑗, and 𝐫_𝑜𝑖 is the center vectors of the groups. According to [13], 𝑅^𝛼 in the formulae (10) and (11) can be approximated using Taylor series expansion: 𝑅^𝛼=𝐫_𝑗𝑖+𝐫_𝑚𝑗+𝐫_𝑛𝑖^𝛼=𝑟^𝛼_𝑗𝑖2𝐫̂1+^𝑗𝑖⋅𝐫_𝑚𝑗𝑟_𝑗𝑖+𝑟²_𝑚𝑗𝑟²_𝑗𝑖+2𝐫̂_𝑖𝑗⋅𝐫_𝑛𝑖𝑟_𝑖𝑗+𝑟²_𝑛𝑖𝑟²_𝑖𝑗−2𝐫_𝑚𝑗⋅𝐫_𝑛𝑖𝑟²_𝑗𝑖^𝛼/2.≈𝑅_𝑚^(𝛼)+𝑅_𝑛^(𝛼), (13) where 𝑅_𝑚^(𝛼)=𝑟^𝛼_𝑗𝑖12𝐫̂+𝛼^𝑗𝑖⋅𝐫_𝑚𝑗𝑟_𝑗𝑖+𝑟²_𝑚𝑗+(𝛼−2)𝐫̂_𝑗𝑖⋅𝐫_𝑚𝑗²2𝑟²_𝑗𝑖𝑅_𝑛^(𝛼)=𝑟^𝛼_𝑖𝑗12𝐫̂+𝛼^𝑖𝑗⋅𝐫_𝑛𝑖𝑟_𝑖𝑗+𝑟²_𝑛𝑖+(𝛼−2)𝐫̂_𝑖𝑗⋅𝐫_𝑛𝑖²2𝑟²_𝑖𝑗. (14) And 𝐑𝐑 can be approximated as 𝐑𝐑≈𝐓_𝑚+𝐓_𝑛, (15) where 𝐓_𝑚=1𝑟²_𝑗𝑖12𝐫_𝑗𝑖𝐫_𝑗𝑖+𝐬_𝑚𝐫_𝑗𝑖+𝐫_𝑗𝑖𝐬_𝑚𝐓_𝑛=1𝑟²_𝑖𝑗12𝐫_𝑖𝑗𝐫_𝑖𝑗+𝐬_𝑛𝐫_𝑖𝑗+𝐫_𝑖𝑗𝐬_𝑛𝐬_𝑚=𝐫_𝑚𝑗−𝐫_𝑚𝑗⋅𝐫̂_𝑗𝑖𝐫̂_𝑗𝑖𝐬_𝑛=𝐫_𝑛𝑖−𝐫_𝑛𝑖⋅𝐫̂_𝑖𝑗𝐫̂_𝑖𝑗. (16) Then substituting (13) and (15) into (9), the formulae of FDM for electromagnetic materials can be obtained as follows: 𝑍^𝑢𝑢_𝑚𝑛≈𝑗𝑘𝜂𝐌4𝜋^′𝑚⋅𝐈𝐴_𝑚+𝐴_𝑛−𝐓_𝑚+𝐓_𝑛𝐵_𝑚+𝐵_𝑛⋅𝐌^𝑢_𝑛=𝑗𝑘𝜂𝐴4𝜋^𝑚𝐌^′_𝑚−𝐵_𝑚𝐌^′_𝑚⋅𝐓_𝑚⋅𝐌^𝑢_𝑛+𝐌^′_𝑚⋅𝐴_𝑛𝐌^𝑢_𝑛−𝐵_𝑛𝐓_𝑛⋅𝐌^𝑢_𝑛−𝐌^′_𝑚⋅𝐓_𝑚⋅𝐵_𝑛𝐌^𝑢_𝑛−𝐵_𝑚𝐌^′_𝑚⋅𝐓_𝑛⋅𝐌^𝑢_𝑛,𝑍𝑢=𝑒,𝑚,_{𝑚𝑒𝑚𝑛}≈𝑗𝑘𝜂𝐌4𝜋^′𝑚⋅𝐶_𝑚+𝐶_𝑛𝐫_𝑚𝑝−𝐫_𝑛𝑝×𝐌^𝑒_𝑛=𝑗𝑘𝜂𝐶4𝜋^𝑚𝐌^′_𝑚×𝐫_𝑚𝑝⋅𝐌^𝑒_𝑛+𝐌^′_𝑚×𝐫_𝑚𝑝⋅𝐶_𝑛𝐌^𝑒_𝑛,+𝐶_𝑚𝐌^′_𝑚⋅𝐌^𝑒_𝑛×𝐫_𝑛𝑝+𝐌^′_𝑚⋅𝐶_𝑛𝐌^𝑒_𝑛×𝐫_𝑛𝑝,𝑍^𝑒𝑚_𝑚𝑛≈−𝑗𝑘𝜂𝐌4𝜋^′𝑚⋅𝐶_𝑚+𝐶_𝑛𝐫_𝑚𝑝−𝐫_𝑛𝑝×𝐌^𝑚_𝑛=−𝑗𝑘𝜂𝐶4𝜋^𝑚𝐌^′_𝑚×𝐫_𝑚𝑝⋅𝐌^𝑚_𝑛+𝐌^′_𝑚×𝐫_𝑚𝑝⋅𝐶_𝑛𝐌^𝑚_𝑛,+𝐶_𝑚𝐌^′_𝑚⋅𝐌^𝑚_𝑛×𝐫_𝑛𝑝+𝐌^′_𝑚⋅𝐶_𝑛𝐌^𝑚_𝑛×𝐫_𝑛𝑝, (17) where 𝐫_𝑚𝑝=𝐫_𝑚−𝐫_𝑝, 𝐫_𝑛𝑝=𝐫_𝑛−𝐫_𝑝, 𝐫_𝑝=(𝐫_𝑜𝑗+𝐫_𝑜𝑖)/2, 𝐴_𝑢=𝑅_𝑢⁽⁻¹⁾+𝑗𝑘⁻¹𝑅_𝑢⁽⁻²⁾+𝑗𝑘⁻²𝑅_𝑢⁽⁻³⁾,𝐵_𝑢=𝑅_𝑢⁽⁻¹⁾+3𝑗𝑘⁻¹𝑅_𝑢⁽⁻²⁾+3𝑗𝑘⁻²𝑅_𝑢⁽⁻³⁾,𝐶_𝑢=𝑅_𝑢⁽⁻²⁾+𝑗𝑘⁻¹𝑅_𝑢⁽⁻³⁾,𝐌^′_𝑚=𝐦^′_𝑚𝑒^{−𝑗𝑘𝑅𝑚(1)},𝐌^𝑢_𝑛=𝐦^𝑢_𝑛𝑒^{−𝑗𝑘𝑅𝑛(1)},𝑢=𝑒,𝑚. (18) In the following part, we will present how to use the FDM to calculate the interactions among the equivalent electric or magnetic dipoles in group 𝑖 and group 𝑗. The effect of all the electric and magnetic dipoles in group 𝑖 on the 𝑚th electric dipoles in group 𝑗 can be expressed as _𝑛∈𝑖𝑍^𝑒𝑒_𝑚𝑛𝐼^𝑒_𝑛+𝑍^𝑒𝑚_𝑚𝑛𝐼^𝑚_𝑛≈𝑗𝑘𝜂𝐴4𝜋^𝑚𝐌^′_𝑚−𝐵_𝑚𝐌^′_𝑚⋅𝐓_𝑚⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐌^𝑒_𝑛+𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐴_𝑛𝐌^𝑒_𝑛−𝐵_𝑛𝐓_𝑛⋅𝐌^𝑒_𝑛−𝐌^′_𝑚⋅𝐓_𝑚⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐵_𝑛𝐌^𝑒_𝑛−𝐵_𝑚𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐓_𝑛⋅𝐌^𝑒_𝑛−𝐶_𝑚𝐌^′_𝑚×𝐫_𝑚𝑝⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐌^𝑚_𝑛−𝐌^′_𝑚×𝐫_𝑚𝑝⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐶_𝑛𝐌^𝑚_𝑛−𝐶_𝑚𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐌^𝑚_𝑛×𝐫_𝑛𝑝−𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐶_𝑛𝐌^𝑚_𝑛×𝐫_𝑛𝑝. (19) In a similar way, the effect of all the electric and magnetic dipoles in group 𝑖 on the 𝑚th magnetic dipoles in group 𝑗 can be expressed as _𝑛∈𝑖𝑍^𝑚𝑚_𝑚𝑛𝐼^𝑚_𝑛+𝑍^𝑚𝑒_𝑚𝑛𝐼^𝑒_𝑛≈𝑗𝑘𝜂𝐴4𝜋^𝑚𝐌^′_𝑚−𝐵_𝑚𝐌^′_𝑚⋅𝐓_𝑚⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐌^𝑚_𝑛+𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐴_𝑛𝐌^𝑚_𝑛−𝐵_𝑛𝐓_𝑛⋅𝐌^𝑚_𝑛−𝐌^′_𝑚⋅𝐓_𝑚⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐵_𝑛𝐌^𝑚_𝑛−𝐵_𝑚𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑚_𝑛𝐓_𝑛⋅𝐌^𝑚_𝑛+𝐶_𝑚𝐌^′_𝑚×𝐫_𝑚𝑝⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐌^𝑒_𝑛+𝐌^′_𝑚×𝐫_𝑚𝑝⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐶_𝑛𝐌^𝑒_𝑛+𝐶_𝑚𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐌^𝑒_𝑛×𝐫_𝑛𝑝+𝐌^′_𝑚⋅_𝑛∈𝑖𝐼^𝑒_𝑛𝐶_𝑛𝐌^𝑒_𝑛×𝐫_𝑛𝑝. (20) It can be found that each term in (19) and (20) achieves the separation of 𝑚 and 𝑛. Therefore, the summation results ∑_𝑛∈𝑖(⋅) in (19) and (20) are independent of 𝑚 and can be reused by each dipole in group 𝑗. These summation results ∑_𝑛∈𝑖(⋅) only need to be computed once. If we suppose that groups 𝑖 and 𝑗 contain 𝑁_𝑖 and 𝑁_𝑗 electric or magnetic dipoles, respectively, then the complexity of interactions between the two far groups is reduced from 𝑂(𝑁_𝑖𝑁_𝑗) to 𝑂(𝑁_𝑖+𝑁_𝑗).

4. Numerical Results and Discussion

All the simulations are performed on a personal computer with the Intel Pentium Dual-Core CPU E2200 with 2.0 GHz (only one core was used) and 2.0 GB RAM. The GMRES iterative solver is employed to obtain an identical residual error, 0.001.

First, we consider the scattering problem of a thin plate with 1.0m×1.0m×0.05m and 𝜀_𝑟=2,𝜇_𝑟=2. The plate is divided into 2665 volumetric cells with an average edge length of 0.05 m, and the total number of unknowns is 12420. Totally, 256 nonempty groups with the size of 0.2 are obtained. The bistatic RCS in 𝜃𝜃 polarization is compared with the EDM shown in Figure 1. Both the CPU time and memory requirements are shown in Table 1. We can see from Table 1, it costs 185 seconds and 312 MB of memory using the FDM, whereas 674 seconds and 1239 MB of memory are needed using the EDM. That is to say the FDM yields reductions of 72.6% of the total computation time and 74.8% of the memory requirements using EDM.

Table 1 

Comparison of CPU time, memory cost, and iterations of RCS solutions for EDM and FDM.

Figure 1 

Bistatic RCSs (𝜃𝜃 polarization) of a thin plate with 1.0m×1.0m×0.05m and 𝜀𝑟=2,𝜇𝑟=2 illuminated by a plane wave with the incident direction of (𝜃,𝜙)=(0∘,0∘) at 0.3 GHz.

Furthermore, we consider the scattering problem of a 10 cm sphere with 𝜀_𝑟=𝜇_𝑟=−1−𝑗0.001 as an example. To validate our FDM method, the calculated frequencies span the range from 0.6 GHz to 2.22 GHz. Figure 2 gives the bistatic RCSs for E-plane calculated by FDM. The results are in agreement with those from [14] (the RCSs at frequencies of 0.6 GHz and 2.22 GHz in [14] have been shown in Figure 2 as references). We compared the CPU time of the FDM with that of EDM when the object is at frequency of 1.14 GHz in Table 1. At this frequency, the sphere is divided into 1624 volumetric cells with an average edge length of 0.03 m, and the total number of unknowns is 6822. All the unknowns are divided into 1324 nonempty groups and the size of each group is 0.2. We can see from Table 1 that it costs 376 seconds and 173 MB of memory using the FDM, whereas 1753 seconds and 382 MB of memory are needed using the EDM.

Figure 2 

Bistatic RCS (E-plane) of an LHM sphere with radius 10 cm and 𝜀𝑟=𝜇𝑟=−1−𝑗0.001 operating at frequencies of 0.6 GHz, 1.14 GHz, 1.68 GHz, and 2.22 GHz. The arrow indicates the incidence direction of a plane wave.

Finally, we consider an LHM-RHM composite array. The dimension and parameters of the array are shown in Figure 3. The geometry is discretized into 2413 tetrahedrons using 0.04 m average edge length, thus resulting in total 10340 unknowns. All the unknowns are divided into 1342 nonempty groups and the size of each group is 0.2. 𝜃𝜃 polarized RCS is plotted, as shown in Figure 3. It can be seen from Table 1 that it costs 435 seconds and 272 MB of memory using the FDM, whereas 4204 seconds and 863 MB of memory using the EDM are needed. That is to say the FDM can save much more time and memory requirements than the EDM. From Table 1, we also can find that the method converges much more slowly when it is used to analyze the scattering involving the LHM than the RHM. This is because LHM structures usually present numerical resonances that inhibit quick convergence of iterations [15].

Figure 3 

Bistatic RCS of a sphere array with 𝜀𝑟1=−1, 𝜇𝑟1=−1, 𝜀𝑟2=2, 𝜇𝑟2=1, 𝜀𝑟3=−1, 𝜇𝑟3=−1, 𝜀𝑟4=2, and 𝜇𝑟4=1 and performed at 0.6 GHz excited by a plane wave with the incident direction of (𝜃,𝜙)=(0∘,0∘).

5. Conclusion

In this work, the formulations of FDM for analyzing the electromagnetic scattering from electromagnetic materials are given, which saves CPU time and memory requirements compared with the conventional EDM. Several dielectric and magnetic targets include RHM objects and LHM objects, and LHM-RHM composite objects are calculated to demonstrate the merit of this method. Furthermore, the numerical results show that the convergences may become much slower if the LHM is included in the targets. In further work, we will try to construct an efficient preconditioner to alleviate the disadvantage brought by the LHM.

Acknowledgments

This work was supported by the National Nature Science Foundation of China under Grant no. 61071019, the Joint Funding Projects of the Aerospace Science Foundation Office of China no. 2008ZA52006, and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.