Showing papers on "Basis function published in 1980"
TL;DR: In this paper, a reduced basis technique and a computational algorithm are presented for predicting the nonlinear static response of structures, where a total Lagrangian formulation is used and the structure is discretized by using displacement finite element models.
Abstract: A reduced basis technique and a computational' algorithm are presented for predicting the nonlinear static response of structures. A total Lagrangian formulation is used and the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of basis vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The Rayleigh-Ritz approximation functions (basis vectors) are chosen to be those commonly used in the static perturbation technique namely, a nonlinear solution and a number of its path derivatives. A procedure is outlined for automatically selecting the load (or displacement) step size and monitoring the solution accuracy. The high accuracy and effectiveness of the proposed approach is demonstrated by means of numerical examples.
414 citations
TL;DR: In this paper, the authors describe a fully relativistic augmented plane wave (APW) method where the basis functions are pure spin functions in the large component, which allows spin-mixing interactions to be separated and treated more efficiently than in the standard APW method.
Abstract: The authors describe a fully relativistic augmented-plane-wave (APW) method where the basis functions are pure spin functions in the large component. This feature allows spin-mixing interactions to be separated and treated more efficiently than in the standard relativistic APW method. These basis functions are constructed by solving an approximate relativistic radial equation. In addition, the energy derivative is used in the construction of the basis functions so that one obtains adequate variational freedom solving a linear secular equation. Both the utility and the limitations of the technique are discussed.
409 citations
01 Mar 1980
TL;DR: In this article, the electric field integral equation (EFIE) is used with the moment to develop a simple and efficient numerical procedure for treating problems of scattering by arbitrarily-shaped objects.
Abstract: : The electric field integral equation (EFIE) is used with the moment to develop a simple and efficient numerical procedure for treating problems of scattering by arbitrarily-shaped objects. The objects are modeled for numerical purposes by planar triangular surface patch models. Because the EFIE formulation is used, the procedure is applicable to both open and closed bodies. Crucial to the formulation is development of a set of special subdomain basis functions defined on pairs of adjacent triangular patches. The basis functions yield a current representation which is free of line or point charges at subdomain boundaries. A second approach using the magnetic field integral equation (MFIE) and employing the same basis functions is also developed. Although the MFIE applies only to closed bodies, the moment matrix of the MFIE is also needed in dielectric scattering problems and in the so-called combined field integral equation used to eliminate difficulties with internal resonances present in the MFIE and EFIE formulations. The EFIE approach is applied to the scattering problems of plane wave illumination of a flat square plate, a bent square plate, a circular disk, and a sphere. Comparisons of surface current densities are made with previous computations or exact formulations and good agreement is obtained in each case. The MFIE approach is also applied to the sphere and reasonable agreement with exact calculations is obtained.
198 citations
TL;DR: In this paper, the spectral domain analysis is applied for deriving despersion characteristics of dominant and higher order modes in fin-line structures, and the characteristic impedance is calculated based on the powervoltage definition.
Abstract: The spectral domain analysis is applied for deriving despersion characteristics of dominant and higher order modes in fin-line structures. In addition to the propagation constantant, the characteristic impedance is calculated based on the power-voltage definition. Numerical results are compared for different choices of basis functions and allow to estimate the accuracy of the solution.
143 citations
TL;DR: In this article, it is argued that it is not necessary for the two-electron molecular orbital integrals and configuration interaction expansion coefficients to be stored in the computer as accurately as they are in present configuration interaction calculations, so these numbers may be kept in the central memory, even for large basis set calculations.
131 citations
TL;DR: In this paper, the matrix elements of the translation operator with respect to a complete orthonormal basis set of the Hilbert space L2(R3) are given in closed form as functions of the displacement vector.
Abstract: The matrix elements of the translation operator with respect to a complete orthonormal basis set of the Hilbert space L2(R3) are given in closed form as functions of the displacement vector. The basis functions are composed of an exponential, a Laguerre polynomial, and a regular solid spherical harmonic. With this formalism, a function which is defined with respect to a certain origin, can be ’’shifted’’, i.e., expressed in terms of given functions which are defined with respect to another origin. In this paper we also demonstrate the feasibility of this method by applying it to problems that are of special interest in the theory of the electronic structure of molecules and solids. We present new one‐center expansions for some exponential‐type functions (ETF’s), and a closed‐form expression for a multicenter integral over ETF’s is given and numerically tested.
94 citations
TL;DR: In this article, an iterative approach using the Schwinger variational principle was proposed to solve the Lippmann-Schwinger equation for electron-molecule scattering.
Abstract: We present an iterative approach which uses the Schwinger variational principle to solve the Lippmann-Schwinger equation for electron-molecule scattering. This method combines the use of discrete basis functions to describe the effects of the noncentral molecular potential with an iterative procedure which provides systematic convergence of the scattering solutions. Results for electron-H2 scattering in the static-exchange approximation show that the method converges rapidly and gives very accurate results.
70 citations
TL;DR: In this paper, a mathematical formulation for solving the three-dimensional nonlinear hydrodynamic equations, using the Galerkin method with an arbitrary set of basis functions, is presented, and an explicit time splitting method is used to integrate these equations through time.
59 citations
TL;DR: In this article, the relativistic Hartree-Fock equations for closed-shell polyatomic systems are derived by expanding each component of the molecular one-electron spinors separately into a scalar basis set.
46 citations
TL;DR: In this paper, a group theory method for deriving a complete and non redundant canonical set of basis functions adapted to the description of Orientational Disorder In (plastic molecular) Crystal (ODIC) is developed.
Abstract: A group theory method for deriving a complete and non redundant canonical set of basis functions adapted to the description of Orientational Disorder In (plastic molecular) Crystal (ODIC) is developed. The method takes full advantage of the properties of the site and molecular symmetry groups. In particular, it is shown that, when both groups contain improper rotations, the canonical basis includes functions which have not been previously considered.
45 citations
TL;DR: In this article, a finite element model which uses triangular, isoparametric elements with quadratic basis functions for the two velocity components and linear basis function for water surface elevation is used in the computation of shallow water wave motions.
Abstract: A finite element model which uses triangular, isoparametric elements with quadratic basis functions for the two velocity components and linear basis functions for water surface elevation is used in the computation of shallow water wave motions. Specifically addressed are two common uncertainties in this class of two-dimensional hydrodynamic models: the treatment of the boundary conditions at open boundaries and the treatment of lateral boundary conditions. The accuracy of the models is tested with a set of numerical experiments in rectangular and curvilinear channels with constant and variable depth. The results indicate that errors in velocity at the open boundary can be significant when boundary conditions for water surface elevation are specified. Methods are suggested for minimizing these errors. The results also show that continuity is better maintained within the spatial domain of interest when ‘smooth’ curve-sided elements are used at shoreline boundaries than when piecewise linear boundaries are used. Finally, a method for network development is described which is based upon a continuity criterion to gauge accuracy. A finite element network for San Francisco Bay, California, is used as an example.
TL;DR: In this paper, a matrix analytic continuations of the Born-Oppenheimer Hamiltonian is used to compute the parallel component of the photoionization cross section for H/sub 2/sup + + +.
Abstract: Procedures for computing molecular photoabsorption cross sections via finite expansions in sets of L/sup 2/ complex basis functions are discussed. Two recently proposed schemes for analytically continuing a matrix representation of the Born-Oppenheimer Hamiltonian are investigated. Both of these matrix analytic continuations are used to compute the parallel component of the photoionization cross section for H/sub 2//sup +/. It is found that to obtain numerically stable results it is necessary to use complex-basis functions which are capable of describing cusps in the molecular wave function at complex values of the coordinates. The application of this technique to larger molecules is also discussed.
TL;DR: In this paper, a method for solving the linear hydrodynamic equations for the sea, using an expansion of the horizontal component of current in terms of depth-varying functions (the basis functions) with coefficients that vary with time and horizontal position, is presented.
TL;DR: In this article, the first application of the Schwinger variational principle to electron-molecule scattering was reported, and results for electron-H2 scattering in the static exchange approximation were shown to converge far more quickly with respect to the size of the basis than any other algebraic expansion technique considered to date.
Abstract: The authors report the first application of the Schwinger variational principle to electron-molecule scattering. Results for electron-H2 scattering in the static-exchange approximation show that the Schwinger method can provide accurate solutions of the scattering problem with small discrete basis sets. The Schwinger variational expression is found to converge far more quickly with respect to the size of the basis than any other algebraic expansion technique considered to date. Results are also presented for hybrid trial scattering wave functions containing both continuum and discrete basis functions.
TL;DR: In this paper, the dimensionless magnetic polarizabilities of a few characteristic apertures are given for the moment method and the relevant integral equations are solved by the moment-based method.
Abstract: Curves are given for the dimensionless magnetic polarizabilities
u_{mx} and
u_{my} of a few characteristic apertures. The relevant integral equations are solved by the moment method. The subareas are triangular, and the basis functions for the triangles touching an edge take the edge singularity into account. Some data are included for a few typical ring-shaped apertures.
TL;DR: An analytic theory for the dispersion of the fundamental mode on wide open microstrip is presented in this paper, where only a single basis function is needed to accurately represent each of the change and current distributions on the strip, thus allowing more efficient determination of the propagation constant as compared to moment-method solutions requiring a larger number of basis functions.
Abstract: An analytic theory for the dispersion of the fundamental mode on wide open microstrip is presented. Only a single basis function is needed to accurately represent each of the change and current distributions on the strip, thus allowing more efficient determination of the propagation constant as compared to moment-method solutions requiring a larger number of basis functions. The results obtained blend smoothly into results of high-frequency (Wiener-Hopf) theories, and still retain the appealing physical interpretation in terms of capitance and inductance of the narrow strip theory previously obtained by the authors.
TL;DR: The Rayleigh–Ritz–Galerkin method with bicubic Hermites is almost always as efficient as the collocation method; the contrast appears to be due to more efficient assembly phase techniques.
Abstract: We compare the computer costs for several finite element methods for linear, self -adjoint, elliptic boundary-value problems on two-dimensional rectangular domains. We consider the efficiency of the assembly phase in each method, and in some cases point out improvements to previous work. Moreover, we develop approximate operation counts for the solve phase which indicate that it is advantageous to use basis functions which are as smooth as possible.We present the results of testing fourth-order versions of the methods on some of the problems from Houstis and Rice (CSD-TR 263, Computer Science Department, Purdue University, West Lafayette IN 1978). Our results contrast with those of the study of Houstis et al. (J. Comp. Phys. 27 (1978), pp. 323–350). First, the Rayleigh–Ritz–Galerkin method with bicubic Hermites is almost always as efficient as the collocation method; the contrast appears to be due to more efficient assembly phase techniques. Second, we present an example of a problem for which evaluating ...
TL;DR: In this paper, the authors compared convergence rates for adiabatic basis sets and conventional free-molecule basis sets for the expansion of the scattering wave function in the quantal treatment of a collinear model of vibrationally inelastic He-H2 scattering.
TL;DR: In this article, the authors apply the Schwinger method, two Kohn-type methods, and three Harris-Michels-type method to electron scattering by the same potential with the same sets of basis functions.
TL;DR: In this paper, the ground-state energy of simple models of the bare two-nucleon interaction, assumping a correlated trial function of Jastrow type, is calculated.
TL;DR: In this article, an alternative method for resolving the classical Liouville operator L into "raising" and "lowering" suboperators is introduced, where a given resolution corresponds to describing the dynamical variable of interest in terms of a specified set of dynamical functions, i.e., within a specified basis.
Abstract: An alternate method for resolving the classical Liouville operator L into ’’raising’’ and ’’lowering’’ suboperators is introduced. A given resolution corresponds to describing the dynamical variable of interest in terms of a specified set of dynamical functions, i.e., within a specified basis. The Gram–Schmidt basis, in which the basis functions are orthogonal, generates a particularly simple resolution of L, which is utilized in conjunction with the Zassenhaus formula for the propagator exp(Lt) to construct a variety of useful approximate expressions for the time‐autocorrelation function (TACF) of the variable of interest. These approximate formulas, which constitute partial summations of the Maclaurin series expansions of the TACF, are used to analyze the computer‐simulation data of Levesque and Verlet [Phys. Rev. A 2, 2514 (1970)] for the single‐particle velocity TACF φ of a Lennard‐Jones liquid. This analysis provides a firm theoretical basis for the phenomenological memory function used by Levesque a...
TL;DR: In this paper, various transformation rules that are due to two different bases of the same trial function space kept over an arbitrary finite element are presented in simple matrix form, and it is further demonstrated that the finite element solutions are theoretically independent of the basis choice unless the assembly admissible conditions are violated.
TL;DR: In this paper, the symmetry analysis of magnetic structures and methods of determining them from neutron-diffraction data is presented. But the authors do not consider the problem of determining a large number of magnetic-moment vectors of the crystal to finding a small number of mixing coefficients.
Abstract: The contemporary state of neutron diffraction of magnetic structures is analyzed from the standpoint of the theory of symmetry of crystals. It is shown that the varied and numerous structures determined in neutron-diffraction studies can be classified and described by the theory of representations of space groups of crystals. This approach is based on expanding the spin density of the crystal in terms of basis functions of the irreducible representations of its space group. Thus the magnetic structure can be specified by the mixing coefficients of the basis functions. Analysis of a large number of different kinds of magnetic structures shows that they arise in the overwhelming majority of cases, in accord with Landau's hypothesis, from a phase transition that follows a single irreducible representation. This means that the number of parameters that fully fix the magnetic structure of an arbitrarily complex crystal is small and equal to the dimensionality of the responsible irreducible representation. This offers great advantages in employing the symmetry approach in deciphering neutron-diffraction patterns of a crystal under study. This is because it reduces the problem of determining a large number of magnetic-moment vectors of the crystal to finding a small number of mixing coefficients. This review presents the fundamentals of such a symmetry analysis of magnetic structures and methods of determining them from neutron-diffraction data. The described method, which is closely allied to Landau's general theory of phase transitions, is illustrated by the most recent neutron-diffraction studies of magnetic structures. They included the so-called multi-k-structures, which are characterized simultaneously by several wave vectors, and structures described simultaneously by several irreducible representations of the space group of the crystal. The article gives a physical explanation of the existence of such structures. The experimental studies of crystal-lattice distortions accompanying the onset of magnetic ordering are reviewed. It is shown how symmetry arguments allow one to determine these distortions as well as the unknown magnetic structure. This review presents in condensed but accessible form the symmetry approach to describing the magnetic structures of crystals and analyzes on this basis the feasibility and degree of reliability of deciphering them by employing the scattering of unpolarized and polarized neutrons.
TL;DR: The Tri-Symmetric Chorin (TSC) estimator as discussed by the authors uses orthonormal Hermite polynomials as basis functions and uses an adaptive series selection algorithm so that only those coefficients determined with sufficient precision are retained in the Hermite expansion.
Book•
01 Jan 1980
TL;DR: In this paper, the problem of finding all critical points of a complicated function is solved by partitioning the domain into cubes and approximating the density function on each cube by a polynomial of three variables, chosen so that all first derivatives are continuous across the cube boundaries.
Abstract: The three-dimensional structure of proteins is commonly determined from x-ray diffraction data by manually inspecting the electron density, displayed as a stack of two dimensional contour maps. A network connecting peaks and passes of the density function has been suggested as a representation more suitable for automatic interpretation. However, locating all critical points of such a complicated function is a difficult problem.
The solution proposed here is to partition the domain into cubes and approximate the density function on each cube by a polynomial of three variables, chosen so that all first derivatives are continuous across the cube boundaries. Two algorithms are examined for this approximation: one is least squares fitting with smooth tent-shaped basis functions (known as tensor product B-splines) by repeated univariate least squares spline fits, and the other scales Fourier coefficients of the electron density function. Critical points can then be sought independently on each cube. One variable is solved for immediately in terms of the other two; the resulting two-dimensional problems are approximated on a subgrid and finally reduced to one-dimensional quadratic equations. This method locates critical points more accurately than discrete pattern matching and more quickly than repeated Newton searches.
TL;DR: In this paper, the notion of relative and absolute invariants and the related basis concepts (module basis and integrity basis) were introduced for physical properties which are represented by analytic functions of an equivalent algebraic structure.
Abstract: Introducing the notion of relative and absolute invariants and the related basis concepts (module basis and integrity basis), it is demonstrated that these algebraic concepts can be of advantage in the investigation of physical properties which are represented by analytic functions of an equivalent algebraic structure. In particular, the discussion centers on the case of a module basis containing only a single element and on analytic absolute invariants depending only on the differences of the independent variables, for which the related integrity basis turns out to be redundant.
TL;DR: HAL as mentioned in this paper is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, which may come from teaching and research institutions in France or abroad, or from public or private research centers.
Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. GROUND-STATE PROPERTIES OF 3He↑ AND D↑ WITHIN THE METHOD OF CORRELATED BASIS FUNCTIONS J. Clark, E. Krotscheck, R. Panoff
TL;DR: Galerkin's method is used to approximate the transient solutions of intial value problems in which a steady state or advanced time state is known as mentioned in this paper, and a convergence theorem is established and choices of basis functions are discussed.
TL;DR: In this paper, a crucial conjecture relating to construction of rational basis functions for finite element computation is examined, and a more sophisticated application of algebraic geometry to numerical analysis is presented.