Numerical Evaluation Of Singular Integrals Research Articles

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

A fast multipole boundary element method for half-space acoustic problems in a subsonic uniform flow

This paper presents a fast multipole boundary element method (FMBEM) for half-space acoustic problems in a subsonic uniform flow. First, the fundamental solution, i.e., Green's function for half-space acoustic wave problems in a subsonic uniform flow is derived. By employing the Burton-Miller method, a convected hybrid boundary integral formula is achieved to overcome the non-uniqueness difficulty. The developed formula does not need to consider the integral on the infinite plane and it is convenient for numerical implementation. The numerical evaluation of singular integrals in the convected boundary integral formulae is also introduced. Then, the fast multipole method (FMM) for convected BEM is derived and adopted to improve the calculation efficiency and increase the computational scale. Different from the previous FMM for full-space problems, the multipole expansion and translation of the mirror-image model are the key to the FMM for half-space problems. The impact of the infinite plane on the results can be computed by building the mirror-image model. And only the structural boundary in the real domain needs to be discretized and constructed for the tree structure. Numerical examples including a pulsating half-sphere problem, a sphere model and an aircraft model in moving flows are given to validate the accuracy and efficiency of the developed method.

Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals

An efficient method for constructing a class of generalized quadrature formulae of Gaussian type on (−1,1) for integrands having logarithmic singularities is developed. That kind of singular integrals are very common in the boundary element method. Several special cases for n-point quadratures, which are exact on both of the spaces P2n−2ℓ−1[−1,1] (the space of algebraic polynomials of degree at most 2n−2ℓ−1) and L2ℓ−1[−1,1]=span{xklog|x|}k=02ℓ−1 (the logarithmic space), where 1≤ℓ≤n, are presented. Regarding a direct connection of these 2m-point quadratures with m-point quadratures of Gaussian type with respect to the weight function t↦t−1/2 over (0,1), the method of construction is significantly simplified. Gaussian quadratures on (0,1) are exact for integrands of the form t↦p(t)+q(t)logt, where p and q are algebraic polynomials of degree at most 2m−ℓ−1 and ℓ−1 (1≤ℓ≤2m), respectively. The obtained quadratures can be used in a software implementation of the boundary element method.

A modified Gauss quadrature formula with special integration points for evaluation of Quasi-singular integrals

It is well known in the boundary element method that integration rules fail when the integrand presents a nearby singularity. This drawback arises when the field point is near the source point, i.e. in the case of a domain with very narrow boundaries or when the field point where we try to calculate stresses or any other field variables, is near the boundaries. In the present paper a quadrature formulas for ℓ isolated singularities near the integration interval, based on ordinary or special Langrange interpolatory polynomials, is obtained. This interpolatory formulas present similarities with known formulas for the numerical evaluation of singular integrals. Quadrature formulas for regular and singular integrals with conjugate poles are also derived. Numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy.

ON QUADRATURE FORMULAE FOR SINGULAR INTEGRALS OF ARBITRARY ORDER

Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.

On The numerical evaluation of singular integrals with coth-kernel

This problem was first tackled by W. Gautschi et al. [2] using Gauss-Christoffel and interpolating quadrature methods. In this paper we present two alternative numerical methods, one based on complex contour integration and the other on a recurrence relation [Modified Olver's Algorithm] derived from a modified moment associated with a given classical orthogonal polynomial. Our methods involve less computational effort and yet are comparable in accuracy to the methods of Gautschi et al. [2]. We give a few numerical examples to prove this assertion.

Improved numerical method for the Traction Boundary Integral Equation by application of Stokes' theorem

This paper concerns the direct numerical evaluation of singular integrals arising in Boundary Integral Equations for displacement (BIE) and displacement gradients (BIDE), and the formulation of a Traction Boundary Integral Equation (TBIE) for solving general elastostatic crack problems. Subject to certain continuity conditions concerning displacements and tractions at the source point, singular integrals in the BIE and the BIDE corresponding to coefficients of displacement and displacement gradients at the source point are shown to be of a form that allows application of Stokes' theorem. All the singular integrals in 3-D BIE and BIDE are reduced to non-singular line integrals, and those in 2-D BIE and BIDE are evaluated in closed form. Remaining terms involve regular integrals, and no references to Cauchy or Hadamard principal values are required. Continuous isoparametric interpolations used on continuous elements local to the source point are modified to include unique displacement gradients at the source point which are compatible with all local tractions. The resulting numerical BIDE is valid for source points located arbitrarily on the boundary, including corners, and a procedure is given for constructing a TBIE from the BIDE. Some example solutions obtained using the present numerical method for the TBIE in 2-D and 3-D are presented. © British Crown Copyright 1997/DERA.

A note on spline approximation for Hadamard finite-part integrals

In this paper we construct product quadrature rules, based on spline interpolation, for the numerical evaluation of singular integrals in the sense of Hadamard. We give a convergence result and examine the behaviour of the stability factor. We also present some numerical tests.

Boundary Integral Equation Method Applied to the Strength Design of Gear. 3rd Report. Corner Problems and Efficient Evaluation of Integrals.

In this paper, in order to develop a more efficient and more convenient Procedure for the strength design of a spur gear, substantial effort is devoted to improving the accuracy in the boundary integral equation analyses. First, corner problems associated with a gear tip are mathematically avoided by adopting Priwalow's formulae. Next, a systematic technique using higher-order shape functions is proposed for the numerical evaluation of singular integrals. Then, a Stieltjes integral is used to deal with a discontinuous change of the boundary direction a corners. The accuracy is investigated using some numerical examples and comparison is made with the results in previous reports.

A comparison between various boundary integral formulations of the exterior acoustic problem

Classical boundary integral formulations of the Helmholtz equation in the exterior domain either via the Helmholtz formulae (leading to direct formulations) or via layer potential representation of the solution (leading to indirect formulations) fail to have a unique solution for certain real values of the wavenumber. In this paper we consider three of the most commonly used formulations which are valid for all wavenumbers. We carry out a systematic comparison between these formulations, investigating their suitability for use in a general boundary element code. We study the effect of parameters in these formulations on the stability and the attainable accuracy of the discretised equations. The computational aspects of our boundary element methods, such as the numerical evaluation of singular integrals and the efficient solution of the resulting linear systems are also discussed.

The numerical evaluation of singular integrals with coth-kernel

Singular integrals with hyperbolic cotangent kernel present their own numerical problems because of the poles of the kernel located in the complex plane. We write such integrals as ordinary Cauchy principal value integrals involving an appropriate (nonclassical) weight function and apply quadrature methods of Gaussian and interpolatory type. The most accurate one is based on Gauss-Christoffel quadrature relative to the weight function in question. Its error is studied both by real-and complex-variable techniques. Numerical examples are given to illustrate the theory.

A new method of numerical evaluation of singular integrals occurring in two-dimensional BIEM

A new quadrature rule for integrands having logarithmical singularities has been developed. This rule proved to be efficient especially in the context of the BIEM.

On the Numerical Evaluation of Singular Integrals of Cauchy Type

Previous article Next article On the Numerical Evaluation of Singular Integrals of Cauchy TypeCharles E. StewartCharles E. Stewarthttps://doi.org/10.1137/0108021PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. Lomax and , P. F. Byrd, Theoretical aerodynamic characteristics of a family of slender wing-tail-body combinations, National Advisory Committee for Aeronautics, TN 2554 (1951), Google Scholar[2] H. Lomax and , M. A. Heaslet, Damping-in-roll calculations for slender swept-back wings and slender wing-body combinations, National Advisory Committee for Aeronautics, TN 1950 (1949), Google Scholar[3] H. Multhopp, Methods for calculating the lift distribution of wings (subsonic lifting-surface theory), Aero. Res. Council, Rep. and Memo., 1950 (1950), 96 pp. (1955), British Royal Aircraft Establishment, January MR0072628 Google Scholar[4] T. Theodorsen and , I. E. Garrick, General potential theory of arbitrary wing sections, National Advisory Committee for Aeronautics, Report 452, (1932), Google Scholar[5] Eric Reissner and , John E. Stevens, Effect of finite span on the airload distributions for oscillating wings. II. Methods of calculation and examples of application, Tech. Notes Nat. Adv. Comm. Aeronaut., 1947 (1947), 97 pp. (34 plates) MR0021804 Google Scholar[6] S. G. Mikhlin, Integral equations and their applications to certain problems in mechanics, mathematical physics and technology, Pergamon Press, New York, 1957xii+338, (Translation from the Russian edition by A. H. Armstrong) MR0087877 0077.09903 Google Scholar[7] N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1946, (Translation from the Russian edited by J. R. M. Radok, P. Noordhoff, N. V., Holland, 1953.) Google Scholar[8] F. G. Tricomi, Integral equations, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York, 1957viii+238 MR0094665 0078.09404 Google Scholar[9] R. C. MacCamy, The numerical solution of singular integral equations, Air Force Office of Scientific Research, TN 57-515, 1957 Google Scholar[10] I. Flügge-Lotz, Mathematical improvement of method for computing Poisson integrals involved in determination of velocity distribution on airfoils, Tech. Notes Nat. Adv. Comm. Aeronaut., 1951 (1951), 84– MR0046750 Google Scholar[11] M. D. Van Dyke, Second order subsonic airfoil theory including edge effects, National Advisory Committee for Aeronautics, Report 1274, (1956), Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Piecewise-Polynomial Quadratures for Cauchy Singular Integrals14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 23, No. 4AbstractPDF (1201 KB)The Numerical Evaluation of the Cauchy Transform on Simple Closed Curves3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 9, No. 2AbstractPDF (1057 KB)Differentiation of Singular Integrals of Cauchy TypeRaimond A. Struble10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 2AbstractPDF (321 KB) Volume 8, Issue 2| 1960Journal of the Society for Industrial and Applied Mathematics History Submitted:20 April 1959Published online:10 July 2006 InformationCopyright © 1960 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0108021Article page range:pp. 342-353ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics