An approximate numerical method for solving Cauchy singular integral equations composed of multiple implicit parameter functions with unknown integral limits in contact mechanics. (English) Zbl 1437.74028
The authors propose an iterative numerical method to solve Cauchy singular integral equations composed of multiple implicit parameter functions, with unknown integral limits, and which are associated with the contact problem between two solid bodies with irregular surfaces. It is based on the quadrature method and uses the constraint function of the boundary by formulating the convergence criterion. Five examples are analysed by using the proposed method.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 74M15 | Contact in solid mechanics |
| 65R20 | Numerical methods for integral equations |
Keywords:
Cauchy singular integral; unknown integral limit; implicit parameter function; quadrature methodReferences:
| [1] | Abdulkawi, M., Solution of Cauchy type singular integral equations of the first kind by using differential transform method, Appl. Math. Model., 39, 2107-2118 (2015) · Zbl 1443.65437 |
| [2] | Akel, M. S.; Hussein, H. S., Numerical treatment of solving singular integral equations by using Sinc approximations, Appl. Math. Comput., 218, 3565-3573 (2011) · Zbl 1244.65235 |
| [3] | Alinia, Y.; Hosseini-nasab, M.; Güler, M. A., The sliding contact problem for an orthotropic coating bonded to an isotropic substrate, Eur. J. Mech. A Solids, 70, 156-171 (2018) · Zbl 1406.74515 |
| [4] | Balci, M. N.; Dag, S., Dynamic frictional contact problems involving elastic coatings, Tribol. Int., 124, 70-92 (2018) |
| [5] | Biabanaki, S. O.R.; Khoei, A. R.; Wriggers, P., Polygonal finite element methods for contact-impact problems on non-conformal meshes, Comput. Methods Appl. Mech. Engrg., 269, 198-221 (2014) · Zbl 1296.74105 |
| [6] | Bonis, M. C.D.; Laurita, C., A quadrature method for systems of Cauchy singular integral equations, J. Integral Equations Appl., 24, 241-271 (2012) · Zbl 1256.65106 |
| [7] | Bougoffa, L.; Mennouni, A.; Rach, R. C., Solving Cauchy integral equations of the first kind by the Adomian decomposition method, Appl. Math. Comput., 219, 4423-4433 (2013) · Zbl 1416.65529 |
| [8] | Boussinesq, J., Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques (1885), Gauthier-Villars, Imprimeur-Libraire · JFM 17.0952.01 |
| [9] | Cai, H., The numerical integration scheme for a fast Petrov-Galerkin method for solving the generalized airfoil equation, Calcolo, 50, 141-162 (2013) · Zbl 1358.65076 |
| [10] | Cai, Y.; Cui, X.; Li, G.; Liu, W., A parallel finite element procedure for contact-impact problems using edge-based smooth triangular element and GPU, Comput. Phys. Commun., 225, 47-58 (2018) · Zbl 07694770 |
| [11] | Castro, L. P.; Rojas, E. M., Explicit solutions of Cauchy singular integral equations with weighted Carleman shift, J. Math. Anal. Appl., 371, 128-133 (2010) · Zbl 1198.45004 |
| [12] | Chakrabarti, A.; Vanden Berghe, G., Approximate solution of singular integral equations, Appl. Math. Lett., 17, 553-559 (2004) · Zbl 1061.65141 |
| [13] | Chidlow, S. J.; Chong, W. W.F.; Teodorescu, M., On the two-dimensional solution of both adhesive and non-adhesive contact problems involving functionally graded materials, Eur. J. Mech. A Solids, 39, 86-103 (2013) · Zbl 1348.74254 |
| [14] | Cohen, J. W.; Boxma, O. J., Boundary Value Problems in Queueing System Analysis (1983), North-Holland Pub. Co.: North-Holland Pub. Co. Amsterdam/New York · Zbl 0515.60092 |
| [15] | Çömez, İ., Continuous and discontinuous contact problem of a functionally graded layer pressed by a rigid cylindrical punch, Eur. J. Mech. A Solids, 73, 437-448 (2019) · Zbl 1406.74516 |
| [16] | Costabel, M.; Darrigrand, E.; Sakly, H., Volume integral equations for electromagnetic scattering in two dimensions, Comput. Math. Appl., 70, 2087-2101 (2015) · Zbl 1443.45003 |
| [17] | De Bonis, M. C.; Laurita, C., Nyström method for Cauchy singular integral equations with negative index, J. Comput. Appl. Math., 232, 523-538 (2009) · Zbl 1185.65232 |
| [18] | De Bonis, M. C.; Laurita, C., Numerical solution of systems of Cauchy singular integral equations with constant coefficients, Appl. Math. Comput., 219, 1391-1410 (2012) · Zbl 1291.65378 |
| [19] | De Lorenzis, L.; Wriggers, P., Computational homogenization of rubber friction on rough rigid surfaces, Comput. Mater. Sci., 77, 264-280 (2013) |
| [20] | Dezhbord, A.; Lotfi, T.; Mahdiani, K., A new efficient method for cases of the singular integral equation of the first kind, J. Comput. Appl. Math., 296, 156-169 (2016) · Zbl 1342.65240 |
| [21] | Du, H.; Shen, J., Reproducing kernel method of solving singular integral equation with cosecant kernel, J. Math. Anal. Appl., 348, 308-314 (2008) · Zbl 1152.45007 |
| [22] | Eid, H.; Adams, G. G.; Mcgruer, N. E.; Fortini, A.; Buldyrev, S.; Srolovitz, D., A combined molecular dynamics and finite element analysis of contact and adhesion of a rough sphere and a flat surface, Tribol. Trans., 54, 920-928 (2011) |
| [23] | Elloumi, R.; Kallel-Kamoun, I.; El-Borgi, S.; Guler, M. A., On the frictional sliding contact problem between a rigid circular conducting punch and a magneto-electro-elastic half-plane, Int. J. Mech. Sci., 87, 1-17 (2014) |
| [24] | Eshkuvatov, Z. K.; Nik Long, N. M.A.; Abdulkawi, M., Approximate solution of singular integral equations of the first kind with Cauchy kernel, Appl. Math. Lett., 22, 651-657 (2009) · Zbl 1161.65370 |
| [25] | Fajdiga, G.; Sraml, M., Fatigue crack initiation and propagation under cyclic contact loading, Eng. Fract. Mech., 76, 1320-1335 (2009) |
| [26] | Fischer, K. A.; Wriggers, P., Mortar based frictional contact formulation for higher order interpolations using the moving friction cone, Comput. Methods Appl. Mech. Engrg., 195, 5020-5036 (2006) · Zbl 1118.74047 |
| [27] | Gay Neto, A.; Pimenta, P. M.; Wriggers, P., A master-surface to master-surface formulation for beam to beam contact. Part I: frictionless interaction, Comput. Methods Appl. Mech. Engrg., 303, 400-429 (2016) · Zbl 1425.74340 |
| [28] | Gay Neto, A.; Pimenta, P. M.; Wriggers, P., A master-surface to master-surface formulation for beam to beam contact. Part II: frictional interaction, Comput. Methods Appl. Mech. Engrg., 319, 146-174 (2017) · Zbl 1439.74218 |
| [29] | Gay Neto, A.; Pimenta, P.d. M.; Wriggers, P., Contact between spheres and general surfaces, Comput. Methods Appl. Mech. Engrg., 328, 686-716 (2018) · Zbl 1439.74223 |
| [30] | Gay Neto, A.; Wriggers, P., Computing pointwise contact between bodies: a class of formulations based on master-master approach, Comput. Mech., 585-609 (2019) · Zbl 1468.74042 |
| [31] | Gu, C.; Hwang, I. S.; Kang, D.-O.; Lee, W. Y., Normal singular Cauchy integral operators with operator-valued symbols, J. Math. Anal. Appl., 447, 289-308 (2017) · Zbl 1356.47053 |
| [32] | Guz, A. N., Contact problems of the theory of elasticity for a half-plane with initial stresses, Sov. Appl. Mech., 16, 691-699 (1980) · Zbl 0466.73142 |
| [33] | Guz, A. N., On dynamic contact problems for an elastic half-plane with initial stresses in the case of a moving rigid punch, Internat. Appl. Mech., 35, 515-521 (1999) · Zbl 1073.74604 |
| [34] | Guz, A. N.; Zozulya, V. V., Fracture dynamics with allowance for crack edge contact interaction, Int. J. Nonlinear Sci. Numer. Simul., 2, 173-233 (2001) · Zbl 1241.74032 |
| [35] | Guz, A. N.; Zozulya, V. V., Elastodynamic unilateral contact problems with friction for bodies with cracks, Internat. Appl. Mech., 38, 895-932 (2002) · Zbl 1094.74543 |
| [36] | Johnson, K. L., Contact Mechanics (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0599.73108 |
| [37] | Karczmarek, P.; Pylak, D.; Sheshko, M. A., Application of Jacobi polynomials to approximate solution of a singular integral equation with Cauchy kernel, Appl. Math. Comput., 181, 694-707 (2006) · Zbl 1148.65320 |
| [38] | Kim, S., Solving singular integral equations using Gaussian quadrature and overdetermined system, Comput. Math. Appl., 35, 63-71 (1998) · Zbl 0999.65149 |
| [39] | Kim, J. H.; Jang, Y. H., Frictional Hertzian contact problems under cyclic loading using static reduction, Int. J. Solids Struct., 51, 252-258 (2014) |
| [40] | Kravchenko, V. V., On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions, J. Math. Anal. Appl., 339, 1103-1111 (2008) · Zbl 1135.30020 |
| [41] | Ladopoulos, E., Singular Integral Equations: Linear and Non-linear Theory and Its Applications in Science and Engineering (2000), Springer-Verlag Berlin Heidelberg: Springer-Verlag Berlin Heidelberg New York · Zbl 0963.65118 |
| [42] | Lin, T.; Ou, H.; Li, R., A finite element method for 3D static and dynamic contact/impact analysis of gear drives, Comput. Methods Appl. Mech. Engrg., 196, 1716-1728 (2007) · Zbl 1173.74433 |
| [43] | Majzoobi, G. H.; Abbasi, F., On the effect of contact geometry on fretting fatigue life under cyclic contact loading, Tribol. Lett., 65 (2017), 125-1-17 |
| [44] | Mikhlin, S. G., Integral equations, (Mikhlin, S. G., Singular Integral Equations (1957), Elsevier Ltd), 335-338 |
| [45] | Mlika, R.; Renard, Y.; Chouly, F., An unbiased Nitsche’s formulation of large deformation frictional contact and self-contact, Comput. Methods Appl. Mech. Engrg., 325, 265-288 (2017) · Zbl 1439.74220 |
| [46] | Nayak, M. M.; Acharya, M.; Acharya, B. P., Numerical evaluation of two-dimensional Cauchy principal value integrals, Appl. Math. Comput., 225, 807-810 (2013) · Zbl 1334.65062 |
| [47] | Pandey, R. K.; Singh, O. P.; Singh, V. K., Numerical solution of system of Volterra integral equations using Bernstein polynomials, Int. J. Nonlinear Sci. Numer. Simul., 10, 891-896 (2009) |
| [48] | Paripour, M., Note on numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method, Appl. Math. Comput., 250, 530-532 (2015) · Zbl 1328.65155 |
| [49] | Salem, H. A.H., On the quadratic integral equations and their applications, Comput. Math. Appl., 62, 2931-2943 (2011) · Zbl 1232.45004 |
| [50] | Setia, A., Numerical solution of various cases of Cauchy type singular integral equation, Appl. Math. Comput., 230, 200-207 (2014) · Zbl 1410.65505 |
| [51] | Shadimetov, K. M.; Hayotov, A. R.; Akhmedov, D. M., Optimal quadrature formulas for Cauchy type singular integrals in Sobolev space, Appl. Math. Comput., 263, 302-314 (2015) · Zbl 1410.65072 |
| [52] | Sharma, V.; Setia, A.; Agarwal, R. P., Numerical solution for system of Cauchy type singular integral equations with its error analysis in complex plane, Appl. Math. Comput., 328, 338-352 (2018) · Zbl 1427.65427 |
| [53] | Sidi, A.; Israeli, M., Quadrature methods for periodic singular and weakly singular Fredholm integral equations, J. Sci. Comput., 3, 201-231 (1988) · Zbl 0662.65122 |
| [54] | Söhngen, H., Zur Theorie der endlichen Hilbert-Transformation, Math. Z., 60, 31-51 (1954) · Zbl 0055.09502 |
| [55] | Su, J.; Ke, L.; El Sami, B.; Xiang, Y.; Wang, Y., An effective method for the sliding frictional contact of a conducting cylindrical punch on FGPMs, Int. J. Solids Struct., 141-142, 127-136 (2018) |
| [56] | Turhan, İ.; Oğuz, H.; Yusufoğlu, E., Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind, Int. J. Comput. Math., 90, 944-954 (2013) · Zbl 1278.65204 |
| [57] | Yang, Z.; Boogaard, A.; Wei, Z.; Liu, J.; Dollevoet, R.; Li, Z., Numerical study of wheel-rail impact contact solutions at an insulated rail joint, Int. J. Mech. Sci., 138-139, 310-322 (2018) |
| [58] | Zeng, G.; Huang, J.; Jia, H., The high accuracy algorithm for Cauchy singular integral and Cauchy singular integral equation, (2010 4th International Conference on Bioinformatics and Biomedical Engineering (2010), IEEE: IEEE Chengdu, China), 1-4 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
