The present paper studies an optimization problem of dynamically loaded cylindrical tubes. This is a problem of linear elasticity theory. As we search for the optimal thickness of the tube which minimizes the displacement under forces, this is a problem of shape optimization. The mathematical model is given by a differential equation (ODE and PDE, respectively); the mechanical problem is described as an optimal control problem. We consider both the stationary (time independent) and the transient (time dependent) case. P. Nestler derives the model-equations from the Mindlin and Reissner hypotheses. Then, necessary optimality conditions for the optimal control problem are given. Numerical solutions are obtained by FEM, numerical examples are presented.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1123,
author = {Peter Nestler and Werner H. Schmidt},
title = {Optimal design of cylindrical shells},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {30},
year = {2010},
pages = {253-267},
zbl = {1238.49058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1123}
}
Peter Nestler; Werner H. Schmidt. Optimal design of cylindrical shells. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 30 (2010) pp. 253-267. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1123/
[000] [1] V. Azhmyakov, J. Lellep and W. Schmidt, On the optimal design of elastic beams, Struct. Multidisc. Optim. 27 (2004), 80-88. doi: 10.1007/s00158-003-0352-1 | Zbl 1243.74135
[001] [2] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen (Vieweg Verlag, Wiesbaden, 2005).
[002] [3] D. Braess, Finite Elemente (Springer Verlag, Berlin, Heidelberg, New York, 1997).
[003] [4] D. Chapelle and K.J. Bathe, The Finite Element Analysis of Shells - Fundamentals (Springer Verlag, Berlin, Heidelberg, 2003). doi: 10.1007/978-3-662-05229-7 | Zbl 1103.74003
[004] [5] J. Wloka, Partielle Differentialgleichungen (Teubner Verlag, Stuttgart, 1982). doi: 10.1007/978-3-322-96662-9
[005] [6] St.P. Timoshenko, Schwingungsprobleme der Technik (J.-Springer-Verlag, 1932).
[006] [7] R. Hill, The mathematical theory of plasticity (Oxford Clarendon Press, 1950). | Zbl 0041.10802
[007] [8] G. Olenev, On optimal location of the additional support for an impulsively loaded rigid-plastic cylindrical shell, Trans. Tartu Univ. 772 (1987), 110-120. | Zbl 0664.73065
[008] [9] Ü. Lepik and T. Lepikult, Automated calculation and optimal design of rigid-plastic beams under dynamic loading, Int. J. Impact Eng. 6 (1987), 87-99. doi: 10.1016/0734-743X(87)90012-1
[009] [10] J. Lellep, Optimization of inelastic cylindrical shells, Eng. Optimization 29 (1997), 359-375. doi: 10.1080/03052159708941002
[010] [11] T. Lepikult, W.H. Schmidt and H. Werner, Optimal design of rigid-plastic beams subjected to dynamical loading, Springer Verlag, Structural Optimization 18 (1999), 116-125. doi: 10.1007/BF01195986
[011] [12] J. Lellep, Optimal design of plastic reinforced cylindrical shells, Control-Theory and Advanced Technology 5 (2) (1989), 119-135.
[012] [13] P. Nestler, Calculation of deformation of a cylindrical shell, Preprint Mathematik 4/2008.
[013] [14] J. Sprekels and D. Tiba, Optimization problems for thin elastic structures, in 'Optimal Control of Coupled System of PDE', ISNM 158 Birkhaeuser (2009), 255-273. | Zbl 1197.49047