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High Accuracy Eigensolutions in Elasticity for Boundary Integral Equations by Nyström Method
Abstract:Elastic boundary eigensolution problems are converted into boundary integral equations by potential theory. The kernels of the boundary integral equations have both the logarithmic and Hilbert singularity simultaneously. We present the mechanical quadrature methods for solving eigensolutions of the boundary integral equations by dealing with two kinds of singularities at the same time. The methods possess high accuracy O(h3) and low computing complexity. The convergence and stability are proved based on Anselone-s collective compact theory. Bases on the asymptotic error expansion with odd powers, we can greatly improve the accuracy of the approximation, and also derive a posteriori error estimate which can be used for constructing self-adaptive algorithms. The efficiency of the algorithms are illustrated by numerical examples.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070723Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 3520
 A.R.Hadjesfandiari, G.F.Dargush, Boundary eigensolutions in elasticity 1: theoretical development, J.Solids Structure, 38(2001), 6589-6625.
 P.K.Banerjee, The boundary element methods in engineering, McGraw- Hill, London, 1994.
 Y.Z.Chen, Z.X.Wang, X.Y.Lin, Eigenvalue and eigenfunction analysis arising from degenerate scale problem of BIE in plane elasticity, Eng.Anal.Bound. Element, 31(2007), 994-1002.
 P.M.Anselone, Singularity subtraction in the numerical solution of integral equations, J.Austral.Math.Soc. (Series B), 22(1981), 408-418.
 C.A.Brebbia, S.Walker, Boundary elements techniques in engineering, Butter worth and Co., Boston, 1980.
 M.Willian, Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000.
 G.H.Paulino, G.Menom, S.Mukherjee, Error estimation using hypersingular integrals in boundary element methods for linear elasticity, Engineering Analysis with Boundary Elements, 25(2001), 523-534.
 V.Z.Parton, P.I.Perlin, Integral equations in elasticity, Mir Publishers, Moscow, 1977.
 A.R.Hadjesfandiari, G.F.Dargush, Boundary eigensolutions in elasticity II. Application to computational mechanics, J.Solids Structure, 40(2003), 1001-1031.
 A.R.Hadjesfandiari, G.F.Dargush, Computational mechanics based on the theory of boundary eigensolutions, Int.J.Numer.Mech.Eng., 50(2001), 325-346.
 G.Cohen, S.Fauqueux, Mixed spectral finite elements for the linear elasticity system in unbounded domains, SIAM J.Numer.Anal., 3(2005), 864-884.
 L.F.Pavarino, Preconditioned mixed spectral element methods for elasticity and stokes problems, SIAM J. Sci. Comput., 6(1998), 1941-1957.
 C.J.Talbot, A.Crampton, Application of the pseudo-spectral method to 2D eigenvalue problems in elasticity, Numer.Algortihms, 38(2005), 95C110.
 Z.Cai, T.A.Manteuffel, S.F.Mccormick, First-order system least squares for the stokes equations with application to linear elasticity, SIAM J.Numer.Anal., 5(1997), 1727-1741.
 Z.Cai, G.Starke, Least-squares methods for linear elasticity , SIAM J. Numer. Anal., 2(2004), 826-842.
 J.Huang, T.L┬¿u, The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems, J. Comp. Math., 5(2004), 719-726
 T.L┬¿u, J.Huang, High accuracy Nystr┬¿om approximations and their extrapolation for solving boundary weakly integral equations of the second kind, J. of Chinese Comp. Phy., 3(1997), 349-355.
 C.B.Lin, T.L┬¿u, T.M.Shih, The splitting extrapolation method, World Scientific, Singapore (1995).
 A.Sidi, M.Israrli, Quadrature methods for periodic singular Fredholm integral equations, J. Sci. Comput. , 3(1988),201-231.
 J.Huang, Z.Wang, Extrapolation algorithm for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods. SIAM J. Sci. Comput. , Vol. 31, 6(2009), 4115-4129.
 P.M.Anselone, Collectively compact operator approximation theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.