×
Zhang, Yaoming; Sladek, Vladimir; Sladek, Jan; Liu, Zhaoyan

A new boundary integral equation formulation for plane orthotropic elastic media. (English) Zbl 1252.74060

Appl. Math. Modelling 36, No. 10, 4862-4875 (2012).
Summary: A new boundary integral equation formulation for solving plane elasticity problems involving orthotropic media is presented. Based on the real variable fundamental solutions of the considered problems, a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) and a novel decomposition technique to the fundamental solutions, the regularized BIEs with indirect unknowns, which do not involve the direct calculation of CPV and HFP integrals, are established. The limiting process is done in global coordinates and no separate numerical treatment for strong and weak singular integrals was necessary. The current method does not need to transform the considered problems into isotropic ones as is normally done in the existing literature, so no inverse transform is required. The numerical implementation is carried out using both discontinuous quadratic elements and exact elements, which is developed to model its boundary with negligible error. The validity of the proposed scheme is demonstrated by three numerical examples. Excellent agreement between the numerical results and exact solutions was obtained even with using small amounts of element.

Cited in 5 Documents

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
74B05 Classical linear elasticity

Keywords:

BEM; orthotropic; elastic problems; IBIE; nonsingular IBIE
Cite Review PDF
Full Text: DOI

References:

[1] Owen, D. R.J.; Figuerias, J. A., Anisotropic elasto-plastic finite element analysis of thick and thin plates and sheets, Int. J. Numer. Methods Eng., 19, 541-566 (1983) · Zbl 0509.73081
[2] Vaziri, R.; Olson, M. D.; Anderson, D. L., Finite element analysis of fibrous composite structures: a plasticity approach, Comput. Struct., 44, 103-116 (1992) · Zbl 0825.73734
[3] Karakuzu, R.; Sayman, O., Elasto-plastic finite element analysis of orthotropic rotating discs with holes, Comput. Struct., 51, 695-703 (1994) · Zbl 0875.73244
[4] Brebbia, C. A.; Telles, J. C.F.; Wrobel, L. C., Boundary Element Techniques: Theory and Applications in Engineering (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0556.73086
[5] Du, Q. H.; Cen, Z. Z., The Boundary Integral Equation Method-Boundary Element Method (1989), Higher Education Press: Higher Education Press Beijing · Zbl 0669.73055
[6] Aliabadi, M. H., The Boundary Element Method-Volume 2: Applications in Solids and Structures (2002), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0994.74003
[7] Green, G., A note on stress systems in aeolotropic materials, Phil. Mag., 34, 416-418 (1934)
[8] Rizzo, F. J.; Shippy, D. J., A method for stress determination in plane anisotropic elastic bodies, J. Compos. Mater., 4, 36-61 (1970) · Zbl 0237.65074
[9] Padhi, G. S.; Shenoi, R. A.; Moy, S. S.J., A quadratic boundary element implementation in orthotropic elasticity using the real variable approach, Commun. Numer. Methods Eng., 16, 257-266 (2000) · Zbl 0968.74075
[10] Perez, M. M.; Wrobel, L. C., An integral equation formulation for anisotropic elastostatics, ASME J. Appl. Mech., 63, 891-902 (1996) · Zbl 1002.74514
[11] John, W.; Tsay, T. K., Analytical evaluation and application of the singularities in boundary element method, Eng. Anal. Bound. Elem., 29, 241-256 (2005) · Zbl 1182.76915
[12] Padhi, G. S.; Shenoi, R. A.; Moy, S. S.J., Analytical Integration of kernel shape function product integrals in the boundary element method, Comput. Struct., 79, 1325-1333 (2001)
[13] M.A. Lifeng, Alexander M. Korsunsky, A note on the Gauss-Jacobi quadrature formulae for singular integral equations of the second kind, Int. J. Fracture, 126 (2004) pp. 339-405.; M.A. Lifeng, Alexander M. Korsunsky, A note on the Gauss-Jacobi quadrature formulae for singular integral equations of the second kind, Int. J. Fracture, 126 (2004) pp. 339-405. · Zbl 1187.65138
[14] Karami, G.; Derakhshan, D., An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis, Eng. Anal. Bound. Elem., 23, 317-326 (1999) · Zbl 0940.65139
[15] Telles, J. C.F., A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int. J. Numer. Methods Eng., 24 (1987), 959-937 · Zbl 0622.65014
[16] Cerrolaza, M.; Alarcón, E., A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods, Int. J. Numer. Methods Eng., 28, 987-999 (1989) · Zbl 0679.73040
[17] Kutt, H. R., The numerical evaluation of principal value integrals by finite-part integration, Numer. Math., 24, 205-210 (1975) · Zbl 0306.65010
[18] Dominguez et al., Flux and traction boundary elements without hypersingular or singular integrals, Int. J. Numer. Methods Eng., 48 (2000) pp. 111-135.; Dominguez et al., Flux and traction boundary elements without hypersingular or singular integrals, Int. J. Numer. Methods Eng., 48 (2000) pp. 111-135. · Zbl 0987.74074
[19] Jorge et al., Self regular boundary integral equation formulations for Laplace’s equation in 2-D, Int. J. Numer. Methods Eng., 51 (2001) pp. 1-29.; Jorge et al., Self regular boundary integral equation formulations for Laplace’s equation in 2-D, Int. J. Numer. Methods Eng., 51 (2001) pp. 1-29. · Zbl 0987.65124
[20] Guiggiani, M.; Casalini, P., Direct computation of Cauchy principal value integrals in advanced boundary elements, Int. J. Numer. Methods Eng., 24, 1711-1720 (1987) · Zbl 0635.65020
[21] M. Guiggiani, G. Krishnasamy, T.J. Rudolphi, et al., General algorithm for the numerical solution of hyper-singular boundary integral equations, (ASME) J. Appl. Mech, 59 (1992) pp. 604-614.; M. Guiggiani, G. Krishnasamy, T.J. Rudolphi, et al., General algorithm for the numerical solution of hyper-singular boundary integral equations, (ASME) J. Appl. Mech, 59 (1992) pp. 604-614. · Zbl 0765.73072
[22] Gao, X. W.; Yang, K.; Wang, J., An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals, Eng. Anal. Bound. Elem., 32, 692-696 (2008) · Zbl 1244.65199
[23] Y.M. Zhang, H.C. Sun, Analytical treatment of boundary integrals in direct boundary element analysis of plan potential and elasticity problems, Appl. Math. Mech., 22(2001) pp. 593-601. (in Chinese).; Y.M. Zhang, H.C. Sun, Analytical treatment of boundary integrals in direct boundary element analysis of plan potential and elasticity problems, Appl. Math. Mech., 22(2001) pp. 593-601. (in Chinese).
[24] Y.M. Zhang, H.C. Sun, Theoretic analysis on virtual boundary element, Chinese J. Comput. Mech., 17(2000) pp. 56-62 (in Chinese).; Y.M. Zhang, H.C. Sun, Theoretic analysis on virtual boundary element, Chinese J. Comput. Mech., 17(2000) pp. 56-62 (in Chinese).
[25] Chen, J. T.; Shen, W. C.; Wu, A. C., Null-field integral equations for stress field around circular holes under antiplane shear, Eng. Anal. Bound. Elem., 30, 205-217 (2006) · Zbl 1195.74220
[26] Chen, J. T.; Hsiao, C. C.; Leu, S. Y., Null-field integral equation approach for plate problems with circular boundaries, ASME J. Appl. Mech., 73, 619-693 (2006) · Zbl 1111.74358
[27] Liu, Y. J., On the simple solution and non-singular nature of the BIE/BEM-a review and some new results, Eng. Anal. Bound. Elem., 24, 286-292 (2000) · Zbl 0974.65110
[28] Rudolphi, T. J., Higher order elements and element enhancement by combined regular and hypersingular boundary integral equations, Bound. Elem. Methods Eng., 448-455 (1990)
[29] Shiah, Y. C.; Tan, C. L., BEM treatment of two-dimensional anisotropic field problems by direct domain mapping, Eng. Anal. Bound. Elem., 20, 347-351 (1997)
[30] J. Rungamornrat, M.F. Wheeler. Weakly singular integral equations for Darcy’s flow in anisotropic porous media, Eng. Anal. Bound. Elem., 30 (2006) pp. 237-246.; J. Rungamornrat, M.F. Wheeler. Weakly singular integral equations for Darcy’s flow in anisotropic porous media, Eng. Anal. Bound. Elem., 30 (2006) pp. 237-246. · Zbl 1195.76296
[31] J. Rungamornrat. Modeling of flow in three-dimensional, multi-zone, anisotropic porous media with weakly singular integral equation method, J. Eng. Mech., 135 (2009) pp. 828-838.; J. Rungamornrat. Modeling of flow in three-dimensional, multi-zone, anisotropic porous media with weakly singular integral equation method, J. Eng. Mech., 135 (2009) pp. 828-838.
[32] Y.M. Zhang, Y. Gu, V. Sladek, J. Sladek. A novel displacement gradient boundary element method for 2D elasticity problems (submitted).; Y.M. Zhang, Y. Gu, V. Sladek, J. Sladek. A novel displacement gradient boundary element method for 2D elasticity problems (submitted).
[33] Tanaka, M.; Sladek, V.; Sladek, J., Regularization techniques applied to boundary element methods, Appl. Mech. Rev., 47, 457-499 (1994)
[34] Y.M. Zhang, Z.Y. Liu, J.T. Chen, Y. Gu. A novel boundary element approach for solving the anisotropic potential problems, Engineering Analysis with Boundary Elem. (in press).; Y.M. Zhang, Z.Y. Liu, J.T. Chen, Y. Gu. A novel boundary element approach for solving the anisotropic potential problems, Engineering Analysis with Boundary Elem. (in press).
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.