Multipole-Free Fast Inhomogeneous Plane Wave Algorithm

Authors


Abstract

[1] Solving radiation and scattering problems in the vicinity of a half-space or stratified medium with integral equation methods is complicated, and slowed by computation of expensive Sommerfeld integrals. In this paper, a new formulation of the spectral layered media Green's function is presented. It is demonstrated for a simple scattering problem in the new fast algorithm called the Multipole-Free Fast Inhomogeneous Plane Wave Algorithm (MF-FIPWA). MF-FIPWA is shown to scale as equation image(N log N) in memory usage and processing time. Additional advantages include rigorous treatment of reflected contributions, simplicity in design, and mathematical insight into the fast algorithm.

1. Introduction

[2] Solving radiation and scattering problems in the vicinity of a half-space or stratified medium with integral equation methods is complicated, and slowed by computation of expensive Sommerfeld integrals. In addition, large scattering problems often require millions of unknowns, thus motivating the fast algorithm. Several methods have been developed for this class of problem, and most rely on the fast multipole algorithm (FMA) [Rokhlin, 1990; Coifman et al., 1993] and multilevel paradigm, as first proposed by Brandt [1991], and implemented for electromagnetics in Lu and Chew [1993] and the multilevel fast multipole algorithm (MLFMA) [Lu and Chew, 1994]. Recent work includes use of hierarchical basis functions JorgensenGRS2004, the discrete complex image method (DCIM) with MLFMA [Geng et al., 2000; Li et al., 2003; Ginste et al., 2004], and steepest descent path integrals with the fast inhomogeneous plane wave algorithm (FIPWA) [Hu and Chew, 2001; Hu et al., 2001].

[3] Much of this work stems from MLFMA and FIPWA; both are multipole-based algorithms that are beyond the scope of introductory discussion. Hence, efforts have been made throughout this article to introduce sufficient detail from the seminal papers on FMA [Rokhlin, 1990; Coifman et al., 1993], MLFMA [Brandt, 1991; Lu and Chew, 1993, 1994] and FIPWA [Hu and Chew, 2001; Hu et al., 2001, 1999; Hu and Chew, 2000], and it is likely that the interested reader will find them quite useful.

[4] In Jorgensen et al. [2004], Legendre polynomial basis functions help reduce the number of unknowns N to achieve a faster solution. Meshing the scatterer with curvilinear quadrilateral patches requires fewer unknowns than Rao, Wilton, Glisson (RWG) basis functions [Rao et al., 1982], and the higher-order basis functions can span multiple media. Yet, the computational complexity is higher than MLFMA with RWG bases (equation image(N2)), and the method is only beneficial to geometries that can be represented with few curvilinear patches. In Geng et al. [2000] and Li et al. [2003], MLFMA extends earlier work [Geng et al., 1999] with the fast multipole algorithm and DCIM, where complex images were computed with approximate reflection coefficients. Using multipole expansions requires a prohibitive number of terms for sources that reside in highly lossy media. The method achieves equation image(N log N) complexity of MLFMA. Whiles parallelization efforts enable novel integration of multiple scatterers in very large problems, the cost to compute the complex images is equal to the free space case, and increases memory requirements up to 50%. In addition, the multipole expansion is still needed.

[5] Yet, FIPWA established a simpler formulation for removing the multipole expansion [Hu and Chew, 2001; Hu et al., 2001, 1999]. Hu et al. developed FIPWA [Hu and Chew, 2001; Hu et al., 2001, 1999; Hu and Chew, 2000] as an alternative to the fast multipole algorithm (FMA) and multilevel fast multipole algorithm (MLFMA) (thoroughly discussed and cited in Chew et al. [2000]). FIPWA evolves from the fast steepest descent path algorithm (FASDPA) developed by Michielssen and Chew [1996], which was the first fast algorithm (equation image(N4/3 ln N)) to use the spectral integral representation of the Green's function, i.e., propagating and evanescent plane waves.

[6] In FIPWA, the spectral dyadic Green's function for layered media is computed with high accuracy because it rigorously treats the reflection coefficients, and it also accelerates computation of the dyadic components using modified steepest descent path (SDP) integrals. Unlike the DCIM approaches of Geng et al. [2000] and Li et al. [2003], FIPWA only requires one image term, making the setup and computation of FIPWA faster than the DCIM approach. Following the MLFMA paradigm of factoring and diagonalizing the Green's function operator, FIPWA constructs a diagonal operator with extrapolation of evanescent waves from propagating waves. In doing so, FIPWA achieves equation image(N log N) efficiency for various two- and three-dimensional scattering problems, and in the case of the layered media problem, FIPWA only costs up to 20% more in memory than the free space case.

[7] While FIPWA still relies on the multipole expansion, the formulation is highly adaptable to a completely multipole-free form. Previous work [Saville and Chew, 2006a, 2006b] laid the groundwork for the multipole-free fast algorithm, with the implication that one can simply substitute the 2-D FIPWA translator for the 2-D FMA translator. Hence, this paper presents the multipole-free expansion of the spectral layered media Green's function (LMGF) in a new fast algorithm called the Multipole-Free Fast Inhomogeneous Plane Wave Algorithm (MF-FIPWA). MF-FIWPA is validated by comparing the scattering solution of a moderately sized problem to the solutions from FIPWA and a full matrix solver. Finally, MF-FIPWA is demonstrated for a larger complex target and shown to retain equation image(N log N) complexity in total memory cost and processing time of the matrix-vector product.

[8] This paper is organized as follows. First, the integral equation formulation is presented for the spectral LMGF. Then, the Green's function operator is factorized and diagonalized without using multipoles. The computational cost to constuct the 3-D translation matrix is shown to be equation image(N) with controllable accuracy. Finally, numerical results demonstrate the accuracy, complexity, and stability of the new multipole-free algorithm.

2. Formulation

2.1. Method of Moments

[9] The selected demonstration problem is to compute the scattering from an impenetrable object that resides above a layered medium, and is posed as an integral equation that is solved with the method of moments (MOM). Specifically, the MOM matrix is formed with the combined field integral equation (CFIE), and is solved using the conjugate gradient method within the multilevel paradigm of MLFMA. The CFIE represents contributions from the electric and magnetic field integral equations (EFIE and MFIE, respectively) as CFIE = αEFIE + (1 − α)MFIE, where the coupling parameter α is typically 0.2 to 0.5. Although the details were reported in previous work [Hu and Chew, 2001; Zhao et al., 1998], they are repeated for convenience to the reader and to identify the contributing component of this work.

[10] The EFIE and MFIE are succinctly written in terms of integral operators equation image and equation image as

equation image
equation image

where J(r) is the unknown electric current density, equation image is the unit normal to the scatterer's surface S, and

equation image
equation image

where the kernel is the symmetric dyadic Green's function equation image(r, r′) [Hu and Chew, 2001; Zhao et al., 1998].

[11] The components of equation image(r, r′) for layered media represent direct, TE, and TM scalar Green's functions [Zhao et al., 1998], and can be derived using the (Ez, Hz) formulation of Chew et al. [2006] and Chew [1995]. For convenience to the layered media problem, each dyadic component is represented with spectral scalar Green's functions as

equation image

with

equation image

and

equation image

where gEM = (equation imagegTEgTM), and Γβ is the Sommerfeld integration path (SIP), or in the case of FIPWA, a modified steepest descent path (MSDP) in the complex β-plane [Hu and Chew, 2001]. Equation (7) identifies the spectral form for the primary, TM-reflected, and TE-reflected contributions in terms of Sommerfeld integrals. The weight functions are

equation image
equation image
equation image

where kz = k cos β, kiz = equation image and equation image is the generalized reflection coefficient for the N-layer medium [Chew, 1995].

[12] The well-known Rao, Wilton, Glisson (RWG) basis functions [Rao et al., 1982] form testing functions and expand the unknown current of equations (3) and (4) as,

equation image

to form the MOM matrix equation

equation image

The matrix and vector elements are

equation image
equation image
equation image

[13] The dominant expense in filling the impedance matrix equation image is computation of gγ, but is reduced with the MSDP of FIPWA and the fast algorithm paradigm of MLFMA. Hence, gγ is factored into a vector-matrix-vector product where the matrix is diagonal. Combining this factorization with the conjugate gradient iterative matrix solver reduces computation and memory costs to equation image(N log N). The next sections shows the simplicity of the factorization and diagonalization without using a multipole expansion.

2.2. Fast Algorithm for Layered Media

[14] The spectral Green's function for layered media includes additive contributions from the direct and reflected interactions, and guided and surface waves. The direct interaction is implemented with free space FIPWA, and of the remaining contributions, only the reflected contribution is presented because the formulations for the guided and surface waves are the same as the reflected term.

[15] The reflected interaction gγ(rr′) is derived from the Sommerfeld integral and represents the source image. (Note that guided and surface waves are included with appropriate pole and branch cuts in the integration.) Using coordinate rotation, where kz = k cos β and kρ = k sin β,

equation image

where γ = TM or TE, Wγ (β) = equation imagek sin βequation imageγ(k1z,…, kNz), equation imageγ is the generalized reflection coefficient [Chew, 1995], kiz = equation image, and rr′ = equation image(xx′) + equation image(yy′) + equation image(z + z′).

[16] To construct the fast algorithm, equation (16) is factored into a vector-matrix-vector product, where the matrix is diagonal; and the differences between MLFMA, FIPWA, and MF-FIPWA are the specific approaches in constructing the matrix expansion. Using MLFMA, spherical Hankel functions and Legendre polynomials form a multipole expansion of the free-space Green's function and complex images approximate the reflected contribution [Geng et al., 2000; Li et al., 2003]. In FIPWA, the integral path of equation (16) is defined on a steepest descent path and Hankel functions expand the monopole into multipoles. Yet, in this work, the monopole is expanded with a second steepest descent path integral [Hu et al., 1999].

[17] Under the MLFMA paradigm, nearby basis functions are grouped as shown in Figure 1. In the illustration, both the source and test groups lie above the surface, but it has been shown that one or both can lie beneath the reference surface [Hu and Chew, 2000]. From the Sommerfeld integral, reflections appear to emanate from source images below the surface, as illustrated with source image group I (Figure 1). Note that only far removed interactions are computed using the fast algorithm, while near interactions are computed with the traditional MOM.

Figure 1.

Illustration of reflection contribution in layered media Green's function. Source and test groups containing RWG [Rao et al., 1982] bases are above the reference surface, and the source image group is shown below the surface.

[18] The interaction between the ith source-basis function in group I and the jth test-basis function in group J (Figure 1) is factored into

equation image

from which, the scalar Green's function is expressed in matrix notation as a sum of plane waves. The vector-matrix-vector expansion

equation image

allows efficient storage and computation of many interactions because βjJ(θ, ϕ) = equation image and βIi(θ, ϕ) = equation image are vectors that represent radiation patterns of groups of bases. The matrix ��JIγ is a diagonal matrix representing coordinate translations in plane wave directions k(θ, ϕ).

[19] Further efficiency is gained by factoring gγ in a recursive fashion with lmax levels. The multilevel paradigm is

equation image

where gγ represents the TM, TE, or TEM component, β are the shifted radiation and receiving patterns, equation imageequation image is the diagonalized translation matrix between patterns, equation image and equation imaget represent interpolation and anterpolation matrices, and (θ, ϕ) dependency is implied. Complete details of MLFMA are given in Chew:yellow. FIPWA and MF-FIPWA, as will be shown, formulate (18) directly from (16), which is a distinct advantage of the inhomogeneous plane wave approach.

2.3. MF-FIPWA

[20] Expanding the dyadic Green's function in equation (5) revealed that the dominant costs in computing LMGF are the scalar spectral components gγ. MF-FIPWA expands gP in the same manner as FIPWA, but the reflected components gTE and gTM are presented below.

[21] Upon factoring rji = rjri in the spectral representation of gγ (equation (16)),

equation image
equation image

where ρji = ∣ρjρi∣ and Γβ is a contour in the complex β-plane.

[22] The path Γβ is a modified steepest descent path, and the Hankel function in (20) and (21) is expanded with the 2-D form of equation (18) using 2-D FIPWA for complex media. Details of 2-D FIPWA are given in Hu et al. [1999], Hu and Chew [2000], and Saville and Chew [2006b]. Here, it is outlined in a slightly different fashion to clarify its contribution to this work.

[23] Using kρ = k sin β[equation image cos(α) + equation image sin(α)], and ρji = ρjJ + ρJI + ρIi, the Hankel function is expressed as a sum of inhomogeneous plane waves with the 2-D Weyl identity and coordinate rotation

equation image

where Γα is a contour in the complex α-plane.

[24] Following substitution of (22) into (21), the exponential components are collected as

equation image

where

equation image
equation image
equation image
equation image
equation image

and Wγ is defined in equations (9) and (10).

[25] Equation (23) represents a sum of weighted plane waves to include both propagating and evanescent waves according to contours Γβ and Γα. It is easy to see that the bracketed term represents weighted plane waves that are directed from the center of group I to the center of group J. Yet, to express the first and last exponential terms of (23) in the form of equation (18), it is inefficient to evaluate the complex exponentials along all possible contours Γβ and Γα.

[26] Instead, the exponential terms containing rjJ and rIi are interpolated from a set of real samples as

equation image
equation image

where I(x) = sinc(x) for exact interpolation. In the cases where β and α are complex-valued, I(x) performs extrapolation into the complex plane, which is possible because βjJ = equation image and βIi = equation image are analytic functions [Brown and Churchill, 2001]. Various interpolation functions were studied for extrapolation [Saville and Chew, 2006b; Hu, 2001; Ohnuki and Chew, 2003; Saville, 2006], but the Dirichlet function, or cosine-windowed sinc function, was determined to be the most versatile for both interpolation and extrapolation with at least 6 digits of precision and ��(N) efficiency [Chew et al., 2000; Schanze, 1995; Candocia and Principe, 1998; Dooley and Nandi, 2000].

[27] Upon substituting equations (29) and (30) into (23), and swapping the order of interpolation and integration

equation image
equation image

where I(θ, ϕ) = I(β − θ)I(α − ϕ) and

equation image

[28] Equation (32) compactly represents a sum of propagating plane waves that are weighted by inhomogeneous plane waves as stored in vectors βjJ and βIi, and the translation matrix ��JIγ, respectively. Under the MLFMA paradigm, equation (33) is often called the translator and is defined on the sphere of size ∣kD∣. However, it represents the translation directions between two groups of equal size. In other words, it is proportional to the Green's function interaction between two groups.

[29] Equation (33) is also the chief difference between MF-FIPWA, FIPWA, and MLFMA with DCIM. Full details of the latter two are given in the references, but it is clear that MF-FIPWA does not use a multipole expansion, implying that the translation matrix ��JIγ(θ, ϕ) can be constructed faster in MF-FIPWA than in FIPWA. There currently is no direct comparison between MLFMA with DCIM and MF-FIPWA, but MF-FIPWA only uses a single source image and a multipole-free expansion making MF-FIPWA simpler to derive and implement in the layered medium problem. Furthermore, MF-FIPWA is faster than FIPWA according to the specific contours Γβ and Γα, which are choosen as modified steepest descent paths (MSDPs). Previous reports on FIPWA and the multipole-free algorithm discussed the M-SDPs in detail [Hu and Chew, 2001; Hu et al., 2001; Saville and Chew, 2006b], so the M-SDPs are presented briefly for the general case of complex media.

2.4. Modified Steepest Descent Paths

[30] Integration paths Γβ and Γα are modified steepest descent paths (M-SDPs) as described by Hu and Chew [2001], Hu et al. [2001], and Saville and Chew [2006b]. Examples of the fundamental and modified paths are shown in Figure 2, where the fundamental-SDP (F-SDP) represents the paths used to construct the modified path. For the single interaction gγ(rjri), F-SDP represents the true steepest descent path for the integrand of equations (21) and (22) that passes through the saddle point and requires constant phase and exponential decay of the integrand. Many interactions are possible for the bases contained in groups I and J, requiring a continuum of F-SDPs. It would be highly inefficient to compute each possible F-SDP, so the M-SDP is constructed from the two F-SDPs that pass through the saddle points shown in Figure 2. Hence, integration along the M-SDP is performed in lieu of all possible fundamental paths with high accuracy and efficiency of the steepest descent path integral.

Figure 2.

Fundamental and modified steepest descent paths. Saddle points indicate the center of the two fundamental paths used to construct the modified path.

[31] The general forms of Γβ and Γα are the same, but special consideration of complex background media is needed for Γα, so it is considered the more general path and Γβ can be derived for arbitrary group size and separation, background media, and path length as

equation image

where s is a parameterizing variable on the real s-axis and θ0 = sin−1(kD/krJI). Path Γα is defined in similar fashion with β = α, k = kρ, θJI = ϕJI, and θ0 = ϕ0. Paths II of the double integral in equation (33) represent the angular field of view between the groups, and Paths I and III represent the directions of the contributing evanescent waves that lie outside the field of view.

2.4.1. Surface Wave Contribution

[32] When a scatterer is placed on the surface, or slightly above the layered medium, lateral surface waves [Chew, 1995] are excited. In most cases, the contribution is small and can be dismissed, as in Geng et al. [2000] and Geng et al. [1999]. Yet, the contribution is easily managed by defining paths around the appropriate branch cut that occurs in the β-plane and adding the contribution as gγ + gbranch. By defining the weight function W′ for the correct Riemann sheet, the additional computational burden is negligible. This approach was demonstrated in FIPWA for layered media and buried object problems. Complete details are given in Hu and Chew [2001] and Hu et al. [2001] and the same method applies to MF-FIPWA. Hence, only the salient features are presented.

[33] The weight functions WTM(β) and WTE(β) in equations (9) and (10), respectively, include generalized reflection coefficients that have pole and branch point singularities in the complex β-plane. As discussed in Hu and Chew [2001], the single branch point corresponds to kρ = kN. Hence, a contour of integration ΓB can be designed to include the branch point and have constant phase and exponential decay. The resulting expansion of gbranch and corresponding translator are only different from equations (31) and (33) by choice of Γβ = ΓB and the new weight function &#55349;&#56498;(β) = W+(β) − W(β), where ± denotes upper (ℛe{kNz} > 0) and lower (ℛe{kNz} < 0) Riemman sheets [Hu and Chew, 2000]. Although the translator for the branch point contribution includes a double integral in (β, α), the computational burden is less than the SDP contribution because ΓB is typically shorter than Γβ. In addition, only translators corresponding to source images near the upper half space are strong enough to excite surface waves requiring proportionally fewer branch cut contributions.

2.4.2. Guided Wave Contribution

[34] Similarly, pole contributions due to equation imageTE,TM are also easily included as gγ + gbranch + gpole. When poles are present, residue theory is used to add the contribution. Of course, an appropriate pole finding algorithm [Hu, 2001] must first locate the poles, but storage and computational costs are minimal. Once the residues are found, gpole reduces from a double integral (β, α-planes) to a single integral in the α-plane, resulting in translators for the poles that require less computational memory and processing time than the SDP components.

[35] In terms of fast algorithms, these approaches for including contributions of surface waves and guided modes are unique to MF-FIPWA and FIPWA. Each is mentioned here to show how MF-FIPWA rigorously captures physical phenomena of the layered media problem. Furthermore, the added computation is performed “fast” with proper construction of the radiation and receiving patterns, and translation matrices.

3. Cost Analysis

3.1. Scaling With Problem Size

[36] In terms of the number of unknowns, N, of the MOM matrix, the construction cost of the translation matrix (equation (33)) scales as &#55349;&#56490;(N). The size of the surface scattering problem is proportional to the spatial bandwidth of the scatterer. Hence, the number of k(θ, ϕ) samples needed to represent the patterns and store the translation matrix scales according to N.

[37] For surface scattering problems, if P2 represents the number of samples needed to construct the translation matrix, and P2 ∝ (kD)2, where kD is the size of a group of bases, then by the excess bandwidth formula of Chew et al. [2000], NP2. In fact, it is well known that N is related to the spatial bandwidth (kD)2 [Brandt, 1991; Bucci et al., 1991], so that construction of &#55349;&#56495;JIγ(θ, ϕ) with local interpolation scales as &#55349;&#56490;(N).

3.2. Construction of Translation Matrix

[38] The total cost to construct &#55349;&#56495;JIγ(θ, ϕ) depends on the number of samples in each angular direction. By letting Nθ equal the number of samples in the θ coordinate, and Nϕ equal the number of samples in the ϕ coordinate, the construction of &#55349;&#56495;JIγ(θ, ϕ) is shown to scale as &#55349;&#56490;(N), as N → ∞.

[39] Following a straightforward method to construct the diagonal matrix &#55349;&#56495;JIγ(θ, ϕ)

equation image

where &#55349;&#56495;JIγj, ϕk,j) is defined by equation (33).

[40] Upon substituting equation (33) into (35), each matrix element is computed as

equation image
equation image
equation image

where

equation image
equation image
equation image

[41] Finally, interpolation can be performed with a finite number of local samples so that Nequation image and Nϕ do not scale with N. Hence, only Nβ and Nα,m scale as equation image, and the translation matrix is constructed efficiently where

equation image
equation image

[42] Depending on the number of integration points on Γα, MF-FIPWA is constructed faster than FIPWA and vice-versa for a given accuracy setting. Typically, large boxes require such a large number of multipoles and quadrature points that MF-FIPWA is computed faster at the highest levels in the MLFMA tree. On the other hand, when boxes are very small, few multipoles and integration points are needed so that FIPWA is faster than MF-FIPWA. Results of the accuracy and setup cost are shown in section 4.

4. Error Control

[43] Common to each multilevel algorithm are error sources that can be exponentially decreased, or controlled, such as numerical integration and interpolation error [Saville and Chew, 2006b; Ohnuki and Chew, 2003; Song and Chew, 2001; Hastriter et al., 2003]. In addition, MLFMA and FIPWA, have controllable error due to truncation of the multipole expansion, and MF-FIPWA and FIPWA have controllable error from interpolation and extrapolation of the radiation patterns in equations (29) and (30). For each individual source, error analysis has been provided in the references, but unique to this work, here, the accuracy of the spectral scalar Green's function is shown for MF-FIPWA and FIPWA.

[44] In equation (33), the integrand represents weighted inhomogeneous plane waves, and in Figure 2, real-valued points on path II represent propagating waves, and points on paths I and III represent evanescent waves. To accurately compute the double integral while retaining computational efficiency, each path must be of the appropriate length, and special attention must be given to hyperbolic paths I and III of Γα [Saville and Chew, 2006b].

[45] While the infinite-length paths are parameterized in equation (33) with the real variable s ∈ (0, ∞), the domain of s is truncated in accordance with the integration rule and desired accuracy. Empirical studies showed that the Gauss-Laguerre rule with 15 points is most suitable for 6-digits of accuracy along paths I and III, while Gauss-Legendre quadrature is used on path II with the number of points set by the refined excess bandwidth formula. The lengths of paths I and III are truncated at smax when the magnitude of the integrand decays at least equation image as

equation image

which is slightly different for each complex plane:

equation image

[46] For efficiency, the number of integration points can be decreased for ∣kρρJI ≫ 1, which corresponds to large group sizes or far interactions, and 15 quadrature points proved to be sufficient for cases when ∣kρρJI ≪ 1, e.g., when the near field behavior becomes dominant.

[47] To demonstrate error control, the accuracy of the plane wave expansions are presented for various group sizes in the multilevel approach. Table 1 lists the corresponding problem sizes, frequencies, and number of levels for the various cases. In each case, the half-space has ε = (6.5, 0.6), the scatterer is located 0.2 m above the half-space, and the smallest box in the tree is 0.10 λ. Perfect electrically conducting spheres were used as the scatterers to maximize the translation directions stored in the translation matrices.

Table 1. List of Scattering Examples and Setup Parameters
 Number of Unknowns
10 092101 568252 300504 3001 002 2522 007 372
Frequency, GHz0.2970.9431.4822.0992.9594.184
Maximum levels567889

[48] Each scalar component of LMGF contains gTM, or gTE, or both. Also, the TM and TE translation matrices are easily shown to represent gTM and gTE when the radiation patterns are omnidirectional. Therefore, gSIPγ represents the Sommerfeld integration path (SIP), and gFIPWAγ and gMFFIPWAγ represent the FIPWA and MF-FIPWA approximations to gSIPγ, respectively, where γ is TM, or TE or d for direct.

[49] To compute the error in FIPWA and MF-FIPWA, the scattering cases in Table 1 are used to simulate a variety of translators of different sizes and orientations. In each case, the box size is normalized by the wavelength so that translators of different problems can be compared. The error estimate of FIPWA and MF-FIPWA is based on the special case when the generalized reflection coefficients are set to unity. In this case, gSIPd = gFIPWAd = gMFFIPWAd = g0 = eikr/r. This method is similar to calibrating a measurement system where the system error is accounted for and understood to provide a limitation to the accuracy. Using the root mean square error (RMSE), the error is determined for each box size that corresponds to a different level in the MLMFA tree. The error is

equation image

where Na is the number of boxes of size a, and γ is either SIP, FIPWA, or MF-FIPWA.

[50] Table 2 summarizes the error estimate for box sizes of 0.10–12.8 λ. For each box size, the translators from all cases in Table 1 are grouped together regardless of separation or angular orientation. Each group has Na boxes. The Sommerfeld integral is independent of the box size and results in a nearly constant RMS error. The RMS error represents the overall integration error in computing gSIPd. FIPWA (gFIPWAd) depends on the box size and shows better accuracy than SIP for large boxes, but it loses accuracy as the box size decreases, indicating low frequency breakdown. These results establish the usable range of box sizes for FIPWA. The smallest box size should be larger than or equal to 0.10 λ.

Table 2. RMS Error of Layered Media Translator by Box Sizea
a (λ)NagSIPdgFIPWAdgMFFIPWAd
  • a

    SIP, FIPWA and MF-FIPWA are compared to g0 = eikr/r. The box size a is normalized to wavelengths, and Na is the number of translators per box size.

12.801057.82e-042.75e-062.37e-04
6.407273.32e-052.23e-061.10e-03
3.201 8723.75e-052.66e-057.93e-03
1.604 3348.37e-051.90e-041.90e-04
0.808 8731.03e-049.12e-049.12e-04
0.4018 0331.27e-043.23e-033.22e-03
0.2036 2661.34e-047.12e-037.06e-03
0.1072 3701.35e-044.32e-011.13e-02

[51] MF-FIPWA has better accuracy for smaller boxes, and the results of large boxes are equivalent to the Sommerfeld integral. Large boxes require more plane wave directions for higher accuracy, but for proper comparison, the same number of waves are used for both FIPWA and MF-FIPWA. In addition, the better accuracy for the smallest box size suggests that low frequency breakdown of MF-FIPWA occurs at smaller boxes than FIPWA. This last point indicates that there is potential to use MF-FIPWA for even smaller boxes before low frequency breakdown occurs.

5. Numerical Results

5.1. Validating the Multipole-Free Algorithm

5.1.1. Setup for Validation Problem

[52] MF-FIPWA is validated by comparison to a full matrix solution along with a comparison to FIPWA (previously validated by comparison to a full matrix solution [Hu, 2001; Hu and Chew, 2001; Hu et al., 2001]. In the benchmark problem, an impenetrable right circular cylinder is placed above a lossy two-layer medium. The cylinder is discretized at 600 MHz with 9708 unknowns, and 5 levels are used in the multilevel approach where the smallest box size is λ/10. Excitation is incident from θinc = 60°, ϕinc = 0°, and the observation angles are θobs = 60°, ϕ ∈ [0, 2π). The cylinder is placed 0.2 m above the layered medium, and the medium parameters are ε0 = 1.0, ε1 = 2.56 with a thickness of 0.3 m, and ε2 = (6.5, 0.6).

5.1.2. Validation of MF-FIPWA

[53] Figure 3 shows the comparison between the two algorithms and a full matrix solution. MF-FIPWA has excellent agreement for both vertical and horizontal polarizations.

Figure 3.

Comparison of MF-FIPWA and FIPWA for the validation problem.

5.2. Scaling and Efficiency

[54] Before studying a complex target, the scaling of the multilevel multipole-free algorithm is shown to be equivalent to FIPWA for moderate to large size spheres. As FIPWA, MF-FIPWA achieves &#55349;&#56490;(N log N) complexity in memory and processing time per matrix-vector multiply for problem sizes of N = 10002, 101568, 252300, 504300, and 1002252 unknowns. Figure 4 illustrates the scaling for moderately sized problems.

Figure 4.

Scaling of memory and processing time for matrix-vector product. Memory is measured in MBytes, and time is measured in cpu seconds.

[55] Because the multilevel paradigm is used, the scaling is not a surprise. Note that all of these results were run on a single Sun-Blade 1000 processor with 8 GB of RAM.

5.3. Moderate Sized Complex Target

[56] To demonstrate the code for a complex target, the bistatic cross section is computed for the fictitious VFY-218 above a lossy half-space. The frequency is 100 MHz, N = 163344 at 7 levels, and the dielectric constant is ε = (6.5, 0.6). Excitation is at θinc = 60°, ϕinc = 0°. The agreement between FIPWA and MF-FIPWA is excellent, as shown in Figure 5, and the total computational cost is 476 MB of RAM and less than 2.5 hours for both VV and HH polarization.

Figure 5.

Bistatic radar cross section of VFY-218 above lossy earth.

5.4. Mixed-Form Algorithm

[57] FIPWA and MF-FIPWA were also used together in a mixed-form algorithm to reduce construction time of the layered medium translation matrices. Based on work with various size scatterers, MF-FIPWA was set to compute the translation matrix when the box edge was larger than 2.0 λ, and FIPWA computed the matrix for smaller boxes. The mixed-form showed a 17%-speedup as compared to the original FIPWA.

6. Summary

[58] The multilevel multipole-free fast algorithm is demonstrated for layered media problems with &#55349;&#56490;(N log N) complexity of memory and processing time. Starting with the Sommerfeld integral, a new formulation was derived and implemented under the fast algorithm paradigm of MLFMA, but without using a multipole expansion. By carefully choosing steepest descent paths for the double integral in the scalar Green's function, controlled accuracy was achieved with high efficiency, and the simpler construction of MF-FIPWA revealed potential for faster set-up time of the translation matrices and higher accuracy for small boxes. Furthermore, by combining MF-FIPWA and FIPWA in a mixed-form algorithm, a 17% improvement in set-up time was achieved.

Acknowledgments

[59] The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, the Department of Defense, or the U.S. Government.

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