×

zbMATH — the first resource for mathematics

Nearly singular approximations of CPV integrals with end- and corner-singularities for the numerical solution of hypersingular boundary integral equations. (English) Zbl 1181.74147
Summary: A local numerical approach to cope with the singular and hypersingular boundary integral equations (BIEs) in non-regularized forms is presented in the paper for 2D elastostatics. The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors. The nearly singular approximations in the present work, including the normal and the tangential distance transformations, are designed for the numerical evaluation of boundary integrals with end-singularities at junctures between two elements, especially at corner points where sufficient continuity requirements are met. The approach is very general, which makes it possible to solve the hypersingular BIE numerically in a non-regularized form by using conforming \(C^0\) quadratic boundary elements and standard Gaussian quadratures without paying special attention to the corner treatment.
With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations.

MSC:
74S15 Boundary element methods applied to problems in solid mechanics
74K20 Plates
65N38 Boundary element methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Richardson, J.D.; Cruse, T.A., Weakly singular stress-BEM for 2D elastostatics, Int J numer meth engng, 45, 13-35, (1999) · Zbl 0960.74073
[2] Mukherjee, S., CPV and HFP integrals and their applications in the boundary element methods, Int J solids struct, 37, 6623-6634, (2000) · Zbl 0992.74075
[3] Martin, P.A.; Rizzo, F.J., Hypersingular integrals: how smooth must the density be?, Int J numer meth engng, 39, 687-704, (1996) · Zbl 0846.65070
[4] Aliabadi, M.H.; Martin, D., Boundary element hyper-singular formulation for elastoplastic contact problems, Int J numer meth engng, 48, 995-1014, (2000) · Zbl 0974.74072
[5] Krishnasamy, G.; Schmerr, L.W.; Rudolphi, T.J.; Rizzo, F.J., Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering, ASME J appl mech, 57, 404-414, (1990) · Zbl 0729.73251
[6] Hui, C.Y.; Shia, D., Evaluations of hypersingular integrals using Gaussian quadrature, Int J numer meth engng, 44, 205-214, (1999) · Zbl 0948.65019
[7] Karami, G.; Derakhshan, D., An efficient method to evaluate hypersingular and supersingular integrals in boundary integrals in boundary integral equations analysis, Engng anal bound elem, 23, 317-326, (1999) · Zbl 0940.65139
[8] Mukherjee, S., Finite parts of singular and hypersingular integrals with irregular boundary source points, Engng anal bound elem, 24, 767-776, (2000) · Zbl 0991.74079
[9] Guiggiani, M.; Casalini, P., Direct computation of Cauchy principal value integrals in advanced boundary elements, Int J numer meth engng, 24, 1711-1720, (1987) · Zbl 0635.65020
[10] Guiggiani, M.; Gigante, A., A general algorithm for multi-dimensional Cauchy principal value integrals in the boundary element method, ASME J appl mech, 57, 906-915, (1990) · Zbl 0735.73084
[11] Guiggiani, M.; Krishnasamy, G.; Rudolghi, T.J.; Rizzo, F.J., A general algorithm for the numerical solution of hypersingular boundary integral equations, ASME J appl mech, 59, 604-614, (1992) · Zbl 0765.73072
[12] Huang, Q.; Cruse, T.A., On the non-singular traction-BEM in elasticity, Int J numer meth engng, 37, 2041-2072, (1994) · Zbl 0832.73076
[13] Cruse, T.A.; Richardson, J.D., Non-singular somigliana stress identities in elasticity, Int J numer meth engng, 39, 3273-3304, (1996) · Zbl 0886.73005
[14] Johnston, P.R., C^2-continuous elements for boundary element analysis, Int J numer meth engng, 40, 2087-2108, (1997) · Zbl 0929.74120
[15] Cruse, T.A.; Aithal, R., Non-singular boundary integral equation implementation, Int J numer meth engng, 36, 237-254, (1993)
[16] Matsumoto, T.; Tanaka, M., Boundary stress calculation using regularized boundary integral equation for displacement gradients, Int J numer meth engng, 36, 783-797, (1993) · Zbl 0825.73909
[17] Liu, Y.J.; Rudolphi, T.J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, Comput mech, 24, 286-292, (1999) · Zbl 0969.74073
[18] Liu, Y.J., On the simple solution and non-singular nature of the BIE/BEM—a review and some new results, Engng anal bound elem, 24, 789-795, (2000) · Zbl 0974.65110
[19] Chen, H.B.; Lu, P.; Schnack, E., Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM, Engng anal bound elem, 25, 851-876, (2001) · Zbl 1051.74050
[20] Mukherjee, S.; Chati, M.K.; Shi, X.L., Evaluation of nearly singular integrals in boundary element contour and node methods for three-dimensional linear elasticity, Int J solids struct, 37, 7633-7654, (2000) · Zbl 0993.74077
[21] Lean, M.H.; Wexler, A., Accurate numerical integration of singular boundary element kernels over boundary with curvature, Int J numer meth engng, 21, 211-228, (1985) · Zbl 0555.65091
[22] Aliabadi, M.H.; Hall, W.S., Taylor expansions for singular kernels in the boundary element method, Int J numer meth engng, 21, 2221-2236, (1985) · Zbl 0599.65011
[23] Telles, J.C.F., A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int J numer meth engng, 24, 959-973, (1987) · Zbl 0622.65014
[24] Cerrolaza, M.; Alarcon, E., A bi-cubic transformation of the Cauchy principal value integrals in boundary methods, Int J numer meth engng, 28, 987-999, (1989) · Zbl 0679.73040
[25] Doblare, M.; Gracia, L., On non-linear transformations for the integration of weakly-singular and Cauchy principal value integrals, Int J numer meth engng, 40, 3325-3358, (1997) · Zbl 1049.74789
[26] Johnston, P.R., Application of sigmoidal transformations to weakly singular and nearly-singular boundary element integrals, Int J numer meth engng, 45, 1333-1348, (1999) · Zbl 0935.65130
[27] Ma, H.; Kamiya, N., Domain supplemental approach to avoid boundary layer effect of BEM in elasticity, Engng anal bound elem, 23, 3, 281-284, (1999) · Zbl 0963.74566
[28] Granados, J.J.; Gallego, G., Regularization of nearly hypersingular integrals in the boundary element method, Engng anal bound elem, 25, 165-184, (2001) · Zbl 1015.74073
[29] Ma, H.; Kamiya, N., A general algorithm for accurate computation of field variables and its derivatives near boundary in BEM, Engng anal bound elem, 25, 843-849, (2001)
[30] Ma, H.; Kamiya, N., Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method, Engng anal bound elem, 26, 329-339, (2002) · Zbl 1003.65133
[31] Ma, H.; Kamiya, N., A general algorithm for the numerical solution of near singular boundary integrals of various orders for two- and three-dimensional elasticity, Comput mech, 29, 277-288, (2002) · Zbl 1128.74343
[32] Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C., Boundary element techniques—theory and applications in engineering, (1984), Springer Heidelberg · Zbl 0556.73086
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.