Papadrakakis, M. A truncated Newton-Lanczos method for overcoming limit and bifurcation points. (English) Zbl 0704.73100 Int. J. Numer. Methods Eng. 29, No. 5, 1065-1077 (1990). Summary: In this study procedures for overcoming limit and bifurcation points in large-scale structural analysis problems are described and evaluated. The methods are based on Newton’s method for the outer iterations, while for the linearized problem in each iteration the preconditioned truncated Lanczos method is employed. Special care is placed upon line search routines for accelerating the convergence properties and enhancing the stability of the outer method. The proposed methodology retains all characteristics of an iterative method by avoiding the factorization of the current stiffness matrix. The necessary eigenvalue information is retained in the tridiagonal matrix of the Lanczos approach. Cited in 3 Documents MSC: 74S30 Other numerical methods in solid mechanics (MSC2010) 65F30 Other matrix algorithms (MSC2010) 74G60 Bifurcation and buckling 74H45 Vibrations in dynamical problems in solid mechanics 65K10 Numerical optimization and variational techniques 74P10 Optimization of other properties in solid mechanics 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:snap-through; snap-back; large-scale structural analysis; Newton’s method for the outer iterations; linearized problem; each iteration; preconditioned truncated Lanczos method; line search routines; accelerating the convergence properties; stability of the outer method; tridiagonal matrix PDFBibTeX XMLCite \textit{M. Papadrakakis}, Int. J. Numer. Methods Eng. 29, No. 5, 1065--1077 (1990; Zbl 0704.73100) Full Text: DOI References: [1] ’Incremental/iterative solution procedures for nonlinear structural problems’, in et al. (eds.), Numerical Methods for Non-linear Problems, Pineridge Press, Swansea, U.K., 1980, pp. 261-290. [2] and , ’Combining quasi-Newton and arc-length methods for the analysis of nonlinear problems in the post limit range’, Proc. First World Congress on Computational Mechanics, Austin, Texas, Sept. 1986. [3] Crisfield, Comp. Struct. 13 pp 55– (1981) [4] and , ’A work control method for geometrically nonlinear analysis’, Proc. NUMETA’ 85 Conference, Swansea, U.K., 1985, pp. 913-921. [5] Papadrakakis, Int. j. numer. methods eng. 29 pp 141– (1990) [6] Papadrakakis, Comp. Struct. 30 pp 705– (1988) [7] and , ’Conjugate gradient algorithms in nonlinear structural analysis problems’, Comp. Methods Appl. Mech. Eng., 11-27 (1986). [8] Papadrakakis, Int.j. numer. methods. eng. 28 pp 1299– (1989) [9] Nash, SIAM J. Sci. Stat. Comp. 6 pp 599– (1985) [10] O’Leary, Math. Program. 23 pp 20– (1982) [11] Crisfield, Int. j. numer. methods eng. 19 pp 1269– (1983) [12] ’Variable step lengths for non-linear structural analysis’, TRRL Laboratory Report 1049, 1982. [13] The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, N.J., 1980, p. 10. · Zbl 0431.65017 [14] Bergan, Comp. Struct. 12 pp 497– (1980) [15] Kani, J. Struct. Div. ASCE 112 pp 1806– (1987) [16] Papadrakakis, Comp. Struct. 14 pp 393– (1981) [17] Dembo, Math. Program. 26 pp 190– (1983) [18] ’A truncated Newton-Lanczos method for overcoming limit and bifurcation points’, ISAAR Report 88-3, NTUA, 1989. [19] Papadrakakis, J. Appl. Mech. ASME 53 pp 291– (1986) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.