Simulation of Complex Multiscale Objects in Half Space ... - IEEE Xplore

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Simulation of Complex Multiscale Objects in Half Space With Calderón Preconditioner and Adaptive Cross Approximation Yongpin P. Chen, Wan Luo, Zaiping Nie, and Jun Hu

Abstract—Simulation of electromagnetic scattering by complex multiscale objects in a half space is addressed in this communication. The objects under investigation can be situated above, embedded in, or even penetrating the interface of the half space. In this analysis, the electric field integral equation is formulated with the kernel of a half-space Green’s function. The Calderón preconditioner is employed to improve the convergence of the Krylov-subspace iterative solvers. A new preconditioning operator is proposed to substantially reduce the construction cost, yet maintain the effectiveness. Moreover, a novel multilevel implementation of the adaptive cross approximation (ACA) algorithm is further adopted to make the method capable for problems with moderate size. The proposed scheme is examined by several multiscale problems, which constitute the most common scenarios in the near-surface applications. Index Terms—Adaptive cross approximation, Calderón preconditioner, Green’s function, half space, method of moments (MoM), near-surface scattering.

I. INTRODUCTION Half space model is of great importance in the simulation of electromagnetic scattering where the proximity of the ground or ocean surface must be taken into account. Such near-surface applications include: geophysical exploration, nondestructive testing, target identification, and remote sensing, etc. The method of moments (MoM) based on surface integral equations is one of the most favorable methods for such analysis since the unknowns can be associated with the boundary of the object. With the help of the half-space (or layered-medium) Green’s function [1], [2], the radiation condition and the half-space boundary condition are both satisfied. However, the MoM analysis of complex multiscale objects is very inefficient, prohibiting its popularity in the near-surface applications. The main reason is that for an -unknown and the computational problem, the memory requirement is for iterative solvers. Meanwhile, the matrix complexity is also system is usually ill-conditioned due to the dense or nonuniform discretization of the complex boundaries, resulting in an extremely slow convergence. The additional difficulty for the half-space modeling is the numerical evaluation of the Green’s function, which consists of computationally expensive Sommerfeld integrals [3]. The electric field integral equation (EFIE) is applied due to its accuracy and versatility. However, the spectrum of the EFIE operator is undesirable, and the resultant matrix system is usually ill-conditioned. For a real near-surface application with multiscale features, this is even a more severe problem. Therefore, preconditioning technique is imperative in this kind of simulation. The recently developed preconditioner Manuscript received April 29, 2014; revised September 22, 2014; accepted September 26, 2014. Date of publication October 02, 2014; date of current version November 25, 2014.This work was supported in part by NSFC grants 61201002, 61231001, 61271033, 6142500243, in part by 111 Project B07046, and in part by IRT1113. The authors are with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2014.2361152

based on Calderón identity turns out to be one of the most promising candidates for such purpose [4]. The identity reveals that the square of plus the EFIE operator equals to a scaled identity operator a compact MFIE operator [5]. Therefore, the unbounded and ill-posed spectrum of the EFIE operator can be manipulated to be well-behaved. This preconditioner has received intensive study in various applications, from frequency domain [4], [6] to time domain [7], and from free space [8], [9] to a layered medium [10]. On the other hand, in order to reduce the computational complexity and memory requirement of MoM, fast integral-equation algorithms are indispensable. Though there are plenty of fast algorithms developed in the last two decades, most of them are mainly designed for free-space Green’s functions [11], [12]. The extension to a half-space problem is by no means trivial due to the very different kernels involved [13]–[15]. One common restriction of these extensions is that the object has to be totally within one region or layer. For general problems, however, the kernel-independent algebraic algorithms are considered to be more promising [16], [17]. The goal of this work is to provide an efficient scheme for complex multiscale simulation in a half space. The objects under investigation can take arbitrary positions; they can be situated above, imbedded in, or even penetrating the interface of the half space. Therefore, this method can be applied in general near-surface applications. In the following, the Calderón preconditioner is first investigated for general half-space problems. A simple preconditioning operator is proposed to substantially reduce the construction cost, and yet maintain the effectiveness. After that, a multilevel ACA implementation suitable for objects with arbitrary locations in a half space is discussed. Finally, several representative examples are presented to validate the efficiency and accuracy of this scheme. II. CALDERÓN PRECONDITIONER FOR HALF-SPACE GREEN’S FUNCTION Consider the scattering of a perfectly electric conductor (PEC) in a half space. The half space consists of a lossless upper region (usually air) and a dielectric lower region, separated by a planar interface. The EFIE reads: (1) is the integral operator and its kernel can be found in [2], [3]. where To solve EFIE in (1) via MoM, the surface of the object is first discretized into planar triangular patches and the surface current is then [18]. expanded by Rao-Wilton-Glission (RWG) basis functions as the testing function, a matrix equation can After applying be obtained: (2) To assemble the matrix, numerical evaluation of the half-space Green’s function is required, which is computationally expensive. In this work, a tabulation and interpolation method is adopted to accelerate the computation [19]. For objects above or below the interface, are spatial functions the reflection Green’s functions . Hence, a series of two-dimensional (2D) sampling of and tables are pre-computed and stored. For general penetrating cases, the are functions of , and , transmission Green’s functions and hence a series of 3D tables are generated and stored. During the matrix assembly, the 2D and 3D Lagrange interpolations are invoked for the reflection and transmission terms respectively.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 12, DECEMBER 2014

As has been introduced, the matrix is ill-conditioned due to the undesirable spectrum of EFIE. This leads to a slow convergence of an iterative solver. When the object contains multiscale features, this problem is even more severe. To model the multiscale geometry properly, very dense mesh is required in some regions with fine structures, while normal discretization density is needed in other parts. Hence, the mesh density may be extremely nonuniform. To mitigate this convergence problem, a Calderón preconditioner for the half-space Green’s function is applied. The Calderón identity is derived as [10]: (3) where the integral operators are defined in the general field-source relations: (4) (5) The right-hand side of (3) is well-behaved since it is a scaled identity operator plus a compact operator. Hence, can be applied to . The preconditioned system of precondition the EFIE operator (1) is hence: (6) The discretization follows the scheme in a standard Calderón multiplicative preconditioner [4], where a Buffa-Christiansen (BC) basis functions is also adopted [21]. The definition and numerical implementation of BC basis is, however, more involved than the RWG basis since a barycentrically refined mesh is required in addition to the original mesh. After discretization, the system becomes (7) where is the Gram matrix. To make the discretized system well-conditioned, the Gram matrix has to be well-conditioned. Unfortunately, this is not the case for an extremely nonuniform discretization. Hence, a diagonal preconditioner is needed to precondition the Gram matrix [4]. However, an alternative solution is to utilize the normalized basis function so that the entries of the Gram matrix are independent of the discretization characteristics, such as the curvilinear basis used in [9]. Similarly, we will apply the normalized RWG [20] to overcome this difficulty. Different from the free space case, another impedance matrix evalis also required in the half space. Hence, the impleuated from mentation of the preconditioner is not compatible with the existing half-space EFIE code and its fast algorithm realization. Based on the instead for the preconanalysis in [10], we can heuristically adopt ditioner, namely (8) In this manner, the implementation of the preconditioned system in a half space is as convenient as in free space. However, the construction cost of the preconditioner is still fairly high due to the Sommerfeld integrals involved (even though with the tabulation and interpolation technique), which is not shared by its free-space counterpart. Since the direct term is always the primary and dominant contribution in the half-space Green’s function, we can further drop the reflection or transmission terms in the preconditioner such that (9) where “D” in analytic [2].

stands for the direct interaction, and its kernel is

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This may slightly sacrifice the compatibility of the code mentioned above but can substantially reduce the construction cost of the preconditioner. As will be shown in the numerical examples, this simple choice saves CPU time in the construction and yet maintains the effectiveness of the preconditioner in typical half-space problems. III. MULTILEVEL IMPLEMENTATION OF ACA HALF-SPACE PROBLEMS

FOR

GENERAL

The direct implementation of the half-space MoM is computationally expansive, limiting its validity within problems of small size. In this section, we will incorporate a multilevel ACA algorithm to make the method suitable for problems with moderate size. The memory and CPU time requirement of the ACA for dynamic problems are both of [16]. Though this complexity is higher than MLMFA [11], ACA provides more flexibilities for general half-space problems due to its purely algebraic nature. The core idea of ACA is to compress a sub-matrix in the well-separated interactions via a low-rank approximation: (10) where is a matrix, is a and is a matrix, satisfied. The matrices and can be with the relation constructed in an adaptive cross manner, and thus requires only partial knowledge of the original matrix . Hence, this acceleration does not depend on a priori knowledge of the Green’s function, and can be achieved for arbitrary source-observation positions, as long as the corresponding sub-matrix has a deficient pseudo-rank. In our implementation of the multilevel ACA algorithm, the oct-tree data structure in MLFMA is also utilized. When the object is completely in one half-space region, the grouping strategy follows that of MLFMA [11]. For nearby interactions in the finest level of the oct-tree, the ranks of the interaction matrices are not deficient, and direct evaluation is required. For well-separated interactions, however, the matrices can be compressed due to the rank deficiency. In matrix notation, the multilevel partitioning of the original MoM matrix is:

(11) denotes the number of nonempty where is the number of levels, is the number of the boxes in the wellboxes in level , and separated interaction list of cube in level . For penetrating or straddling cases, the object is naturally separated into two parts by the interface. Due to the distinct material contrast, different mesh densities are required according to the wavelength in each region. Hence, there are multiscale features from both the geometry of the object and the material contrast of the background. In this situation, the two half-objects in the upper and lower regions are enclosed by two independent big boxes and subdivided individually, leading to two independent oct-trees. For the direct and reflection terms, the ACA is implemented within their own oct-trees. For transmission interactions, however, the well-separated interaction list shall be carefully constructed since the number of levels of the two trees are generally different. For this case, we developed a -dependent grouping strategy to redefine the well-separated interactions and calculate the mutual couplings properly [17]. IV. NUMERICAL RESULTS Several numerical results are presented in this section to validate the efficiency and accuracy of the proposed scheme. The computational cost and the performance of the proposed preconditioner are first tested by the scattering of a sphere in a half space. Three typical examples with multiscale features in the near-surface applications are then

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Fig. 1. Comparison of the filling time in the example of a sphere above the half space.

Fig. 3. Number of iterations versus relative permittivity of the lower region in the example of a sphere above the half space.

Fig. 2. Comparison of the solution (iteration) time in the example of a sphere above the half space.

Fig. 4. Number of iterations in the example of a sphere below the half space.

demonstrated. In all the cases, double precision is adopted and the generalized minimal residual (GMRES) algorithm [22] is chosen as the . All the iterative solver. The relative residue error is set to be simulations are run on an AMD two quad-core 64-bit operation workstation with 3.3-GHz CPU. A. Computational Test of a Sphere above/below a Half Space A sphere of radius 1 m is firstly located 0.25 m above a half space and . An with the lower region material of MHz plane wave is incident from and . The sphere is discretized by several different mesh densities, resulting in the number of unknowns from 6,036 to 52,062. The matrix filling (setup) time of the ACA accelerated preconditioned systems are shown in Fig. 1, compared with that of the traditional MoM system based on EFIE. As can be seen, the filling time can be drastically reduced by the ACA algorithm. Also, the cost of the previously suggested system in (8) [denoted -ACA”] can be further reduced by the proposed system as “CP in (9) since the Sommerfeld integrals are completely absent from the preconditioning operator. Fig. 2 shows the comparison of the solution (iteration) time. The two preconditioned systems with ACA perform similarly and requires much less CPU time than the traditional system without a preconditioner. Next, the effect of the lower region material on convergence is tested to show the robustness of the preconditioner. One set of the mesh (with unknowns) is adopted and the sphere is moved downward to touch the interface. The number of iterations of the three systems versus the relative permittivity of the lower region is shown in Fig. 3. It is shown that both preconditioners perform similarly and is much better than the one without preconditioner. A buried case is further tested by moving the sphere downward to be completely submerged in the lower MHz so that the same sets region. The frequency is scaled to

Fig. 5. A tank model is situated on the ground. The nonuniform mesh is shown in the figure, where the detailed feature of the monopole antenna is magnified.

of mesh can be reused. The number of iterations is shown in Fig. 4 and again the effectiveness of the preconditioner can be observed. B. Scattering from a Tank Model on the Ground The tank model is shown in Fig. 5. It is situated on the ground with and . The tank is 6.2 m in length, 3.74 m in weight, and 1.9 m in height. As shown in the figure, the model contains a thin monopole antenna, and thus requires a refined mesh in this , it is at the region. The mesh size of the main body is for the antenna, where is the wavelength in gun part, and free space. An incident wave with frequency MHz is from and . The object is modeled with 12,128 planar triangular patches, resulting in 18,192 unknowns. The ACA accelerated preconditioned system in (9) and (8) are both tested, with comparison to the traditional MoM implementation. The current distribution is shown in Fig. 6 to show the accurate molding of the localized fine structure.

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Fig. 6. Current distribution of the tank model (dB).

Fig. 9. A landmine model is embedded in the ground. The mesh is shown in the figure, with side, top and bottom views.

Fig. 7. Accuracy test. The bistatic RCS of the tank model calculated from different methods, validated by FEKO.

Fig. 10. Current distribution of the landmine model (dB).

Fig. 8. Convergence comparison of different methods in the example of tank scattering. TABLE I COMPARISON OF COMPUTATIONAL COST

Fig. 11. Accuracy test. Near field distribution of the landmine model calculated from different methods, validated by FEKO.

C. Scattering from a Landmine Embedded in the Ground

The HH polarized bistatic RCS is shown in Fig. 7, where the accuracy is validated by the reference data from FEKO [23] simulation. The convergence history is shown in Fig. 8. The two preconditioned systems quickly converge and have similar performance, while the traditional MoM without a preconditioner can only converge to after 18, 192 steps. The computational cost are listed in Table I. As can be seen, the preconditioned systems with ACA are superior to the traditional method. It is also observed that the newly proposed preconditioner in (9) further reduces the construction cost of the previous one in (8) while maintaining high effectiveness. In the following, only the preconditioned system in (9) will be further tested.

In this example, the object is embedded in the ground, as shown in Fig. 9. The diameter of the object is 27 cm and the height is 8.6 cm. It is embedded 50 cm below the interface. The frequency of the incident and with polarizawave is 600 MHz, coming from tion angle . The material of the ground is the same as the preand vious example. The main body of the model is meshed with , where is the dielectric wavethe fine edges are meshed with length in the ground. The current distribution is show in Fig. 10 and the near field along the observation line is collected in Fig. 11. Again, the results of the proposed scheme agree well with the traditional MoM, and also agree with the reference data from FKEO simulation. The proposed system converges in 14 steps, while it takes 1,414 steps for the traditional method to converge.

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Fig. 12. Current distribution of the partially buried cone sphere with a gap (dB). The current is shown from side views and bottom view.

D. Scattering from a Cone Sphere with a Gap Inserting into the Ground A standard benchmark model, the cone sphere with a gap is further tested in this example. The model is vertically straddling the interface. m in length, and half of the cone is buried in the ground It is (with material the same as before). This object is under the illuminance MHz plane wave with normal incidence. The mesh size of a at the tip to at the main body, resulting varies from in 26,859 unknowns. The computed current distribution is shown in Fig. 12 from side and bottom views, where the smaller wavelength in the lower region can be easily observed due to the denser surrounding material. The accuracy of the near field and far field are also validated by the reference data and are no longer shown here. In the computation, the proposed system converges in 42 steps while the number is 3,573 for the traditional system. V. CONCLUSION This work provides an efficient scheme to address the simulation of complex multiscale objects in a half space. The electric field integral equation with the half-space Green’s function is adopted due to its accuracy and versatility. The Calderón preconditioner is investigated in the half-space problems, where a new simple implementation is proposed to reduce the construction cost and yet maintain the effectiveness. A novel multilevel implementation of the adaptive cross approximation algorithm is further invoked to accelerate the computation for general half-space problems, where the objects can be above, below or penetrating the interface. Representative examples in the near-surface applications are demonstrated to validate this method.

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