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**Sensitivity computation and shape optimization for a nonlinear arch model with limit-points instabilities.**
*(English)*
Zbl 0927.74054

The paper deals with optimal design of beams and arches with geometric nonlinearities. The proposed optimization method is based on a sensitivity gradient which is computed by means of an adjoint state equation. To increase the rate of convergence near critical points, the authors use second derivatives of critical load with respect to state and shape variables. The derivatives are calculated by means of software packages for analytical differentiation. Numerical solutions of linear and nonlinear examples are compared with analytical solutions. The proposed method is proved to be efficient, and it remains stable at the turning point. In most cases, sufficiently good approximations are obtained in the second iteration.

Reviewer: A.Žilinskas (Vilnius)

### MSC:

74P10 | Optimization of other properties in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

65K10 | Numerical optimization and variational techniques |

### Keywords:

geometric nonlinearities; sensitivity gradient; adjoint state equation; rate of convergence; second derivatives of critical load; analytical differentiation
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\textit{P. Aubert} and \textit{B. Rousselet}, Int. J. Numer. Methods Eng. 42, No. 1, 15--48 (1998; Zbl 0927.74054)

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### References:

[1] | et al., ’Shell optimization, some remarks’, in (ed.), Advances in Structural Optimization, Vol. 25, Solid Mechanics and its Applications, Kluwer, Dordrecht, The Netherlands, 1995, pp. 381-412. |

[2] | Doedel, Int. J. Bifurcation Chaos 1 pp 493– (1991) |

[3] | Wriggers, Int. J. Numer. Meth. Engng. 30 pp 155– (1990) |

[4] | and (eds.), Structural and Multidisciplinary Optimization, Vol. 1, Elsevier Science, Oxford, U.K., 1995. |

[5] | ’Optimisation de forme en présence d’instabilités’, Ph.D. Thesis, Université de Nice-Sophia Antipolis, Nice, France, Laboratoire J. A. Dieudonné, Parc Valrose, 06108 Nice Cedex 2, 1996. |

[6] | and , A General Theory of Elastic Stability, Wiley, New York, 1973. |

[7] | Aubert, Appl. Math. Comput. Sci. 6 pp 101– (1996) |

[8] | Bernadou, Struct. Optimiz. 3 pp 7– (1991) |

[9] | Phelan, Int. J. Numer. Meth. Engng. 31 pp 1649– (1991) |

[10] | Habbal, Mech. Struct. Mach. 20 pp 93– (1992) |

[11] | ’Theory of buckling and post-buckling of elastic structures’, Adv. Appl. Mech., 2-63 (1974). |

[12] | Theory of Elastic Stability, North-Holland, Amsterdam, 1980. |

[13] | Nonlinear problems of elasticity, Applied Mathematical Sciences, Vol. 107, Springer, Berlin, 1991. |

[14] | Simo, Comput. Methods Appl. Mech. Engng. 49 pp 55– (1984) |

[15] | Simo, Comput. Methods Appl. Mech. Engng. 58 pp 79– (1986) |

[16] | Rousselet, Mech. Struct. Mach. 20 pp 415– (1992) |

[17] | ’Design sensitivity of critical loads and vibration frequencies of nonlinear structures’, in (ed.), Optimization of Large Structural Systems, NATO ASI Series E: Applied Sciences, Vol. 231, Kluwer, Dordrecht, The Netherlands, 1993, pp. 455-476. |

[18] | and , On numerical approximation in bifurcation theory, Recherche Mathématiques Appliquée, Masson, Paris. Masson, Springer, Berlin, 1989. |

[19] | ’Sensitivity analysis for a geometrically nonlinear arch model’, in Numerical Methods in Engng., Wiley, Eccomas, 1996, pp. 257-263. |

[20] | Crisfield, Comp. Struct. 13 pp 55– (1981) |

[21] | and , Continuation and path following, Chapter 1, Acta Numerica. Press syndicate, Cambridge University Press, Cambridge, 1993, pp. 1-64. · Zbl 0792.65034 |

[22] | and , ’Sensitivity analysis of a nonlinear arch’, in preparation. |

[23] | Fried, Comp. Methods Appl. Mech. Engng. 38 pp 29– (1983) |

[24] | and , ’First and second order design sensitivity at a bifurcation point’, in (ed.), Advances in Structural Optimization, Solid Mechanics and its Applications, Vol. 25, Kluwer, Dordrecht, The Netherlands, 1995, pp. 349-380. |

[25] | The Finite Element Method, Mac-Graw Hill, New York, 1947. |

[26] | Kooper, J. Comp. Phys. 118 pp 320– (1995) |

[27] | Mróz, Int. J. Solids Struct. 31 pp 2071– (1994) |

[28] | Bergan, Comp. Struct. 12 pp 497– (1980) |

[29] | Doedel, Int. J. Bifurcation Chaos 1 pp 745– (1991) |

[30] | Chan, SIAM J. Sci. Statist. Comp. 5 pp 121– (1984) |

[31] | Chan, SIAM J. Sci. Statist. Comp. 5 pp 135– (1984) |

[32] | Rostaing, Tellus 45 pp 558– (1993) |

[33] | et al., A Package for the Automatic Differentiation of Algorithms Written in C/C++, (January 1995). Version 1.6. |

[34] | et al., ADIFOR 2.0 User’s Guide, 1995. |

[35] | et al., User’s Guide for CFSQP, A C Code for Solving Large Scale Constraint Nonlinear Optimization Problems, 2nd edn, Electrical engineering Department, University of Maryland, College Park, MD20742, 1994. |

[36] | et al., PETSc 2.0 Users Manual, Mathematics and Computer Science Division, Argonne National Laboratory, 1997. |

[37] | and , ’Thickness optimization of geometrically non-linear beam structures with inextensibility’, in and (eds.), Structural and Multidisciplinary Optimization, Vol. 1, Elsevier Science, Oxford, U.K., 1995. |

[38] | ’Optimality conditions and analytical methods of shape optimization’, in and (eds.), Optimization of Distributed Parameter Structures, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1981, pp. 973-1004. |

[39] | Problems and methods of optimal structural design, Vol. 26 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, 1983. |

[40] | Introduction to Optimization of Structures, Springer, New York, 1990. |

[41] | Olhoff, Struct. Optim. 3 pp 163– (1991) |

[42] | ’Structural optimization by variational method’, in Computer Aided Optimal Design, Structural and Mechanical Systems, Vol. 1 of NATO/NASA/NSF/USAF, Advanced Study Institute, 7-97. Center of mechanics and materials of the technical university of Lisbon, C.A. Mota Soares, Tria, Portugal, July 1986. |

[43] | et al., ’Finite Element computation of hyper-elastic rods in large displacements’, Tech. rep., Centre de recherche mathématiques de la décision, Université Paris Dauphine, May 1991. |

[44] | and , ’Optimal forms of shallow arches with respect to vibration and stability’, Tech. Rep. 234, The Danish Center for Applied Mathematics and Mechanics, March 1982. |

[45] | Olhoff, Tech. Rep. 242 (1982) |

[46] | Griewank, SIAM J. Numer. Anal. 33 pp 1912– (1996) |

[47] | Waterloo Maple Software, Maple Manual, 1995. |

[48] | Wolfram Research, Mathematica Manual, 1995. |

[49] | MuPAD User’s Manual–MuPAD Version 1.2.2, 1st edn, Wiley, Chichester, March 1996. · Zbl 0877.68069 |

[50] | Analyse fonctionelle, théorie et applications, Collection mathématiques appliquées pour la maitrise, Masson, Paris, 1993. |

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.