# Optimal design of cylindrical shells

• Volume: 30, Issue: 2, page 253-267
• ISSN: 1509-9407

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## Abstract

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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.

## How to cite

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Peter Nestler, and Werner H. Schmidt. "Optimal design of cylindrical shells." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.2 (2010): 253-267. <http://eudml.org/doc/271199>.

@article{PeterNestler2010,
abstract = {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.},
author = {Peter Nestler, Werner H. Schmidt},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {linear elasticity; shell theory; cylindrical tube; optimal control; shape optimization},
language = {eng},
number = {2},
pages = {253-267},
title = {Optimal design of cylindrical shells},
url = {http://eudml.org/doc/271199},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Peter Nestler
AU - Werner H. Schmidt
TI - Optimal design of cylindrical shells
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2010
VL - 30
IS - 2
SP - 253
EP - 267
AB - 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.
LA - eng
KW - linear elasticity; shell theory; cylindrical tube; optimal control; shape optimization
UR - http://eudml.org/doc/271199
ER -

## References

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2. [2] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen (Vieweg Verlag, Wiesbaden, 2005).
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7. [7] R. Hill, The mathematical theory of plasticity (Oxford Clarendon Press, 1950). Zbl0041.10802
8. [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. Zbl0664.73065
9. [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
10. [10] J. Lellep, Optimization of inelastic cylindrical shells, Eng. Optimization 29 (1997), 359-375. doi: 10.1080/03052159708941002
11. [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
12. [12] J. Lellep, Optimal design of plastic reinforced cylindrical shells, Control-Theory and Advanced Technology 5 (2) (1989), 119-135.
13. [13] P. Nestler, Calculation of deformation of a cylindrical shell, Preprint Mathematik 4/2008.
14. [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. Zbl1197.49047

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