Numerical Evaluation Of Integrals Research Articles

Dynamic response of a beam to a train of moving forces driven by a translated Poisson process

Dynamic response of a beam to a random train moving forces driven by a so-called translated Poisson process is considered. The commonly used model of the counting process driving the random train of loads due to highway traffic is the Poisson process. Other, more adequate models of highway traffic loads are renewal processes, for example an Erlang renewal process (Ashton, 1966). In this paper the inter-arrival times of forces (vehicles) are assumed to have a translated (shifted) negative-exponential distribution. Such a renewal counting process, called a translated Poisson process, is known to be an adequate model of highway traffic (Haight, 1963). The problem is converted into that for modal responses by normal mode technique. In time domain approach the integral expressions for a mean value and the mean square of the response involve the renewal densities (Iwankiewicz, 1995; Jabłonka and Iwankiewicz, 2021). However, since for the translated Poisson process no exact solution for the renewal density exists, this renewal density is determined approximately, using a Gaver–Stehfest method (Gaver Jr, 1966; Stehfest, 1970). The mean value and variance of the response are determined by numerical evaluation of integrals and verified against direct Monte Carlo simulations. The results are also compared with those for a train of moving forces driven by a Poisson and by an Erlang process.

Evaluation of numerical integration by using Hermite wavelets

In this research, we present a numerical scheme to evaluate the numerical integration of a given function using Hermite wavelets. The proposed technique is based on the expansion of the given function into a series of Hermite wavelets basis functions. Some numerical experiments have been performed to illustrate the accuracy of the proposed method.

Characterizing the mobile coverage probability in different geometries

This work characterizes the radio frequency (RF) power distribution within a given area via its mean and variance in three particular geometries. Through our statistical approach we have developed computationally simple equations that can be used to compute the mean value and variance of the RF power received in a circular area and in some other geometries without numerical evaluations of integrals and which improve on the results/methods provided in the literature to date. We begin our study with the analysis of the received power on a linear trajectory and then we extend the result for a circular geometry. We have validated our results via Monte Carlo simulations. These results are useful both the study of the coverage of a RF transmitter and to identify spectrum holes.

AN EXTRAPOLATORY QUADRATURE RULEFOR ANALYTIC FUNCTIONS

A quadrature rule for the numerical evaluation of integrals of analytic functions along directed line segments in the complex plane has been formulated using the transformed rule based on Gauss Legendre two point quadrature formulas and an interpolatory three point rule. The degree of precision has been increased from five to seven

APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS

This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

Electromechanical response of (3–0, 3–1) particulate, fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.

A highly accurate boundary treatment for confined Stokes flow

Fluid flow phenomena in the Stokesian regime abounds in nature as well as in microfluidic applications. Discretizations based on boundary integral formulations for such flow problems allow for a reduction in dimensionality but have to deal with dense matrices and the numerical evaluation of integrals with singular kernels. The focus of this paper is the discretization of wall confinements, and specifically the numerical treatment of flat solid boundaries (walls), for which a set of high-order quadrature rules that accurately integrate the singular kernel of the Stokes equations are developed. Discretizing by Nyström’s method, the accuracy of the numerical integration determines the accuracy of the solution of the boundary integral equations, and a higher order quadrature method yields a large gain in accuracy at negligible cost. The structure of the resulting submatrix associated with each wall is exploited in order to substantially reduce the memory usage. The expected convergence of the quadrature rules is validated through numerical tests, and this boundary treatment is further applied to the classical problem of a sedimenting sphere in the vicinity of solid walls.

A Fast Algorithm to Solve the Bitsadze Equation in the Unit Disk

An algorithm is provided for the fast and accurate computation of the solution of the Bitsadze equation in the complex plane in the interior of the unit disk. The algorithm is based on the representation of the solution in terms of a double integral as it shown by Begehr [1,2], some recursive relations in Fourier space, and Fast Fourier Transforms. The numerical evaluation of integrals at points on a polar coordinate grid by straightforward summation for the double integral would require floating point operation per point. Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of per point. The algorithm has the added advantage of working in place, meaning that no additional memory storage is required beyond that of the initial data. This paper is a result of application of many of the original ideas described in Daripa [3].

Variational Approach to Power Evolution in Cascaded Fiber Raman Laser

A variational approach is formulated and implemented for numerically solving a system of nonlinear two-point boundary value problem (BVP) with coupled boundary conditions modeling the power evolution in cascaded fiber Raman laser with the fiber Bragg gratings at the ends of the cavity. The nonlinearity is treated by successive linearization and the coupled boundary conditions are naturally incorporated into the system through integration in the variational setting. A global approximation of the dependent variables in terms of Legendre polynomials is used to provide a stable Lagrangian interpolation representation as well as the Legendre-Gauss quadrature for accurate numerical evaluation of integrals in the variational formulation. An initial approximate solution is constructed for the delicate convergence to the solution. The approach is validated against an approximate analytic solution and some exact integrals of the variables. The numerical experiments show exponential (spectral) accuracy achieved with much lower resolution in comparison to a widely available BVP solver. Further numerical experiments are performed to reveal the physical characteristics of the underlying model.

Magnon excitations in vortex-state nanorings

The magnon mode excitation spectrum is calculated using the linearized Landau–Lifshitz equations applied to the ring in the vortex ground state. A combination of analytical techniques with numerical evaluation of integrals is used to obtain the mode frequency, with the effective magnetostatic volume charge as a small perturbation. In general, the modes can be classified according to radial number n and azimuthal number m. It is shown that the frequency increases as n increases, and the frequency decreases as m increases.

A modification of the numerical integration error method for the zero-finding problem of an analytic function

The numerical integration error method (NIEM) is a zero-finding method which is based on the numerical evaluation of integrals of ƒ′/ƒ NIEM is an iterative method of higher-order, and (therefore) a lot of computational complexities are required. Besides it is necessary to select one from a number of choices of NIEM correction. In this paper we introduce an algorithm that improves computational efficiency of NIEM, yielding better results for the case of multiple or clustered zeros. Some numerical examples demonstrating the efficiency of this algorithm are introduced.

Folding of a plastic fibre under combined shear and compression

Large deflection of an elastic–plastic, nonlinear strain-hardening clamped-sliding beam subjected to a combined compression and shear load is solved analytically. The solution requires only numerical evaluation of integrals. Analytical expressions for applied force, deflection, bending moment, and strain energy are obtained. Numerical examples are given for a single stainless steel fibre of the hybrid stainless steel assembly (HSSA). The contact condition of the beam with end platens is determined. It is shown that the normalized effective crush distance ranges from 0.85 to 1.00, and depends only on loading angles and material hardening exponents. The crushing load has a maximum for the case with a small material hardening exponent and a small loading angle. It is found that no plastic unloading will occur before the beam contacts with the end platens. Finite element simulations are performed to verify the accuracy of the analytical solution, and a rather good agreement is obtained.

A quasi-Newton interior point method for semi-infinite programming

We propose an interior point method for solving nonlinear semi-infinite programming (SIP) problems. The method is based on a transcription of the SIP problem into a nonlinear finite problem. We apply the interior point strategy to the finite problem and obtain an algorithm for nonlinear SIP. The evaluation of the constraint functions and derivatives of the finite problem requires the numerical evaluation of integrals that we compute by an adaptative trapezoid formula. Numerical results with a collection of several problems from the SIPAMPL [A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes. SIPAMPL v2.0: Semi-infinite programming with AMPL. Technical Report ALG/EF/4-2002, Universidade do Minho, Braga, Portugal, December 2002. [http://www.norg.uminho.pt/aivaz/].] database are also shown. †E-mail: emgpf@dps.uminho.pt ‡E-mail: p.gomes@ic.ac.uk

Why Gaussian quadrature in the complex plane

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitudec T A 1 b is an example whereA is a discretization of a differential‐integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use c T x .k/ . A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1=. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.

Geoacoustic Inversion via Genetic Algorithm and Its Application to Manganese Sediment Identification

An acoustic inversion method using a wide-band signal and two near field receivers is proposed and applied to multiple layered seabed models including a manganese sediment. The inversion problem can be formulated into a probabilistic model comprised of signals, a forward model, and additive noise. The forward model simulates wide-band signals, such as chirp signals, and is chosen to be the source-wavelet-convolution plane wave modeling method. The wavelet matching technique, using weighted least-squares fitting, estimates the sediment sound-speed and thickness on which determination of the possible numerical ranges for a priori uniform distribution is based. The genetic algorithm is applied to a global optimization problem to find a maximum a posteriori solution for determined a priori search space. Here the object function is defined by an L2 norm of the difference between measured and modeled signals. Not only the marginal pdf but also its statistics are calculated by numerical evaluation of integrals using the samples selected during importance sampling process of the genetic algorithm.

Neurospectral computation for input impedance of rectangular microstrip antenna

The neural network method evaluates the integrals appearing in the spectral domain analysis of microstrip antennas. Closed form expressions are found for the integral in the indefinite form and then limits are imposed to obtain the definite integral value. This technique eliminates the difficulties of handling the singularity points encountered in the numerical evaluation of integrals. Results are compared with those of the standard spectral domain method.

The 3-D thermoelastic boundary element method: semi-analytical integration for subparametric triangular elements

In this paper a semi-analytical integration scheme is described which is designed to reduce the errors that can result with the numerical evaluation of integrals with singular integrands. The semi-analytical scheme can be applied to quadratic subparametric triangular elements for use in thermoelastic problems. Established analytical solutions for linear isoparametric triangular elements are combined with standard quadrature techniques to provide an accurate integration scheme for quadratic subparametric triangular elements. The use of subparametric elements provides an efficient means for coupling thermal and elastostatic analyses. It is possible for the same mesh to be employed, with linear isoparametric elements used for thermal analysis and quadratic subparametric elements used for deformation analysis. Numerical tests are performed on simple test problems to demonstrate the advantages of the semi-analytical approach which is shown to be orders of magnitude more accurate than standard quadrature techniques. Moreover, the expected increased accuracy with subparametric elements is also demonstrated. © 1998 John Wiley & Sons, Ltd.

The elastostatic three-dimensional boundary element method: analytical integration for linear isoparametric triangular elements

In this paper an analytical integration scheme is described that is designed to reduce the errors resulting from the numerical evaluation of integrals with singular integrands. The analytical scheme can be applied to linear triangular elements for use in elastostatic problems and is particularly useful for predicting distortion, to high accuracy, close to surfaces. It is demonstrated that although the analytical scheme takes longer computationally than the usual quadrature approach it is quicker when element subdivision is required to achieve reasonable accuracy. Numerical tests are performed on a simple test problem to demonstrate the advantages of the analytical approach, which is shown to be orders of magnitude more accurate than standard quadrature techniques.

Numerical evaluation of integrals of analytic functions

Numerical evaluation of integrals of analytic functions

String bowing revisited

This work explores an integral equation formulation of a generalised bowing problem incorporating air resistance and bow width. The Keller/Friedlander results are obtained as a special case. Some illustrative results are presented for a lightly damped string bowed at a point. The related integral equations are solved using Filon's method, a clever but relatively unknown algorithm for the numerical evaluation of integrals with rapidly oscillating trignometric kernels.