Abstract
This article considers methods of weakly singular and hypersingular integral regularization based on the theory of distributions. For regularization of divergent integrals, the Gauss–Ostrogradskii theorem and the second Green’s theorem in the sense of the theory of distribution have been used. Equations that allow easy calculation of weakly singular, singular, and hypersingular integrals in one- and two-dimensional cases for any sufficiently smooth function have been obtained. These equations are compared with classical methods of regularization. The results of numerical calculation using classical approaches and those based of the theory of generalized functions, along with a comparison for different functions, are presented in tables and graphs of the values of divergent integrals versus the position of the colocation point.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of Applied Mathematics Series, Department of Commerce, National Bureau of Standards (1972)
Aliabadi, M.H.: The boundary element method. Application in Solids and Structures, vol. 2. John Wiley (2002)
Aliabadi, M.H., Hall, W.S.: The regularizing transformation integration method for BE. Comparison with Gauss quadrature. Engineering Analysis with Boundary Elements 6, 66–71 (1986)
Ashour, S.A., Ahmed, H.M.: A Gauss quadrature rule for hypersingular integrals. Appl. Math. Comput. 186, 1671–1682 (2007)
Boehme, T.K.: Operational calculus and the finite part of divergent integrals. Trans. Am. Math. Soc. 106, 346–368 (1963)
Bonnet, M., Guiggiani, M.: Galerkin BEM with direct evaluation of hypersingular integrals. Electron. J. Bound. Elem. 1(2), 95–111 (2003)
Bureau, F.J.: Divergent integrals and partial differential equations. Commun. Pur. Appl. Math. 8(1), 143–202 (1955)
Caianiello, E.R.: Generalized integration procedure for divergent integrals. Il Nuovo cimento 15A(2), 145–161 (1973)
Carley, M.: Numerical quadratures for singular and hypersingular integrals in boundary element methods. SIAM J. Sci. Comput. 29(3), 1207–1216 (2007)
Carley, M.: Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods. Mathematical Proceedings of the Royal Irish Academy 109A(1), 49(60) (2009)
Chan, Y.-S., Fannjiang, A.C., Paulino, G.H.: Integral equations with hypersingular kernels—-theory and applications to fracture mechanics. Int. J. Eng. Sci. 41, 683–720 (2003)
Chen, G., Zhou, J.: Boundary Element Method, p 646. Academic Press, London (1992)
Chen, J.T., Hong, H.-K.: Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Appl. Mech. Rev. 52(1), 17–33 (1999)
Chen, M.-C.: Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part II: Hypersingular integral equation and theoretical analysis. Int. J. Fract. 121, 133–148 (2003)
Chen, M.-C.: Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part II: Numerical analysis. Int. J. Fract. 121, 149–161 (2003)
Chen, Y.Z., Lin, X.Y., Peng, Z.Q.: Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation. Arch. Appl. Mech. 68, 271–280 (1998)
Cheng, A.H.-D., Cheng, D.T.: Heritage and early history of the boundary element method. Engineering Analysis with Boundary Elements 29, 268–302 (2005)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. II. John Wiley, New York (1968)
Davey, K., Hinduja, S.: Analytical integration of linear three-dimensional triangular elements in BEM. Applied Mathematical Modeling 13, 450–461 (1989)
Davis, P.J., Rabinowitz, P., 2nd edn: Methods of Numerical Integration. Academic Press, New York (1984)
de Klerk, J.H.: Hypersingular integral equations—past, present, future. Nonlinear Anal. 63, 533–540 (2005)
Diligenti, M., Monegato, G.: Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains. J. Comput. Appl. Math. 81, 29–57 (1997)
Dumont, N.A.: On the efficient numerical evaluation of integrals with complex singularity poles. Engineering Analysis with Boundary Elements 13, 155–168 (1994)
Dumont, N.A.: Cauchy principal values, finite-part integrals and interval normalization: some basic considerations. In: Atluri, S. N., Yagawa, G., Cruse, T. A. (eds.) Computational Mechanics 95, pp 2830–2835. Springer-Verlag, Berlin (1995)
Dumont, N.A, Noronha, M.A.M.: A simple, accurate scheme for the numerical evaluation of integrals with complex singularity poles. Comput. Mech. 22, 42–49 (1998)
Erdogan, F.E., Gupta, G.D., Cook, T.S.: The Numerical Solutions of Singular Integral Equations. Methods of Analysis and Solutions of Crack Problems, pp 368–425. Noordhoff Intern. Publ., Leyden (1973)
Fata, S.N.: Explicit expressions for 3D boundary integrals in potential theory. Int. J. Numer. Methods Eng. 78, 32–47 (2009)
Frangi, A., Guiggiani, M.: Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals. Int. J. Numer. Methods Eng. 46(11), 1845–1863 (1999)
Frangi, A., Guiggiani, M.: A direct approach for boundary integral equations with high-order singularities. Int. J. Numer. Methods Eng. 49, 871–898 (2000)
Gao, X.-W.: An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals. Comput. Methods Appl. Mech. Eng. 199, 2856–2864 (2010)
Gautschi, W.: Orthogonal Polynomials.. Computation and Approximation, p 312. Oxford University Press, Oxford (2004)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Academic Press, New York (1964)
Gray, L.J., Glaeser, J., Kaplan, T.: Direct evaluation of hypersingular Galerkin surface integrals. SIAM J. Sci. Comput. 25, 1534–1556 (2004)
Gray, L.J., Salvadori, A., Phan, A.-V., Mantic, V.: Direct Evaluation of Hypersingular Galerkin Surface Integrals. II. Electron. J. Bound. Elem. 4(3), 105–130 (2006)
Guiggiani, M.: Computing principal value integrals in 3D BEM for time-harmonic elastodynamics. A direct approach. Communications in Applied Numerical Methods 8, 141–149 (1992)
Guiggiani, M.: Formulation and numerical treatment of boundary integral equations with hypersingular kernels. In: Sladek, V., Sladek, J (eds.) Singular Integrals in Boundary Element Methods. WIT Press, Southampton (1998)
Guiggiani, M., Gigante, A.: A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. ASME J. Appl. Mech. 57, 906–915 (1990)
Guiggiani, M., Krishnasamy, G., Rudolphi, T.J., Rizzo, F.J.: A general algorithm for the numerical solution of hypersingular boundary integral equations. ASME J. Appl. Mech. 59, 604–14 (1992)
Gunter, N.M.: Potential Theory And Its Application to Basic Problems of Mathematical Physics. Fredric Ungar, New York (1967)
Gurtin, M.E.: The linear theory of elasticity, In: Encyclopedia of Physics. In: Fluegge, S. (ed.) Vol. IVa-2. Springer, New York–Berlin (1972)
Guz, A.N., Zozulya, V.V.: Brittle Fracture of Constructive Materials Under Dynamic Loading. Kiev, Naukova Dumka (1993). (in Russian)
Guz, A.N., Zozulya, V.V.: Fracture dynamics with allowance for a crack edges contact interaction. International Journal of Nonlinear Sciences and Numerical Simulation 2(3), 173–233 (2001)
Guz, A.N., Zozulya, V.V.: Elastodynamic unilateral contact problems with friction for bodies with cracks. Int. Appl. Mech. 38(8), 3–45 (2002)
Hadamard, J.: Lectures on Cauchy’s Problem in Linear Hyperbolic Differential Equations. Dover, New York (1953)
Hardy, G.H.: Divergent Series. Oxford At The Clarendon Press (1949)
Hörmander, L.: On the theory of general partial differential operators. Acta Mathematica 94, 161–248 (1955)
Hong, H.-K., Chen, J.T.: Derivations of Integral Equations of Elasticity. ASCE J. Eng. Mech. 114(6), 1028–1044 (1988)
Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer-Verlag, Berlin Heidelberg (2008)
Hussain, F., Karim, M.S., Ahamad, R.: Appropriate Gaussian quadrature formulae for triangles. International Journal of Applied Mathematics and Computation 4(1), 24–38 (2012)
Ioakimidis, N.I.: Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mech. 45, 31–47 (1982)
Iovane, G.: Hypersingular integral equations, Kähler manifolds and Thurston mirroring effect in \(\epsilon ^{(\infty )}\) Cantorian spacetime. Chaos, Solitons and Fractals 31, 1041–1053 (2007)
Iovane, G, Lifanov, I.K., Sumbatyan, M.A.: On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics. Acta Mech. 162, 99–110 (2003)
Johnston, B.M., Johnston, P.R., Elliott, D.: A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method. J. Comput. Appl. Math. 245, 148–161 (2013)
Kantor, B.J, Strel’nikova, E.A.: Hypersingular integral equations in the problems of solid mechanics. Kharkiv, Novoe slovo (2005). (in Russian)
Kantorovich, L.V.: On approximate calculation of some type of definite integrals and other applications of the method of singularities extraction. Matematicheskii Sbornik 41(2), 235–245 (1934). (in Russian)
Karami, G., Derakhshan, D.: An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis. Engineering Analysis with Boundary Elements 23, 317–326 (1999)
Karasalo, I.: On evaluation of hypersingular integrals over smooth surfaces. Comput. Mech. 40, 617–625 (2007)
Kaya, A.C., Erdogan, F.: On the solution of integral equations with strongly singular kernels. Q. Appl. Math. 45, 105–122 (1987)
Kieser, R., Schwab, C., Wendland, W.L.: Numerical Evaluation of Singular and Finite-Part Integrals on Curved Surfaces Using Symbolic Manipulation. Computing 49, 279–301 (1992)
Kim, P., Choi, U.J.: Two trigonometric quadrature formulae for evaluating hypersingular integrals. Int. J. Numer. Methods Eng. 56, 469–486 (2003)
Kolm, P., Rokhlin, V.: Numerical quadratures for singular and hypersingular integrals. Computers and Mathematics with Applications 41, 327–352 (2001)
Krylov, V.I.: Approximate Calculation of Integrals, p 386. Dover Publications, Inc. (2005)
Kupradze, V.D.: Potential Methods in the Theory of Elasticity, p 339. Daniel Davey & Co., Inc. (1965a)
Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, p 929. North-Holland Publ. Comp., Amsterdam (1979)
Kythe, P.K., Schaferkotter, M.R.: Handbook of Computational Methods for Integration, p 604. Chapman & Hall/CRC Press (2005)
Lebedev, L.P., Vorovich, I.I., Gladwell, G.K.: Functional Analysis. Applications in Mechanics and Inverse Problems. Springer, Netherlands (1996)
Lighthill, M.: Introduction to Fourier Analysis and Generalized Functions. Cambridge Univ. Press, Cambridge (1959)
Lifanov, I.K., Lifanov, P.I.: On some exact solutions of singular integral equations in the class of generalized functions and their numerical computation. Diff. Equat. 40(12), 1770–1780 (2004)
Lifanov, I.K., Poltavskii, L.N., Vainikko, G.M.: Hypersingular Integral Equations and Their Applications. Chapman & Hall/CRC, London (2004)
Linkov, A.M.: Complex hypersingular integrals and integral equations in plane elasticity. Acta Mech. 105, 18–205 (1994)
Martin, P.A., Rizzo, F.J.: Hypersingular integrals: How smooth must the density be? Int. J. Numer. Methods Eng. 39, 687–704 (1996)
Michlin, S.G.: Singular integral equations. Uspehi Maematiceskih Nauk 3(25), 29–112 (1948). (in Russian)
Michlin, S.G.: Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford (1965)
Monegato, G.: Numerical evaluation of hypersingular integrals. J. Comput. Appl. Math. 50, 9–31 (1994)
Monegato, G.: Definitions, properties and applications of finite-part integrals. J. Comput. Appl. Math. 229, 425–439 (2009)
Moore, M.N.J., Gray, L.J., Kaplan, T.: Evaluation of supersingular integrals: Second-order boundary derivatives. Int. J. Numer. Methods Eng. 69, 1930–1947 (2007)
Mousavi, S.E., Xiao, H., Sukumar, N.: Generalized Gaussian quadrature rules on arbitrary polygons. Int. J. Numer. Methods Eng. 82(1), 99–113 (2010)
Mukherjee, S.: CPV and HFP integrals and their applications in the boundary element method. Int. J. Solids Struct. 37, 6623–6634 (2000)
Mukherjee, S.: Finite parts of singular and hypersingular integrals with irregular boundary source points. Engineering Analysis with Boundary Elements 24, 767–776 (2000)
Mukherjeea, S., Mukherjee, Y.X., Wenjing, Y.: Cauchy principal values and finite parts of boundary integrals—revisited. Engineering Analysis with Boundary Elements 29, 844–849 (2005)
Mukherjee, Y.X., Mukherjee, S.: Error analysis and adaptivity in three-dimensional linear elasticity by the usual and hypersingular boundary contour method. Int. J. Solids Struct. 38, 161–178 (2001)
Mukherjee, S., Mukherjee, Y.X.: Boundary Methods: Elements, Contours, and Nodes. CRC/Taylor & Francis, Boca Raton (2005)
Muskhelishvili, N.I.: Singular Integral Equations, p 452. P, Noordhoff Ltd. (1953)
Nagarajan, A., Lutz, E.D., Mukherjee, S.: A novel boundary element method for linear elasticity with no numerical integration for 2D and line integrals for 3D problems. J. Appl. Mech. 61, 264–269 (1994)
Ninham, B.W.: Generalised Functions and Divergent Integrals. Numer. Math. 8, 44–57 (1966)
Niu, Z., Wendland, W.L., Wang, X., Zhou, H.A: semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Comput. Methods Appl. Mech. Engrg. 194, 1057–1074 (2005)
Ortner, V.N.: Regularisierte Faltung von Distributionen. Teil l. Zur Berechnung von Fundamentallösungen. Z. Angew. Math. Phys. (ZAMP) 31, 133–154 (1980)
Ortner, V.N.: Regularisierte Faltung von Distributionen. Teil 2. Eine Tabelle von Fundamentallösungen. Z. Angew. Math. Phys. (ZAMP) 31, 155–173 (1980)
Polyzos, D., Tsepoura, K.G., Tsinopoulos, S.V., Beskos, D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part I: Integral formulation. Comput. Methods Appl. Mech. Eng. 192, 2845–2873 (2003)
Qin, T.Y., Yu, Y.S., Noda, N.A.: Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials. Int. J. Solids Struct. 44, 4770–4783 (2007)
Rathod, H.T., Karim, M.S.: An explicit integration scheme based on recursion for the curved triangular finite elements. Comput. Struct. 80, 43–76 (2002)
Saez, A., Gallego, R., Dominguez, J.: Hypersingular quarter-point boundary elements for crack problems. Int. J. Numer. Methods Eng. 38, 1681–1701 (1995)
Salvadori, A.: Analytical integrations in 2D BEM elasticity. Int. J. Numer. Methods Eng. 53, 1695–1719 (2002)
Salvadori, A.: Analytical integration of hypersingular kernel in 3D BEM problems. Comput. Methods Appl. Mech. Eng. 190, 3957–3975 (2001)
Salvadori, A.: Analytical integrations in 3D BEM for elliptic problems: evaluation and implementation. Int. J. Numer. Methods Eng. 84, 505–542 (2010)
Salvadori, A., Temponi, A.: Analytical integrations for the approximation of 3D hyperbolic scalar boundary integral equations. Engineering Analysis with Boundary Elements 34, 977–994 (2010)
Samko, S.G.: Hypersingular iIntegrals and Their Applications. Taylor & Francis (2002)
Sanz, J.A., Solis, M., Dominguez, J.: Hypersingular BEM for piezoelectric solids: formulation and applications for fracture mechanics. CMES: Comput. Model. Eng. Sci. 17(3), 215–229 (2007)
Schwab, C, Wendland, W.L.: Kernel properties and representations of boundary integral operators. Mathematische Nachrichten 156, 187–218 (1992)
Schwartz, L.: Theorie des distributions, Vols. I and II. Hermann, Paris (1957)
Sladek, V., Sladek, J. (eds.): Singular Integrals in Boundary Element Methods. WIT Press, Southampton (1998)
Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, vol. 7 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA (1963)
Tanaka, M., Sladek, V., Sladek, J.: Regularization techniques applied to boundary element methods. Appl. Mech. Rev. 47(10), 457–499 (1994)
Tomioka, S., Nishiyama, S.: Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method. Engineering Analysis with Boundary Elements 34, 393–404 (2010)
Tsepoura, K.G., Tsinopoulos, S., Polyzos, D., Beskos, D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part II: Numerical implementation. Comput. Methods Appl. Mech. Eng. 192, 2875–2907 (2003)
Vainikko, G.: Multidimensional Weakly Singular Integral Equations, p 169. Springer-Verlag, Berlin Heidelberg (1993)
Wendland, W.L., Stephan, E.P.: A hypersingular boundary integral method for two-dimensional screen and crack problems. Arch. Ration. Mech. Anal. 112, 363–390 (1990)
Young, A.: Improved numerical method for the Traction BIE by application of Stokes’ theorem. Int. J. Numer. Methods Eng. 40, 3141–3161 (1997)
Zemanian, A.H.: Distribution Theory and Transform Analysis. Dover, New York (1987)
Zozulya, V.V.: Integrals of Hadamard type in dynamic problem of the crack theory. Doklady Academii Nauk. Ukrainskoy SSR, Ser. A. Physical Mathematical & Technical Sciences 2, 19–22 (1991). (in Ukrainian)
Zozulya, V.V.: Contact interaction between the edges of a crack and an infinite plane under a harmonic loading. Int. Appl. Mech. 28(1), 61–65 (1992)
Zozulya, V.V.: Regularization of the divergent integrals. I. General consideration. Electron. J. Bound. Elem. 4(2), 49–57 (2006)
Zozulya, V.V.: Regularization of the divergent integrals. II. Application in Fracture Mechanics. Electron. J. Bound. Elem. 4(2), 58–56 (2006)
Zozulya, V.V. In: Skerget, P., Brebbia, C.A. (eds.) : Regularization of the hypersingular integrals in 3-D problems of fracture mechanics. In: Boundary Elements and Other Mesh Reduction Methods XXX, pp 219–228. WIT Press, Southampton (2008)
Zozulya, V.V.: Divergent Integrals in Elastostatics. Regularization in 2-D case. Electron. J. Bound. Elem. 7(2), 50–88 (2009)
Zozulya, V.V.: Regularization of hypersingular integrals in 3-D fracture mechanics: triangular BE, and piecewise-constant and piecewise-linear approximations. Engineering Analysis with Boundary Elements 34(2), 105–113 (2010)
Zozulya, V.V. In: Constanda, C., Perez, M.E. (eds.) : Regularization of the divergent integrals in boundary integral equations for elastostatics. In: Integral Methods in Science and Engineering. Vol. 1. Analytic Methods, pp 333–347, Birkhäuser (2010)
Zozulya V.V. Regularization of the divergent integrals in boundary integral equations. In: Advances in Boundary Element Techniques. In: Zhang, Ch., Aliabadi, M.H., Schanz, M. (eds.) , pp 561–568. Published by EC, Ltd, UK (2010)
Zozulya, V.V.: Divergent integrals in elastostatics: regularization in 3-D case. CMES: Comput. Model. Eng. Sci. 70(3), 253–349 (2010)
Zozulya, V.V.: Divergent integrals in elastostatics: general considerations. ISRN Applied Mathematics. Article ID 726402, 25 (2011)
Zozulya, V.V.: An approach based on generalized functions to regularize divergent integrals, Engineering Analysis with Boundary Elements 40, 162–180 (2014). (submitted)
Zozulya, V.V., Gonzalez-Chi, P.I.: Weakly singular, singular and hypersingular integrals in elasticity and fracture mechanics. J. Chin. Inst. Eng. 22(6), 763–775 (1999)
Zozulya, V.V., Lukin, A.N.: Solution of three-dimensional problems of fracture mechanics by the method of integral boundary equations. Int. Appl. Mech. 34(6), 544–551 (1998)
Zozulya, V.V., Men’shikov, V.A.: Solution of three dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals. Int. Appl. Mech. 36(1), 74–81 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Leslie Greengard
Rights and permissions
About this article
Cite this article
Zozulya, V.V. Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches. Adv Comput Math 41, 727–780 (2015). https://doi.org/10.1007/s10444-014-9399-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-014-9399-3