Improved hybrid discretisation of volume integral equation with the characteristic basis function method for analysing scattering from complex dielectric objects

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.

A novel implementation of the characteristic basis function method (CBFM) is given, in which the high-level basis functions, called characteristic basis functions (CBFs), are represented in terms of curved rooftops generated from nonuniform rational B-splines (NURBS) in the parametric (u,v) domain. The associated macro-testing functions are defined by using curved razor-blade functions corresponding to each rooftop. The underlying objective of the CBFM is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments (MoM). The result is, therefore, an approach which can handle many complex cases via direct solvers, without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill-conditioned matrices. As a result of the combination of the CBFM with the special class of low-level basis and testing functions directly located over NURBS surfaces, complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the CPU time as well as required memory resources.

The Characteristic Basis Function Method (CBFM) is applied to rapidly compute the impedance and radiation characteristics of electrically large aperture-fed antenna arrays. A stationary formula for the antenna input admittance matrix is expressed in terms of a product of matrix blocks that are readily available from a method of moment formulation. Numerical results are shown for large arrays of waveguide antennas requiring more than 2 million basis functions, which is reduced by a factor of 9000, so that the solution for the currents are still obtainable in-core on a single desktop computer, while being orders faster than commercial software codes or a standard MoM approach, provided that sufficient memory is available for the Gaussian elimination.

A computationally efficient algorithm, which combines the characteristic basis function method (CBFM), the physical optics (PO) approach (when applicable) with the forward backward method (FBM), is applied for the investigation of electromagnetic scattering fromand propagation overlarge-scale rough terrain problems. The algorithm utilizes high-level basis functions defined on macro-domains (blocks), called the characteristic basis functions (CBFs) that are constructed by aggregating low-level basis functions (i.e., conventional sub-domain basis functions). The FBM as well as the PO approach (when applicable) are used to construct the aforementioned CBFs. The conventional CBFM is slightly modified to handle large-terrain problems, and is further embellished by accelerating it, as well as reducing its storage requirements, via the use of an extrapolation procedure. Numerical results for the total fields, as well as for the path loss are presented and compared with either measured or previously published reference solutions to assess the efficiency and accuracy of the algorithm.

The characteristic basis function method (CBFM) is an efficient approach to analyze electromagnetic problems through the size reduction of the original matrix in the method of moments equation. However, adequate plane waves (PWs) must be set in each sub-block to construct characteristic basis functions (CBFs), thus increasing the number of CBFs and causing higher time consumption in singular value decomposition. In addition, the reduced matrix calculation procedure is time-consuming because numerous vector–matrix–vector products (VMVPs) are contained. To mitigate these problems, an improved CBFM is presented. This method fully considers the mutual coupling effects among sub-blocks to obtain the secondary level characteristic basis function (SCBF). Therefore, the number of PWs, as well as the number of CBFs, is significantly reduced. The fast dipole method is also used to accelerate the matrix–vector products in the construction of SCBFs and VMVPs in the calculation of the reduced matrix. Numerical results demonstrate that the proposed method is accurate and efficient.

This paper presents an interpolation scheme to reduce the computation time needed to generate the reduced matrix in the Characteristic Basis Function Method (CBFM). The first step in this approach is to identify the observation points and calculate the electric fields at those points that are produced by the induced currents represented by the macro-basis functions called the Characteristic Basis Functions (CBFs). Next, we interpolate the above field values to obtain the same at other points on the structure. Finally, the reduced matrix is formed by testing the electric field with the CBFs. Numerical results are presented to show the accuracy and computational efficiency of the method.

This communication proposes a novel method for electrically large-scale open-ended cavities, using the generalized transition matrix (GTM) model and accelerating it by implementing the characteristic basis function method (CBFM). The approach involves the representation of electric surface currents on both the exterior and interior walls of a cavity using characteristic basis functions (CBFs), effectively reducing the number of unknowns in the problem. The proposed method can efficiently enhance the computational complexity of the GTM model construction as well as the system matrix formulation. Consequently, the system matrix is reduced compared with the original matrix system, resulting in the reduction of the matrix-solving times. The detailed formulation processes are introduced in this communication. Furthermore, the proposed method is verified using examples of open-ended cavities and evaluates the computational efficiency compared with the original method through the numerical results.

Reconstruction of unknown objects requires efficient inversion for measured electromagnetic scattering data. In frequency-domain integral equation method for reconstructing dielectric objects, the volume integral equations (VIEs) are involved because the imaging domain with both true unknown objects and part of background is inhomogeneous. When solving the forward scattering integral equation (FSIE), the Nyström method is used since the traditional method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function may not be convenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and lower requirement on geometrical discretization, so it is very suitable for inhomogeneous problems. When solving the inverse scattering integral equation (ISIE), the Gauss-Newton minimization approach (GNMA) with a multiplicative regularization method (MRM) is employed. Numerical examples for reconstructing three-dimensional dielectric objects are presented to illustrate the inversion approach.

In this work, the characteristic basis function method (CBFM) is presented in junction with the characteristic modes (CMs). In this approach, characteristic modes are defined as primary basis functions on each array element. In order to take into account mutual couplings, secondary basis functions are introduced as in the conventional CBFM. Since characteristic modes are independent of the excitation, the reduced matrix can be constructed without the information of the excitation by using characteristic modes as primary basis functions. Once the reduced matrix is calculated and stored, it can be reused in the array problem for different type of excitations.

The multilevel characteristic basis function method (MLCBFM) with the adaptive cross approximation (ACA) algorithm for accelerated solution of electrically large scattering problems is studied in this paper. In the conventional MLCBFM based on Foldy-Lax multiple scattering equations, the improvement is only made in the generation of characteristic basis functions (CBFs). However, it does not provide a change in impedance matrix filling and reducing matrix calculation procedure, which is time-consuming. In reality, all the impedance and reduced matrix of each level of the MLCBFM have low-rank property and can be calculated efficiently. Therefore, ACA is used for the efficient generation of two-level CBFs and the fast calculation of reduced matrix in this study. Numerical results are given to demonstrate the accuracy and efficiency of the method.

This article presents an acceleration technique for the fast calculation of the impedance matrix elements in the method of moments in the case of using characteristic basis functions (CBFs). This technique is based on taking advantage of the smooth amplitude and phase behaviors of the fields generated by a CBF when the distance is sufficiently large. Computational efficiency is realized by calculating the reaction between a source‐CBF and a test‐CBF by first evaluating a sampling of the field at a sparse set of points, and then computing the fields at the rest via the use of interpolation. Thus, this procedure avoids the highly expensive integration in the source domain that is needed to calculate the radiated field in the entire test domain when following the conventional procedure. Phase unwrap prior to interpolation and the fast sample calculation scheme are detailed in the article. The technique has been validated by considering several examples involving Radar Cross Section computations. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1818–1824, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24478

Electromagnetic problems with inhomogeneous or anisotropic penetrable media require using volume integral equations (VIEs) to describe in the integral equation approach. The VIEs are usually solved by the method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function, but the implementation may not be convenient because conforming meshes are required to define the basis function. In this paper, we propose a novel meshless method to discretize the VIEs. The method uses the Green-Gauss theorem to transform a volume integral into a boundary integral so that discretizing volume domains can be removed, resulting in a truly meshless method. For electrically large problems, we incorporate the method with multilevel fast multipole algorithm (MLFMA) to accelerate the solving process. A numerical example is presented to demonstrate the approach and its effectiveness and robustness have been validated.

In this paper, an improved CBFM/p-FFT algorithm is presented, which can be applied to solve electromagnetic scattering problems of large-scale periodic composite metallic/dielectric arrays, even when the array has electrically small periodicity or separating distance. Using characteristic basis function method (CBFM), scattering characteristics of any inhomogeneous targets can be represented by special responses derived from a set of incident plane waves (PWs). In order to reserve the dominant scattering characteristics of the targets and remove the redundancy of the overfull responses, a singular value decomposition (SVD) procedure is applied, then, new series of basis functions are built based on the left singular vectors after SVD whose corresponding singular values beyond a predeflned threshold. However, the algorithm of CBFM combined with method of moments (MoM) still requires a lot of memory and CPU resources to some large scale problems, so the precorrected- fast Fourier transform (p-FFT) method is applied based on the novel built basis functions, with which, the required memory and solve time for solution can be reduced in an extraordinary extent. For a near correction technique is applied to process the interactions between cells placed within a distance less than a predeflned near-far fleld threshold, arrays with electrically small periodicity can be analyzed accurately. Moreover, the incomplete LU factorization with thresholding (ILUT) preconditioner is applied to improve the condition number of the combined algorithm, which improves the convergence speed greatly.

In this paper, the tangential equivalence principle algorithm (T-EPA) combined with characteristic basis functions (CBFs) is presented to analyze the electromagnetic scattering of large-scale antenna arrays. The T-EPA is a kind of domain decomposition scheme for the electromagnetic scattering and radiation problems based on integral equation (IE). CBFs are macrobasis functions which are constructed by conventional local basis functions. By utilizing CBFs together with the T-EPA, the scattering analysis of large-scale arrays will be much more efficient with decreased unknowns compared with the original T-EPA. Further, the multilevel fast multipole algorithm (MLFMA) is applied to accelerate the matrix-vector multiplication in the T-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed technique.

In this paper, a merged ultra-wideband characteristic basis function method (MUCBFM) is presented for high-precision analysis of wideband scattering problems. Unlike existing singular value decomposition (SVD) enhanced improved ultra-wideband characteristic basis function method (SVD-IUCBFM), the MUCBFM reduces the number of characteristic basis functions (CBFs) necessary to express a current distribution. This reduction is achieved by combining primary CBFs (PCBFs) with the secondary level CBFs (SCBFs) to form a single merged ultra-wideband characteristic basis function (MUCBF). As the MUCBF incorporates the effects of PCBFs and SCBFs, the accuracy does not change significantly compared to that obtained by the SVD-IUCBFM. Furthermore, the efficiencies of constructing the CBFs and filling the reduced matrix are improved. Numerical examples verify and demonstrate that the proposed method is credible both in terms of accuracy and efficiency.