Supplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems

  • Nikolaos I. Ioakimidis University of Patras

Abstract

Singular and hypersingular integral equations appear frequently in engineering problems. The approximate solution of these equations by using various numerical methods is well known. Here we consider the case where these equations are supplemented by inequality constraints-mainly parametric inequality constraints, but also the case of singular/hypersingular integral inequalities. The approach used here is simply to employ the computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica and derive the related set of necessary and sufficient conditions for the validity of the singular/hypersingular integral equation/inequality together with the related inequality constraints. The present approach is applied to singular integral equations/inequalities in the problem of periodic arrays of straight cracks under loading- and fracture-related inequality constraints by using the Lobatto-Chebyshev method. It is also applied to the hypersingular integral equation/inequality of the problem of a single straight crack under a parametric loading by using the collocation and Galerkin methods and parametric inequality constraints.

Keywords

singular integral equations/inequalities, hypersingular integral equations/inequalities, boundary integral equations/inequalities, parametric inequality constraints, numerical methods, crack problems,

References

[1] W.-T. Ang. A hypersingular boundary integral formulation for heat conduction across a curved imperfect interface. Commun. Numer. Methods Eng., 24(10): 841–851, 2008. http://dx.doi.org/10.1002/cnm.997.
[2] W.-T. Ang. Hypersingular Integral Equations in Fracture Analysis. Woodhead Publishing (now Elsevier), Cambridge, 2013. http://www.sciencedirect.com/science/book/9780857094797.
[3] W.T. Ang, G. Noone. A hypersingular-boundary integral equation method for the solution of an elastic multiple interacting crack problem. Eng. Anal. Bound. Elem., 11(1): 33–37, 1993. http://dx.doi.org/10.1016/0955-7997(93)90076-W.
[4] S.A. Ashour. Numerical solution of integral equations with finite part integrals. Int. J. Math. Math. Sci., 22(1): 155–160, 1999. http://www.emis.ams.org/journals/HOA/IJMMS/22/1155.pdf.
[5] H.F. Bueckner. Field singularities and related integral representations. In: G.C. Sih, ed. Methods of Analysis and Solutions of Crack Problems (Mechanics of Fracture, Vol. 1). Noordhoff, Leyden, the Netherlands (now Springer, Dordrecht, the Netherlands), chap. 5, pp. 239–314, 1973. http://dx.doi.org/10.1007/978-94-017-2260-5 5.
[6] B.F. Caviness, J.R. Johnson, eds. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien, 1998. http://dx.doi.org/10.1007/978-3-7091-9459-1.
[7] A.C. Chrysakis, G. Tsamasphyros. Numerical solution of Cauchy type singular integral equations with logarithmic weight, based on arbitrary collocation points. Comput. Mech., 7(1): 21–29, 1990. http://dx.doi.org/10.1007/BF00370054.
[8] G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In [6], pp. 85–121. http://dx.doi.org/10.1007/978-3-7091-9459-1 4. Reproduced from the original publication in: H. Brakhage, ed. Automata Theory and Formal Languages, Proceedings of the 2nd GI Conference Kaiserslautern (Lecture Notes in Computer Science, Vol. 33). Springer, Berlin, 1975, pp. 134–183. http://dx.doi.org/10.1007/3-540-07407-4 17.
[9] G.E. Collins, H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput., 12(3): 299–328, 1991. http://dx.doi.org/10.1016/S0747-7171(08)80152-6.
[10] A.P. Datsyshin, M.P. Savruk. Integral equations of the plane problem of crack theory. J. Appl. Math. Mech. (PMM), 38(4): 677–686, 1974. [Translation of Prikl. Mat. Mekh. (PMM), 38(4): 728–737, 1974.] http://dx.doi.org/10.1016/0021-8928(74)90018-5.
[11] P.J. Davis, P. Rabinowitz. Methods of Numerical Integration, 2nd ed. Dover, Mineola, NY, 2007 (reprint of the 2nd ed., Academic Press, Orlando, FL, 1984). http://store.doverpublications.com/0486453391.html.
[12] J.H. de Klerk. Hypersingular integral equations – past, present, future. Nonlinear Anal.: Theory, Methods Applic., 63(5–7): e533–e540, 2005. http://dx.doi.org/10.1016/j.na.2004.12.036.
[13] D. Elliott. Convergence theorems for singular integral equations. In: M.A. Golberg, ed. Numerical Solution of Integral Equations (Mathematical Concepts and Methods in Science and Engineering, Vol. 42). Plenum Press, New York (now Springer, New York), chap. 6, pp. 309–361, 1990. http://dx.doi.org/10.1007/978-1-4899-2593-0 6.
[14] F. Erdogan. Approximate solutions of systems of singular integral equations. SIAM J. Appl. Math., 17(6): 1041–1059, 1969. http://dx.doi.org/10.1137/0117094.
[15] F. Erdogan, G.D. Gupta. On the numerical solution of singular integral equations. Quart. Appl. Math., 29(4): 525–534, 1972. http://dx.doi.org/10.1090/qam/408277.
[16] F. Erdogan, G.D. Gupta, T.S. Cook. Numerical solution of singular integral equations. In: G.C. Sih, ed. Methods of Analysis and Solutions of Crack Problems (Mechanics of Fracture, Vol. 1). Noordhoff, Leyden, the Netherlands (now Springer, Dordrecht, the Netherlands), chap. 7, pp. 368–425, 1973. http://dx.doi.org/10.1007/978-94-017-2260-5 7.
[17] A. Gerasoulis. The use of piecewise quadratic polynomials for the solution of singular integral equations of Cauchy type. Comp. Math. Appl., 8(1): 15–22, 1982. http://dx.doi.org/10.1016/0898-1221(82)90036-0.
[18] A. Gerasoulis. Singular integral equations – the convergence of the Nystr¨om interpolant of the Gauss-Chebyshev method. BIT, 22(2): 200–210, 1982. http://dx.doi.org/10.1007/BF01944477.
[19] A. Gerasoulis, R.P. Srivastav. A method for the numerical solution of singular integral equations with a principal value integral. Int. J. Eng. Sci., 19(9): 1293–1298, 1981. http://dx.doi.org/10.1016/0020-7225(81)90148-8.
[20] M.A. Golberg. The convergence of several algorithms for solving integral equations with finite part integrals. J. Integral Equ., 5(4): 329–340, 1983.
[21] M.A. Golberg. The convergence of several algorithms for solving integral equations with finite part integrals. II. Appl. Math. Comp., 21(4): 283–293, 1987. http://dx.doi.org/10.1016/0096-3003(87)90017-8.
[22] M.A. Golberg. Introduction to the numerical solution of Cauchy singular integral equations. In: M.A. Golberg, ed. Numerical Solution of Integral Equations (Mathematical Concepts and Methods in Science and Engineering, Vol. 42). Plenum Press, New York (now Springer, New York), chap. 5, pp. 183–308, 1990. http://dx.doi.org/10.1007/978-1-4899-2593-0 5.
[23] J.J. Golecki. An elementary approach to two-dimensional elastostatic crack theory. SM Archives, 6: 1–29, 1981.
[24] J.J. Golecki. Numerical evaluation of finite-part singular integrals in crack theory (the new method). Int. J. Fract., 50(1): R3–R8, 1991. http://link.springer.com/article/10.1007%2FBF00035171.
[25] J.J. Golecki. Numerical evaluation of finite-part singular integrals in crack theory (the linear approximation). Eng. Fract. Mech., 46(4): 693–700, 1993. http://dx.doi.org/10.1016/0013-7944(93)90175-R.
[26] J.J. Golecki. Direct displacement method in crack theory (numerical resolution). Meccanica, 42(6): 555–566, 2007. http://dx.doi.org/10.1007/s11012-007-9074-6.
[27] M. Guiggiani. Hypersingular boundary integral equations have an additional free term. Comput. Mech., 16(4): 245–248, 1995. http://dx.doi.org/10.1007/BF00369869.
[28] M. Guiggiani, G. Krishnasamy, T.J. Rudolphi, F.J. Rizzo. A general algorithm for the numerical solution of hypersingular boundary integral equations. ASME J. Appl. Mech., 59(3): 604–614, 1992. http://dx.doi.org/10.1115/1.2893766.
[29] N.I. Ioakimidis. General Methods for the Solution of Crack Problems in the Theory of Plane Elasticity (Doctoral Thesis, in Greek). National Technical University of Athens, Athens, 1976.
[30] N.I. Ioakimidis. On the natural interpolation formula for Cauchy type singular integral equations of the first kind. Computing, 26(1): 73–77, 1981. http://dx.doi.org/10.1007/BF02243425.
[31] N.I. Ioakimidis. Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mech., 45(1–2): 31–47, 1982. http://dx.doi.org/10.1007/BF01295569.
[32] N.I. Ioakimidis. Two methods for the numerical solution of Bueckner’s singular integral equation for plane elasticity crack problems. Comput. Meth. Appl. Mech. Eng., 31(2): 169–177, 1982. http://dx.doi.org/10.1016/0045-7825(82)90022-6.
[33] N.I. Ioakimidis. Validity of the hypersingular integral equation of crack problems in three-dimensional elasticity along the crack boundaries. Eng. Fract. Mech., 26(5): 783–788, 1987. http://dx.doi.org/10.1016/0013-7944(87)90141-X.
[34] N.I. Ioakimidis. Application of quantifier elimination to a simple elastic beam finite element below a straight rigid obstacle. Mech. Res. Commun., 22(3): 271–278, 1995. http://dx.doi.org/10.1016/0093-6413(95)00023-K.
[35] N.I. Ioakimidis. Quantifier elimination in applied mechanics problems with cylindrical algebraic decomposition. Int. J. Solids Struct., 34(30): 4037–4070, 1997. http://dx.doi.org/10.1016/S0020-7683(97)00002-4.
[36] N.I. Ioakimidis. Automatic derivation of positivity conditions inside boundary elements with the help of the REDLOG computer logic package. Eng. Anal. Bound. Elem., 23(10): 847–856, 1999. http://dx.doi.org/10.1016/S0955-7997(99)00049-1.
[37] N.I. Ioakimidis. Finite differences/elements in classical beam problems: derivation of feasibility conditions under parametric inequality constraints with the help of Reduce and REDLOG. Comput. Mech., 27(2): 145–153, 2001. http://dx.doi.org/10.1007/s004660000223.
[38] N.I. Ioakimidis. Quantifier-free formulae for inequality constraints inside boundary elements. In: G.D. Manolis, D. Polyzos, eds. Recent Advances in Boundary Element Methods: A Volume to Honor Professor Dimitri Beskos. Springer, Dordrecht, pp. 209–222, 2009. http://dx.doi.org/10.1007/978-1-4020-9710-2 14.
[39] N.I. Ioakimidis. Derivation of conditions of complete contact for a beam on a tensionless Winkler elastic foundation with Mathematica. Mech. Res. Commun., 72: 64–73, 2016. http://dx.doi.org/10.1016/j.mechrescom.2016.01.007.
[40] N.I. Ioakimidis. Caustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination. Opt. Lasers Eng., 88: 280–300, 2017. http://dx.doi.org/10.1016/j.optlaseng.2016.07.001.
[41] N.I. Ioakimidis, P.S. Theocaris. Array of periodic curvilinear cracks in an infinite isotropic medium. Acta Mech., 28(1–4): 239–254, 1977. http://dx.doi.org/10.1007/BF01208801.
[42] N.I. Ioakimidis, P.S. Theocaris. On the numerical solution of singular integrodifferential equations. Quart. Appl. Math., 37(3): 325–331, 1979. http://dx.doi.org/10.1090/qam/548991.
[43] N.I. Ioakimidis, P.S. Theocaris. On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems. Comput. Struct., 11(4): 289–295, 1980. http://dx.doi.org/10.1016/0045-7949(80)90079-6.
[44] A.I. Kalandiya. On the approximate solution of a class of singular integral equations [in Russian]. Dokl. Akad. Nauk SSSR, 125(4): 715–718, 1959.
[45] A.I. Kalandiya. Mathematical Methods of Two-dimensional Elasticity. Mir Publishers, Moscow, 1975. (Translation of the Russian edition: Nauka, Moscow, 1973.)
[46] A.C. Kaya, F. Erdogan. On the solution of integral equations with strongly singular kernels. Quart. Appl. Math., 45(1): 105–122, 1987. http://dx.doi.org/10.1090/qam/885173.
[47] A.M. Linkov, S.G. Mogilevskaya. Complex hypersingular integrals and integral equations in plane elasticity. Acta Mech., 105(1–4): 189–205, 1994. http://dx.doi.org/10.1007/BF01183951.
[48] R. Loos, V. Weispfenning. Applying linear quantifier elimination. Comput. J., 36(5): 450–462, 1993. http://doi.org/10.1093/comjnl/36.5.450.
[49] S.G. Mogilevskaya. The universal algorithm based on complex hypersingular integral equation to solve plane elasticity problems. Comput. Mech., 18(2): 127–138, 1996. http://dx.doi.org/10.1007/BF00350531.
[50] S.G. Mogilevskaya. Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks. Int. J. Fract., 102(2): 177–204, 2000. http://dx.doi.org/10.1023/A:1007633814813.
[51] H. Multhopp. Methods for Calculating the Lift Distribution of Wings (Subsonic Lifting-Surface Theory). Ministry of Supply, Aeronautical Research Council Reports and Memoranda (R & M) No. 2884. Her Majesty’s Stationery Office, London, 1950. http://naca.central.cranfield.ac.uk/reports/arc/rm/2884.pdf.
[52] M.N. Pavlović. Symbolic computation in structural engineering. Comput. Struct., 81(22–23): 2121–2136, 2003. http://dx.doi.org/10.1016/S0045-7949(03)00286-4.
[53] S. Ratschan. Applications of quantified constraint solving over the reals – bibliography. arXiv: 1205.5571v1, Cornell University Library, Ithaca, NY, 2012. http://arxiv.org/abs/1205.5571v1.
[54] V. Sladek, J. Sladek. Regularization of hypersingular integrals in BEM formulations using various kinds of continuous elements. Eng. Anal. Bound. Elem., 17(1): 5–18, 1996. http://dx.doi.org/10.1016/0955-7997(95)00080-1.
[55] V. Sladek, J. Sladek, M. Tanaka. Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity. Int. J. Numer. Meth. Eng., 36(10): 1609–1628, 1993. http://dx.doi.org/10.1002/nme.1620361002.
[56] A. Strzeboński. Cylindrical algebraic decomposition using local projections. J. Symb. Comput., 76: 36–64, 2016. http://dx.doi.org/10.1016/j.jsc.2015.11.018.
[57] A. Strzeboński. CAD adjacency computation using validated numerics. arXiv: 1704.06856, Cornell University Library, Ithaca, NY, 2017. http://arxiv.org/abs/1704.06856.
[58] P.S. Theocaris, N.I. Ioakimidis. Numerical integration methods for the solution of singular integral equations. Quart. Appl. Math., 35(1): 173–183, 1977. http://dx.doi.org/10.1090/qam/445873.
[59] P.S. Theocaris, G. Tsamasphyros. Numerical solution of systems of singular integral equations with variable coefficients. Appl. Anal., 9(1): 37–52, 1979. http://dx.doi.org/10.1080/00036817908839250.
[60] M. Trott. The Mathematica GuideBook for Symbolics. Springer, New York, 2006. http://dx.doi.org/10.1007/0-387-28815-5.
[61] G. Tsamasphyros, P.S. Theocaris. Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations. Computing, 27(1): 71–80, 1981. http://dx.doi.org/10.1007/BF02243439.
[62] J. Tweed, R.St. John, M.H. Dunn. Algorithms for the numerical solution of a finite-part integral equation. Appl. Math. Lett., 12(3): 3–9, 1999. http://dx.doi.org/10.1016/S0893-9659(98)00163-3.
[63] D.E. Williams. Some Mathematical Methods in Three-dimensional Subsonic Flutter-Derivative Theory. Ministry of Aviation, Aeronautical Research Council Reports and Memoranda (R & M) No. 3302. Her Majesty’s Stationery Office, London, 1961. http://naca.central.cranfield.ac.uk/reports/arc/rm/3302.pdf.
[64] Wolfram Research Inc. Mathematica, version 7.0. Wolfram Research Inc., Champaign, IL, 2008. https://www.wolfram.com.
[65] Wolfram Research Inc. Wolfram Language Tutorial: Real Polynomial Systems. Wolfram Research Inc., Champaign, IL, 2014. https://reference.wolfram.com/language/tutorial/RealPolynomialSystems.html.
[66] Ch. Zhang, J. Sladek, V. Sladek. Antiplane crack analysis of a functionally graded material by a BIEM. Comput. Mater. Sci., 32(3–4): 611–619, 2005. http://doi.org/10.1016/j.commatsci.2004.09.002.
Published
Sep 13, 2017
How to Cite
IOAKIMIDIS, Nikolaos I.. Supplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems. Computer Assisted Methods in Engineering and Science, [S.l.], v. 24, n. 1, p. 41-65, sep. 2017. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/202>. Date accessed: 26 jan. 2022. doi: http://dx.doi.org/10.24423/cames.202.
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