SCATTERING ANALYSIS OF PERIODIC ARRAYS USING COMBINED CBF/P-FFT METHOD

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.

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 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.

Finite arrays of waveguides and horn antennas are widely used both as direct radiators as well as feed clusters for reflector antennas. Accurate and efficient numerical simulation techniques represent a powerful tool for the design of such antennas. Many approaches have been proposed in the literature that employ hybrid techniques to efficiently analyze both the interior waveguide problem and as well as their coupling to the outer free space [1]–[3]. The mode matching (MM) is used to compute the generalized scattering matrix (GSM) for the inner waveguide problem, whereas the Method of Moments (MoM) evaluates the GSM for the outer free space coupling. When the array is electrically large in terms of number of elements or radiating apertures dimensions, the applicability of this approach is limited because the associated MoM matrix is large. Some novel approaches have been proposed in the literature to reduce the matrix size [4]. Recently the Characteristic Basis Function method (CBFM) [5] has been proposed to solve electrically large problems by using a set of high-level basis functions, called CBFs, that are physics-based and tailored to the problem.

This communication presents a comprehensive numerical analysis of single microstrip antenna and antennas array in spherically layered media, using the method of moments (MoM). The curvilinear Rao–Wilton–Glisson (CRWG) triangular basis functions are employed to simulate the equivalent current on the antenna surface. First, the dyadic Green’s functions using Debye potential are deduced. Then, their efficient computation procedures are carried out to improve calculation efficiency. Finally, the characteristic basis function method is employed to save memory and calculation time by reducing dimension of the impedance matrix. The results of this study are in good agreement with those of the FEKO software, and they show that the method proposed in this communication is more efficient than FEKO in the analysis process.

The improved hybrid discretisation of volume integral equation (VIE) combined with the characteristic basis function method is proposed to analyse the scattering from complex dielectric objects in this study. To mitigate the excessive number of discrete elements which arise from the conformal discretisation on the complex dielectric objects, the hybrid discretisation is developed. For the conventional hybrid discretisation, the conformal Schaubert–Wilton–Glisson (SWG) basis functions are used to discretise the homogeneous regions, meanwhile, the non‐conformal half‐SWG basis functions are employed to discretise the boundary regions which separate different materials or multiscale structures. The hybrid discretisation takes advantage of the merits of both conformal and non‐conformal basis functions and improves the computational efficiency for solving the VIE. In our study, a kind of edge basis function defined within boundary tetrahedral elements substitutes for the SWG basis functions to model the boundary regions, leading to the further reduction on the number of basis functions. Moreover, the characteristic basis functions extracted from the low‐level basis functions are employed to construct the reduced matrix of which the size is considerably smaller than that of the system matrix derived from the VIE. Several numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

A pressure deconvolution technique based on ridge regression with existing convolution features was explored and reinterpreted to frame a new optimization problem structure. We found a way to split the feature matrix into a deconvolution part and a smoothing part that handle each effect independently. The deconvolution part is comprised of a convolution matrix (Toeplitz matrix) for discrete-time convolution while the smoothing part is composed of a basis function of the impulse response. Hence, this formulation preserves linearity (a superposition of pressure solution) and the kernel for nonlinear mapping (e.g., polynomial kernel, radial basis function kernel, etc.) is effective only on the smoothing part (a basis function of impulse response). Importantly, the key idea of existing convolution feature is the modelling of the impulse response with a linear combination of a basis functions.For single-well pressure behavior, ridge regression often struggles with high-frequency measurements that contain multistage responses. An impulse response might share similar functional characteristics, for example wellbore storage and pseudosteady state are both linear responses but there is only one linear basis function in the ridge regression which cannot describe them both. In traditional well test interpretation, the engineer is typically focused on certain parts of the impulse response on a log-scale, to identify reservoir behavior (e.g., early-time for wellbore storage, transient-time for infinite acting radial flow, late-time for boundary effect, etc.). Analogously, a basis function could be formed from multiple pressure responses in each log-interval (inherent from a solution of the diffusion equation) then bridged together with a constraint on function value and its derivative at each interval boundary. This is the basic principle of spline regression that has more flexibility in terms of function expression. In addition, it also subsumes ridge regression with Laplacian regularization.For multiwell pressure behavior, additional basis functions are included to handle interference responses. Typically, the exponential integral is approximated with a logarithm function for single-well pressure response which is also one of the basis functions in previous convolution features. Nonetheless, interference pressure response is more subtle and its solution could not be easily approximated, which previously resulted in a poor fitting using earlier methods. By inspection of the convergent series of the exponential integral, its higher-order terms are associated with 1/tm that can be easily included as a basis function to enhance the capability of the model and capture the detail of the interference test that is essential for multiwell problems. A full extension to multiwell problems is also presented with the additional derivative constraint for the impulse response and symmetry constraint. Finally, we can quantify the sensitivity of the deconvolution process using singular value decomposition (SVD) and Monte Carlo simulation (MCS). Using singular value decomposition, we could reveal the orthogonal basis of the feature matrix and its corresponding singular value that helped us identify the effectiveness of the basis functions as well as their susceptibility to noise of certain pattern. Nevertheless, singular value decomposition is only suitable for noisy pressure measurement problems without inequality constraint. Thus, Monte Carlo simulation is required for full quantification of noisy measurements in both pressure and flow rate. We can observe a propagation of error in measurement to a band in pressure prediction and pressure derivative of the impulse response.This study developed a more robust method for deconvolution of pressure measurements. In addition, the mathematical insights obtained in the study allow for a more general understanding of how the deconvolution problem can be deconstructed into its relevant parts – likely allowing for expanded capability of deconvolution algorithms in industrial practice.

The aim of this research work is to discuss the use of the diagonal representation (DR) for the construction of the macro-basis functions (MBFs) when applying the Characteristic Basis Function Method (CBFM) to the problem of modeling of scattering from forests. We have applied the CBFM to improve the performances of a previously studied 3D full-wave model in order to solve electrically large forest simulation scenes. This efficient domain decomposition method has shown good performance both in terms of CPU time and memory while achieving a satisfactory level of accuracy compared to that of a conventional Method of Moments (MoM). Therefore, it enables us to solve forest EM problems of upward of 2 million unknowns by consuming only a reasonable amount of CPU time and using just 48 GB of shared memory (Fenni I.; Roussel H.; Darces M.; Mittra R., AP IEEE Trans., 62–8, 4282–4291, 2014).

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 aim of a correspondence analysis is the graphical representation of the categories of variables in one frame of reference. This visualization is possible due to the decomposition of the basic matrix with the use of Singular Value Decomposition (SVD). There are three matrices used in the process of decomposition: right singular vectors, left singular vectors, and a singular value diagonal matrix. The aim of this paper is to compare four different approaches and algorithms of SVD methods used in a correspondence analysis. In the literature, four approaches are known to singular value decomposition, defined by: R.A. Fisher (1940), M.J. Greenacre (1984), E.B. Anderson (1991), and J.D. Jobson (1992). Those computational procedures will be presented and compared in this paper. Also, methods of determining the coordinates of the category column and line matrix, as well as the values of inertia will be defined for these approaches. A key problem is to compare the well-known approaches, since in the literature only one approach ‒ proposed by Greenacre ‒ is used for singular value decomposition. The reason of the superiority of this algorithm over the others may be the simplicity and ease of the mathematical calculations. Greenacre’s algorithm is also used in R statistical software, making its availability and popularity growing, however, other algorithms are worth presenting and focusing on.

The FORTRAN codes in this Chapter address the question of computing distinct singular values and corresponding left and right singular vectors of real rectangular matrices, using a single-vector Lanczos procedure. For a given real rectangular ℓ × n matrix A, these codes compute nonnegative scalars σ and corresponding real vectors x ≠ 0 and y ≠ 0 such that EquationSource$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGbbGaamiEaiabg2da9iabeo8aZjaadMhaieaacaWFGaGaamyy % aiaad6gacaWGKbGaa8hiaiaadgeapaWaaWbaaSqabeaapeGaamivaa % aakiaadMhacqGH9aqpcqaHdpWCcaWG4bGaaiOlaaaa!4716! $$Ax = \sigma y and {A^T}y = \sigma x.$$$$ ((6.1.1)) Every real rectangular ℓxn matrix, where ℓ ≥ n, has a singular value decomposition, EquationSource$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGbbGaeyypa0Jaamywaiabfo6atjaadIfapaWaaWbaaSqabeaa % peGaamivaaaaieaak8aacaWFGaGaam4DaiaadMgacaWG0bGaamiAai % aa-bcapeGaamiwaiabg2da9iaadIfapaWaaWbaaSqabeaapeGaamiv % aaaak8aacaWGybGaeyypa0JaamysaiaacYcacaWFGaGaamywamaaCa % aaleqabaGaamivaaaakiaadMfacqGH9aqpcaWGjbGaa8hiaiaa-fga % caWFUbGaa8hzaiaa-bcacqqHJoWucaWF9aWaamWaaeaafaqabeGaba % aabaGaeu4Odm1aaSbaaSqaaiaadMeaaeqaaaGcbaGaaGimaaaaaiaa % wUfacaGLDbaaaaa!59A3! $$A = Y\Sigma {X^T} with X = {X^T}X = I, {Y^T}Y = I and \Sigma = \left[ {\matrix {{\Sigma _I}} \\ 0 \\ \endmatrix } \right]$$$$ ((6.1.2)) where Σ is ℓ × n and = diag {σ1,..., σn} with σi, 1 ≤ i ≤ n, the singular values of A. X is a n × n orthogonal matrix, Y is a ℓ × ℓ orthogonal matrix, and the columns of X and of Y are respectively, right and left singular vectors of A. There are many applications for this type of decomposition. Singular values and vectors are discussed in detail for example in Stewart [1973].KeywordsError EstimateSingular VectorMain ProgramVector ComputationReal Symmetric MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The compressed beamforming weights feedback is used in the precoded MIMO-OFDM system, e.g., IEEE 802.11n/ac WLAN, to reduce the amount of feedback information. The precoded sub-system at the receiver of this system needs to perform singular value decomposition (SVD) on the estimated channel matrices at all sub-carriers. The computed angles of the right singular vectors are to be feedback to the transmitter. Also the computed singular values and left singular vectors are to be used for designing the optimal detector. Under the exemplary scenario of four antennas at the access point and two antennas at the station, our designed architectures for the precoding sub-system can provide SVD throughput rates 16.8 and 50.4 M matrices/sec for the uplink and downlink transmissions, respectively.

Although the Method of Moment(MoM) for Bodies of Revolution(BoR) to solve the electromagnetic scattering and radiation problems has been studied for several decades. The research of scattering from Multi-BoR is merely referred.For the purpose of solve scattering from Multi-BoR efficiently a method combine the traditional BoR-MoM with Characteristic Basis Function Method is developed in this paper. Some numerical results are given to demonstrate the validity and accuracy of this method.

Singular value decomposition has served as a diagnostic tool in optical computed tomography by using its capability to provide insight into the condition of ill-posed inverse problems. Various tomographic geometries are compared to one another through the singular value spectrum of their weight matrices. The number of significant singular values in the singular value spectrum of a weight matrix is a quantitative measure of the condition of the system of linear equations defined by a tomographic geometery. The analysis involves variation of the following five parameters, characterizing a tomographic geometry: 1) the spatial resolution of the reconstruction domain, 2) the number of views, 3) the number of projection rays per view, 4) the total observation angle spanned by the views, and 5) the selected basis function. Five local basis functions are considered: the square pulse, the triangle, the cubic B-spline, the Hanning window, and the Gaussian distribution. Also items like the presence of noise in the views, the coding accuracy of the weight matrix, as well as the accuracy of the accuracy of the singular value decomposition procedure itself are assessed.

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.