Research Article  Open Access
D. Z. Ding, Y. Shi, Z. N. Jiang, R. S. Chen, "Augmented EFIE with Adaptive Cross Approximation Algorithm for Analysis of Electromagnetic Problems", International Journal of Antennas and Propagation, vol. 2013, Article ID 487276, 9 pages, 2013. https://doi.org/10.1155/2013/487276
Augmented EFIE with Adaptive Cross Approximation Algorithm for Analysis of Electromagnetic Problems
Abstract
The augmented electric field integral equation (AEFIE) with charge neutrality enforcement provides a stable formulation to conquer lowfrequency breakdown characteristic of conventional EFIE. It is augmented with additional charge unknowns through current continuity equation. The AEFIE combined with the multilevel adaptive crossapproximation (MLACA) algorithm is developed to further reduce the memory requirement and computation time for analyzing electromagnetic problems. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.
1. Introduction
Electromagnetic integral equations are often discretized with the method of moments (MoM) [1, 2], one of the most widespread and generally accepted techniques for electromagnetic problems. The integral equation is first discretized into a matrix equation using the Galerkinbased MoM with subdomain basis functions such as rooftop functions [3] for curvilinear quad patches and RaoWiltonGlisson (RWG) functions [4] for triangular patches. It is convenient to model objects with arbitrary shape using triangular patches; hence, RWG functions are widely used for representing unknown current distributions. When iterative solvers are used to solve the MoM matrix equation, the fast multipole algorithm (FMA) or multilevel fast multipole algorithm (MLFMA) [5–8] can be used to accelerate the calculation of matrixvector multiplies in iterative algorithm.
However, the electric field integral equation (EFIE) only works well at midfrequencies due to the wellknown lowfrequency breakdown. To circumvent this disadvantage, some methods have been proposed to remedy lowfrequency characteristic of electrical field integral equation (EFIE) [9–12]. Wilton and Glisson proposed looptree methods for the Helmholtz decomposition feature at low frequency regime [10]. Mautz and Harrington proposed a loopstar method and analyzed the reason of instability for integral equation in lowfrequency regime [11]. Lim et al. introduced loopstar method into RWG functions [12]. Wu et al. provided the comparison between the looptree method and loopstar method for the EFIE [13]. However, the loopstar or looptree basis functions only partly remove the problem at very low frequencies. Furthermore, the resulting system matrix in MoM is still illconditioned leading to slowly converging iterative solutions. Zhao and Chew utilized a basis rearrangement technique [14] to speed up convergence speed of the iterative solver, but it is also time consuming.
A wellconditioned EFIE (WEFIE) to overcome the shortcomings of the EFIE has been proposed in [15–23]. Due to the fact that the square of the EFIE operator does not have eigenvalues accumulating at zero or infinity, the WEFIE can give rise to a wellconditioned EFIE system independently of the discretization density [15, 17–19]. An efficient Calderón preconditioner is introduced based on the Calderón identities for scattering by perfect electric conductor (PEC) [16, 20–23]. The Calderón preconditioner is made purely multiplicative by using the BuffaChristiansen (BC) basis functions. It has also been used to accelerate the convergence rate of the iterative solution of the PoggioMillerChangHarringtonWuTsai (PMCHWT) equations for wave scattering by homogeneous dielectric objects [24].
Some other fast methods have also been proposed to cover the regime from low frequency to mid frequency, such as the plane wave methods based on the generalized Gaussian quadrature rules [25] and the lowfrequency fast inhomogeneous plane wave algorithm (LFFIPWA) [26]. Since evanescent waves are highly direction dependent, much memory is required for LFFIPWA. The fast multipole algorithm is numerically unstable due to the oscillatory characteristic of the spherical Hankel function for small arguments. As an attempt for a possible remedy, the mixedform fast multipole algorithm (MFFMA) [27] is proposed to cover a wide band from the low frequency to mid frequency. However, it is noticed that the MFFMA need to construct multipole expansions for each nonempty box at the lowfrequency regime. This results in a source distribution of highdensity for the whole object. The adaptive MFFMA (AMFFMA) is proposed for efficient analysis of electromagnetic scattering from objects containing fine structure [28]. The AMFFMA applies MFFMA for some parts of the object which belongs to the lowfrequency regime and the rest of the object which belongs to the midfrequency regime is still analyzed by MLFMA. Hence, for a cuboid with many fine structures in [28], AMFFMA takes advantage of adaptive grouping scheme to set the finest level box size as 0.05λ for fine structure, while the finest level box size is 0.2λ for the rest of the object. Much memory can be saved for AMFFMA compared to MLFMA.
The augmented electric field equation (AEFIE) [16, 29–33] method has been proposed to solve the lowfrequency problem without the looptree decomposition. As a new fullwave method, AEFIE provides a stable formulation over a wide frequency band. In AEFIE, the contributions of the vector potential and the scalar potential are separated; thus the imbalance between them in the conventional EFIE is avoided. It is augmented with additional charge unknowns through current continuity equation. Qian and Chew emphasize the charge neutrality enforcement (CNE) to remove the rankdeficient probelm in AEFIE [30]. The complex interconnect packaging problem with over one million unknowns is solved without the help of parallel computers. Qian and Chew also proposed a perturbation method for solving the possible lowfrequency inaccuracy problem of AEFIE [31]. Chen et al. extended the AEFIE method into the layered medium problem [33]. Yan et al. proposed a Calderón preconditioned AEFIE algorithm (CPAEFIE) for analysis of scattering by electrically large objects [16]. The resonance problem of the Calderón preconditioned EFIE is removed by introducing a reasonable complex wavenumber into the Calderón preconditioner [16].
Though the Calderón preconditioning method can eliminate the lowfrequency breakdown and the densegrid breakdown, it is highly technical. For the MLFMA and its lowfrequency extensions, however, a priori knowledge of Green’s function is needed. As a result, it cannot be easily applied to analyze the layered media problems. Compared to the MLFMA, the multilevel adaptive crossapproximation (MLACA) algorithm [34–36] is another popular technique for analyzing scattering/radiation problems. It makes use of the wellknown fact that the approximate rank of the submatrices is low when the subscatterers are sufficiently separated. The MLACA is purely algebraic and therefore does not depend on a priori knowledge of Green’s function. In this paper, the MLACA algorithm is introduced into the AEFIE with CNE to accelerate the matrixvector multiplies in iterative algorithm. The ACA algorithm is an adaptive and onthefly rankrevealing block factorization of the rankdeficient submatrices. Compared to the singular value decomposition (SVD) technique, the ACA algorithm only requires partial knowledge of the original matrix. It has been shown that for moderate electrical size problems the memory and CPU time requirements for the ACA algorithm scale as O( log N) [35]. However, the setup time of the MLFMA is less than that of ACA due to the fact that the MLFMA reuses multipole and local expansion information across levels. Jiang et al. utilize the predetermined interaction list supported octree (PILOT) [36, 37] to reduce the setup time and the memory consumption of the ACA algorithm.
The aim of this paper is to apply the MLACA algorithm to improve the efficiency of the AEFIE with CNE for analyzing electromagnetic problems. This paper is organized as follows. Section 2 describes the theory and implementation of the AEFIE combined with the ACA algorithm in more details. Section 3 gives some numerical examples to demonstrate the accuracy and efficiency of our approach. Section 4 gives some conclusions.
2. AEFIE with MLACA Algorithm
2.1. AEFIE Formulation
The conventional EFIE is given as where the is free space wave impedance and is the wavenumber. denotes the righthand vector (excitation). is the unknown current. The and correspond to the magnetic vector potential and electric scalar potential: where the subscripts , denote the interaction between the th line and the th line. denotes free space scalar Green’s function. denotes RWG basis function. and represent the relative permeability and the relative permittivity, respectively. In AEFIE, we use a normalized RWG function to expand the surface current by removing the edge length from the original RWG basis function: where the is the area of the triangle and is the free vertices of the triangle pair. Then, we use pulse function to expand the charge on each element which is defined as follows: The expansion equations of surface current and charge are given as where the is the number of inner lines and is the number of mesh elements. and are the unknown expansion coefficients for surface current and charge, respectively. According to the current continuity equation: we define a transform matrix as follow [29]: Then, the scalar potential matrix can be factorized as where the denotes a patchtopatch scalar potential matrix as And we substitute (5) and (6); into (7) we get where the is the light speed in vacuum. Substituting the above equations into the EFIE matrix equation and enforcing the current continuity equation explicitly, we can arrive at the following AEFIE system. So, we get the form of AEFIE as where the is an identity matrix with dimension of .
At very low frequencies, charge neutrality causes rank deficiency in the above forms of the AEFIE [30]. Therefore, we drop a charge unknown from each unattached objects due to the charge neutrality. Then, the vector of charge is reduced to be . is the dimension of reduced charge vector and equal to ( denotes the number of disconnected objects for a problem). We define two mapping matrixes and : where the represents the previous charge density coefficient in (6). The matrix maps the full vector forward to the reduced one, and the matrix projects the reduced vector backward to the full one. Both of them are highly sparse matrices. Then, the AEFIE in (12) is given as where the is an identity matrix with dimension of . The linear system of equations in (14) can be solved by the restart GMRES iterative method with preconditioning technique to accelerate the convergence of the iterative solver [5, 6, 31]. A simple and efficient preconditioning matrix has been given as [31] where the is the diagonal matrix of . Since is a sparse matrix, its inverse matrix can be solved by fast sparse matrix solver such as UMFPACK solver in this paper.
For the analysis of microstrip antenna and microwave integrated circuit (MIC) in layered medium, free space scalar Green’s function in (2) should be substituted by layered media Green’s function [38]. In general, spatial domain Green’s functions are expressed in terms of Sommerfeld integrals. Due to the highly oscillatory nature of the integrand, numerical integration is very time consuming. In this paper, a twolevel generalized pencil of function (GPOF) method is utilized to realize DCIM [39]. Then, spatial domain Green’s functions can be obtained in closed forms from their spectraldomain counterparts via the Sommerfeld identity.
2.2. MLACA Acceleration
The MLACA employs the same octree data structure as in the MLFMA [7]. The octaltree algorithm is used to subdivide a box that encloses an object into smaller boxes. Figure 1 shows the decomposition of the problem domain at different levels. With reference to Figure 1, far interactions exit at levels 2 and higher. Far interactions can be computed using the MLACA [40].
(a)
(b)
(c)
(d)
According to the above octree data structure, the impedance matrix in (14) can be decomposed into nearfield interactions and farfield interactions . It can be simply written as where the and are computed directly and compressed by the MLACA algorithm, respectively. When the two boxes are sufficiently separated, the impedance matrix associated with them can be expressed using lowrank representations. This feature is utilized by the MLACA. In the MLACA implementation, the impedance matrix between two sufficiently distant boxes can be expressed in terms of two small matrices where the is the interaction matrix between the observation and source boxes. and denote the number of the basis functions in the observation and source boxes, respectively. The index denotes the rank of and is much smaller than and . The impedance matrix in (16) can be rewritten in multilevel operations as where the is the number of nonempty groups at level and denotes the number of far interaction groups of the th nonempty group for each observer group at level . Since the matrices and generated by the MLACA are usually not orthogonal, they may contain redundancies that can be removed by the SVD technique [41].
Finally, the linear system of equations in (14) can be solved by preconditioned restated generalized minimal residual algorithm (GMRES) using the MLACA to accelerate the matrixvector multiplies in iterative algorithm.
3. Numerical Experiments
In this section, we show some numerical results for electromagnetic structures that illustrate the effectiveness of the proposed AEFIE with CNE accelerated by the MLACA algorithm. All numerical experiments are performed on Intel(R) Core (TM) 2 Quad CPU at 2.83 GHz and 3.5 GB of RAM in double precision and the truncating tolerance of the ACA is (relative to the largest singular value). The restart number of the generalized minimal residual (GMRES) is set to be 30. The innerouter flexible GMRES algorithm (FGMRES) is used as the iterative solver for microstrip circuit problems.
Additional details and comments on the implementation are given below:(i)zero vector is taken as initial approximate solution for all examples and all systems in each example;(ii)the iteration process is terminated when the normalized backward error is reduced by for all the examples.
First, electromagnetic scattering from a perfect electric conducting (PEC) sphere is analyzed in order to demonstrate the advantages of the proposed AEFIE with the ACA algorithm (denoted as AEFIEACA). The sphere has a radius of 2 m. It is discretized with 7332 triangular patches leading to 18330 unknowns. The angle of incident plane wave is , and with the frequency KHz. Figure 2 shows the VVpolarization bistatic RCS solved by Mie series and AEFIEACA. It can be found that the AEFIEACA gives right results at low frequency. The condition numbers versus frequency are also demonstrated in Figure 3. It is observed from Figure 3 that the AEFIE with charge neutrality enforcement makes the condition number almost constant when decreasing the frequency to 1 Hz. The condition number grows slowly when increasing the frequency from 1 GHz to 10 GHz. Figure 4 further shows the VVpolarization bistatic RCS when the number of unknowns for the above PEC sphere becomes larger. It can be found that the proposed AEFIEACA works well when the number of unknowns increases from 19435 to 65130. However, when the number of unknowns is increased to 76480, the procedure for AEFIEACA cannot work due the densegrid breakdown problem. Table 1 shows the comparison of memory requirement and solution time of GMRES between the AEFIEACA and AEFIE without the ACA algorithm (denoted as AEFIEMoM) in bistatic RCS computation for the sphere. The 2level ACA is used to reduce the solution time and memory requirement in AEFIEACA. It can be found that the AEFIEACA can save much time by a factor of 7.9 than that of the AEFIEMoM. The AEFIEACA can also save much solution time by a factor of 3.0 than that of the AEFIEMoM.

Next, we investigate the performance of the proposed AEFIEACA on electromagnetic scattering at high frequency. The mathematic description of NASA almond is defined in [42] and shown in Figure 5. The total length of the almond is 9.936 inches. The almond is discretized with 5216 triangular patches with 13040 unknowns. The angle of incident plane wave is , and . Figure 5 gives the VVpolarization bistatic RCS for an almond at 3 GHz. It can be found that the results obtained by the AEFIEACA agree well with ones from traditional EFIE with ACA algorithm (denoted as EFIEACA). The 2level ACA is used in the AEFIEACA and EFIEACA. Table 2 shows the comparison of memory requirement and solution time of GMRES among different methods in bistatic RCS computation for the almond. It can be found that the AEFIEACA can save much memory requirement by a factor of 2.1 than that of the AEFIEMoM, 6.6 than that of the EFIE ACA. It can be also found that the AEFIEACA can save much memory requirement by a factor of 2.9 than that of the AEFIEMoM. However, the memory requirement for the AEFIEACA is more than that for the EFIEACA due to additional charge unknowns introduced into the AEFIE.

The third example is a microstrip bandpass filter (BPF) based on five intercoupled splitring resonators (SRR) proposed in [43]. The periodic SRR structure is shown in Figure 6. The distance between each SRR’s arms is 0.05 mm. The relative dielectric constant of substrate is 3.9. The structure is implemented on the substrate Corning 1059 with a relative dielectric constant of 3.9 and thickness of 0.37 mm. Microstrip line width is 0.8 mm. The width of the interdigital and the gap in each SRR structure are both 0.114 mm. The length of each SRR structure is 0.5 wavelengths at center frequency of 12 GHz. The structure is discretized with 924 triangular patches with 1077 unknowns. Frequency is from 5 GHz to 20 GHz. The sampled frequency interval is 0.1 GHz and a total of 151 frequencies need to be calculated in the proposed AEFIEACA. Figure 7 gives scattering parameters S11 and S12 obtained from the AEFIEACA method for the above BPF structure. It can be found that the results from AEFIEACA agree well with those from Ansoft Designer software. It is observed from Figure 6 that the BPF exhibits a passband region from 9.0 to 15.2 GHz, which corresponds to a fractional bandwidth of 49.6%. The insertion losses in the passband region are around 1.2 dB. Therefore, the simulated results confirm the feasibility of utilizing the intercoupled splitring resonator structure in microwave bandpass filter applications. The BPF is a typical electrical small structure when the wavelength is 0.015 m at 20 GHz. The condition number of impedance matrix is very bad due to the fine structure in the BPF with SRR structure. Therefore, the innerouter Flexible GMRES algorithm (FGMRES) is used as the iterative solver. In FGMRES, the nearfield impedance matrix is used as the preconditioner in the inner iteration. In the FGMRES algorithm, the stop precision is 1.E4. Table 3 shows the comparison of memory requirement and solution time among the AEFIEACA, AEFIEMoM, and EFIEACA methods for the BPF with SRR structure. It can be found that the proposed AEFIEACA can save much memory requirement by a factor of 2.54 than that of the AEFIEMoM. The AEFIEACA can also save much time for average solution time at each frequency by a factor of 1.48 than that of AEFIEMOM, by a factor of 39.4 than that of EFIEACA. Therefore, the AEFIEACA can save much time for the total iterative solution time by a factor of 39.5 than that of the EFIEACA.

The last example is a dualmode dualband BPF [44] as shown in Figure 8. The dualmode dualband BPF is implemented on the substrate Duroid 6010 with a relative dielectric constant of 10.2, loss tangent of 0.0023, and thickness of 1.27 mm. The related dimensions as shown in Figure 6 are determined as follows: mm, mm, mm, mm, mm, mm, mm, mm, and mm. The horizontal and vertical lengths of the slot are 9.2 mm and 6.2 mm, respectively. The size of the patch including stubs is , where is 48 mm, the guided wavelength at the lower passband. The simulated results obtained from the AEFIEACA method are given in Figure 9 compared to the results from Ansoft HFSS software. The BPF structure is discretized with 1476 triangular patches with 2070 unknowns. Frequency is from 1.5 GHz to 6.0 GHz. The interval of frequency is 0.1 GHz. Good agreement is achieved between these two kinds of results. It is observed from Figure 8 that the lower and upper passbands have the fractional bandwidth of 12.8% and 6.5% centered at 2.3 and 5.05 GHz, respectively. The simulated insertion loss is 0.73 dB and 0.86 dB, respectively. The return loss is better than 10 dB. Two transmission zeros are generated near the passband edges, guaranteeing high selectivity. It also can be found that two transmission poles in both passbands demonstrate the effectiveness of the dualmode dualband filter. The 3level ACA is used. The SVD is used to recompress the far interaction matrix in the AEFIEACA method. Table 4 shows the comparison of memory requirement and solution time among the AEFIEACA, AEFIEMoM, and EFIEACA methods for the BPF structure. It can be found that the proposed AEFIEACA can save much memory requirement by a factor of 2.44 than that of the AEFIEMoM. The average solution time at each frequency for the AEFIEMOM is about 1.34 times than that of of the AEFIEACA. Furthermore, the AEFIEACA can save much time by a factor of 31.3 than that of the EFIEACA. It can be concluded that the proposed AEFIEACA is very efficient compared to other methods.

4. Conclusions
In this paper, the AEFIE with CNE accelerated by the MLACA algorithm is presented for solving electromagnetic scattering and microstrip circuit problems. Numerical experiments are performed and the comparison is made with the EFIEACA and AEFIEMoM methods. It can be found that the proposed AEFIEACA method is more efficient and can significantly reduce the overall computational cost. Due to the fact that the setup time of the ACA is much more than that of MLFMA, it is known that the ACA does not maintain its favorable complexity once the problem becomes electrically large. The PILOT algorithm can be used to reduce the setup time and the memory consumption of ACA. The PILOT algorithm utilizes the idea that the farfield interaction lists of siblings share many common cubes to regroup a new farfield interaction list based on an octal tree data structure. Higher compression is achieved by using the PILOT algorithm when the dimension of the matrix is large. Further investigations deserve to be undertaken to study efficiency of the PILOT algorithm when the proposed AEFIEACA method is used for the analysis of electromagnetic problems from electrically large objects.
Acknowledgments
The authors would like to thank Jiangsu Natural Science Foundation (BK2012034) and Natural Science Foundation (61171041) for support.
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Copyright © 2013 D. Z. Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.