High Accuracy Eigensolutions in Elasticity for Boundary Integral Equations by Nyström Method
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.
[1] A.R.Hadjesfandiari, G.F.Dargush, Boundary eigensolutions in elasticity
1: theoretical development, J.Solids Structure, 38(2001), 6589-6625.
[2] P.K.Banerjee, The boundary element methods in engineering, McGraw-
Hill, London, 1994.
[3] 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.
[4] P.M.Anselone, Singularity subtraction in the numerical solution of integral
equations, J.Austral.Math.Soc. (Series B), 22(1981), 408-418.
[5] C.A.Brebbia, S.Walker, Boundary elements techniques in engineering,
Butter worth and Co., Boston, 1980.
[6] M.Willian, Strongly elliptic systems and boundary integral equations,
Cambridge University Press, 2000.
[7] 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.
[8] V.Z.Parton, P.I.Perlin, Integral equations in elasticity, Mir Publishers,
Moscow, 1977.
[9] A.R.Hadjesfandiari, G.F.Dargush, Boundary eigensolutions in elasticity
II. Application to computational mechanics, J.Solids Structure, 40(2003),
1001-1031.
[10] A.R.Hadjesfandiari, G.F.Dargush, Computational mechanics based on
the theory of boundary eigensolutions, Int.J.Numer.Mech.Eng., 50(2001),
325-346.
[11] G.Cohen, S.Fauqueux, Mixed spectral finite elements for the linear
elasticity system in unbounded domains, SIAM J.Numer.Anal., 3(2005),
864-884.
[12] L.F.Pavarino, Preconditioned mixed spectral element methods for elasticity
and stokes problems, SIAM J. Sci. Comput., 6(1998), 1941-1957.
[13] C.J.Talbot, A.Crampton, Application of the pseudo-spectral method
to 2D eigenvalue problems in elasticity, Numer.Algortihms, 38(2005),
95C110.
[14] 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.
[15] Z.Cai, G.Starke, Least-squares methods for linear elasticity , SIAM J.
Numer. Anal., 2(2004), 826-842.
[16] 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
[17] 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.
[18] C.B.Lin, T.L¨u, T.M.Shih, The splitting extrapolation method, World
Scientific, Singapore (1995).
[19] A.Sidi, M.Israrli, Quadrature methods for periodic singular Fredholm
integral equations, J. Sci. Comput. , 3(1988),201-231.
[20] 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.
[21] P.M.Anselone, Collectively compact operator approximation theory,
Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
[1] A.R.Hadjesfandiari, G.F.Dargush, Boundary eigensolutions in elasticity
1: theoretical development, J.Solids Structure, 38(2001), 6589-6625.
[2] P.K.Banerjee, The boundary element methods in engineering, McGraw-
Hill, London, 1994.
[3] 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.
[4] P.M.Anselone, Singularity subtraction in the numerical solution of integral
equations, J.Austral.Math.Soc. (Series B), 22(1981), 408-418.
[5] C.A.Brebbia, S.Walker, Boundary elements techniques in engineering,
Butter worth and Co., Boston, 1980.
[6] M.Willian, Strongly elliptic systems and boundary integral equations,
Cambridge University Press, 2000.
[7] 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.
[8] V.Z.Parton, P.I.Perlin, Integral equations in elasticity, Mir Publishers,
Moscow, 1977.
[9] A.R.Hadjesfandiari, G.F.Dargush, Boundary eigensolutions in elasticity
II. Application to computational mechanics, J.Solids Structure, 40(2003),
1001-1031.
[10] A.R.Hadjesfandiari, G.F.Dargush, Computational mechanics based on
the theory of boundary eigensolutions, Int.J.Numer.Mech.Eng., 50(2001),
325-346.
[11] G.Cohen, S.Fauqueux, Mixed spectral finite elements for the linear
elasticity system in unbounded domains, SIAM J.Numer.Anal., 3(2005),
864-884.
[12] L.F.Pavarino, Preconditioned mixed spectral element methods for elasticity
and stokes problems, SIAM J. Sci. Comput., 6(1998), 1941-1957.
[13] C.J.Talbot, A.Crampton, Application of the pseudo-spectral method
to 2D eigenvalue problems in elasticity, Numer.Algortihms, 38(2005),
95C110.
[14] 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.
[15] Z.Cai, G.Starke, Least-squares methods for linear elasticity , SIAM J.
Numer. Anal., 2(2004), 826-842.
[16] 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
[17] 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.
[18] C.B.Lin, T.L¨u, T.M.Shih, The splitting extrapolation method, World
Scientific, Singapore (1995).
[19] A.Sidi, M.Israrli, Quadrature methods for periodic singular Fredholm
integral equations, J. Sci. Comput. , 3(1988),201-231.
[20] 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.
[21] P.M.Anselone, Collectively compact operator approximation theory,
Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57428", author = "Pan Cheng and Jin Huang and Guang Zeng", title = "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.", keywords = "boundary integral equation, extrapolation algorithm, aposteriori error estimate, elasticity.", volume = "5", number = "2", pages = "132-5", }
