Abstract
Reduction methods using eigenvectors and Ritz vectors as basis vectors are empolyed to reduce the finite element nonlinear system of equations using a 48 D.O.F. doubly curved thin plate/shell element. With and without basis updating, the solutions obtained by reduction methods are compared with the direct solutions. It is observed that basis updating is essential to obtain accurate solutions. The present reduction methods need a large number of basis vectors (eigenvectors and Ritz vectors) to account for the impact load which has high frequency characteristics. Furthermore, for nonlinear analysis, the reduction achieved in the CPU time are only marginal since most of the CPU time was spent in the calculation of the internal nodal force vector. These considerations indicate that reduction methods may not be efficient for the impact response analysis.
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Akay, H. U. (1980): Dynamic large deflection analysis of plates using mixed finite elements. Comput. and Struct. 11, 1–11
Almroth, B. O.; Stern, P.; Brogan, F. A. (1978): Automatic choice of global shape functions in structural analysis. AIAA J. 16, 525–528
Bathe, K. J.; Gracewski, S. (1981): On nonlinear dynamic analysis using substructuring and mode superposition. Comput. and Struct. 13, 699–707
Bathe, K. J.; Ramm, E.; Wilson, E. L. (1975): Finite element formulations for large deformation dynamic analysis. Int. J. Numer. Methods Eng. 9, 353–386
Bayles, D. J.; Lowery, R. L.; Boyd, D. E. (1973): Nonlinear vibrations of rectangular plates. ASCE, Journal of the Structural Division 99, 853–864
Bayo, E. P.; Wilson, E. L. (1984): Use of Ritz vectors in wave propagation and foundation response. Earthquake Eng. and Structural Dynamics 12, 499–505
Byun, C. (1991): Free vibration and nonlinear transient analysis of imperfect laminated structures. Ph.D. dissertation. Virginia Polytechnic Institute and State University, Blacksburg, VA 24061
Camarda, C. J. (1990): Development of advanced modal methods for calculating transient thermal and structural response. Ph.D. dissertation. Virginia Polytechnic Institute and State University, Blacksburg, VA 24061
Camarda, C. J.; Haftka, R. H.; Riley, M. F. (1987): An evaluation of higher-order modal methods for calculating transient structural response. Comput. and Struct. 27, 89–101
Chan, A. S. L.; Hsiao, K. M. (1985): Nonlinear analysis using a reduced number of variables. Comp. Meth. in Appl. Mech. and Eng. 52, 899–913
Chan, A. S. L.; Lau, T. B. (1987): Further development of the reduced basis method for geometric nonlinear analysis. Comp. Meth. in Appl. Mech. and Eng. 62, 127–144
Chang, C.; Engblom, J. J. (1991): Nonlinear dynamical response of impulsively loaded structures: A reduced basis approach. AIAA J. 29, 613–618
Chang, P.; Haley, T. (1988): Computation of symmetry modes and exact reduction in nonlinear structural analysis. Comput. and Struct. 28, 135–142
Clough, R. W.; Wilson, E. L. (1979): Dynamic analysis of large structural systems with local nonlinearties. Comp. Meth. in Appl. Mech. and Eng. 17, 18, 107–129
Cornwell, R. E.; Graig, R. R., Jr.; Johnson, C. P. (1983): On the application of the mode-acceleration method to structural engineering problems. Earthquake Eng. and Structural Dyanmics 11, 679–688
Das, S. K.; Uktu, S.; Wada, B. K. (1990): Use of reduced basis technique in the inverse dynamics of large space cranes. Computing Systems in Engineering 1, 577–589
Horri, K.; Kawahara, M. (1969): A numerical analysis on the dynamic response of structures. Proceedings of 19th Japan National Congress for Applied Mechanics, 1969, pp. 17–22.
Idelsohn, S. R.; Cardona, A. (1985a): A reduction method for nonlinear structural dynamic analysis. Comp. Meth. in Appl. Mech. and Eng. 49, 253–279
Idelsohn, S. R.; Cardona, A. (1985b): A load-dependent basis for reduced nonlinear structural dynamics. Comput. and Struct. 20, 203–210
Kapania, R. K.; Yang, T. Y. (1986): Formulation of an imperfect quadrilateral doubly-curved shell element for post-buckling analysis. AIAA J. 24, 310–311
Kapania, R. K.; Yang, T. Y. (1987): Buckling, postbuckling, and nonlinear vibrations of imperfect plates. AIAA J. 25, 1338–1346
Kline, K. A. (1986): Dynamic analysis using a reduced basis of exact modes and Ritz vectors. AIAA J. 24, 2022–2029
Lanczos, C. (1950): An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal of Research of the National Bureau of Standards 45, 255–282
Leung, Y. T. (1983): Fast response method for undamped structures. Eng. Struct. 5, 141–149
McGowan, D. M.; Bostic, S. W. (1991): Comparison of advanced reduced-basis methods for transient structural analysis. Proc. of 32nd AIAA/ASME/ACE/AHS/ACE Structures, Structural Dynamics and Materials Conference. Baltimore, Maryland, 2523–2535
Moharz, B.; Elghadamsi, F. E.; Chang, C. (1991): An incremental mode superposition for nonlinear dynamic analysis. Earthquake Eng. and Structural Dynamics 20, 471–481
Morris, N. F. (1977): The use of modal superposition in nonlinear dynamics. Comput. and Struct. 7, 65–72.
Nagy, D. A.; König, M. (1979): Geometrically nonlinear finite element behaviour using buckling mode superposition. Comp. Meth. in Appl. Mech. and Eng. 19, 447–484
Nickell, R. E. (1976): Nonlinear dynamics by mode superposition. Comp. Meth. in Appl. Mech. and Eng. 7, 107–129
Noor, A. K.; Peters, J. M. (1980): Reduced basis technique for nonlinear analysis of structures. AIAA J. 18, 455–462
Noor, A. K.; Anderson, C. M.; Peters, J. M. (1981a); Reduced basis technique for collapse analysis of shells. AIAA J 19, 393–397
Noor, A. K. (1981b): Recent advances in reduction methods for nonlinear problems. Comput. and struct. 13, 31–44
Noor, A. K. (1982). On making large nonlinear problems small. Comp. Meth. in Appl. Mech. and Eng. 34, 955–985
Noor, A. K.; Peters, J. M. (1983): Instability analysis of space trusses. Comp. Meth. in Appl. Mech. and Eng. 40, 199–218
Nour-Omid, B.; Clough, R. W. (1984): Dynamic analysis of structures using Lanczos co-ordinates. Earthquake Eng. and Structural Dynamics 12, 565–577
Przybylo, W. (1985): An application of the concept of the increasing series of the marcomodels (ISM) to the mechanics of the large discrete nonlinear systems. Proc. 1st Int. Conf. Advances in Numerical Methods in Engineering, NUMETA '85. Swansea, Wales, 253–262
Remesth, S. N. (1979): Nonlinear static and dynamic analysis of framed structures. Comput. and Struct. 10, 879–897
Safjan, A. (1988): Nonlinear structural analysis via reduced basis technique. Comput. and Struct. 29, 1055–1061
Safjan, A. (1990): Some advances in reduction methods for nonlinear structural statics. Comput. and Struct. 34, 753–763
Saigal, S.; Kapania, R. K.; Yang, T. Y. (1986): Geometrically nonlinear finite element analysis of imperfect Laminated shells. J. of Composite Materials 20, 197–214
Saigal, S.; Yang, T. Y.; Kapania, R. K. (1987): Dynamic Buckling of imperfection sensitive shell structures. J. of Aircraft 24, 718–724
Shah, V. N.; Bohm, G. J.; Nahavandi, A. N. (1979): Modal superposition method for computationally economical nonlinear structural analysis. J. of Pressure Vessel Technology 101, 134–141
Wilson, E. L.; Yuan, M.-W.; Dickens, J. M. (1982): Dynamic analysis by direct superposition of Ritz vectors. Earthquake Eng. and Structural Dynamics 10, 813–821
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Communicated by S. N. Atluri, May 4, 1992
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Kapania, R.K., Byun, C. Reduction methods based on eigenvectors and Ritz vectors for nonlinear transient analysis. Computational Mechanics 11, 65–82 (1993). https://doi.org/10.1007/BF00370072
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DOI: https://doi.org/10.1007/BF00370072