Abstract
The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex source beam (MLCSB) method. As an effective solution of the low-frequency problem, A-EFIE includes both current and charge as unknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE. However, dense impedance submatrices are involved in the A-EFIE system, and the computational cost becomes extremely high for problems with a large number of unknowns. As an exact solution to Maxwell’s equations, the complex source beam (CSB) method can be well tailored for A-EFIE to accelerate the matrix-vector products in an iterative solver. Different from the commonly used multilevel fast multipole algorithm (MLFMA), the CSB method is free from the problem of low-frequency breakdown. In our implementation, the expansion operators of CSB are first derived for the vector potentials and the scalar potentials. Consequently, the aggregation and disaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity. The accuracy and efficiency of the proposed method are discussed in detail through a variety of numerical examples. It is observed that the numerical error of the MLCSB-AEFIE keeps constant for a broad frequency range, indicating the good stability and scalability of the proposed method.
1. Introduction
The method of moments (MoM) [1] for solving electric field integral equation (EFIE) has received intensive study in the analysis of electromagnetic (EM) radiation, scattering, and circuit problems in recent years. However, the commonly used EFIE with RWG basis function suffers from a “low-frequency breakdown” problem [2] when the frequency decreases and/or when the mesh is refined. In this situation, EFIE is dominated by the scalar potentials while the contribution from the vector potentials is overwhelmed due to the finite machine precision. Since the scalar potential term is singular, the EFIE matrix system becomes extremely ill-conditioned and results in convergence issue when the iterative solver is applied.
To address this difficulty, several methods have been proposed in the past years. A widely used remedy is the loop-tree or loop-star decomposition [3] which decouples the magneto- and electrostatic physics. However, the extraction of global loops is difficult for complex interconnecting structures, and the system matrix becomes ill-conditioned as the frequency increases. On the other hand, the convergence issue can be improved by a Calderón preconditioner [4] when iterative solvers are applied. To avoid the involved loop searching process, alternative formulations have been proposed by separating current and charge, such as the current and charge integral equation (CCIE) [5], split potential integral equation (SPIE) [6, 7], and augmented electric field integral equation (A-EFIE) [8–11]. In the A-EFIE method, the contributions of the vector potentials and the scalar potentials are separated by adding charge as extra unknowns, where the current continuity equation is explicitly enforced. Frequency scaling is then implemented to stabilize the system equation. The resulting impendence matrix of A-EFIE has the characteristics of a saddle point matrix and is well-conditioned at low frequencies. Recently, the perturbation method is introduced to enhance the accuracy of A-EFIE at extremely low frequencies [12]. The augmented equivalence principle algorithm (A-EPA) [13] and discontinuous Galerkin (DG) method [14] are combined with the A-EFIE for domain decomposition problems.
As the number of unknowns increases, a fast algorithm has to be incorporated into the iterative solver to reduce the operation complexity of the matrix-vector product (MVP). Such fast algorithms should be low-frequency stable; for instance, the low-frequency multilevel fast multipole algorithm (LF-MLFMA) [15], the multilevel accelerated Cartesian expansion algorithm (MLACEA) [16], the multilevel adaptive cross-approximation (MLACA) algorithm [17], and fast Fourier transform (FFT) [18] are eligible candidates for such purpose. Recently, a complex source beam-method of moment (CSB-MoM) is proposed to accelerate the far-field interactions of MoM at midfrequencies [19]. The object is first divided into groups and complex source beams (CSBs) are used to expand the fields of the basis functions residing in each group [20, 21]. A multilevel version of this method is developed in [22]. To further improve the computational efficiency, a nested complex source beam (NCSB) method is proposed by utilizing an equivalent relationship between adjacent levels [23]. Since the CSBs are exact solutions of Maxwell’s equations, any arbitrary EM fields can be expanded in terms of a set of CSBs [20, 24]. Therefore, this method can be extended to solve the low-frequency problems without any theoretical barriers.
This paper is organized as follows. The basis formulations of CSB-MoM and A-EFIE are briefly reviewed in Section 2. In Section 3, the CSB-MoM is integrated into the A-EFIE system to remedy the low-frequency breakdown problem. The detailed derivation of the CSB expansions for A-EFIE is first presented. The aggregation/disaggregation, translation operators for a multilevel CSB method are then discussed. Finally, numerical examples are summarized in Section 4 to demonstrate the validity and efficiency of the proposed method.
2. Theory Background
Given a 3D perfectly electrical conducting (PEC) body defined by its surface, the conventional MoM formulation can be applied to the electric field integral equation (EFIE), leading to a matrix equation of the formwhere is the unknown vector for the surface current density, is the excitation vector, and is the dense impedance matrix. The impedance matrix can be expressed aswhereis the dyadic Green function and is the scalar Green function. Moreover,anddenote the RWG basis functions for expanding surface current and Galerkin testing, respectively. This equation suffers from the low-frequency breakdown problem, because the contribution of the vector potential is swamped by that of the scalar potential at low frequencies due to the finite machine precision.
2.1. Formulation of CSB-MoM for EFIE
In CSB-MoM method, the MVP acceleration of is achieved through a FMM-like near-field and far-field decomposition. For the far-field interactions between well separated groups, the MVP are carried out by a series of CSBs launched on a complex equivalence surface enclosing each group:where and denote the observation and source groups and indicates the finest level (single level in CSB-MoM). and are the expansion matrices for both and components, which can be obtained by the far-field matching of basis functions in groups and , respectively [19]. Superscript stands for the transpose of the corresponding matrix. It needs to be emphasized that only and components are considered in CSBs, since the far-field radiation fields contain only and components.
By using the expansion matrices, the CSB expansion coefficients for the source group can be expanded from the surface currents as
in (4) is the translation matrix, of which the elements are expressed aswhere and denote the unit vectors of th and th CSB, respectively. The complex position vectors and are the launch points of CSBs associated with groups and . is the dyadic Green function with complex arguments. The directional property of CSBs can be used to reduce the computational cost in the translation procedure.
2.2. Formulation of A-EFIE
In the augmented electric field integral equation, the surface current is discretized by the normalized RWG basis function, which is modified by removing the length of the common edgewhere are the area of triangles, are the free vertices of the two triangles. On the other hand, the surface charge density is approximated by the pulse function defined on each triangle, which is expressed as
Combining EFIE in (1) and the current continuity condition between current and charge, we can arrive at the A-EFIE equationwithwhere and denote the unknown coefficients for current and charge, and are the wave number and the wave impedance in free space, andis the light speed in free space. The dense matrix represents the vector potentials and depicts the current interactions between inner edges. On the other hand, denotes the scalar potentials, which describes charges interactions between triangles. The sparse matrix represents the relationship between edge and the triangles patch, and is the identity matrix. In the matrix system (9), the vector potential and scalar potential are balanced by using a proper frequency scaling, which is critical for low-frequency problems.
3. Implementation of MLCSB for A-EFIE
3.1. CSB Expansions for A-EFIE
Motivated by the idea of CSB expansion method for EFIE, we can derive the expansions for A-EFIE in a similar way. The inner integrals of (10) and (11) can be written as where and represent the th normalized RWG basis function and the th pulse function in group . The above integral equation is equivalent to the summation of CSB vector and scalar potentials: where is the vector CSB weight for current , is the weight for charge, and represents the total number of beams in this discretization. Combining (16) with (14) and (17) with (15) and testing on a far-field matching point, we getwhere () is the matching point. Those linear equations can be converted into a matrix form and combined as a multiple right-hand side problem by all the current and charge basis in the group :where
By using the expansion matrices , and , the CSB expansion coefficients for vector potentials of group can be expanded from as
Similarly, the coefficients for scalar ones are
Hence, the far-field MVP procedure for (10) and (11) can be obtained with a similar process in (4):where the translation operator is expressed asIt is noticed that the scalar Green function with complex arguments is used here, and the translation for the vector and scalar expansion coefficients shares the same operators.
3.2. Multilevel Algorithm of CSB for A-EFIE
So far, we have derived the CSB expansion method for A-EFIE and have presented the far-group MVP process with the help of translation operators in a single level algorithm. In the following, we will obtain the aggregation and disaggregation operators of CSB expansion coefficients for both (10) and (11), to realize a multilevel algorithm. Firstly, the vector and scalar potentials by the th CSB in a group is where means the th level of the octree in the multilevel algorithm. By applying the summation and testing procedures similar to (16)–(18), a linear system can be set up to obtain the equivalent relationship of CSBs between two adjacent levels:where is the matching matrix in parent level . This matching matrix connects the equivalent CSB sources to the scalar potentials . Different from the right-hand side in (19), the right-hand side here for each vector potential component is the same as the scalar one, which is
Once and are assembled, the aggregation matrix for level can be numerically solved. By using the aggregation matrix, the CSB expansion coefficients of the parent group in levelcan be obtained efficiently from its child groups in level + :
Similar to the multilevel fast multipole algorithm (MLFMA) [25], the CSB expansion coefficients of a receiving group in level are obtained from the translation in the same level as well as the disaggregation from its parent level . The disaggregation matrix can be easily obtained from the transpose of the aggregation matrices.
4. Numerical Results
In this section, the accuracy, error analysis, computational complexity, and the efficiency of the method are investigated through several numerical examples. All the examples were run on a computer of 2 processors, each with 14 cores at 2.6 GHz, 512 GB memory, and OpenMP parallelization.
4.1. Small Sphere
To show the accuracy of MLCSB-AEFIE at low frequencies, the electromagnetic scattering by a PEC sphere of 1 m radius is analyzed at 100 Hz. The sphere is discretized with 1,764 triangular patches, which corresponds to 2,646 inner edges. A three levels’ MLCSB algorithm is used with a group size of at the finest level. The incident angle of a plane wave is , and the observed azimuth angle is fixed at . The residual error threshold is set to be for GMRES-30. It takes 80 iteration steps to converge, with the help of the saddle point preconditioner in [9]. A good agreement of the bistatic RCS is observed in Figure 1, as compared with the analytical solution of Mie series.
4.2. Computational Complexity
To demonstrate the computational complexity of the proposed method, the plane wave scattering of a PEC cube with a side length of 0.1 m is calculated at 300 MHz. The electric size of the cube is . The surface of the cube is discretized into six different meshes: Mesh A, Mesh B, Mesh C, Mesh D, Mesh E, and Mesh F. The coarsest Mesh A has 1262 planar triangles and 1893 interior edges, and the average edge length is . Then we refine Mesh A by halving the edge length recursively until Mesh F. Mesh F comprises 1,534,536 planar triangles and 2,301,804 interior edges, and the average edge length is .
The CPU times for each MVP and the memory usage is plotted in Figure 2. It can be found that the computational complexity and the memory requirement of MLCSB-AEFIE both scale as , where are the unknown numbers. The detailed computational statistics are summarized in Table 1, and the iteration histories are shown in Figure 3 for comparison. By using the preconditioner in [9], the iteration converges quickly to . A good agreement of the bistatic radar cross sections (RCS) of the five cases is demonstrated in Figure 4.
4.3. Error Analysis
In the proposed MLCSB-AEFIE, the numerical error is mainly from expansion and aggregation. Since the expansion and aggregation processes are similar for the vector and scalar potentials, we only show the scalar potential case in the following. For the expansion, the relative errors related to a group with the size of 0.005 m and average discretization length of 0.001 m are studied. The relative error of expansion is defined aswhere is the scalar potential obtained via the expansion process in (17), is the exact data calculated by (15), and the observation point is fixed at the center of the nearest cousin group. Figure 5 shows the relative errorsfor different numbers of CSBs , 42, 58, and 82 with respect to the frequencies from 30 Hz to 300 MHz. It is noticed that the expansion errors are almost constant for the same in a very wide frequency range, indicating an excellent stability and scalability of the proposed method for broadband computations.
To investigate the interpolation error in the aggregation process, we compare the scalar potentials related to a group with the size of 0.005 m and the parent group with the size of 0.01 m. The scalar potentials are calculated by CSBs, while the CSB coefficients of the parent group are aggregated from the child group. The RMS relative error is shown in Figure 6, which is defined asand here , are the scalar potentials of the parent group and the child group, respectively, and is set on azimuth circle centered at the parent group. Figure 6 also shows a constant accuracy level for different frequencies with the same .
4.4. Scattering of Multiscale Cone
The performance of the MLCSB-AEFIE for multiscale structures is evaluated by a nonuniformly meshed cone illustrated in Figure 7. The radius of the cone is 5 m at the bottom and the height is 10 m. We mesh the cone with a length of 0.1 m at the bottom and gradually reduce it to 0.001 m at the sharp point. Finally, the cone is discretized with 191,976 planar patches, and hence 287,964 interior edges. Figure 8 shows the bistatic RCS of the cone excited by a -polarized plane wave incident from thedirection atKHz and. In this example, MLCSB-AEFIE converges to relative residual error of within and iterations, for 10 KHz and 10 MHz,respectively.
5. Conclusion
In this paper, we have proposed a MLCSB-AEFIE method for the well-known low-frequency problem. The vector and scalar potentials from the current and charge unknowns are expended with CSBs. An aggregation matrix is obtained for the CSB expansion coefficients to form an efficient multilevel algorithm. Numerical examples have validated the good accuracy, efficiency, and scalability of the proposed method for low-frequency problems.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported partly by the National Excellent Youth Fund by NSFC no. 61425010, Programme of Introducing Talents of Discipline to Universities under Grant no. b07046, the Chang Jiang Scholar Project of MOE, and the National Natural Science Foundation of China under Grant no. 61490695.