Linear embedding via Green’s operators and Arnoldi basis
functions for analyzing complex structures
Citation for published version (APA):
Lancellotti, V., Hon, de, B. P., & Tijhuis, A. G. (2011). Linear embedding via Green’s operators and Arnoldi basis functions for analyzing complex structures. In Proceedings of the 5th European Conference on Antennas and Propagation (EUCAP), 11-15 April 2011, Rome, Italy (pp. 3363-3367). Institute of Electrical and Electronics Engineers.
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Linear Embedding via Green’s Operators and
Arnoldi Basis Functions for Analyzing Complex
Structures
V. Lancellotti, B. P. de Hon, A. G. Tijhuis
Department of Electrical Engineering, Technical University of Eindhoven P. O. Box 513, 5600 MB Eindhoven, The Netherlands
{v.lancellotti, b.p.d.hon, a.g.tijhuis}@tue.nl Abstract—We have extended the linear embedding via Green’s
operators (LEGO) method to the solution of complicated struc-tures comprised of many different penetrable and conducting media. By combining LEGO with the Arnoldi basis functions, we are able to substantially reduce the size of the relevant algebraic system that arises from the application of the Method of Moments. We review the basics of the overall approach and we provide examples of validation. Finally, the advantages of LEGO are demonstrated through the solution of the electromagnetic scattering from a complicated two-layer finite-size frequency selective surface.
I. INTRODUCTION
Complex structures made of many different conducting and penetrable media are of great importance in practical applications. In principle, the scattering of electromagnetic (EM) waves from these structures may be formulated through a set of (global) integral equations (IEs). However, the latter may grow unnecessarily complicated even for situations when just few different media are concerned. More importantly, the relevant IEs most likely lead to algebraic systems with full matrices and lots of unknowns, when one applies the conven-tional Method of Moments (MoM). In this respect, domain decomposition methods (DDMs), such as linear embedding via Green’s operators (LEGO), represent a better (if not the only) choice, considering they try and divide the original complicated problem into a number of small sub-problems, generally more amenable to numerical solution.
LEGO was initially devised for (though not limited to) the solution of scattering problems that involve composite struc-tures comprised of many objects immersed in a background medium [1–3]. Specifically, in LEGO a complex structure is separated into “small” parts (i.e., small as compared to the whole structure) which are then included in simple-shaped domains 𝒟𝑘 (bricks), 𝑘 = 1, . . . , 𝑁𝐷. The EM behavior of 𝒟𝑘 is captured by its scattering operator S𝑘𝑘, whereas the interaction between two bricks 𝒟𝑘 and 𝒟𝑛, 𝑛 ∕= 𝑘, is accounted for by the transfer operatorT𝑘𝑛.
The bricks so defined are then used to obtain the EM model of the original structure. Since no restrictions apply to the bricks’ shape and content and position in space, with LEGO we can tackle quite complicated problems. For instance, in [4] we solved the problem of a real antenna radiating in the presence of a frequency selective surface (FSS). Furthermore,
PEC or dielectric inclusions (medium 3)
piecewise homogeneous dielectric (medium 2)
background (medium 1)
Fig. 1. Example of complex structure: (left) a piecewise homogenous dielectric slab (medium 2) immersed in a homogeneous background (medium 1) and containing PEC or penetrable objects (medium 3); (right) LEGO model of the structure comprising 𝑁𝐷 = 25 rectangular bricks (colors denote different content as per both the object’s shape and the filling host medium’s EM properties); both objects’ and bricks’ surfaces are represented by 3-D surface triangular meshes with which RWG functions are associated.
in [5, 6] we extended and employed LEGO along with the Arnoldi basis functions (ABFs) for solving the scattering from an aggregate of identical objects (PEC or dielectric) enclosed in a homogeneous host medium (tagged 2) in turn immersed in the background (tagged 1). The structure was modelled with 𝑁𝐷 identical bricks (as all the objects were equal). We demonstrated that the ABFs are well suited for reducing the order of the algebraic system which stems from the application of the baseline MoM to the integral equations of LEGO.
In this communication we briefly discuss the usage of LEGO to solve a problem more general than those outlined above, namely, the case when the inclusions in the host medium possess different shapes or EM properties (Fig. 1). Such an extension — albeit formally straightforward in the light of the inherent modularity of LEGO — in practice required substantive modifications to the developed numerical code. In fact, in the application to be described, we allow the bricks to have different content, even though they possess the same shape (Fig. 1). Thereby, sinceS𝑘𝑘 depends on a brick’s content, in the most general scenario we need to compute and store 𝑁𝐷 scattering operators.
In what follows, after reviewing the fundamentals of LEGO and the order reduction technique hinging on the ABFs, EuCAP 2011 - Convened Papers
we present two examples of validation of the newly de-veloped code. Finally, we show how a real-life complex structure, namely, a finite two-layer FSS, can be handled with LEGO/ABFs easily and efficiently.
II. OVERVIEW OFLEGOANDABFS
As detailed in [5], the EM problem in Fig. 1 can be formulated through the integral equation
(I − T diag{S𝑘𝑘})𝑞itot= 𝑞i, (1)
where the unknown total incident currents 𝑞i
tot, the known
incident currents𝑞iand the total transfer operatorT are defined
in [1]. To solve (1) we begin by applying the standard MoM (in Galerkin’s form) with 2𝑁𝐹 Rao-Wilton-Glisson (RWG) basis functions over∂𝒟𝑘. This yields
([𝐼] − [𝑇 ] diag{[𝑆𝑘𝑘]})[𝑞itot
]
=[𝑞i], (2)
where the system matrix (of rank2𝑁𝐹𝑁𝐷) may be relatively large when 𝑁𝐷 ≫ 1. As a consequence, on the one hand storing the matrix may become memory demanding, whereas, on the other hand, inverting the system may perforce require resorting to iterative methods.
To circumvent these hurdles, we devised a two-step proce-dure [5, 6] based on:
1) Compression of the off-diagonal blocks of [𝑇 ], i.e., the transfer matrices [𝑇𝑛𝑘] [1], via the adaptive cross approximation (ACA) algorithm [7].
2) Compression of the matrix [𝐼] − [𝑇 ] diag{[𝑆𝑘𝑘]} by formally expanding [𝑞i
tot
]
on a basis of entire-domain orthonormal functions{[𝜓𝑠]}, 𝑠 = 1, . . . , 𝑁𝐴.
The ACA algorithm (which exploits the inherent rank-deficient nature of [𝑇𝑛𝑘]) enables us to reduce the memory occupancy of[𝑇 ] as well as to fill it faster.
To generate the vectors[𝜓𝑠], we apply the Arnoldi iteration [8] to the sequence{[𝑇 ] diag{[𝑆𝑘𝑘]}𝑠−1[𝑞i]}𝑁𝑠=1𝐴. Apparently, the entries of[𝜓𝑠] define a function whose support is ∪𝑘∂𝒟𝑘. Then, we use{[𝜓𝑠]} to span the dominant space of[𝑞itot
] , viz., [ 𝑞i tot ] = [Ψ𝑁𝐴] [𝑎] , with [Ψ𝑁𝐴] = [ [𝜓1] . . . [𝜓𝑁𝐴] ] , (3)
and turn (2) into the final reduced-order system
([𝐼𝑁𝐴] − [𝐻𝑁𝐴]) [𝑎] = [Ψ𝑁𝐴]𝐻[𝑞i], (4)
where [𝐼𝑁𝐴] is the identity matrix of order 𝑁𝐴 and[𝐻𝑁𝐴] is the upper Hessemberg matrix yielded by the Arnoldi iteration [5]. Since in practice𝑁𝐴≪ 2𝑁𝐹𝑁𝐷, the reduced system can be inverted through LU factorization.
III. NUMERICAL RESULTS
A. Validation
In order to accommodate for the occurrence of 𝑁𝐷 bricks with different content in the LEGO model of a structure, as implied in (2), we had to update our numerical code. As the required modifications, though, do not affect the core (i.e., low-level matrix filling, calculation of algebraic scattering and transfer operators and ACA decomposition thereof), we feel
0 1 2 3 -0.5 0 0.5 -0.5 0 0.5 x [m] y [m] z [ m ] d 4d 2a −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 x [m] y [m] z [m]
Fig. 2. For LEGO validation: a sample complex structure comprised of four dielectric spheres embedded in a rectangular dielectric slab (left) is modelled with𝑁𝐷= 4 cubic bricks (right) each one embedding a sphere.
entitled to validate the new code against the results provided by its previous version, since the latter was thoroughly validated against MoM [5, 6].
For instance, as a mere check of consistency, let us consider the plane wave scattering (𝑬i= ˆ𝒚 exp(−j2𝜋𝑥/𝜆0) V/m) from a finite dielectric slab (𝜀2= 2𝜀0, dimensions𝑑 × 𝑑 × 4𝑑, with
𝑑 = 0.333𝜆0) immersed in free-space (𝜀1=𝜀0) and containing four dielectric spheres (𝜀3 = 3𝜀0, radius 𝑎 = 1.730𝜆3 =
0.083𝜆0), as illustrated in Fig. 2. This toy problem well fits our needs, in that it can be modelled by means of 𝑁𝐷 = 4 identical bricks (with 2𝑁𝐹 = 648 RWG functions over ∂𝒟𝑘 and 𝑁𝑂 = 588 RWG functions over a sphere). The case of
𝑁𝐷 identical bricks constitutes a trivial limit situation for the new code but also exactly the kind of problems handled by our old code. Hence, we expect the two of them to yield the same results, if no mistakes have unwittingly been made.
The calculations were performed using 𝑁𝐴 = 20 ABFs with the threshold for stopping the ACA set to 10−5 (10−3) for adjacent (well separated) bricks. For a start, with the aid of Fig. 3, we can visually ascertain that the total tangential
Magnitude nxH @ f = 0.1 GHz [A/m] 2 4 6 x 10-3
(a)
Magnitude nxH @ f = 0.1 GHz [A/m] 2 4 6 x 10-3(c)
Magnitude Exn @ f = 0.1 GHz [V/m] 0.5 1 1.5(b)
Magnitude Exn @ f = 0.1 GHz [V/m] 0.5 1 1.5(d)
Fig. 3. For LEGO validation: total (twisted) tangential fields over the surface of the bricks modelling the structure in Fig. 2; (a), (b) results computed with previous version of the LEGO code, (c), (d) results computed with the new version of LEGO code.
(twisted) electric and magnetic fields over the bricks’ surfaces are correctly calculated (from [𝑞toti
]
− [𝑞s]). It is worth
re-minding that in this example a brick’s boundary∂𝒟𝑘is a real interface (see [5, Section 2.1 and Appendix A]), not just a
mathematical separation surface (as was in [1]), as background
and host media are endowed with different EM properties. Secondly, if the currents over the bricks are correct, then most likely the radar cross section (RCS) computed with the new code also perfectly agrees with the results yielded by the old one, as confirmed by Fig. 4 where the RCSs in three relevant planes are plotted versus the elevation angle (𝜃).
The next numerical experiment is specifically devised to challenge the new code’s capabilities a bit. In fact, the structure under study, shown in Fig. 5, is a (finite) dielectric slab (𝜀2= 2𝜀0, dimensions 𝑑 × 𝑑 × 4ℎ, with 𝑑 = 0.333𝜆0 and ℎ =
0.0667𝜆0) immersed in free space (𝜀1 = 𝜀0) and containing
two PEC crosses (dimensions 𝑤1 = 0.0943𝜆2 = 0.0667𝜆0 and 𝑤2 = 0.283𝜆2 = 0.2𝜆0) and two PEC circular patches (radius 𝑎 = 0.1415𝜆2 = 0.1𝜆0). The relevant LEGO model is made of 𝑁𝐷 = 4 bricks whose scattering operators are not equal to one another (as the embedded object may be either a cross or a circle). The number of RWG functions over ∂𝒟𝑘 is 2𝑁𝐹 = 768 for all the bricks, whereas 𝑁𝑂 = 201 and 𝑁𝑂 = 223 for the cross and the circle, respectively. The calculations were performed using𝑁𝐴= 50 ABFs again with the ACA threshold set to10−5(10−3) for adjacent (well separated) bricks. To obtain a reference solution, we tackled the problem by modelling the structure by means of a single large brick whose boundary coincides with the slab’s surface. In this case,2𝑁𝐹 = 1920 and 𝑁𝑂 = 782.
We have solved the scattering problem for an incident plane
wave𝑬i= ˆ𝒚 exp(−j2𝜋𝑥/𝜆0) V/m with the two approaches.
The RCS provided by our new code and the reference solution are plotted in Fig. 6 for comparison. As can be seen, the new results are in excellent agreement with the reference values. It should be noted that comparing the total (twisted) tangential
0 50 100 150 -20 -15 -10 -5 0 5 10 15 20 Elevation (θ) [deg] bi st at ic R C S /d 2 [d B ] φ = 0° φ = 180° φ = 90°
k
E
ix
y
z
Fig. 4. For LEGO validation: bistatic RCS of the structure in Fig. 2; the results obtained with the previous version of the LEGO code (–) are compared with the results provided by the extended code (∘). Inset: structure, incident plane wave and reference system.
0 0.05 0.1 -0.05 0 0.05 -0.05 0 0.05 x [m] y [m] z [ m ] d 4h w2 w1 2a 0 0.05 0.1 -0.05 0 0.05 -0.05 0 0.05 x [m] y [m] z [ m ]
Fig. 5. For LEGO validation: a sample complex structure comprised of 2 PEC crosses and 2 PEC circular patches embedded in a rectangular-shaped dielectric slab (left) is modelled with𝑁𝐷= 4 rectangular bricks (right), each one embedding either a cross or a circle.
fields over the bricks’ surface is not straightforward (as in the previous experiment), because the number of bricks (and hence
∪𝑘∂𝒟𝑘) in the two approaches we followed do not match. B. Application example
We now consider a real-life problem, viz., the assessment of the transmitting and reflecting properties of a finite-size two-layer FSS immersed in free space and made of two types of double square loops (denoted type-a and type-b) outlined in Figs. 7(a)-(b). More precisely, the FSS consists of a planar rectangular arrangement of10×10 type-a loops plus a similar parallel arrangement of as many type-b loops deployed at a distance𝑎 = 6.4 mm from the other one (see insets of Figs. 8, 10 for a sketch). The FSS is modelled with𝑁𝐷= 10 × 10 × 2 = 200 bricks of dimensions 𝑑 × 𝑑 × ℎ, with 𝑑 = 17.6 mm and ℎ = 6.4 mm. Each brick is in turn represented by a regular triangular patching with which 2𝑁𝐹 = 1080 RWG functions are associated [Fig. 7(c)]. Besides, the number of RWG functions over the loops’ surface is 𝑁𝑂 = 222 and
𝑁𝑂 = 120 for type-a and type-b, respectively. Finally, the dimensions of the loops are listed in Table I, where 𝑤𝑖, 𝑖 =
0 50 100 150 5 10 15 20 25 30 Elevation (θ) [deg] bi st at ic R C S /d 2 [d B ] φ = 0° φ = 90° φ = 180°
k
E
ix
y
z
Fig. 6. For LEGO validation: bistatic RCS of the structure in Fig. 5; the results computed with the previous version of the LEGO code (–) are compared with the results provided by the extended code (∘). Inset: structure, incident plane wave and reference system.
−5 0 5 x 10−3 −5 0 5 x 10−3 −2 0 2 x 10−3 z [m] x [m] y [m] (a) −5 0 5 x 10−3 −5 0 5 x 10−3 −2 0 2 x 10−3 z [m] x [m] y [m] (b) −5 0 5 x 10−3 −5 0 5 x 10−3 −2 0 2 x 10−3 x [m] y [m] z [m] (c) d h
w
1w
2w
3w
4 (d)Fig. 7. Example of application: unit cells of a finite two-layer FSS comprised of10×10×2 PEC loops either in free space or in two dielectric host media; (a), (b) triangular patching of type-a and type-b loops, respectively, along with enclosing LEGO brick, (c) triangular mesh of a brick and (d) cartoon of the loops for defining their sizes.
TABLE I
DIMENSIONS OF THE LOOPS INFIG. 7
𝑤1 (mm) 𝑤2 (mm) 𝑤3(mm) 𝑤4(mm) type-a 10.6 8.7 5.4 1.0 type-b 15.7 12.1 7.2 1.2
1, . . . , 4, are the geometrical quantities defined in Figs. 7(d). We have addressed two cases (referred to as 1 and FSS-2, from now on) that differ for the EM properties assigned to the host medium. Specifically, for FSS-1 the host medium in both brick types is free-space, whereas for FSS-2 the host medium has permittivity𝜀2= 2.1𝜀0and𝜀2= 2.3𝜀0for type-a type-and type-b loops, respectively. Hence, the solution of FSS-2 can only be performed using the full capabilities of the newly developed code. The ACA threshold is chosen adaptively as in the examples discussed in Section III-A, whereas the number of ABFs is given in Table II. We have computed the transmitted and reflected RCS and the radiations patterns (both for normal incidence) in two frequency ranges which we carefully chose so as to include the (expected) resonances. The results are shown in Figs. 8-11: both FSSs exhibit two resonances at which they reflect back most of the incident power. At such resonances the transmitted RCS conspicuously drops down to a local minimum, whereas the radiation pattern exhibits a pronounced main lobe in the backward direction (𝜃 = 180∘).
From Table II the advantage of using the ABFs to solve (2) is apparent, since the size of the reduced system (𝑁𝐴) is far smaller than the rank of the original system (2𝑁𝐹𝑁𝐷). We also observe that the number of ABFs employed to analyze FSS-1 is likely an overkill. In fact, from the plot of𝑎𝑠 (that is, the entries of vector[𝑎]) in Fig. 12, it is seen that for FSS-1 the Arnoldi coefficients decay exponentially right from the beginning, whereas they do so from about 𝑠 ≈ 300 on for
6 8 10 12 14 30 32 34 36 38 40 42 44 46 48 f [GHz] bi st at ic R C S /d 2 [d B ] Back scattered (θ = 180°) Transmitted (θ = 0°) Type-2 loops Type-1 loops
k
E
ix
y
z
d 0 2 ε ε = 0 2 ε ε = aFig. 8. Example of application: bistatic RCS versus frequency for FSS-1; here host and background media are free space. Inset: schematic cross-sectional view of the FSS, incident plane wave and reference system.
FSS-2. Thereby, to attain a satisfactory level of accuracy of both near- and far-fields results, we could have generated just as many ABFs as necessary for the Arnoldi coefficients to drop down to≈ 10−4max ∣𝑎𝑠∣, as we did for FSS-2 (in order to keep the overall computation time at bay).
Fig. 12 also shows that the convergence rate of𝑎𝑠is greatly affected by the EM properties of the host medium [6]. By and large, a sharper contrast between the host and background medium (a feature captured by [𝑆𝑘𝑘]) results in stronger and more involved multiple scattering inside the structure. As a consequence, a larger number of ABFs is required to well represent the currents [𝑞i
tot
]
. This topic is addressed in details in a forthcoming paper [9].
Finally, it is apparent from Table II that the average times spent computing one ABF for FSS-1 and FSS-2 are not exactly equal, as one might expect, knowing that the matrices [𝑇 ] of the two problems have same size and block-structure. Nevertheless, the simulation campaign was carried out for different frequency ranges, and the effective rank of [𝑇𝑘𝑛] does depend on the frequency. Accordingly, the ranks of the matrices involved in the ACA decomposition of a given[𝑇𝑘𝑛]
0.2 0.4 0.6 0.8 1 30 150 60 120 90 90 120 60 150 30 180 0 φ = 0° φ = 180°
Fig. 9. Example of application: radiation pattern (normal incidence, E-plane) of FSS-1 at𝑓 = 13 GHz.
4 5 6 7 8 32 34 36 38 40 42 44 f [GHz] bi st at ic R C S /d 2 [d B ] Back scattered (θ = 180°) Transmitted (θ = 0°) Type-2 loops Type-1 loops
k
E
ix
y
z
d 0 2 2.1ε ε = 0 2 2.3ε ε =Fig. 10. Example of application: bistatic RCS versus frequency for FSS-2; here the background medium is free space, whereas the loops are embedded in two different host media, as indicated. Inset: schematic cross-sectional view of the FSS, incident plane wave and reference system.
[5] may easily not be the same at different frequencies. This ultimately affects the time taken to perform the matrix-vector product[𝑇 ] diag{[𝑆𝑘𝑘]} [𝜓𝑠] (see [5, Section 3.3]) and thereby the calculation of the ABFs.
IV. CONCLUSIONS
We have outlined a numerical procedure for efficiently solving the scattering from complex structures in which di-electric and conducting media are assumed to coexist. Our strategy relies on two steps: firstly, we formulate the problem with LEGO (by modelling the structure with a set of EM bricks), and secondly, we invert the relevant IEs by applying the MoM with a set of ABFs. LEGO allows reducing the complexity of the original structure, as the EM problems within each brick are simpler than the original one. Since the bricks (i.e., their scattering operators) can be re-used to model other structures, LEGO turns out very efficient, as compared to a direct approach. Furthermore, we have shown through selected examples that the number of ABFs (required to accurately solve a problem) is far smaller than the number of
0.2 0.4 0.6 0.8 1 30 150 60 120 90 90 120 60 150 30 180 0 φ = 0° φ = 180°
Fig. 11. Example of application: radiation pattern (normal incidence, E-plane) of FSS-2 at𝑓 = 7.25 GHz.
TABLE II
SIZE OF SYSTEMS(2), (4)AND COMPUTATION TIMES RELATIVE TO THE
FSSAPPLICATION EXAMPLE
2𝑁𝐹𝑁𝐷 𝑁𝐴 𝑡𝐴𝐵𝐹/𝑁𝐴(†) FSS-1 (Fig. 8) 216,000 90 32.1 s FSS-2 (Fig. 10) 216,000 320 29.8 s (†) On a Linux-based x86 64 workstation endowed with an Intel Xeon 2.66-GHz processor and 8-GB RAM.
RWG functions introduced in conjunction with the underlying MoM. Therefore, we can invert the reduced system by LU factorization and thus get rid of slow-convergence issues (if any) that may afflict iterative solvers.
ACKNOWLEDGEMENTS
This research was supported by the TU/e project no. 36/363450, and performed in the framework of the MEMPHIS project (http://www.smartmix-memphis.nl).
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0 50 100 150 200 250 300 10−15 10−10 10−5 100 Index of ABFs (s) |a s |/max|a s | FSS−2 @ f = 7.25 GHz FSS−1 @ f = 13 GHz
Fig. 12. Example of application: ABFs coefficients for FSS-1 and FSS-2 described in the text (see Figs. 8, 10) at one of their respective resonances.
