Some infinite integrals involving products of exponential and bessel functions

This paper is concerned with the evaluation of some infinite integrals involving products of exponential and Bessel functions. These integrals are transformed, through some identities, into the expressions containing modified Bessel functions. In this way, the difficulties associated with the computations of infinite integrals with oscillating integrands are eliminated.     

Application of the modified Taylor approximation

In this paper we introduce and describe a new scheme for the numerical integration of smooth functions. The scheme is based on the modified Taylor expansion and is suitable for functions that exhibit near-sinusoidal or repetitive behaviour. We discuss the method and its rate of convergence, then implement it for the approximation of certain integrals. Examples include integrands involving Airy wave, Bessel, Gamma, and elliptic functions. The results, from the data in the tables, demonstrate that the method converges rapidly and approximates the integral as well as some well-known numerical integration methods used with sufficiently small step sizes.     

Evaluation of integrals for a ten-node isoparametric tetrahedral finite element

The use of isoparametric finite elemts in solving three-dimensional problems typically requires the numerical evaluation of a large number of integrals over individual element domains. The evaluation of these integrals by numerical quadrature, which is the traditional approach, can be computationally expensive. For certain problems the present study provides a more efficient method for evaluation of the needed integrals. For these problems some or all of the desired integrals can be evaluated as linear combinations of basic integrals whose integrands are either (i) products of shape (interpolation) functions or (ii) a derivative of a shape function times a product of one or more shape functions. Basic integrals of these two types (when written in terms of local coordinate systems) have integrands which are polynomial both in the variables of integration and in the nodal coordinates and, thus, can be expressed as linear combinations (with rational number coefficients) of a set of polynomial functions of the nodal coordinates. Group theoretic techniques can be employed to select appropriate sets of polynomial functions for use in these expansions and to reduce substantially the number of basic integrals which need to be explicitly evaluated.

The details for the approach have been worked out for a ten-node isoparametric tetrahedral element through the use of MACSYMA, a computer system for algebraic manipulation. The symmetry group for this element has order 24. The basic integrals of types (i) and (ii) are expressed as linear combinations of 20 and 26 terms, respectively. The special case of a straight-edged tetrahedral element with mid-edge nodes is also discussed.     

Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadratures, which ensures that the final quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.     

SINGINT: Automatic numerical integration of singular integrands

We explore the combination of deterministic and Monte Carlo methods to facilitate efficient automatic numerical computation of multidimensional integrals with singular integrands. Two adaptive algorithms are presented that employ recursion and are runtime and memory optimized, respectively. SINGINT, a C implementation of the algorithms, is introduced and its utilization in the calculation of particle scattering amplitudes is exemplified.     

Numerical integration based on quasi-interpolating splines

In this paper product quadrature rules based on quasi-interpolating splines are proposed and convergence results are proved for bounded integrands. Convergence results are also proved for sequences of Cauchy principal value integrals of these quasiinterpolating splines. Some comparisons with other methods and numerical examples are given.     

Algorithm 32 automatic computation of integrals with singular integrand,over a finite or an infinite range

A numerical quadrature algorithm is developed, for integrands which may exhibit some kind of singular behaviour within the finite of infinite integration range. Using the automatical FORTRAN IV integration program, one should provide the abscissae the function is not “smooth” at. The quadrature formula has been obtained by applying the trapezoidal rule after transformation of the integrand. Standing severe tests which were based on the test functions of Casaletto et al. and on Kahaner's sample set, the integration scheme turned out to be of a remarkable reliability, efficiency and accuracy.     

May the singular integrals in BEM be replaced by zero?

When approximating the singular integrals arising in the boundary element method by quadrature techniques, it is important to keep the quadrature error consistent with the discretization error in order to reach the optimal order of convergence. In classical approaches, this means that the order of the quadrature grows logarithmically in the number of degrees of freedom. We present a quadrature scheme based on alternative representations of the singular integrands that allows us to use a constant quadrature order without giving up consistency.     

Adaptive quadratures over volumes

We consider the numerical approximation of volume integrals over bounded domainsD:={D∈R ³:H(x>≤0}, whereH:R ³→R is a suitable decidability function. The integrands may be smooth maps or singular maps such as those arising in the volume potentials for boundary integral methods. An adaptive extrapolation method is described which is based on some simple quadrature rules. It utilizes an automatic simplicial subdivision of the domain. The method offers improvements over recently given approaches. A special version is offered for the important application of the numerical computation of volume potentials in boundary integral methods. Several examples illustrate the performance of the method.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

Numerical quadratures for singular and hypersingular integrals

We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.     

Computer algebra for the calculus of variations,the maximum principle,and automatic control

This paper describes how a computer-algebra system can solve variational optimization problems analytically. For a calculus-of-variations problem, users provide functional integrands and constraints. A program derives corresponding Euler-Lagrange equations, together perhaps with first integrals. Other programs attempt analytic solution of these equations. For an optimal control problem, users provide analytic expressions for the differential constraints on the state variables. A program determines the corresponding Hamiltonian and differential equations for the auxiliary variables, together with solutions to any trivial auxiliary equations. Other programs attempt analytic solution of the remaining equations while maximizing the Hamiltonian.This material is based upon work supported by the National Science Foundation under Grant No. MCS75-22983 A01.     

An Adaptive Numerical Integration Algorithm with Automatic Result Verification for Definite Integrals

We present a verified numerical integration algorithm with an adaptive strategy for smooth integrands. Verified representations of the remainder term are derived and a new adaptive strategy is given which delivers a desired integral enclosure with an error usually bounded by a specified error bound. Here, we discuss the distribution of the specified error bound onto the subintervals used in the algorithm more closely and present numerical results depending on the distribution of the error. Received March 15, 1999; revised December 1, 1999     

A natural quadrature formula for the numerical evaluation of the Macgregor-Westergaard complex potentials es crack problems

The method of singular integral equations can be used for the numerical solution of crack problems in plane and antiplane elasticity. Here we consider the problem of the subsequent numerical evaluation of the stress components in the whole cracked medium by using the MacGregor-Westergaard complex potentials. To this end we use a modified quadrature formula for Cauchy type (but not principal value) integrals and their derivatives, where the poles of the integrands are properly taken into consideration. This is achieved by using a natural interpolation-extrapolation formula for singular integral equations and, for this reason, the new term ‘natural quadrature formula’ is proposed. Two simple applications to specific crack problems, based on the Gauss- and Lobatto-Chebyshev quadrature formulas, show the efficiency of the suggested quadrature formula.     

Optimal Time Substitution in a Control Process

Control for the current state of a process by observation is investigated. The aim functional is defined by the ratio of integrals of integrands dependent on control functions. By this functional, control is interpreted as a time substitution control. Optimal control functions for certain classes of control functions and a method for computing them are determined. Relay control functions and related functions are shown to be optimal.__________Translated from Avtomatika i Telemekhanika, No. 8, 2005, pp. 64–83.Original Russian Text Copyright © 2005 by Kharlamov.This work was supported by the Program “Control for Mechanical Systems” no. 19 of the Presidium of the Russian Academy of Sciences and National School Grant, no. 2258.2003.1.     

On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems

A modification of the collocation method for the numerical solution of Cauchy type singular integral equations with generalized kernels is proposed. In accordance with this modification, although the abscissas and weights used in the numerical integration rule for the approximation of the integrals of the integral equation remain unaltered, yet the collocation points are selected in such a way that the poles of the integrands due not only to the Cauchy principal value part of the kernel, but also to the singularities of the generalized part of the kernel are taken into account. This modification assures the convergence of the method to the correct results since the error terms, usually neglected for the collocation points nearest to the end-points of the integration interval and generally tending to infinity, are now taken into consideration for the selection of the collocation points. The method was applied to the singular integral equations derived for the antiplane and plane elasticity problems of a crack terminating at a bimaterial interface.     

Quadrature for parabolic Galerkin BEM with moving surfaces

The successful implementation of the Galerkin Boundary Element Method hinges on the accurate and effective quadrature of the influence coefficients. For parabolic boundary integral operators quadrature must be performed in space and time where integrals have singularities when source- and evaluation points coincide. For problems where the surface is fixed, the time integration can be performed analytically, but for moving geometries numerical quadrature in space and time must be used. For this case a set of transformations is derived that render the singular space–time integrals into smooth integrals that can be treated with standard tensor product Gauss quadrature rules. This methodology can be applied to the heat equation and to transient Stokes flow.     

Four-index integral transformation exploiting symmetry

We present a Fortran implementation of four-index integral transformation in the LCAO-MO (linear combination of atomic orbitals-molecular orbitals) framework that exploits symmetry. Electron correlation calculations, such as configuration interaction (CI) calculations, usually require electron repulsion integrals to be transformed to a molecular orbital basis from a basis using atomic orbitals. In large molecular systems it is vital to exploit the sparsity of integrals in making this transformation. By exploiting symmetry, the sparsity of integrals is fully utilized, the size of intermediate file is minimized, and the computational cost is reduced. The present algorithm is simple and can readily be added to existing quantum chemistry program packages.

Program summary

Title of program: SYM4TR (symmetry adapted 4-index integral transformation)Catalogue identifier: ADUWProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUWProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: IBM/AIX, HP Alpha server/Tru64, PC's/LinuxProgram language used: Fortran 95Number of lines in distributed program, including test data, etc.: 4519No. of bytes in distributed program, including test data, etc.: 32 095Distributed format: tar gzip fileNature of physical problem: Molecular orbital calculations including electron correlation effects usually require electron repulsion integrals to be transformed from an atomic orbital (AO) basis to a molecular orbital (MO) basis. By exploiting the sparsity of molecular integrals, the computational cost and memory needed for the transformation are minimized.Method of solution: The sparsity of molecular integrals is exploited. The program treats only nonzero integrals. The length of running indices in DO loops is reduced using the block-diagonal form of the MO coefficient matrix. In the present program, the point group is limited to D_2h and its subgroups.     

High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions

This paper presents the mechanical quadrature methods (MQMs) for solving the boundary integral equations of steady-state anisotropic heat conduction equation on the smooth domains and polygons, respectively. The costless and high-accurate Sidi–Israeli quadrature formula are applied to deal with the integrals in which the kernels have a logarithmic singularity. Especially, the Sidi transformation is used for the polygon cases in order to obtain a rapid convergence by degrading the singularity at the corners on the boundary. The convergence and stability of the MQMs solution are proved based on Anselone's collective compact theory. In addition, asymptotic error expansion of the MQMs shows that the approximation order is of O(h³), where h is the partition size of the boundary. Finally, numerical examples are tested and results verify the theoretical analysis.     

A program to generate a basis set adaptive radial quadrature grid for density functional theory

We present an automatic quadrature routine (AQR) which generates an atomic basis set adaptive radial quadrature grid for the numerical evaluation of molecular integrands in density functional theory. Unlike the popular radial grids that are tuned to a particular class of integrands and rely on a fixed selection of points, our grid adapts itself automatically to the atomic shell structure of any radial integrand and determines the best number of quadrature points that provides user specified accuracy. We evaluate the performance of our radial grid on various tight, diffuse, and noble gas atom radial integrands. We conclude that the radial quadrature grid generated by our AQR is generally comparable to and sometimes better than the best ranked popular radial grids in efficiency and reliability.