Computation of the attractive force of an ellipsoid

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.     

Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations

Quadrature formulae are established for product integration rules based on discrete spline quasi-interpolants on a bounded interval. The integrand considered may have algebraic or logarithmic singularities. These formulae are then applied to the numerical solution of integral equations with weakly singular kernels.     

The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals.     

Multi-Dimensional Quadrature of Singular and Discontinuous Functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

The numerical evaluation of cauchy principal values of integrals by Romberg integration

The problem considered is that of evaluating numerically an integral of the form where the integrand has one or more simple poles in the interval (O,p). Modified forms of the trapezoidal and mid-ordinate rules, taking account of the singularities, are obtained; it is then shown that the resulting approximations can be extrapolated by Romberg's method. Further modifications to deal with the case when the integrand has an integrable branch singularity at one or both ends of the interval of integration are also briefly discussed.     

薄体结构温度场的高阶边界元分析

分析了二维问题边界元法3节点二次单元的几何特征,区分和定义了源点相对高阶单元的Ⅰ型和Ⅱ型接近度.针对二维位势问题高阶边界元中奇异积分核,构造出具有相同Ⅱ型几乎奇异性的近似核函数,在几乎奇异积分单元上分离出积分核中主导的奇异函数部分.原积分核扣除其近似核函数后消除几乎奇异性,成为正则积分核函数,并采用常规Gauss数值方法计算该正则积分;对奇异核函数的积分推导出解析公式,从而建立了一种新的边界元法高阶单元几乎奇异积分半解析算法.应用该算法计算了二维薄体结构温度场算例,计算结果表明高阶单元半解析算法能充分发挥边界元法优势,显著提高计算精度.     

A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element

A general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations (BIEs). The method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located. The distance functions are redefined in two local coordinate systems. A new system denoted as (α,β) is introduced here firstly. Its implementation is simpler than that of the polar system and it also performs efficiently. Then a new distance transformation is developed to remove or weaken the near singularities. To perform integration on irregular elements, an adaptive integration scheme is applied. Numerical examples are presented for both planar and curved surface elements. The results demonstrate that our method can provide accurate results even when the source point is very close to the integration element, and can keep reasonable accuracy on very irregular elements. Furthermore, the accuracy of our method is much less sensitive to the position of the projection point than the conventional method.     

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.     

Hypersingular kernel integration in 3D Galerkin boundary element method

We consider Cauchy singular and Hypersingular boundary integral equations associated with 3D potential problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. In particular, for constant and linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations and suggest suitable quadrature schemes to evaluate the outer integrals required to form the Galerkin matrix elements. These numerical indications are given after an analysis of the singularities arising in the whole integration process, which is valid also for shape functions of higher degrees.     

A singular function boundary integral method for elliptic problems with singularities

In this work we present a singular function boundary integral method for elliptic problems with boundary singularities. In this method, the approximation is constructed from the truncated asymptotic expansion for the solution near the singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of singularity. We are able to show that the method approximates the generalized stress intensity factors, i.e. the coe cients in the asymptotic expansion, at an exponential rate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional integral to the contour integral representation. We then express the integrand as a combination of gamma functions so that we can easily find all poles of the integrand. We obtain various asymtotics of the kernels by using Cauchys Residue Theorem with shifting contour integration. As a byproduct, exact coefficients are also obtained. We apply this method to general Lévy processes whose characteristic functions are radial and satisfy some regularity and size conditions. Our approach is based on the Fourier analytic point of view.     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 2: Collocation method for the radon equation

The non-uniquely solvable Radon boundary integral equation for the two-dimensional Stokes-Dirichlet problem on a non-smooth domain is transformed into a well posed one by a suitable compact perturbation of the velocity double-layer potential operator. The solution to the modified equation is decomposed into a regular part and a finite linear combination of intrinsic singular functions whose coefficients are computed from explicit formulae. Using these formulae, the classical collocation method, defined by continuous piecewise linear vector-valued basis functions, which converges slowly because of the lack of regularity of the solution, is improved into a collocation dual singular function method with optimal rates of convergence for the solution and for the coefficients of singularities.     

On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients

This paper represents the third part of a contribution to the “dictionary” of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other. In the first part (cf. [4]) we studied the case that the differential equation (D) has at most regular singularities at O and at ∞, and arbitrary singularities in the rest of the complex plane. We constructed fundamental systems of solutions of a corresponding difference equation (A), using integral transforms of microsolutions of (D) at its singular points in ?. In the second part ([5]) we considered differential equations having at most a regular singularity at O and an irregular one at O. We used integral transforms of asymptotically flat solutions of (D) to define it fundamental system of solutions of (Δ), holomorphic in a right half plane, and integral transforms of sections of the sheaf of solutions of (D) modulo solutions with moderate growth as t → 0 in some sector, to define a fundamental system of (Δ), holomorphic in a left half plane. In this final part we combine the techniques and results of the preceding papers to deal with the general case.     

On the boundaries of the parametric resonance domain

The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is considered. The singularities of the general position arising at the boundaries of the stability and instability (parametric resonance) domains in the case of two and three parameters are listed. A constructive approach is proposed which enables one, in the first approximation, to determine the stability domain in the neighbourhood of a point of the boundary (regular or singular) from the information at this point. This approach enables one to eliminate a tedious numerical analysis of the stability region in the neighbourhood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stability of the oscillations of an articulated pipe conveying fluid with a pulsating velocity is considered. In the space of three parameters (the average fluid velocity and the amplitude and frequency of pulsations) a singularity of the boundary of the stability domain of the “dihedral angle” type is obtained and the tangential cone to the stability domain is calculated.     

Gaussian quadrature and acceleration of convergence

The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration considered in this paper are only real or purely imaginary. This revised version was published online in August 2006 with corrections to the Cover Date.     

A High-Order Piecewise Quartic Spline Rule for Hadamard Integral and Its Application of the Cavity Scattering

We develop a fourth-order piecewise quartic spline rule for Hadamard integral. The quadrature formula of Hadamard integral is obtained by replacing the integrand function with the piecewise quartic spline interpolation function. We establish corresponding error estimates and analyze the numerical stability. The rule can achieve fourth-order convergence at any point in the interval, even when the singular point coincides with the grid point. Since the derivative information of the integrand is not required, the rule can be easily applied to solve many practical problems. Finally, the quadrature formula is applied to solve the electromagnetic scattering from cavities with different wave numbers, which improves the whole accuracy of the solution. Numerical experiments are presented to show the efficiency and accuracy of the theoretical analysis.     

浮体与自由面交线附近流场的奇异性

本文研究了浮体与自由面交线附近势流流场的奇异性。结果表明,线性时域解在交线附近的奇异特征是d2lnd.线性频域解在交线附近的奇异特征也是d2lnd,但若采用无穷大频率自由面条件φ=0,交线附近流场的奇异特征是d1nd,这里的d表示交线上的点与场点的距离。     

Quadpack computation of Feynman loop integrals

The paper addresses a numerical computation of Feynman loop integrals, which are computed by an extrapolation to the limit as a parameter in the integrand tends to zero. An important objective is to achieve an automatic computation which is effective for a wide range of instances. Singular or near singular integrand behavior is handled via an adaptive partitioning of the domain, implemented in an iterated/repeated multivariate integration method. Integrand singularities possibly introduced via infrared (IR) divergence at the boundaries of the integration domain are addressed using a version of the Dqags algorithm from the integration package Quadpack, which uses an adaptive strategy combined with extrapolation. The latter is justified for a large class of problems by the underlying asymptotic expansions of the integration error. For IR divergent problems, an extrapolation scheme is presented based on dimensional regularization.     

A Chebyshev polynomial method for line integrals with singularities

In this paper we consider a Chebyshev polynomial method for the calculation of line integrals along curves with Cauchy principal value or Hadamard finite part singularities. The major point we address is how to reconstruct the value of the integral when the parametrization of the curve is unknown and only empirical data are available at some discrete set of nodes. We replace the curve by a near‐minimax parametric polynomial approximation, and express the integrand by means of a sum of Chebyshev polynomials. We make use of a mapping property of the Hadamard finite part operator to calculate the value of the integral. This revised version was published online in June 2006 with corrections to the Cover Date.