Volumetric MLFMA formulation for dielectric targets in the presence of a half-space

Authors


Abstract

[1] The fast multipole method (FMM) and the multilevel fast multipole algorithm (MLFMA) are extended to the analysis of volumetric electric field integral equations (VEFIE) for targets in the presence of a half-space, to calculate the electromagnetic fields scattered from dielectric scatterers situated above or within a half-space environment. The “near” interactions are handled via rigorous evaluation of the dyadic Green's functions, by using the discrete complex image technique (DCIT); the “far” interactions are calculated by considering image source elements and groups. Numerical results are presented for comparisons between the VEFIE and surface combined-field integral equation (SCFIE) MLFMA formulations.

1. Introduction

[2] The method of moments (MoM) is a powerful technique for the solution of integral equations. It is widely used in the analysis of electromagnetic scattering problems [Harrington, 1968; Peterson et al., 1998; Rao et al., 1982; Schaubert et al., 1984]. For a problem with N unknowns (number of basis functions for expansion of the unknown currents), the MoM results in a matrix equation with N2 matrix elements. The computational complexity for solving this matrix equation is O(N3), with a direct matrix solver (LU-decomposition), or O(N2) with an iterative matrix solver [Peterson et al., 1998; Sarkar and Arvas, 1985]. The computational complexity of the iterative solution is based on the O(N2) complexity to fill the matrix and for computation of the matrix-vector product. The fast multipole method (FMM) and its multilevel extension, the multilevel fast multipole algorithm (MLFMA), have been developed to reduce the memory and computation time of MoM, so that it may be applied to the analysis of scattering from electrically large targets [Rokhlin, 1983, 1990; Coifman et al., 1993; Brandt, 1991; Song and Chew, 1994, 1995; Song et al., 1997, 1998; Chew et al., 2001]. It is important to note that the FMM and its multilevel implementation are not limited to high-frequency applications; in fact the FMM was originally developed for potential theory [Rokhlin, 1983]. Since the FMM is a special case of the MLFMA, in the remainder of the text we only mention the MLFMA, for which all results are presented.

[3] Integral equations in electromagnetic scattering problems are based on either a surface or volumetric formulation (or a combination of the two). The surface-integral formulation is applicable to perfectly conducting or homogeneous dielectric targets, and is formulated using the boundary conditions on the surface of the scatterers. The volume formulation is applicable to dielectric and/or permeable targets, and for a non-magnetic target the unknowns are equivalent volume electric currents defined in terms of the target permittivity, which may be heterogeneous. Volume integral equations use a volumetric mesh, and therefore the number of unknowns is proportional to D3, where D is the principal dimension of the scatterer. As a result, the number of unknowns increases rapidly with the size of the scatterer and the volume integral equation technique is typically only applied to electrically small targets. In order to make volume integral equation techniques applicable to electrically large scatterers, the MLFMA has been exploited [Lu et al., 2000]. However, this previous research has focused on free-space scattering problems. There are many applications for which the free-space models are not appropriate. For example, there is significant interest in radar-based remote sensing of buried targets such as land mines, unexploded ordnance (UXO), drums, tunnels, etc. The MLFMA has been extended previously to the half-space problem for the case of surface electric and combined-field integral equations [Geng et al., 1999, 2000, 2001]. In this paper, we extend the MLFMA to volume integral equations, for scatterers in the vicinity of a half-space. The principal advantage of the volumetric formulation is found in its ability to handle general target heterogeneity.

[4] The main numerical challenge in addressing the half-space problem, vis-à-vis the free-space case, is found in evaluation of the half-space dyadic Green's function. The MLFMA clusters the basis and testing functions into a set of groups. Depending on the distance between these groups, the intergroup interactions are characterized as “near” or “far”, with this discussed in greater detail below. “Near” terms are handled in a manner analogous to a traditional MoM analysis, and here the “near” terms are calculated by rigorous evaluation of the dyadic half-space Green's function, using the complex-image technique (DCIT) [Michalski and Zheng, 1990; Chow et al., 1991; Yang et al., 1991; Shubair and Chow, 1993]. The “far” interactions are addressed approximately, but accurately, using image terms. The latter formulation is necessitated by the fact that the underlying relationship associated with the MLFMA is based on the free-space Green's function, augmented here via image theory.

[5] We present a detailed description of the implementation of the half-space MLFMA for a volumetric electric field integral equation (VEFIE) formulation. The volumetric formulation is compared to a half-space MLFMA analysis based on surface combined-field integral equations (SCFIE) of the first kind. The two approaches are compared using several numerical examples.

2. Theory

2.1. VEFIE and MoM Formulation

[6] The VEFIE is formulated by representing the total electric field in the region occupied by the scatterer as the sum of the scattered field and the incident field. For most cases, we can assume that the materials of both the target (scatterer) and the background medium are non-magnetic, i.e., μt = μb = μ0. For this case the VEFIE for a dielectric scatterer situated entirely above or under the interface of the half-space environment (Figure 1) may be written in vector-potential form as

equation image

for observation points within the volume of the target (equation imageV), or alternatively we may use the mixed-potential form

equation image

where the subscript or superscript b indicates the background half-space in which the scatterer is located (upper half-space b = 1, lower half-space b = 2, see Figure 1); εb = εb′ − jσb/ω and kb represent the permittivity and wavenumber of the background medium; ω is the angular frequency; μo is the permeability; and equation image represents the unit dyadic. Details on the dyadic Green's functions, equation imageAJbb and equation imageAJbb, and the scalar Green's function Kϕebb can be found in Michalski and Zheng [1990], where we use their formulation C.

Figure 1.

Geometry for source and observation groups in three-dimensional MLFMA, for half-space environment. Real image source elements and groups are introduced to account for the reflected contributions from the interface.

[7] The equivalent volume current equation image(equation image) has the following relationship with the total electric field inside the scatterer

equation image

or

equation image

then the integral equations (1) become the equation with unknown vector function equation image(equation image) or equation image(equation image)

equation image

[8] By applying the MoM to the integral equation (3), the dielectric volume occupied by the target is discretized into small tetrahedral volume cells and the unknown equivalent volume current equation image(equation image′) is expanded by a set of basis functions equation image(equation image′) = jω equation imageInκ(equation image′)equation imagen(equation image′), where In are the expansion coefficients; equation imagen (equation image′) are the three dimensional RWG tetrahedral-pair basis functions introduced in Schaubert et al. [1984] (Figure 2); and κ(equation image′) = [ε(equation image′) − εb]/εb is constant within each of the tetrahedral cells

equation image

where Tn± represent the two tetrahedrons supporting the n′th basis function equation imagen. Note that we have chosen three-dimensional RWG tetrahedral-pair basis functions, although the formulation is general and may be applied to alternative volumetric basis functions.

Figure 2.

Definition of the three dimensional RWG (Rao-Wilton-Glisson [Rao et al., 1982]) vector basis function. The basis function is defined in two neighbored tetrahedron volume cells with a shared surface.

[9] Testing the VEFIE eq. (1) with a set of weighting functions equation imagen(equation image), we obtain the MoM matrix equation Z · I = V for the unknown current coefficients In, where the N × 1 vector V = {vn}n = 1,2, …,N is the incident field tested by the weighting functions

equation image

and element znn of the N × N square impedance matrix Z is the interactions between the nth testing function equation imagen and the n′th basis function equation imagen, i.e.,

equation image

in vector potential form, or

equation image

in mixed potential form, where Vn = Tn+Tn and Vn = Tn+Tn are the supporting tetrahedrons of the nth testing function equation imagen and the n′th basis function equation imagen, respectively. Here we use the RWG tetrahedron-pair functions as the testing functions as well, i.e., equation imagen = equation imagen, and thus Galerkin testing is applied.

[10] In the MLFMA analysis, the basis and testing functions equation imagen are clustered into cubic groups of the same size (for a single-stage FMM) or with different sizes at multiple levels (for MLFLA). The interactions between the basis functions equation imagen and the testing functions equation imagen are divided into “far” terms and “near” terms according to the distances between their supporting volumes Vn and Vn. The impedance matrix is divided into two parts, Z = Znear + Zfar. The “far” term interactions are best handled via the formula (6a), while the “near” term interactions are calculated via the rigorous evaluation of the dyadic and scalar Green's functions, for which the matrix-vector product for these elements is calculated element-by-element as in the MoM. The near-term Green's function for the half-space case is evaluated via the complex image technique (DCIT), thereby avoiding direct numerical evaluation of the oscillatory Sommerfeld integrals. The impedance matrix elements representing these “near” term interactions are stored in the sparse matrix Znear. The detailed formulas and techniques for evaluation of these “near” term elements are found in Wilton et al. [1984].

[11] For the “far” term interactions, instead of explicitly calculating the elements of the matrix Zfar, we directly calculate the matrix-vector product related to these matrix elements, via the spectral method that constitutes the MLFMA.

2.2. Free Space and Half-Space Multipole Method

[12] The fundamentals of the MLFMA are based on a series representation of the free-space Green's function [Coifman et al., 1993; Abromowitz and Stegun, 1970],

equation image
equation image

where Rmm = ∣equation imagemequation imagem∣ and equation image are the distance and the unit vector linking two fixed points equation imagem and equation imagem, respectively (the center of the source and observation group); TL is the translation factor linking the fields of the two fixed points equation imagem and equation imagem; hl(2) is the second kind, l-th order Hankel function; and Pl is the l-th order Legendre function (polynomial). The addition theorem represents interactions between the source point equation image′ and the observing point equation image in three steps. The first step is the aggregation from the source point equation image′ to an aggregation center equation imagem; the second step is the translation from the source aggregation center equation imagem to a testing disaggregation center equation imagem; and the last step is the disaggregation from the disaggregation center equation imagem to the observing point equation image.

[13] If equation imagem and equation imagem are the source and testing group centers, respectively, we can perform the matrix-vector product corresponding to the basis and testing functions within these two groups (with number of elements being Nm and Nm, respectively) via the aggregation-translation-disagregation procedures

equation image

where N(m) is the set containing all the indices of the basis elements within the mth group, and the plane wave (Fourier) spectrum of the basis and weighting functions are

equation image
equation image

[14] For the half-space case, the contribution of the half-space interface is characterized by the reflection terms for both the “near” and “far” interactions [Michalski and Zheng, 1990]

equation image
equation image
equation image

where the first terms are the Green's functions in free space, which represent the direct contributions, and Δequation imageJA, Δequation imageJA and ΔKϕe are the reflection terms due to the half-space. For the “near” interactions, the reflection terms are calculated by using DCIT.

[15] For the “far” interactions, the contributions of the reflection terms are characterized by approximate image sources. Considering the fact that the matrix-vector product in MLFMA is evaluated by summing up the propagation (or translation) of each plane wave components, we will use the far-field reflection coefficient approximation to represent the reflection contribution. For each cluster of real basis elements, there is a single cluster of image basis elements. Assuming that the interface is the z = 0 plane, the position of the image source for a real source equation image(equation image′) located at equation image′ = (x′, y′, z′) is

equation image

The vector amplitude distribution of the image basis function equation imageI (equation image′) is

equation image

The contribution of these image sources is weighted by the reflection coefficients. In order to account for the polarization dependence of the plane wave reflection coefficients, we use the asymptotic reflection dyadic introduced in [Lindell, 1992]

equation image

with equation image and θ is the angle between the ray-optical reflection path and the unit vector equation image normal to the interface (Figure 1). Note that the far interactions typically involve basis/testing functions separated by at least a wavelength, making the asymptotic reflection coefficient an accurate approximation [Geng et al., 1999, 2000, 2001].

[16] By introducing the image source elements and image source groups, the extension of the free-space MLFMA to the half-space case is straightforward. We include an additional set of image sources, and calculate the contribution of these image sources as in the treatment of the original real sources (Figure 1). Geng et al. [1999, 2000, 2001] have validated the effectiveness and accuracy of this approximate image technique for MLFMA analysis of surface integral equations.

3. Implementation of MLFMA for Half-Space VEFIE

3.1. Calculation/Storage of the Fourier Spectra of the Basis and Testing Function

[17] Because we apply Galerkin testing to the MoM formulation, the basis functions equation imagen and the testing functions equation imagen are the same 3D-RWG vector functions equation imagen, and therefore we may exploit this property to save computation time and memory for analysis of their Fourier spectra equation image(equation image) and equation image(equation image). From (9) we find that because of the material parameter κ(equation image) in the integrand, the Fourier spectra of the basis functions and the weighting functions are no longer conjugates of one another. Direct calculation and storage of these Fourier spectra requires independent processing for the basis and testing functions.

[18] Considering the fact that the material parameter κ(equation image) is constant within each of the tetrahedron cells, we can divide the volume integration domain Vn = Tn+Tn of the basis or testing function equation imagen into two tetrahedrons Tn+ and Tn. Then the material parameter κ(equation image) inside each tetrahedron is a constant and may be extracted. According to the definition of 3D-RWG vector functions (Figure 2)

equation image

where an is the area of the surface shared by the two dual tetrahedrons, Vn± are the volumes of the two tetrahedrons Tn±, equation imagen+ = equation imageequation imagen+ is the vector from the vertex of Tn+ opposite to the common surface to the observing point, equation imagen = equation imagenequation image is the vector from the observing point to the vertex of Tn opposite to the common surface. These terms constitute the local coordinates of the observation point.

[19] We may define the Fourier spectra of the tetrahedron cells

equation image

where the subscript t, i represents the ith vertex or surface of the tth tetrahedron cell. If the two tetrahedron cells supporting the nth basis functions are Tt1 and Tt2, respectively. The shared surface of these two tetrahedrons are the i1th surface of Tt1 and the i2th surface of Tt2, then the Fourier spectra of the nth basis function and the nth testing function are

equation image

and

equation image

If we represent the direction equation image by the elevation angle θ and the azimuth angle φ, i.e.,

equation image

then (15) becomes

equation image

and

equation image

and the Fourier spectra of the image source element equation imagenI is

equation image

that is,

equation image

It is clear that calculating and storing the Fourier spectra of the tetrahedron cells saves computation and memory vis-à-vis direct calculation and storage of the basis, image basis and testing functions. If we further make use of the shape and orientation similarities of the volume tetrahedron cells, much more computation and memory may be saved for the Fourier spectra.

[20] Unlike the 2D-RWG basis functions (used in surface formulations), the Fourier transform of 3D-RWG basis functions has no analytic expression. They have to be calculated via numerical integration. If the mesh density is high enough (usually less than λ/8 ∼ λ/6), then the integrands of (14) are smooth varying functions. Gaussian-quadrature can ensure sufficient integration accuracy.

3.2. Calculation/Storage of the Translation Operators for the Far Interactions

[21] In the MLFMA, the translation operators represent the far-term interactions between groups. The calculation and storage of translation operators must cover all the possible far interaction group combinations. Direct calculation and storage of the translation operators corresponding to all intercluster combinations leads to unnecessary computation and memory requirements. From (7b), we find that the translation operator TL (kRmm, equation image · equation imagemm) for a plane wave component equation image is decided by the intergroup distance Rmm and the angle between equation image and equation imagemm. Let α = cos−1 (equation image · equation imagemm), i.e., TL (kRmm, equation image · equation imagemm) = TL (kRmm, α) then we just need to calculate and save the translation operators for all the unique distances with a set of sampling of 0 ≤ α ≤ π. For the far-term interactions, the distances between the groups are always greater than the group diameters, and the translation operator as a function of angle α is smoothly varying (Figure 3). As a result, we just need to calculate the translation operators for a small number of sampling of α, and the translation operator with arbitrary angle α is evaluated by simple interpolation of these sampled values. Similar issues to those reported in Figure 3 have been discussed by Song and Chew [2001].

Figure 3.

The properties of the translation operators for a fixed intergroup distance as function of the angle α = cos−1 (equation image · equation imagemm), where the unit of the angle is degrees. When the distance between the groups is bigger than the group diameters, i.e., kR > L, the translation operators varies smoothly with the change of angle α. (a) L = 20, kR = 50; (b) L = 5, kR = 10.

4. Comparison With SCFIE and Numerical Examples

[22] Both the volume electric-field integral equation (VEFIE) and surface combined-field integral equation (SCFIE) may be applied to calculate the scattered fields from homogeneous or discretely inhomogeneous dielectric targets. A discretely heterogeneous dielectric target may be decomposed into a finite set of homogeneous components. A complete discussion of the SCFIE for dielectric scatterers may be found in He et al. [2000] and He [2001]. Table 1 provides a comparison between the VEFIE and SCFIE for scattering from a single homogeneous dielectric target.

Table 1. Comparison Between VEFIE and SCFIEa
ItemsVEFIESCFIE
  • a

    Note that one could use a second-kind SCFIE formulation, although here we have used a first-kind integral equation. In Table 1, D represents the principal dimension of the scatterer, and λ is the wavelength.

Mesh type and number of mesh cellsVolume mesh results in number of tetrahedron mesh cells Nt ∝ (D/λ)3Surface mesh results in number of triangle mesh cells Ns ∝ (D/λ)2
Number of unknownsOnly electric field (or current) is the unknown vector function, number of unknowns NNt/2 for 3D-RWG basis functionsBoth electric and magnetic field (or currents) are the unknown vector functions, number of unknowns N = NJ + NM ∼ 3Ns for 2D-RWG basis functions
Integral equation typeVEFIE is second-kind integral equation with lower singularitySCFIE is first-kind integral equation with higher singularity
Iteration convergenceBetter spectral properties leads to faster iteration convergencePoorer spectral properties leads to slower iteration convergence
Applicable scatterersApplicable to scatterers with both homogenous and inhomogeneous material, for both continuous and discontinuous structuresApplicable only to scatterers with homogenous material or discrete inhomogeneity

[23] In the following, we present several numerical examples. The scattered field associated with the VEFIE are computed via the half-space MLFMA developed here; the scattered field associated with SCFIE are computed via the half-space MLFMA developed in Geng et al. [1999, 2000, 2001]. In all computations, we have assumed linearly polarized plane-wave incidence. The scattered field is evaluated in the far-zone, via an asymptotic approximation to the half-space dyadic Green's function.

4.1. One Homogeneous Cylinder Above the Interface

[24] We first consider a cylinder above the interface of the half-space environment. The diameter and height of the cylinder are 0.25 m and 1.00 m, respectively, and the bottom of the cylinder is 0.5 m above the interface. The cylinder axis is normal to the interface plane. The lossless dielectric constants of the scatterer and the lower half-space environment are εr = 4.0 and εr = 2.0, respectively. The frequency of the incident wave is 600 MHz. The incident plane-wave angles are θi = 45° and ϕi = 0°, respectively, where θi is the elevation angle leaving from the equation image direction (normal to the interface), and φi is the azimuth angle leaving from the equation image direction. The scattering angles are θs = 45° and ϕs = 0° ∼ 360°. Figure 4 presents the bistatic scattered fields for both VV and HH polarization. Excellent agreement is observed between the results of VEFIE and SCFIE. Table 2 shows the mesh parameters and convergence performance of these two techniques (with a preconditioner).

Figure 4.

The bistatic scattering from a cylinder above the interface of half-space environment: V-polarization; H-polarization.

Table 2. Comparison of VEFIE and SCFIE for Scattering From a Cylinder Above the Half-Space Interface
 VEFIESCFIE
Number of unknowns36533960
Zfar percentage52%55%
CG steps38(V-pol) 35(H-pol)110(V-pol) 114(H-pol)

[25] Because the size of this cylinder is not very large compared with the wavelength, the difference between the numbers of mesh cells of VEFIE modeling and SCFIE modeling is not significant. The number of unknowns of the SCFIE (including both electric and magnetic currents) is actually larger than that of the VEFIE (only including the electric current). While the scattered field agrees very well, the convergence properties of VEFIE is much better than SCFIE.

4.2. Rough Surface

[26] The second example is the scattering from a sinusoidal rough surface. The rough surface is slightly above the interface of the half-space environment (5 cm, which is 1/20λ). The dimension of this surface is LX × LY = 6 m × 4 m. The maximum height (amplitude) of the roughness is 0.25 m. The incident frequency is 300 MHz, and the dielectric properties of the lossless rough-surface target and lower half-space are εr = 4.0. Figure 5 presents the bistatic scattered field. The incident and scattering angles are θi = 45° and ϕi = 90°, θs = 45° and ϕs = 0° ∼ 360°, respectively. The comparison between the VEFIE and SCFIE shows good agreement. The difference between the H-polarization results is bigger than that of the V-polarization, and the differences are attributed to differences in the explicit target realized by the surface and volumetric meshes (these differences are diminished as the mesh is refined, but at significant computation cost). Table 3 presents the comparisons between the mesh parameters and convergent properties of these two models. All results were computed on a Pentium IV PC, with 2 GB of memory and 1.5 GHz clock speed.

Figure 5.

The bistatic scattering from a sinusoidal rough surface. The dimension of this surface is LX × LY = 6 m × 4 m. (a) Geometrical description of the problem; (b) V-polarization; (c) H-polarization.

Table 3. Comparison of VEFIE and SCFIE for Scattering From a Rough Surface
 VEFIESCFIE
Number of mesh cells1399010094
Number of unknowns251412 × 15141 = 30282 (J + M)
MLFMA level54
Zfar percentage80.1%79.2%
Znear memory (MB)19161280
Total memory (MB)20352037
CG iteration steps113(V-pol) 52(H-pol)167(V-pol) 206(H-pol)
Total CPU time (min)23004860

[27] Because this rough surface is a very thin and extended surface, the number of volume mesh cells for VEFIE modeling is not much bigger than that of the surface mesh cells for SCFIE. In fact, the number of unknowns for the VEFIE is actually smaller than that of SCFIE. Due to the shape and geometrical distribution of the rough surface scatterer, the convergence speed of SCFIE modeling is very slow. For SCFIE modeling, the number of plane wave components is decided by the dielectric coefficients of both the scatterer material and background medium, while the number of plane wave components for VEFIE modeling is decided by the dielectric coefficient of the background medium. If the scatterer is made of high-dielectric material compared with the background medium, the number of plane waves for SCFIE will be much larger than that for VEFIE. A larger number of plane waves leads to more operations and longer CPU time for each of the matrix-vector product operations.

4.3. A Buried Inhomogeneous Cylinder

[28] This example is for scattering from inhomogeneous cylinder buried under a lossy interface. The diameter and height of the cylinder are 0.25 m and 1.00 m, respectively, and the top of the target is buried 0.5 m below the interface (the target axis is perpendicular to the interface). The lower half-space is model soil, characterized by complex dielectric constant εr = 5.0 − j0.2 and conductivity σ = 0.01 S/m. The axis of the cylinder is normal to the interface plane. The frequency of the incident wave is 200 MHz. The incident angles are θi = 45° and ϕi = 0°, respectively, and the scattering angles are θs = 45° and ϕs = 0° ∼ 360°. We calculate the bistatic scattered field with three different dielectric-constant distributions inside the cylinder. The first is with a homogenous average dielectric constant and conductivity are εrt = 7.0 − j0.2 and σt = 0.01 S/m, respectively; the second is with a stepped increase in the dielectric constant distribution, from εrt = 5.0 − j0.2 at the top to εrt = 9.0 − j0.2 at the bottom of the target, and the number of piecewise constant layers is five; the third is the same as the second except that the number of layers is twenty, which represents a smoother variation of the target's dielectric properties. The conductivity of the layered target is fixed as σt = 0.01 S/m. Figure 6 shows the bistatic scattered fields for both VV and HH polarization. We see excellent agreement between the results of VEFIE and SCFIE, where the SCFIE are for the average homogeneous target. Table 4 shows the mesh parameters and convergence performance of these two techniques.

Figure 6.

The bistatic scattering from buried inhomogeneous cylinder. Different approximations are made to the target heterogeneity: the target is treated as homogeneous, using the average dielectric constant, and the target is decomposed in terms of discrete dielectric steps (here with 5 and 20 layers). (a) V-polarization; (b) H-polarization.

Table 4. Comparison of VEFIE and SCFIE for Scattering From a Buried Cylinder
 VEFIESCFIE
Number of mesh cells74771890
Number of unknowns140092 × 2835 = 5670 (J + M)
MLFMA level43
Zfar percentage43.3%55.8%
Znear memory (MB)1699136
Total memory (MB)1830226
CG iteration steps49(V-pol) 52(H-pol)131(V-pol) 128(H-pol)
Total CPU time (min)1755215

[29] From Figure 6 we see that the difference between cylinder with homogenous average dielectric coefficient and that with the stepped dielectric coefficient distribution is significant; while the difference between the two different step-increased dielectric coefficient distribution is relatively small. We also compare the memory requirement of this cylinder for VEFIE modeling and SCFIE modeling. Because the size of the cylinder is large compared with the wavelength, the number of mesh cells and number of unknowns for the surface model is much smaller than those of the volume model. VEFIE modeling requires a memory of 1830 MB, while SCFIE modeling only requires a memory 226 MB. Due to great difference of number of unknowns, the CPU time of SCFIE modeling is also much shorter than VEFIE modeling.

4.4. RAM and CPU Requirements of Volumetric MLFMA

[30] It is of interest to examine the RAM and CPU requirements of the volumetric MLFMA, for the free-space and half-space cases. Recall that all results were computed on a Pentium IV PC, with 2 GB of memory and 1.5 GHz clock speed. We consider scattering from a lossless dielectric cube in free space and above a lossy dielectric half-space. The dielectric constant of the cube is εr = 2.0 and the electrical properties of the half-space are εr = 5 − j0.2. In the half-space case the target is situated 0.2 m above the interface (from the bottom of the cube). For the example results presented here, the target is discretized using tetrahedrons with principal dimension approximately equal to λ/6, where λ represents the wavelength within the dielectric. In these results the frequency of operation is fixed at f = 600 MHz, and the dimension of the cube increases sequentially from 0.6 m to 1.0 m, manifesting a change in the number of volumetric unknowns ranging from N = 4,398 to N = 17,158. We present RAM and CPU results as a function of N.

[31] In Figure 7 we plot the RAM requirements for the volumetric MLFMA for the half-space and free-space cases. In Figure 7a we plot RAM requirements for the near terms, and in Figure 7b total RAM requirements are plotted. We observe that, by using the procedure discussed in Sec. IIIB, the RAM requirements of the far terms are infinitesimal relative to the near terms. The total RAM requirements are approximately linear in N, for the range of N considered. We also note that the total RAM requirements of the free-space and half-space MLFMA are almost identical. In Figure 7 the kinks in RAM requirements occur at transitions from i to i + 1 MLFMA levels, with increasing N.

Figure 7.

RAM requirements as a function of the number of unknowns N, for a dielectric sphere in a free-space and half-space environment. (a) RAM requirements for near terms; (b) total RAM requirements.

[32] In Figure 8 we plot the CPU requirements as a function of N, for the free-space and half-space cases. In Figure 8a are plotted CPU requirements for computation of the near interactions (performed once), and in Figure 8b the total CPU requirements are plotted (the number of CG iterations are approximately 60 for the half-space case and 55 for the free-space case, to achieve 1% accuracy for each N considered). It is observed that the near-term computations dominate for the range of N considered, with this term requiring O(N) CPU. The CPU requirements of the half-space case are approximately eight times that of the free-space case, with this attributed primarily to addressing the dyadic nature of the half-space Green's function. The results in Figure 8 are for computation of the VV scattered fields.

Figure 8.

CPU requirements as a function of the number of unknowns N, for the example in Figure 7. (a) CPU requirements for near terms; (b) total CPU requirements.

4.5. Discussion

[33] From the above numerical examples, we may summarize general properties observed through numerous numerical experiments. We have found that in addition to its ability to handle general heterogeneous scatterers, the VEFIE model has much better convergence performance than SCFIE modeling (for homogeneous targets). However, for scatterers for which the internal volume is large relative to its area (e.g., a cube, sphere or cylinder with comparable length and radius), the number of volume mesh cells and the corresponding number of unknowns is much larger for the VEFIE formulation vis-à-vis the SCFIE modeling, and consequently VEFIE computation and memory requirements increase significantly. However, VEFIE modeling is appropriate for scatterers with extended volume distribution, such as a thin surface, for which the difference in the number of unknowns between the volume and surface model is often not significant. For such targets we have found the SCFIE formulation to often be undermined by poor convergence of the iterative solver (poor conditioning).

5. Conclusions

[34] The MLFMA has been extended to three-dimensional volumetric electric-field integral equations, for dielectric scatterers situated above or within a half-space. The “near” interactions and the corresponding Znear elements are calculated rigorously via the complex-image technique, for the dyadic and scalar half-space Green's function. The “far” interactions are divided into the direct (free space) contribution and a reflection contribution from the interface. The reflected contributions are calculated by adding a set of image elements and groups. The position and distribution of those image sources are determined by the approximate image techniques. We have also discussed important practical issues for increasing the efficiency of the half-space VEFIE MLFMA.

[35] Numerical comparisons have been presented for the volumetric (VEFIE) and surface (SCFIE) integral-equation formulations, and good agreement in the calculated results is observed (for homogeneous targets, for which the SCFIE is applicable). It has also been demonstrated that there are often significant differences between the scattered fields from a heterogeneous target and its homogeneous (average) approximate, for realistic buried targets.

Ancillary