The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions
Tóm tắt
In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented.
Tài liệu tham khảo
Adams, R.A.: Sobolev Spaces. Academic, New York, San Francisko, London (1975)
Allgower, E.L., Chien, C.S.: Continuation and local perturbation for multiple bifurcations. SIAM J. Sci. Comput. 7, 1265–1281 (1986)
Allgower, E.L., Georg, K.: Numerical Continuation Methods. Springer, Berlin (1990)
Caloz, G., Rappaz, J.: Numerical analysis for nonlinear and bifurcation problems. Handbook of numerical analysis, North Holland, Amsterdam, vol. V, pp. 487–637 (1997)
Chien, C.S., Chang, S.L., Mei, Z.: Tracing the buckling of a rectangular plate with the Block GMRES method. J. Comput. Appl. Math. 136, 199–218 (2001) [Online] http://www.elsevier.com/locate/cam
Chien, C.S., Gong, S.Y., Mei, Z.: Mode jumping in the von Kármán equations. SIAM J. Sci. Comput. 22(4), 1354–1385 (2000) [Online] http://epubs.siam.org/sam-bin/dbq/article/30732
Chien, C.S., Weng, Z.L., Shen, C.L.: Lanczos-type methods for continuation problems. Numer. Linear Algebra Appl. 4, 23–41 (1997)
Ciarlet, P., Rabier, P.: Les equations de von kármán. Springer-Verlag, Berlin Heidelberg, New York (1980)
Dossou, K.: Résolution numérique des équations de von Karman. Thèse, Départment de mathématiques et de statistique. FacultTé des Sciences et Génie Université Laval Québeg (2000)
Dossou, K., Pierre, R.: A Newton-GMRES approach for the analysis of the postbuckling behavior of the solutions of the von Kármán equations. SIAM J. Sci. Comput. 24(6), 1994–2012 (2003) [Online] http://epubs.siam.org/sam-bin/dbq/article/37614
Harrar, D.L., Osborne, M.R.: Computing eigenvalues of ordinary differential equations. ANZIAM J. 44(E), C313–C334 (2003) [Online] http://anziamj.austms.org.au/V44/CTAC2001/Harr
Holder, E.J., Schaeffer, D.G.: Boundary conditions and mode jumping in the von Kármán’s equations. SIAM J. Math. Anal. 15, 446–458 (1984)
Keller, H.: Numerical solution of bifurcation and nonlinear eigenvalue problem. In: Rabinovitz, P.H. (ed.) Applications of Bifurcation Theory, pp. 359–384. Academic, New York (1977)
Muradova, A.D.: Numerical techniques for linear and nonlinear eigenvalue problems in the theory of elasticity. ANZIAM J. 46(E), C426–C438 (2005) [Online] http://anziamj.austms.org.au/V46/CTAC2004/Mura
Schaeffer, D.G., Golubitsky, M.: Boundary conditions and mode jumping in the buckling of a rectangular plate. Comm. Math. Phys. 69, 209–236 (1979)
Vashakmadze, T.S.: The Theory of Anisotropic Elastic Plates. Kluwer, Dordrecht, Boston, London (1999)