Summary
This paper presents a novel method called the Hypersingular Boundary Contour Method (HBCM) for two-dimensional (2-D) linear elastostatics. This new method can be considered to be a variant of the standard Boundary Element Method (BEM) and the Boundary Contour Method (BCM) because: (a) a regularized form of the hypersingular boundary integral equation (HBIE) is employed as the starting point, and (b) the above regularized form is then converted to a boundary contour version based on the divergence free property of its integrand. Therefore, as in the 2-D BCM, numerical integrations are totally eliminated in the 2-D HBCM. Furthermore, the regularized HBIE can be collocated at any boundary point on a body where stresses are physically continuous. A full theoretical development for this new method is addressed in the present work. Selected examples are also included and the numerical results obtained are uniformly accurate.
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References
Bonnet, M.: Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity. Eng. Anal. Bound. Elem.15, 93–102 (1995).
Cruse, T. A., Richardson, J. D.: Non-singular Somigliana stress identities in elasticity. Int. J. Num. Meth. Eng.39, 3273–3304 (1996).
Gray, L. J., Martha, L. F., Ingraffea, A. R.: Hypersingular integrals in boundary element fracture analysis. Int. J. Num. Meth. Eng.29, 1135–1158 (1990).
Gray, L. J., Balakrishna, C., Kane, J. H.: Symmetric Galerkin fracture analysis. Eng. Anal. Bound. Elem.15, 103–109 (1995).
Gray, L. J., Paulino, G. H.: Crack tip interpolation, revisited. SIAM J. Num. Anal.58, 428–455 (1998).
Gray, L. J., Paulino, G. H.: Symmetric Galerkin boundary integral formulation for interface and multy-zone problems. Int. J. Num. Meth. Eng.40, 3085–3101 (1997).
Guiggiani, M., Krishnasamy, G., Rudolphi, 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).
Krishnasamy, G., Schmerr, L. W., Rudolphi, T. J., Rizzo, F. J.: Hypersingular boundary integral equations: some applications in acoustics and elastic wave scattering. ASME J. Appl. Mech.57, 404–414 (1990).
Krishnasamy, G., Rizzo, F. J., Rudolphi, T. J.: Hypersingular boundary integral equations: their occurrence, interpretation, regularization and computation. In: Developments in boundary element methods, vol. 7 (Banerjee, P. K., Kobayashi, S., eds.), pp. 207–252. London: Elsevier 1992.
Lutz, E. D., Ingraffea, A. R., Gray, L. J.: Use of ‘simple solutions’ for boundary integral methods in elasticity and fracture analysis. Int. J. Num. Meth. Eng.35, 1737–1751 (1992).
Martin, P. A., Rizzo, F. J.: Hypersingular integrals: how smooth must the density be? Int. J. Num. Meth. Eng.39, 687–704 (1996).
Menon, G.: Hypersingular error estimates in boundary element methods. M. S. Thesis, Cornell University, Ithaca, 1996.
Menon, G., Paulino, G. H., Mukherjee, S.: Analysis of hypersingular residual error estimates for potential problems in boundary element methods. Comput. Meth. Appl. Mech. Eng. (in press).
Mukherjee, S.: Boundary element methods in creep and fracture. New York: Elsevier 1982.
Mukherjee, Y. X., Mukherjee, S., Shi, X., Nagarajan, A.: The boundary contour method for three-dimensional linear elasticity with a new quadratic boundary element. Eng. Anal. Bound. Elem.20, 35–44 (1997).
Nagarajan, A., Lutz, E. D., Mukherjee, S.: A novel boundary element method for linear elasticity with no numerical integration for two-dimensional and line integrals for three-dimensional problems. ASME J. Appl. Mech.61, 264–269 (1994).
Nagarajan, A., Lutz, E. D., Mukherjee, S.: The boundary contour method for three-dimensional linear elasticity. ASME J. Appl. Mech.63, 278–286 (1996).
Paulino, G. H.: Novel formulations of the boundary element method for fracture mechanics and error estimation. Ph. D. dissertation, Cornell University, Ithaca, 1995.
Paulino, G. H., Gray, L. J., Zarkian, V.: Hypersingular residuals — A new approach for error estimation in the boundary element method. Int. J. Num. Meth. Eng.39, 2005–2029 (1996).
Phan, A.-V., Mukherjee, S., Mayer, J. R. R.: The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements. Comput. Mech.20, 310–319 (1997).
Phan, A.-V., Mukherjee, S., Mayer, J. R. R.: A boundary contour formulation for design sensitivity analysis in two-dimensional linear elasticity. Int. J. Solids Struct.35, 1981–1999 (1998).
Phan, A.-V., Mukherjee, S., Mayer, J. R. R.: Stresses, stress sensitivities and shape optimization for two-dimensional linear elasticity by the boundary contour method. Int. J. Num. Meth. Eng. (in press).
Rizzo, F. J.: An integral equation approach to boundary value problems of classical elastostatics. Q. Appl. Math.25, 83–95 (1967).
Timoshenko, S. P., Goodier, J. N.: Theory of elasticity. New York: McGraw-Hill 1970.
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Phan, A.V., Mukherjee, S. & Mayer, J.R.R. The hypersingular boundary contour method for two-dimensional linear elasticity. Acta Mechanica 130, 209–225 (1998). https://doi.org/10.1007/BF01184312
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DOI: https://doi.org/10.1007/BF01184312