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Residue approach for evaluating the 3D anisotropic elastic Green’s function: multiple roots. (English) Zbl 1182.74025

Summary: A numerical algorithm based upon the residue calculus for computing the three-dimensional anisotropic elastic Green’s function and its derivatives has been presented by, among the others, Sales and Gray. Although this residue approach is in general faster than the standard Wilson-Cruse interpolation scheme, the convergence rate and accuracy can seriously degrade in the neighborhood of a non-simple pole. In this paper, explicit expressions, also based on the residue calculus, are obtained for computing the Green’s function and its first-order derivatives in the presence of a multiple root. Further, the computation time for the residue algorithm proposed here has been significantly reduced by implementing the double-subscript-notation for the elastic constants that define the Christoffel tensor.

MSC:

74B05 Classical linear elasticity
74E10 Anisotropy in solid mechanics

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References:

[1] Bonnet, M., Boundary integral equation method for solids and fluids (1995), Wiley: Wiley England
[2] París, F.; Cañas, J., Boundary element method: fundamentals and applications (1997), Oxford University Press: Oxford University Press New York · Zbl 0868.73083
[3] Gray, L. J.; Graffith, A.; Johnson, L.; Wawrzynek, P. A., Evaluation of Galerkin singular integrals for anisotropic elasticity: displacement equation, Electron J Bound Elem, 1, 68-94 (2003)
[4] Nakamura, G.; Tanuma, K., A formula for the fundamental solution of anisotropic elasticity, Q J Mech Appl Math, 50, 179-194 (1997) · Zbl 0885.73005
[5] Ting, T. C.T.; Lee, V.-G., The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids, Q J Mech Appl Math, 50, 407-426 (1997) · Zbl 0892.73006
[6] Wang, C.-Y., Elastic fields produced by a point source in solids of general anisotropy, J Eng Math, 32, 41-52 (1997) · Zbl 0907.73008
[7] Sales, M. A.; Gray, L. J., Evaluation of the anisotropic Green’s function and its derivatives, Comput Struct, 69, 247-254 (1998) · Zbl 0939.74007
[8] Pan, E.; Yuan, F. G., Three-dimensional Green’s functions in anisotropic bimaterials, Int J Solids Struct, 37, 5329-5351 (2000) · Zbl 0992.74022
[9] Tonon, F.; Pan, E.; Amadei, B., Green’s functions and boundary element method formulation for 3D anisotropic media, Comput Struct, 79, 469-482 (2001)
[10] Phan, A.-V.; Gray, L. J.; Kaplan, T., On the residue calculus evaluation of the 3-D anisotropic elastic Green’s function, Commun Numer Meth Eng, 20, 335-341 (2004) · Zbl 1280.74008
[11] Lifschitz, I. M.; Rozentsweig, L. N., Zh Eksperim Teor Fiz, 17, 783 (1947)
[12] Barnett, D. M., The precise evaluation of derivatives of the anisotropic elastic Green’s functions, Phys Status Solidi (b), 49, 741-748 (1972)
[13] Wilson, R. B.; Cruse, T. A., Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis, Int J Numer Meth Eng, 12, 1383-1397 (1978) · Zbl 0377.73054
[14] Dederichs, P. H.; Liebfried, G., Elastic Green’s function for anisotropic cubic crystals, Phys Rev, 188, 1175-1183 (1969)
[15] Hildebrand, F. B., Advanced calculus for applications (1976), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0333.00003
[16] Burden, R. L.; Faires, J. D., Numerical Analysis (1989), PWS-Kent Publishing: PWS-Kent Publishing Boston · Zbl 0671.65001
[17] (Seitz, F.; Turnbull, D., Solid state physics, vol. 7 (1958), Academic Press: Academic Press New York) · Zbl 0085.23004
[18] Portela, A.; Aliabadi, M. H.; Rooke, D. P., Dual boundary element incremental analysis of crack propagation, Comput Struct, 46, 237-247 (1993) · Zbl 0825.73888
[19] Bonnet, M.; Maier, G.; Polizzotto, C., Symmetric Galerkin boundary element method, ASME Appl Mech Rev, 51, 669-704 (1998)
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