×

zbMATH — the first resource for mathematics

Prediction and control of buckling: the inverse bifurcation problems for von Karman equations. (English) Zbl 1425.74197
Dutta, Hemen (ed.) et al., Applied mathematical analysis: theory, methods, and applications. Cham: Springer. Stud. Syst. Decis. Control 177, 353-381 (2020).
Summary: The chapter presents novel approaches to predict and control buckling of thin-walled structures; mathematically, these approaches are formalized as the first and second inverse bifurcation problems for von Karman equations. Both approaches are based upon the method employed to solve the direct bifurcation problem for the equations in question. The approach considered was applied to several difficult problems of actual practice, viz., for the first inverse problem, to the problems of optimal thickness distribution and optimal external pressure distribution for a cylindrical shell, optimal curvature for a cylindrical panel as well; for the second inverse problem, to the problem to predict buckling of a cylindrical shell under an external pressure.
For the entire collection see [Zbl 1417.00004].
MSC:
74G60 Bifurcation and buckling
37N35 Dynamical systems in control
35Q74 PDEs in connection with mechanics of deformable solids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gilmore, R.: Catastrophe Theory for Scientist and Engineers. Dover Publications, Inc., New York (1981) · Zbl 0497.58001
[2] Vorovich, I.I.: Nonlinear Theory of Shallow Shells. Springer, New York (1999) · Zbl 0916.73002
[3] Guarracino, F.: Considerations on the numerical analysis of initial post-buckling behaviour in plates and beams. Thin-Walled Struct. 45, 845-848 (2007)
[4] Wang, C.M., Tay, Z.Y., Chowdhuary, A.N.R., Duan, W.H., Zhang, Y.Y., Silvestre, N.: Examination of cylindrical shell theories for buckling of carbon nanotubes. Int J Struct Stability Dynam. 11(6), 1035-1058 (2011)
[5] Guarracino, F., Walker, A.: Some comments on the numerical analysis of plates and thin-walled structures. Thin-Walled Struct. 46, 975-980 (2008)
[6] Lee, M.C.W., Mikulik, Z., Kelly, D.W., Thomson, R.S., Degenhardt, R.: Robust design—a concept for imperfection insensitive composite structures. Compos. Struct. 92, 1469-1477 (2010)
[7] Obodan, N.I., Lebedeyev, O.G., Gromov, V.A.: Nonlinear Behaviour and Stability of Thin-Walled Shells. Springer, New York (2013) · Zbl 1295.74002
[8] Bendsøe, M.P., Sigmund, O.: Topology Optimization—Theory, Methods and Applications. Springer, N.-Y (2003)
[9] Lindgaard, E., Dahl, J.: On compliance and buckling objective functions in topology optimization of snap-through problems. Struct. Multidisc. Optim. 47, 409-421 (2013) · Zbl 1274.74363
[10] Lindgaard, E., Lund, E., Rasmussen, K.: Nonlinear buckling optimization of composite structures considering “worst” shape imperfections. Int. J. Solids Struct. 47, 3186-3202 (2010) · Zbl 1203.74118
[11] Henrichsen, S.R., Lindgaard, E., Lund, E.: Buckling optimization of composite structures using a discrete material parametrization considering worst shape imperfections. In: E. Onãte, X. Oliver, A. Huerta (eds.) Proceedings of 11th World Congress on Computational Mechanics, N.-Y. (2014)
[12] Smołka, M.: Differentiability of the objective in a class of coefficient inverse problems. Comput. Math Appl. 73, 2375-2387 (2017) · Zbl 1375.58008
[13] Engl, H.W., Kügler, P.P.: Nonlinear inverse problems: theoretical aspects and some industrial applications. In: Capasso and Periaux (eds.), Multidisciplinary Methods for Analysis, Optimization and Control of Complex Systems, Springer Heidelberg, Series Mathematics in Industry, pp. 3-48 (2005)
[14] Dierkes, T., Dorn, O., Natterer, F., Palamodov, V., Sielschott, H.: Fréchet derivatives for some bilinear inverse problems. SIAM J. Appl. Math. 62(6), 2092-2113 (2002) · Zbl 1010.35115
[15] Yildirim Aksoy, N.: Variational method for the solution of an inverse problem. J. Comput. Appl. Math. 312, 82-93 (2017) · Zbl 1348.35306
[16] Babaniyi, O.A., Oberai, A.A., Barbone, P.E.: Direct error in constitutive equation formulation for plane stress inverse elasticity problem. Comput. Methods Appl. Mech. Eng. 314, 3-18 (2017)
[17] Obodan, N.I., Guk, N.A:. The Inverse Problems of Thin-Walled Shell Theory. Berlin: Lambert academic publishing; (2012) (in Russian)
[18] Gowda, K., Kuehn, C.: Early-warning signs for pattern-formation in stochastic partial differential equations. Commun. Nonlin. Sci. Numer. Simulat. 22, 55-69 (2015) · Zbl 1331.60119
[19] Wiesenfeld, K.: Virtual Hopf phenomenon: a new precursor of period-doubling bifurcations. Phys. Rev. A 32(3), 1744-1751 (1985)
[20] Zulkuparov, M-G.M., Malinetskii, G.G., Podlazov, A.V.: The inverse bifurcation problem for noisy dynamic systems. Preprint of Keldysh Institute of Applied Mathematics (Russian Academy of Science) (in Russian) (2005). http://keldysh.ru/papers/2005/prep39/prep2005_39.pdf
[21] Lim, J., Epureanu, B.I.: Forecasting a class of bifurcations: theory and experiment. Phys. Rev. E 83(1), 162-165 (2011)
[22] Stull, C.J., Nichols, J.M., Earls, C.J.: Stochastic inverse identification of geometric imperfections in shell structures. Comput. Methods Appl. Mech. Eng. 200, 2256-2267 (2011) · Zbl 1230.74085
[23] Abramovich, H., Govich, D., Grunwald, A.: Damping measurements of laminated composite materials and aluminum using the hysteresis loop method. Prog. Aerosp. Sci. 78, 8-18 (2015)
[24] Obodan, N.I., Adlucky, V.J., Gromov, V.A.: Rapid identification of pre-buckling states: a case of cylindrical shell. Thin-Walled Struct. 124, 449-457 (2018)
[25] Thompson, J.M.T., Hunt, G.W.: A General Theory of Elastic Stability. Wiley, London (1978)
[26] Obodan, N.I., Lebedev, A.G..: The Secondary Branching in the Nonlinear Theory of Thin-Walled Shells. Reports of the Academy of Science of Ukranian SSR. Series 2, vol. 12, pp. 38-41 (1980) (in Russian)
[27] Obodan, N.I., Gromov, V.A.: Numerical analysis of the branching of solutions to nonlinear equations for cylindrical shells. Int. Appl. Mech. 42(1), 90-97 (2006) · Zbl 1114.74461
[28] Zhou, Y., Stanciulescu, I., Eason, T., Spottswood, M.: Nonlinear elastic buckling and postbuckling analysis of cylindrical panels. Finite Elem. Anal. Des. 96, 41-50 (2015)
[29] Fujii, F., Noguchi, H.: Symmetry-breaking bifurcation and post-buckling strength of a compressed circular cylinder. In: Solid Mechanics and Fluid Mechanics: Computational Mechanics for the Next Millennium, pp. 563-568. Pergamon: Amsterdam
[30] Fujii, F., Noguchi, H., Ramm, E.: Static path jumping to attain postbuckling equilibria of a compressed circular cylinder. Comp. Mech. 26, 259-266 (2000) · Zbl 0989.74023
[31] Hu, N., Burgueño, R.: Elastic postbuckling response of axially-loaded cylindrical shells with seeded geometric imperfection design. Thin-Walled Struct. 96, 256-268 (2015)
[32] Zhao, Y., Chen, M., Yang, F., Zhang, L., Fang, D.: Optimal design of hierarchical grid-stiffened cylindrical shell structures based on linear buckling and nonlinear collapse analyses. Thin-Walled Struct. 119, 315-323 (2017)
[33] Obodan, N.I., Gromov, V.A.: The complete bifurcation structure of nonlinear boundary problem for cylindrical panel subjected to uniform external pressure. Thin-walled Struct. 107, 612-619 (2016)
[34] Grigolyuk, E.I., Lopanitsyn, E.A.: Axisymmetric post-buckling behaviour of shallow spherical domes. Appl. Math. Mech. 66(4), 621-633 (2002) · Zbl 1094.74637
[35] Grigolyuk, E.I., Lopanitsyn, E.A.: Nonaxisymmetric post-buckling behaviour of shallow sperical domes. Appl. Math. Mech. 67(6), 921-932 (2003) · Zbl 1150.70322
[36] Krysko, A.V., Awrejcewicz, J., Pavlov, S.P., Zhigalov, M.V., Krysko, V.A.: On the iterative methods of linearization, decrease of order and dimension of the Karman-Type PDEs. Scientif. World J 2014 (Article ID 792829) 1-15 (2014)
[37] Gromov, V.A.: On an approach to solve nonlinear elliptic equations of von Karman type. Bulletin of Dnepropetrovsk university. Math. Models 8, 122-142 (2017)
[38] Hunt, G.W.: Buckling in space and time. Nonlin. Dyn. 43, 29-46 (2006) · Zbl 1138.74332
[39] Hunt, G.W., Lord, G.J., Champneys, A.R.: Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells. Comput. Methods Appl. Mech. Eng. 170, 239-251 (1999) · Zbl 0958.74021
[40] Lord, G.J., Champneys, A.R., Hunt, G.W.: Computation of localized post buckling in long axially compressed cylindrical shells. Phil. Trans. R. Soc. Lond. A. 355, 2137-2150 (1997) · Zbl 0894.73040
[41] Knobloch, E.: Spatial localization in dissipative systems. Ann. Rev. Condensed Mat. Phys. 6, 325-359 (2015)
[42] Kreilos, T., Schneider, T.M.: Fully localized post-buckling states of cylindrical shells under axial compression. Proc. R. Soc. A. 473, 2017.0177 (2017) · Zbl 1402.74072
[43] Awrejcewicz, J., Erofeev, N.P., Krysko, V.A.: Non-symmetric and chaotic vibrations of Euler-Bernoulli beams under harmonic and noisy excitations. J Phys. Conf. Ser. 721(012003) (2016)
[44] Awrejcewicz, J., Krysko, A.V., Papkova, I.V., Zakharov, V.M., Erofeev, N.P., Krylova, EYu., Mrozowski, J., Krysko, V.A.: Chaotic dynamics of flexible beams driven by external white noise. Mech. Syst. Signal Process. 79, 225-253 (2016)
[45] Awrejcewicz, J., Krysko, V.A.: Chaos in Structural Mechanics. Springer, New York (2008) · Zbl 1144.74002
[46] Awrejcewicz, J., Krysko, V.A., Krysko, A.V.: Spatio-temporal chaos and solitons exhibited by von Karman model. Int J Bifurcat. Chaos. 12(7), 1465-1513 (2002)
[47] Awrejcewicz, J., Krysko, V.A., Narkaitis, G.G.: Bifurcations of a thin plate-strip excited transversally and axially. Nonlin. Dyn. 32, 187-209 (2003) · Zbl 1062.74530
[48] Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-posed problems. Winston, New York (1977) · Zbl 0354.65028
[49] Arbelo, M.A., Degenhardt, R., Castro, S.G.P., Zimmermann, R.: Numerical characterization of imperfection sensitive composite structures. Compos. Struct. 108, 295-303 (2014)
[50] Hao, P., Wang, B., Du, K., Li, G., Tian, K., Sun, Y., Ma, Y.: Imperfection-insensitive design of stiffened conical shells based on perturbation load approach. Compos. Struct. 136, 405-413 (2016)
[51] Hao, P., Wang, B., Tian, K., Li, G., Du, K., Niu, F.: Efficient optimization of cylindrical stiffened shells with reinforced cutouts by curvilinear stiffeners. AIAA J. 54(4), 1350-1363 (2016)
[52] Mania, R.J., Madeo, A., Zucco, G., Kubiak, T.: Imperfection sensitivity of post-buckling of FML channel section column. Thin-Walled Struct. 114, 32-38 (2017)
[53] Krasovsky, V.L., Marchenko, V.A., Schmidt, R.: Deforming and buckling of axially compressed cylindrical shells with local load in numerical simulation and experiments. Thin-walled Struct. 49, 576-580 (2011)
[54] Yamaki, N.: Elastic stability of circular cylindrical shells. North-Holland, Amsterdam, The Netherlands (1984) · Zbl 0544.73062
[55] Virot, E., Kreilos, T., Schneider, T.M., Rubinstein, S.M.: Stability landscape of shell buckling. Phys. Rev. Lett. 119(22), 224101(5) (2017)
[56] Litvinov, W.G.: Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics. Springer (2000) · Zbl 0947.49001
[57] Kiss, I.Z., Hudson, J.L.: Experiments on coherence resonance: noisy precursors to Hopf bifurcations. Phys. Rev. E 67, 15-19 (2003)
[58] Omberg, L., Dolan, K., Neiman, A., Moss, F.: Detecting the onset of bifurcations and their precursors from noisy data. Phys. Rev. E 61(5), 4848-4853 (2000)
[59] Lapko, A.V., Chentsov, S.V.: Nonparametric Information Processing Systems. Nauka, Novosibirsk (2000) · Zbl 0998.68558
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.