A characteristic mode basis function method for solving wide-angle electromagnetic scattering problems

The ultra‐wideband characteristic basis function method (UCBFM) is known as an effective technique for analyzing the wideband scattering problems, where the ultra‐wideband characteristic basis functions (UCBFs) can be reused for any sampling frequency in the range of interest. However, these UCBFs need to be constructed via an external excitation source, and the number of UCBFs is generally large for the increasing size of target, which leads to an increased size of the reduced matrix. To mitigate these problems, a universal characteristic mode basis function method (UCMBFM) is proposed in this study. Unlike the existing UCBFM, the construction of universal characteristic mode basis functions (UCMBFs) proposed in this study are only dependent on the impedance matrix, which is essentially determined by the material and shape of objects. Therefore, these UCMBFs are independent of frequency and can be reused at lower frequency band. Moreover, in proposed scheme, the number of UCMBFs and the size of reduced matrix are both less, when compared to traditional UCBFM. Finally, the corresponding numerical results demonstrate that efficiency and accuracy can be achieved by the proposed method.

This paper presents a hybrid approach for efficient analysis of electromagnetic (EM) scattering from large-scale aperiodic structures (e.g., aperiodic Penrose and Danzer tilings), which integrates the characteristic basis function method (CBFM) and the adaptive integral method (AIM). By performing a domain decomposition, a series of characteristic basis functions (CBFs) that are defined on a macro block and comprised of a relatively large number of sub-domain basis functions facilitate a substantial reduction in the method of moments (MoM) matrix size, enabling the use of a direct solver for large problems. The AIM is applied to accelerate the calculation of CBFM-reduced MoM matrices, significantly decreasing the CPU time and memory required for solving large-scale problems. As the size of one block becomes electrically large, the original CBFM combined with the AIM is employed to generate the initial CBFs by solving a large problem with multiple excitations, which results in efficiently constructing the final CBFs afforded by the singular value decomposition (SVD) procedure. This methodology produces a two-level “CBFM + AIM” hybrid algorithm for efficiently characterizing large-scale objects. The numerical results presented demonstrate the accuracy and efficiency of the proposed hybrid algorithm. Then, the developed solver is applied to investigate EM scattering properties of large-scale aperiodic tilings. The numerical results show that Penrose/Danzer tilings exhibit significantly improved grating lobe suppression as compared to their periodic counterparts.

During the last few years, the Characteristic Basis Function Method (CBFM) has been introduced to solve large-scale electromagnetic problems. The CBFM is a non-iterative domain decomposition approach that employs characteristic basis functions (CBFs), called the high-level physics-based basis functions, to represent the fields inside each sub-domain. This technique was first introduced to solve time-harmonic electromagnetic problems in the context of the Method of Moments (MoM) [1]. Quite recently, the CBFM procedure has been utilized for the first time in the Finite Element Method (FEM), and has been named the “Characteristic Basis Finite Element Method (CBFEM)” [2–4]. This method, which is different from the previous MoM-based CBFM, has been used in both the quasi-static [2] and the time-harmonic regimes [3–4], by generating the CBFs via point charges and dipole-type sources, respectively. Two major features of the CBFEM are: (i) it leads to a reduced-matrix that can be handled by using direct—as opposed to iterative—solvers; and (ii) its parallelizable nature can be taken advantage of to reduce the overall computation time. The basic steps of the CBFEM algorithm are summarized as follows: (i) Divide the computational domain into a number of subdomains; (ii) Generate the CBFs that are tailored to each individual subdomain; (iii) Express the unknowns as a weighted sum of CBFs; (iv) Transform the original matrix into a smaller one (called reduced-matrix) by using the Galerkin procedure, which uses the CBFs as both basis and testing functions; (v) Solve the reduced matrix for the weight coefficients, and substitute the coefficients into the series expressions to find the unknowns inside the entire computational domain.

The characteristic basis function method is known as an effective method to solve the electromagnetic scattering problems, but the convergence of the iterative solution of the reduced matrix equation is slow when the characteristic basis function method is used to analyze the electromagnetic scattering characteristics of the electrically large target. In order to mitigate this problem, a new reduced matrix construction method is proposed to improve the iterative solution efficiency of characteristic basis function method in this paper. Firstly, the singular value decomposition technique is used to compress the incident excitations, and the characteristic basis functions of each sub-domain under the new excitations are solved. Then, the new excitations and the characteristic basis functions are defied as the testing and basis functions to construct the reduced matrix. The diagonal sub-matrices of the reduced matrix constructed by the new testing and basis functions are all identity matrices, thereby improving the condition of reduced matrix. Thus, the total number of iterations to achieve reasonable results is significantly reduced. Numerical simulations are conducted to validate the performance of the proposed method. The results demonstrate that the efficiency of the iterative solution of the reduced matrix equation constructed by the new method is significantly improved. Furthermore, the characteristic basis functions’ generation time required by the proposed method is noticeably less than that by the traditional characteristic basis function method due to the reduced number of matrix equation solutions.

The Characteristic Basis Finite Element Method (CBFEM) [1–3] is a new descendant of the Characteristic Basis Function Method (CBFM), which is a non-iterative domain decomposition approach for solving large-scale electromagnetic problems by using characteristic basis functions (CBFs). The CBFM was originally introduced in the context of Method of Moments [4], but has rapidly evolved, with appropriate modifications, into a general-purpose approach that is also applicable to Finite Element Method (FEM). Some of the features that are common to all of the CBFM-based approaches are: (i) utilization of high-level physics-based basis functions — called CBFs — to represent the fields inside each sub-domain; (ii) reduced-matrix that can be handled by using direct solvers; (iii) parallelization to reduce the overall CPU time and memory. The basic steps of the CBFEM algorithm are as follows: (i) computational domain is divided into a number of sub-domains; (ii) CBFs are generated for each sub-domain; (iii) unknowns are expressed as a weighted sum of CBFs; (iv) original matrix is transformed into a reduced-matrix by using the Galerkin procedure, which uses the CBFs as both basis and testing functions; (v) reduced matrix is solved for the weight coefficients, which are substituted into the series expressions to solve for the unknowns in the entire computational domain.

We introduce a novel technique that combines the AIM algorithm with the characteristic basis function method (CBFM) to solve the problem of electromagnetic scattering by large but finite periodic arrays. An important advantage of using the CBFM for this problem is that we only need to analyze a single unit cell to construct the characteristic basis functions (CBFs) for the entire array. The CBFs are generated by illuminating a single unit cell with a plane wave incident from different angles, for both the θ- and φ-polarizations. The initial set of CBFs, generated in the manner described above, are then downselected by applying a singular value decomposition (SVD) procedure and retaining only the left singular vectors whose corresponding singular values fall above a threshold. Next, in the conventional CBFM, we derive a reduced matrix by applying the Galerkin procedure and solve it directly if its size is manageable. However, when solving an array problem, which precludes the direct-solve option, we can utilize the adaptive integral method (AIM) algorithm, detailed below, not only to accelerate the solution but to reduce memory requirements as well. Numerical examples are included in this communication to demonstrate the accuracy and the numerical efficiency of the proposed technique.

The block size of the characteristic basis function method (CBFM) based on volume integral equation (VIE) is optimized to improve the efficiency on analyzing the scattering from dielectric objects in this study. Several literatures have discussed the optimum block size of surface integral equation (SIE)‐based CBFM, which achieved the minimum CPU time for analyzing conductor objects. However, due to the differences of basis functions, the conclusions derived from SIE‐based CBFM are not applicable for VIE‐based CBFM. This study presents the relationship between the Schaubert–Wilton–Glisson (SWG) basis functions and characteristic basis functions (CBFs), which paves the way for evaluating the computational complexity of VIE‐based CBFM. In addition, the Sherman–Morrison–Woodbury formula‐based algorithm (SMWA) and adaptive cross approximation (ACA) algorithm are combined to accelerate the CBFs generation and reduced matrix construction. Based on the computational complexity of the VIE‐based CBFM combined with SMWA and ACA, the optimal number of blocks for VIE‐based CBFM is derived theoretically. Numerical results of homogeneous and inhomogeneous dielectric objects scattering problems are presented to validate the feasibility and accuracy of the proposed optimization method.

The characteristic basis function method (CBFM) (V.V.S. Prakash and R. Mittra, 2003) embodies an efficient way to handle large electromagnetic problems in a rigorous manner by defining a reduced set of precomputed characteristic basis functions (CBFs), which represent the currents along the target surface. By utilizing the CBFs, the number of unknowns which arise in the application of the conventional method of moments can be decreased by an order of magnitude, enabling the use of direct solvers even for large problems. In the CBFM, the CPU-time is mainly consumed in carrying out the following two operations: calculating the characteristic basis functions and computation of the matrix-vector products required to generate the reduced matrix elements. After splitting the geometry into several blocks, the CBFs for each is generated from a set of solutions for the induced currents in the subdomains that are illuminated by plane waves incident from different angles. The orthogonalization of these solutions, via an SVD procedure (M. DeGiorgi et al., 2005), is an important step that reduces the final number of CBFs by eliminating the redundancy in these functions, that also serves to improve the condition number of the reduced matrix.

In this paper, the characteristic basis function method is adapted to analyze the problem of scattering from multiple and multi-scale impenetrable targets in conjunction with the discontinuous Galerkin method, and the monopolar RWG functions are chosen as the basis functions. This method enables us to analyze multiple and multi-scale targets using nonconforming discretizations. The use of the CBFs helps reduce the size of the impedance matrix of the associated method of moment significantly, enabling us to employ direct solvers as opposed to iterative solvers. The process of generating the reduced matrix is naturally parallel, and the reduced matrix is well conditioned, which obviates the need for pre-conditioning. In addition, the adaptive cross-approximation algorithm is implemented to reduce the complexity of the computation. Numerical results are included to demonstrate the accuracy and efficiency of the present approach when analyzing scattering from multiple and multi-scale targets using nonconforming discretizations.

The Asymptotic Waveform Evaluation (AWE) technique is an extrapolation method that provides a reduced-order model of linear system and has already been successfully used to analyze wideband electromagnetic scattering problems. As the number of unknowns increases, the size of Method Of Moments (MOM) impedance matrix grows very rapidly, so it is a prohibitive task for the computation of wideband Radar Cross Section (RCS) from electrically large object or multi-objects using the traditional AWE technique that needs to solve directly matrix inversion. In this paper, an AWE technique based on the Characteristic Basis Function (CBF) method, which can reduce the matrix size to a manageable size for direct matrix inversion, is proposed to analyze electromagnetic scattering from multi-objects over a given frequency band. Numerical examples are presented to illustrate the computational accuracy and efficiency of the proposed method.

In this paper, the efficient analysis of electromagnetic scattering from objects is performed utilizing the characteristic basis function method (CBFM) with the compressed sensing (CS) technique. CBFM generates sets of high-level basis functions, i.e. the characteristic basis function (CBF), defined over macro domains of the object under discussion, to significantly reduce the matrix size and storage and make the reduced linear system easily manageable and solvable. In the conventional fulfillment, to generate the CBF, a series of matrix equations need to be solved repeatedly with various excitations of multiple incident angles and different polarizations. With the help of CS, much fewer excitations are necessary to solve the matrix equations, and one can obtain the measurements of the induced currents accordingly. Finally, the orthogonal matching pursuit (OMP) method is used to reconstruct the real induced currents and from which the CBF is generated. Numerical examples are included to demonstrate the performance of the proposed method.

The characteristic basis function method (CBFM) has been hybridized with the adaptive cross approximation (ACA) algorithm to construct a reduced matrix equation in a time-efficient manner and to solve electrically large antenna array problems in-core, with a solve time orders of magnitude less than those in the conventional methods. Various numerical examples are presented that demonstrate that the proposed method has a very good accuracy, computational efficiency and reduced memory storage requirement. Specifically, we analyze large 1-D and 2-D arrays of electrically interconnected tapered slot antennas (TSAs). The entire computational domain is subdivided into many smaller subdomains, each of which supports a set of characteristic basis functions (CBFs). We also present a novel scheme for generating the CBFs that do not conform to the edge condition at the truncated edge of each subdomain, and provide a minor overlap between the CBFs in adjacent subdomains. As a result, the CBFs preserve the continuity of the surface current across the subdomain interfaces, thereby circumventing the need to use separate ldquoconnectionrdquo basis functions.

In this paper, the Adaptive Modified Characteristic Basis Function Method (AMCBFM) is proposed for quickly simulating electromagnetic scattering from a one-dimensional perfectly electric conductor (PEC) rough surface. Similar to the traditional characteristic basis function method (CBFM), Foldy-Lax multiple scattering equations are applied in order to construct the characteristic basis functions (CBFs). However, the CBFs of the AMCBFM are different from those of the CBFM. In the AMCBFM, the coefficients of the CBFs are first defined. Then, the coefficients and the CBFs are used to structure the total current, which is used to represent the induced current along the rough surface. Moreover, a current criterion is defined to adaptively halt the order of the CBFs. The validity and efficiency of the AMCBFM are assessed by comparing the numerical results of the AMCBFM with the method of moments (MoM). The AMCBFM can effectively reduce the size of the matrix, and it costs less than half the CPU time used by the MoM. Moreover, by comparing it with the traditional CBFM, the AMCBFM can guarantee the accuracy, reduce the number of iterations, and achieve better convergence performance than the CBFM does. The second order of the CBFs is set in the CBFM. Additionally, the first order of the CBFs of the AMCBFM alone is sufficient for this result.

A novel technique for an efficient and rigorous solution of EM scattering and radiation problems from electrically large objects and large arrays is introduced in this paper. Higher order basis functions are used in the characteristic basis function method (CBFM) to model the source and the geometries. The characteristic basis functions are high level basis functions that are constructed by using the piecewise subdomain higher order basis functions, which leads to a much smaller matrix. This method reduces the requirement of the memory and CPU time and can be used to analysis of large antennas and scatters. Numerical results validate the accuracy of this method.

The method of moments is one of the most commonly used algorithms for analyzing the electromagnetic scattering problems of conductor targets. However, it is difficult to solve the matrix equation when analyzing the electromagnetic scattering problem of the electric large target. In recent years, the theory of the compressed sensing was introduced into the method of moments to improve the computation efficiency. The random selected impedance matrix is used as a measurement matrix, and the excitation voltage is used as a measurement value when using compressed sensing theory. The recovery algorithm is used to solve the induced current of target. The method can avoid the inverse problem of matrix equation and improve the computational efficiency of the method of moments, but it can be applied only to 2-dimensional or 2.5-dimensional target. The application of compressed sensing needs to know the sparse basis of the current in advance, but the induced current of three-dimensional target which is expressed by an Rao-Wilton-Glisson basis function is not sparse on the commonly used sparse basis, such as discrete cosine transform basis and discrete wavelet basis. To solve this problem, a method of combining compressed sensing with characteristic basis functions is proposed to analyze the electromagnetic scattering problem of three-dimensional conductor target in this paper. The characteristic basis function method is an improved method of moments. The target is divided into several subdomains, the main characteristic basis functions are comprised of current bases arising from the self-interactions within the subdomain, and the secondary characteristic basis functions are obtained from the mutual coupling effects of the rest of the subdomains. Then a reduction matrix is constructed to reduce the order of matrix equation, and the current can be expressed by the characteristic basis function and its weighting coefficient. In the method presented in this paper, the weighting coefficient is considered as a sparse vector to be solved when the characteristic basis function is used as sparse basis. The number of weighting coefficients is less than the number of unknown ones, so it can be obtained from the compressed sensing recovery algorithm. At the same time, the generalized orthogonal matching pursuit algorithm is used as the recovery algorithm to speed up the recovery process. Finally, the proposed method is used to calculate the radar cross sections of a PEC sphere, nine discrete PEC targets and a simple missile model. The numerical results validate the accuracy and efficiency of the method.