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Singular Integrals (singular + integral)

Distribution by Scientific Domains
Engineering83%
Mathematics and Statistics16%

Terms modified by Singular Integrals

  • singular integral equation
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    Selected Abstracts

    Crack edge element of three-dimensional displacement discontinuity method with boundary division into triangular leaf elements

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2001
    H. Li
    Abstract In this paper, the existing Displacement Discontinuity Method (DDM) for three-dimensional elastic analysis with boundary discretized into triangular elements, which is purely based on analytical integrals, is extended from the constant element to the square-root crack edge element. In order to evaluate the singular integral when the receiver point falls into the remitter element, i.e., the observed point (x,y) ,,, a part analytical and part numerical integration procedure is adopted effectively. The newly developed codes prove valid in estimating the Stress Intensity Factor (SIF) KI. Furthermore, for the sake of keeping the advantages of high-speed and high-accuracy in developing the numerical system, a novel method to realize pure analytical integration of influence function is found by the aid of symbolic computation technology of Mathematica. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    Computational aspects in 2D SBEM analysis with domain inelastic actions

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010
    T. Panzeca
    Abstract The Symmetric Boundary Element Method, applied to structures subjected to temperature and inelastic actions, shows singular domain integrals. In the present paper the strong singularity involved in the domain integrals of the stresses and tractions is removed, and by means of a limiting operation, this traction is evaluated on the boundary. First the weakly singular domain integral in the Somigliana Identity (S.I.) of the displacements is regularized and the singular integral is transformed into a boundary one using the Radial Integration Method; subsequently, using the differential operator applied to the displacement field, the S.I. of the tractions inside the body is obtained and through a limit operation its expression is evaluated on the boundary. The latter operation makes it possible to substitute the strongly singular domain integral in a strongly singular boundary one, defined as a Cauchy Principal Value, with which the related free term is associated. The expressions thus obtained for the displacements and the tractions, in which domain integrals are substituted by boundary integrals, were utilized in the Galerkin approach, for the evaluation in closed form of the load coefficients connected to domain inelastic actions. This strategy makes it possible to evaluate the load coefficients avoiding considerable difficulties due to the geometry of the solid analyzed; the obtained coefficients were implemented in the Karnak.sGbem calculus code. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2003
    Seppo Järvenpää
    Abstract A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co-ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high-order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Self-regular boundary integral equation formulations for Laplace's equation in 2-D

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2001
    A. B. Jorge
    Abstract The purpose of this work is to demonstrate the application of the self-regular formulation strategy using Green's identity (potential-BIE) and its gradient form (flux-BIE) for Laplace's equation. Self-regular formulations lead to highly effective BEM algorithms that utilize standard conforming boundary elements and low-order Gaussian integrations. Both formulations are discussed and implemented for two-dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the flux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, flux error results in the potential-BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the flux results converge monotonically to the exact answer. In the flux-BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with some oscillations, when refining the mesh or increasing the order of the interpolating function. The flux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and refinement. Accurate results for the potential and the flux were obtained for coarse-graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self-regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular potential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular potential-BIE is compared with the standard (CPV) formulation, showing the equivalence between these formulations. The self-regular BIE formulations and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    An efficient time-domain damping solvent extraction algorithm and its application to arch dam,foundation interaction analysis

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2008
    Hong Zhong
    Abstract The dynamic structure,unbounded foundation interaction plays an important role in the seismic response of structures. The damping solvent extraction (DSE) method put forward by Wolf and Song has a great advantage of simplicity, with no singular integrals to be evaluated, no fundamental solution required and convolution integrals avoided. However, implementation of DSE in the time domain to large-scale engineering problems is associated with enormous difficulties in evaluating interaction forces on the structure,unbounded foundation interface, because the displacement on the corresponding interface is an unknown vector to be found. Three sets of interrelated large algebraic equations have to be solved simultaneously. To overcome these difficulties, an efficient algorithm is presented, such that the solution procedure can be greatly simplified and computational effort considerably saved. To verify its accuracy, two examples with analytical solutions were investigated, each with a parameter analysis on the domain size and amount of artificial damping. Then with the parameters suggested in the parameter study, the complex frequency,response functions and earthquake time history analysis of Morrow Point dam were presented to demonstrate the applicability and efficiency of DSE approach. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    On the BIEM solution for a half-space by Neumann series

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2007
    M. Y. Antes
    Abstract This paper presents an approach which allows the solution of elastic problems concerning a half-space (half-plane) with cavities by the boundary integral equation methods using Neumann's series. To evaluate the series terms at singular points, the regular representations of singular integrals for the external problems were proven and the regular recurrent relationships for the series terms, which can be calculated by any known quadrature rule, are obtained. The numerical proposed procedure was tested by comparison with known theoretical solution and the method convergence was studied for various depths of a buried cavity. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Complete semi-analytical treatment of weakly singular integrals on planar triangles via the direct evaluation method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2010
    Athanasios G. Polimeridis
    Abstract A complete semi-analytical treatment of the four-dimensional (4-D) weakly singular integrals over coincident, edge adjacent and vertex adjacent triangles, arising in the Galerkin discretization of mixed potential integral equation formulations, is presented. The overall analysis is based on the direct evaluation method, utilizing a series of coordinate transformations, together with a re-ordering of the integrations, in order to reduce the dimensionality of the original 4-D weakly singular integrals into, respectively, 1-D, 2-D and 3-D numerical integrations of smooth functions. The analytically obtained final formulas can be computed by using typical library routines for Gauss quadrature readily available in the literature. A comparison of the proposed method with singularity subtraction, singularity cancellation and fully numerical methods, often used to tackle the multi-dimensional singular integrals evaluation problem, is provided through several numerical examples, which clearly highlights the superior accuracy and efficiency of the direct evaluation scheme. Copyright © 2010 John Wiley & Sons, Ltd. [source]

    Improvement of the asymptotic behaviour of the Euler,Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004
    U. Jin Choi
    Abstract In the recent works (Commun. Numer. Meth. Engng 2001; 17: 881; to appear), the superiority of the non-linear transformations containing a real parameter b , 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so-called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finite-part integrals by using the Euler,Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b. Through the asymptotic error analysis of the Euler,Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Boundary element formulation for 3D transversely isotropic cracked bodies

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004
    M. P. Ariza
    Abstract The boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to Hooke's law. The use of both traction and displacement integral representations leads to a mixed (or dual) formulation of the BEM where the discretization effort for crack problems is much smaller than in the classical formulation. A boundary element analysis of three-dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper. The hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion, at the collocation point. All the remaining integrals are analytically evaluated or transformed by means of Stokes' theorem into regular or weakly singular integrals, which are numerically computed. The method is general and can be used for elements of any shape including quarter-point crack front elements. No change of co-ordinates is required for the integration. The formulation as presented in this paper is something as clear, general and easy to handle as the classical BE formulation. It is used in combination with three-dimensional quadratic and quarter-point elements to obtain accurate results for several different crack problems. Cracks in boundless and finite transversely isotropic domains are studied. The meshes are simple and include only discretization of the crack and the external boundary. The obtained results are in good agreement with those existing in the literature. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    One-sided operators in Lp (x) spaces

    MATHEMATISCHE NACHRICHTEN, Issue 11 2008
    David E. Edmunds
    Abstract Boundedness of one-sided maximal functions, singular integrals and potentials is established in L(I) spaces, where I is an interval in R. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

    New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2005
    Xuan Thinh Duong
    In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John-Nirenberg inequality holds on these spaces and they interpolate with Lp spaces by the complex interpolation method. We then give applications on Lp -boundedness of singular integrals whose kernels do not satisfy the Hörmander condition. © 2005 Wiley Periodicals, Inc. [source]