Abstract
Direct methods provide elegant and efficient approaches for the prediction of the long-term behaviour of engineering structures under arbitrary complex loading independent of the number of loading cycles. The lower bound direct method leads to a constrained non-linear convex problem in conjunction with finite element methods, which necessitates a very large number of optimization variables and a large amount of computer memory. To solve this large-scale optimization problem, we first reformulate it in a simpler equivalent convex program with easily exploitable sparsity structure. The interior point with DC regularization algorithm (IPDCA) using quasi definite matrix techniques is then used for its solution. The numerical results obtained by this algorithm will be compared with those obtained by general standard code Lancelot. They show the robustness, the efficiency of IPDCA and in particular its great superiority with respect to Lancelot.
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References
Melan E. (1936) Theorie Statisch Unbestimmter Systeme aus Ideal-Plastischem Baustoff. Sitber. Akad. Wiss. Wien, Abt. IIA 145, 195–218
Koiter W.T. (1960) General theorems for elastic-plastic solids. In: Sneddon I.N., Hill R. (eds) Progress in Solid Mechanics. North-Holland, Amsterdam, pp. 165–221
Hachemi A., Weichert D. (1998) Numerical shakedown analysis of damaged structures. Comput. Methods Appl. Mech. Eng 160, 57–70
Akoa F., Le Thi Hoai An, Pham Dinh Tao (2004) An interior point algorithm with DC regularisation for nonconvex quadratic programming. In: Le Thi Hoai An, Pham Dinh Tao (eds) Modelling, Computation and Optimization in Information Systems and Management Sciences. Hermes Science, London, pp. 87–96
Hachemi A., Le Thi Hoai An, Mouhtamid S., Pham Dinh Tao. (2004) Large-scale nonlinear programming and lower bound direct method in engineering applications. In: Le Thi Hoai An, Pham Dinh Tao (eds) Modelling, Computation and Optimization in Information Systems and Management Sciences. Hermes Science, London, pp. 299–310
Conn A.R., Gould N.I.M., Toint Ph.L. (1992) LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization (Release A). Springer - Verlag, Berlin
Weichert D., Hachemi A., Schwabe F. (1999) Shakedown analysis of composites. Mech. Res. Comm. 26, 309–318
Hachemi A., Schwabe F., Weichert D. (2000) Failure investigation of fiber-reinforced composite materials by shakedown analysis. In: Weichert D., Maier G. (eds) Inelastic Analysis of Structures under Variable Loads: Theory and Engineering Applications. Kluwer Academic Publishers, Dordrecht, pp. 107–119
König J.A., Kleiber M. (1978) On a new method of shakedown analysis. Bull. Acad. Polon. Sci. Sér. Sci. 26, 165–171
Akoa F.: Approches de point intérieur et de la programmation DC en optimisation non convexe. Codes et simulations numériques industrielles. PhD Thesis, Insa - Rouen, January 2005.
Le Thi Hoai An: Analyse numérique des algorithmes de l’optimization D.C: Approches locale et globale, PhD Thesis, University of Rouen, December 1994.
Le Thi Hoai An, Pham Dinh Tao: DC programming: theory, algorithms and applications. the state of the art (28 pages), In: Proceedings (containing the refereed contributed papers) of The First International Workshop on Global Constrained Optimization and Constraint Satisfaction (Cocos’ 02), Valbonne-Sophia Antipolis, France, October 2–4, 2002.
Le Thi Hoai An: Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, et Applications. Habilitation à Diriger des Recherches (HDR), Université de Rouen, Juin 1997.
Pham Dinh Tao, Le Thi Hoai (1997) An Convex analysis approach to DC programming: theory, algorithms and applications. Acta Mathematica Vietnamica (dedicated to Professor Hoang Tuy on the occasion of his 70th birthday) 22(1), 289–355
Pham Dinh Tao, Le Thi Hoai An (1998) DC optimization algorithm for solving the trust region problem. SIAM J. Optim., 8(2): 476–505
Pham Dinh Tao, Le Thi Hoai An (2003) Large scale molecular optimization from distance matrices by a DC optimization approach. SIAM J. Optim, 14(1): 77–116
Le Thi Haoi An, Pham Dinh Tao (2005) The DC (difference of convex functions) Programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper Res. 133, 23–46
Altman A., Gondzio J.: Regularized symmetric Indefinite Systems in Interior Point Methods for Linear and Quadratic Optimization. Logilab Technical Report 1998.6
Saunders M. (1996) Cholesky-based methods for sparse least squares: the benefits of regularization. In: Adams L., Nazareth L. (eds) Linear and Nonlinear Conjugate Gradient-Related Methods. SIAM Philadelphia, USA, pp. 92–100
Saunders M., Tomlin J.A.: Solving regularized linear programs using barrier methods and KKT systems. Technical Repport SOL 96-4, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, CA 94305, USA, December 1996.
Vanderbei R.J., Shanno D. (1999) An Interior-Point Algorithm for Nonconvex Nonlinear Pro- gramming. Statistics and Operations Research. Princeton SOR-97-21 Comput. Optim. App. 13, 231–252
Vanderbei R.J.: An Interior-Point Algorithm for Quadratic Programming. Statistics and Operations Research, Princeton SOR-94-15 (1994)
Vanderbei R.J. (1995) Symmetric quasi-definite matrices. SIAM J. Optim 5(1): 100–113
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François, A., Abdelkader, H., An, L.T.H. et al. Application of lower bound direct method to engineering structures. J Glob Optim 37, 609–630 (2007). https://doi.org/10.1007/s10898-006-9069-1
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DOI: https://doi.org/10.1007/s10898-006-9069-1