Types Of Basis Functions Research Articles

Comparison between two sets of basis functions for the current modeling in the Galerkin spectral domain solution for microstrips

A frequently used technique for the numerical full wave analysis of planar microwave passive structures is the method of moments (MoM). A very important determinant for the accuracy and efficiency of this technique is the choice of the basis functions used in the approximation of the unknown current components. The application of several types of entire and subsectional domain basis functions has been proposed in the literature. However, little comparison has been made between different types of basis functions. In this paper, a comparison is presented between two sets of frequently used subsectional domain basis functions for the canonical case of the microstrip transmission line. The emphasis in this paper is on the relative accuracy, the convergence and the computational efficiency of both sets of basis functions. The intent of the paper is primarily to be an eye opener for the implications of the choice of the basis functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Scattering from a conductive rectangular plate covered by a thick dielectric layer and excited from a dipole source or a plane wave

The scattering of electromagnetic radiation from a conductive rectangular plate of infinitesimal thickness covered by a dielectric parallelepiped of arbitrary size is analyzed. An integral equation formulation is employed in terms of the electric field inside the dielectric layer and the surface current density on the conductive plate, and is solved approximately by using entire domain Galerkin technique. The electric field inside the dielectric layer is described in terms of a superposition of exponential waves while, in describing the surface current density on the conductive plate, two types of entire domain basis functions are utilized: either Fourier-exponential series or Chebyshev series type finite-term expansions are employed alternatively. Numerical results are computed and presented for several scatterer sizes and excitation types.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Error analysis of the finite‐strip method for parabolic equations

AbstractThe finite‐strip method (FSM) is a hybrid technique which combines spectral and finite‐element methods. Finite‐element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted‐residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time‐dependent, parabolic partial differential equations indicates the numerical solution error is O(h2 + M−r). M represents the Fourier truncation mode number and h represents the finite‐element grid mesh. The exponent r ≥ 2 increases with the exact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite‐layer method, where a two‐dimensional trigonometric basis is used, the numerical solution error is O(h2 + M−r + N−q). The N and q represent the trucation mode number and degree of exact solution smoothness in the additional dimension. © 1993 John Wiley &amp; Sons, Inc.

Ab initio studies of electron transfer. 2. Pathway analysis for homologous organic spacers

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTAb initio studies of electron transfer. 2. Pathway analysis for homologous organic spacersCongxin Liang and Marshall D. NewtonCite this: J. Phys. Chem. 1993, 97, 13, 3199–3211Publication Date (Print):April 1, 1993Publication History Published online1 May 2002Published inissue 1 April 1993https://doi.org/10.1021/j100115a023RIGHTS & PERMISSIONSArticle Views345Altmetric-Citations147LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (2 MB) Get e-Alerts

Pattern recognition using artificial neural networks

In this paper we examine the key features of simple neural networks and their application to pattern recognition. Beginning with a three-layer backpropagation network we examine the mechanisms of pattern classification. We relate the numbers of input, output and hidden nodes to the problem features and parameters. In particular, each hidden neuron corresponds to a discriminant in the input space. We point out that the interactions between number of discriminants, the size and distribution of the training set, and numerical magnitude make it very difficult to provide precise guidelines. We found that the shape of the threshold function plays a major role in both pattern recognition, and quantitative prediction and interpolation. Tuning the sharpness parameter could have a significant effect on neural network performance. This feature is currently under-utilized in many applications. For some applications linear discriminant is poor choice. As an example the conventional BPN finds it almost impossible to separate concentric semicircles or “onion rings”, which are readily separated if the a quadratic discriminant is used. Our work suggests varying performance for different basis functions in different applications. Four different types of basis functions were discussed in this study, including two which, as far as we know, have not been previously published.

Fast fourier transform techniques for solving the electric field integral equation for a periodic body

The authors present a method of applying fast Fourier transform (FFT) techniques to the solution of the electric field integral equation (EFIE) on finite periodic bodies using the methodof- moments (MoM). These techniques are used to speed matrix–vector multiplications in an iterative solution of the discretized EFIE, and to save memory space required for storing the impedance matrix. An individual array element can have arbitrary three-dimensional structure and be electrically connected to its neighbors. The method is independent of the type of basis functions used, of the method of surface patching, and of the field excitation (which need not be periodic). The algorithm can be applied both to scattering and to antenna problems. The authors demonstrate improvements in speed-of-execution and memory required.

Numerical field modelling of objects with edges and corners using a conformal mapping

A numerical method is developed to study electromagnetic scattering from three-dimensional geometries, which is an extension of the method previously used for bodies of revolutions. On the object's surface two orthogonal tangent vectors are defined. Along one, surface currents are represented by discrete triangular functions and along the other by a finite set of entire domain basis functions. To enable the latter expansion the object's cross-section is conformally transformed onto a circle, and Fourier type basis functions are selected in the new space. The technique is then applied to study the scattering from prismatic geometries, such as cubic boxes and plates. The behavior of the solutions with the number of modes is investigated to determine the coupling of the entire domain modes. The formulation is applicable to other geometries, but it needs the transformation function to be useful.

How to calculate orbitals as integrals over the exponential parameters: a new type of basis functions

A new type of exponential and Gaussian basis functions is presented that permits a dual representation of orbitals as both an integral over the domain of exponential parameters and as a linear combination of basis functions. These basis sets permit a systematic and hierarchic computation. Their behaviour has been tested in the atomic-core Hamiltonian problem, which appears to be a stringent test for this type of basis. Very accurate results have been obtained for a wide range of atomic charges with the same basis set. In addition, the accuracy is quite consistent at all computational levels for different atomic charges. The new type of basis functions is particularly promising for ab initio calculations on atoms, and also on molecules if the pseudospectral method is used, and for the density functional method. Application to the traditional ab initio method for molecules, though, is foreseen to be costly.

Numerical analysis of finite frequency selective surfaces with rectangular patches of various aspect ratios

The finite frequency selective surface studied consists of a two-dimensional array, finite in one dimension and infinite in the other, of perfectly electrically conducting plates in free space. The analysis uses a Green's function formulation in the spectral domain, and the resulting equation is solved using the method of moments. Techniques are demonstrated for increasing the speed of the numerical computation so that the problem can be solved on a VAX 11-780 in a reasonable amount of time. Two types of basis functions, one subsectional and the other complete domain, are used to represent the currents on the patches. Both yield similar results and both are suitable for use in this problem. Techniques are demonstrated for approximating the tails of the infinite integrations which allow a more accurate solution than truncation. Results for scattered power density and patch currents illustrate the edge effect and incident angle dependencies. Results showing 3 dB bandwidth, resonance peak versus frequency, and scattered power versus frequency are plotted for patches with aspect ratios varying from 0.1 to 1. >

Accurate multireference configuration interaction calculations of the potential energy function and the dissociation energy of N2

The potential energy function of the N2 molecule is calculated using the internally contracted multireference CI method (CMRCI) and complete active space SCF (CASSCF) reference wave functions. A full CI calculation in a DZP basis set is used to estimate the errors associated with the CMRCI wave function. The dependence of the computed spectroscopic constants and the dissociation energy on the basis set is also investigated. Uncontracted and segmented basis sets are compared with ANO (atomic natural orbital) and other generally contracted basis sets. It is found that the energy optimized ‘‘correlation consistent’’ basis sets of Dunning yield substantially better results than ANO basis sets of the same size. In the largest calculations, which included up to h type basis functions and also accounted for core–core and core–valence correlation effects, the remaining errors are 0. 0003 Å, 8 cm−1, and 0.7 kcal/mol for re, ωe, and De, respectively. The inclusion of an i type basis function reduces the error in the dissociation energy to 0.3 kcal/mol (0.013 eV).

Multiconfigurational electronic wave functions without a reference configuration: Analysis of a simulated annealing strategy

Multiconfigurational floating Gaussian electronic wave functions are obtained for a variety of 2-electron atoms and for H2. These correlated wave functions are unusual in that they are constructed without taking excitations from a reference configuration. The description of electron correlation, the degree to which the proper ground state symmetry is attained, and the behavior of wave functions obtained by simulated annealing as a function of nuclear position are analyzed for two types of basis functions, product Gaussians and a second type of basis function that explicitly includes electron correlation. This work is aimed toward applications in mixed quantum/classical condensed phase simulations. Several recent studies have used simulated annealing to build floating Gaussian electronic wave functions in condensed phase or cluster simulations. We analyze the ability of the simulated annealing method to follow the optimum, or near optimum wave functions as nuclear positions move.

Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems

A preconditioning technique for pseudospectral solutions of elliptic problems based on quadrangular finite-element algorithms is analyzed, which exhibits excellent convergence properties. The pseudospectral technique is implemented through a collocation grid based on Gauss–Lobatto quadrature nodes associated to the Jacobi orthogonal polynomials. Various types of basis functions are used in the finite-element preconditioner (i.e., low-order Lagrange or cubic Hermite elements). Dirichlet and Neumann problems are investigated in one- and two-space dimensions. Numerical results show that the eigenvalue spectrum of the iteration matrix is inside the unit circle and even, close to zero for a wide range of operators. This property ensures convergence until roundoff error level in a few iterations. The differences between finite-element and finite-difference preconditioning are analyzed. Finally, the application of the algorithm to a problem exhibiting geometric induced singularities is discussed.

設計対象の機能記述に基づく設計・製造の統合化

This paper deals with a method for automatic generation of intermediate manufacturing information required for linking the result of design with the tasks in process planning. The method is characterized as that such manufacturing information is derived from the designer's intention expressed in the form of functional descriptions and their related geometries, both of which can be preserved in the product design stages in the CAD system. In order to describe the designer's intention in a unified way, we develop a descriptive frame work to represent the functions of the design object by three types of basic functions about a face element; function of (1) transmission, (2) constraint, (3) fixation, and by two items of functional properties such as the mechanism utilized for realization of the basic function and the condition and direction of the motion. By using this representation, the intermediate manufacturing information can easily be obtained. To confirm the validity of the method proposed in this paper, a prototype system of intermediate manufacturing planning has been fabricated. An appropriate result is obtained. This work is carried out under full consideration to realize the integration of CAD and CAM.

General quantum mechanical operators. An open‐ended approach for one‐electron integrals with Gaussian bases

AbstractA universal computational approach for evaluating integrals over gaussian basis functions for general operators of the form is presented. The implementation is open‐ended with respect to the types of basis functions (s, p, d, f, g, h…) and with respect to the integers that specify the operator. These one‐electron integrals comprise operators associated with electrical and magnetic properties of molecules and include those needed to find multipole polarizabilities, multipole susceptibilities, chemical shifts, and so on. The scheme also generates the usual kinetic, nuclear attraction, and overlap operators.

Numerical approximation of linear operator equations using a generalized Fourier series: ordinary and partial differential equations with boundary conditions

Many important engineering problems fall into the category of being linear operators with supporting conditions. In this paper an inner product and norm are used to numerically approximate the linear operator equation by use of generalized Fourier series. The resulting approximation is the “best” approximation in that a least squares error is minimized simultaneously for both fitting the problem's boundary conditions and satisfying the linear operator relationship (the governing equations) over the problem's domain. Because the numerical approximation technique involves a well-defined inner product, the approximation error can be evaluated by using Bessel's inequality. Minimization of the approximation error is subsequently achieved with respect to a weighting of the inner product components and the addition of basis functions used in the approximation. In this paper the numerical modelling technique is applied toward evaluating linear operation relationships such as ordinary and partial differential equations with fixed boundary equations. Nonlinear operators are also considered that can be approximated as being stepwise linear for portions of the problem domain (space or time). Additionally, different types of basis functions are considered, namely, the usual finite element trial function class, and also basis functions that span the domain with nonzero values (almost everywhere) such as polynomials, among others. Application of the generalized Fourier series approach to solving the problem of coupled heat and soil water flow in freezing soils demonstrates the utility of this numerical modelling technique.

Segmentation of EEG during sleep using time-varying autoregressive modeling.

Time-varying AR modeling is applied to sleep EEG signal, in order to perform parameter estimation and detect changes in the signal characteristics (segmentation). Several types of basis functions have been analyzed to determine how closely they can approximate parameter changes characteristic of the EEG signal. The TV-AR model was applied to a large number of simulated signal segments, in order to examine the behaviour of the estimation under various conditions such as variations in the EEG parameters and in the location of segment boundaries, and different orders of the basis functions. The set of functions that is the basis for the Discrete Cosine Transform (DCT), and the Walsh functions were found to be the most efficient in the estimation of the model parameters. A segmentation algorithm based on an "Identification function" calculated from the estimated model parameters is suggested.

Eigenvalues of the moment-method matrix and their effect on the convergence of the conjugate gradient algorithm (EM scattering)

A theory that relates eigenvalues of a continuous operator to those of the moment-method matrix operator is discussed and confirmed by examples. This theory suggests reasons for ill conditioning when certain types of basis and testing functions are used. In addition, the effect of eigenvalue location on the convergence of the conjugate gradient (CG) method is studied. The convergence rate of the CG method is dependent on the eigenvalues of the iteration matrix as well as on the number of eigenvectors of the iteration matrix needed to represent the right side of the equation. These findings explain the previously reported convergence behavior of the CG method when applied to electromagnetic-scattering problems. >

A comparison of basis functions for the pseudo-spectral method for a model reaction-diffusion problem

Different types of basis functions are considered for the pseudospectral method applied to the equation u t=Δu+ƒ(u) in 1- and 2-space dimensions. For large N, where N is the number of basis functions, the choice of Cheybyshev polynomials gives much greater accuracy than the eigenfunctions of the Laplacian Δ. However for small N (≲ 7), the eigenfunction expansion can be more accurate even with weakly nonperiodic boundary conditions.

A method to extend the spectral iteration technique

A method has been developed to extend the spectral iteration technique to encompass a more general class of subdomain type basis functions. While retaining the advantages of the spectral iteration technique, this method enables the usage of basis functions other than piecewise constant basis functions in expressing the unknown current distribution. Advantages of this technique with the choice of piecewise sinusoidal basis functions as compared to piecewise constant basis functions have been demonstrated for both antenna and scattering problems. This choice of basis functions is found to result in a further reduction in the number of unknowns required to represent the current distribution. Numerical results of the input impedance for a cylindrical antenna and the scattering cross section of a thin fiat plate are presented and discussed.

Multiple d-type basis functions for molecules containing second row atoms

The importance of multiple sets of d functions on the energy of SO2 and other molecules of Si, P, S, and Cl has been tested by comparing total energies from the SWPS set (supplementation by five sets of d functions as recommended by Strömberg, Wahlgren, Pettersson, and Siegbahn) with those obtained with one or two sets of d functions. The energy improvement due to supplementation is very sensitive to the nature of the attached atoms and to valence shell expansion but the optimum d function exponents are not appreciably affected by either of these factors. When used in conjunction with Dunning–Hay basis sets for second row atoms, supplementation by a single set of d functions provides about 80% of the self-consistent field (SCF) energy lowering obtained with five sets; use of a second set generates about 15% more. These results are compared with the results of adding f or replacing nuclear-centered functions with bond functions. Varying d function exponents about the optimum values produces an energy surface with the same general shape for all ten molecules studied. Based on this uniformity within a limited sample of molecules containing second row elements, recommendations are presented for the inclusion of supplementary sets of d functions in basis sets and the level of energy improvement to be achieved by using them is estimated.