Topology optimization of sheets in contact by a subgradient method

International Journal for Numerical Methods in Engineering

Volumn 40, Issue 7, 1997, Pages 1295-1321

  Petersson, Joakim ^a,c   Patriksson, Michael ^a,b  

^a LINKÖPING INSTITUTE OF TECHNOLOGY   (Sweden)

^b UNIVERSITY OF WASHINGTON   (United States)

^c TECHNICAL UNIVERSITY OF DENMARK   (Denmark)

Author keywords

Non differentiable convex programming; Optimum sheet; Subgradient optimization algorithm; Unilateral contact

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; ITERATIVE METHODS; MATHEMATICAL MODELS; PROBLEM SOLVING; STIFFNESS MATRIX; TOPOLOGY;

OPTIMUM SHEET PROBLEMS; SUBGRADIENT METHOD;

OPTIMIZATION;

EID: 0031124360     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0207(19970415)40:7<1295::AID-NME115>3.0.CO;2-P     Document Type: Article

References (40)

1. W. Prager and J. E. Taylor, 'Problems of optimal structural design', Trans. ASME J. Appl. Mech., 35, 102-106(1968).
2. M. P. Rossow and J. E. Taylor, 'A finite element method for the optimal design of variable thickness sheets', AIAA J., 11, 1566-1568 (1973).
3. R. L. Benedict, 'Maximum stiffness design for elastic bodies in contact', J. Mech. Des., 104, 825-830 (1982).
4. J. Céa and K. Malanowski, 'An example of a max-min problem in partial differential equations', SIAM J. Control, 8, 305-316 (1970).
5. J. Petersson, 'On stiffness maximisation of variable thickness sheet with unilateral contact', Quart. Appl. Math., 54, 541-550 (1996),
6. J. Petersson and J. Haslinger, 'An approximation theory for optimum sheets in unilateral contact', Quart. Appl. Math. (to appear).
7. G. I. N. Rozvany, Structural Design via Optimality Criteria, Kluwer, Dordrecht, 1989.
8. M. P. Bendsee and J. M. Guedes, 'Some computational aspects of using extremal material properties in the optimal design of shape, topology and material', Control Cybernet., 3, 327-350 (1994).
9. A. Ben-Tal and M. P. Bendsoe, 'A new method for optimal truss topology design', SIAM J. Optim., 3, 322-358 (1993).
10. U. T. Ringertz, 'On rinding the optimal distribution of material properties', Struct, Optim., 5, 265-267 (1993).
11. R, T. Rockafellar, 'Augmented Lagrangians and applications of the proximal point algorithm in convex programming', Math. Oper. Res., 1, 97-116 (1976).
12. J. M. Danskin, The Theory of Max-Min, Springer, Berlin, 1967.
13. V. F. Dem'yanov and V. N. Malozemov, Introduction to Minimax, Wiley, New York, 1974.
14. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I: Fundamentals, Springer, Berlin, 1993.
15. J. Petersson, 'Stiffness optimization of general structure in Signorini-type contact', in. M. Raus, M. Jean and J. J. Moreau (eds.), Contact Mechanics, Plenum Press, New York, 1995, pp. 41-48.
16. J, Petersson and A. Klarbring, 'Saddle point approach to stiffness optimization of discrete structures including unilateral contact', Control Cybernet., 3, 461-480 (1994).
17. W. Achtziger, M. P. Bendsee, A. Ben-Tal and J. Zowe, 'Equivalent displacement based formulations for maximum strength truss topology design', IMPACT Comput. Sci. Eng., 4, 315-345 (1992).
18. B. N. Pshenichny and Yu. M. Danilin, Numerical Methods in Extremal Problems, MIR Publishers, Moscow, 1978.
19. B. T. Poljak, 'A general method of solving extremum problems', Soviet Math. Dokl. Phys., 8, 593-597 (1967).
20. B. T. Polyak, 'Minimization of unsmooth functionals', U.S.S.R. Comput. Math. Math. Phys., 9, 14-29 (1969).
21. N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer, Berlin, 1985.
22. M. A. Shepilov, 'Method of the generalized gradient for finding the absolute minimum of a convex function', Cybernetics, 12, 547-553 (1976).
23. C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, N.J., 1982.
24. A. V. Aho, J. E. Hopcroft and J. D. Ullman, Data Structures and Algorithms, Addison-Wesley, Reading, MA, 1985.
25. Y. M. Dirickx and L. P. Jennergren, System Analysis by Multilevel Methods, Wiley, Chichester, 1979.
26. G. B. Dantzig and P. Wolfe, 'Decomposition principle for linear programs', Oper. Res., 8, 101-111 (1960).
27. T. Larsson and Z.-W. Liu, 'A Lagrangian relaxation scheme for structured linear programs with application to multicommodity network flows', Optimization (to appear).
28. T. Larsson, Z.-W. Liu and M. Patriksson, 'A dual scheme for traffic assignment problems', Report LiTH-MAT-R-92-21, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden, 1992,
29. M. S, Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edn, Wiley, New York, 1993.
30. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
31. T. Larsson, M. Patriksson and A.-B. Strömberg, 'Ergodic results in subgradient optimization', in G. Di Pillo and F. Giannessi (eds.), Nonlinear Optimization and Applications, Plenum Press, New York, 1996, pp. 229-248.
32. R. T. Rockfellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.
33. M. Patriksson and J. Petersson, 'A subgradient method for contact structural optimization', in M. C. Ferris and J.-S. Pang (eds.), Complementarity and Variational Problems-State of the Art, Proceedings of the International Conference on Complementarity Problems (ICCP-95), SIAM, Philadelphia, PA (to appear).
34. T, J. R. Hughes, The Finite Element Method-Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1987.
35. A. Ben-Tal and A. Nemirovskii, 'An interior point algorithm for truss topology design', in M. P. Bendsøe and C. A. Mota Scares (eds.), Topology Design of Structures, Kluwer, Dordrecht, 1993, pp. 55-70.
36. A. Ben-Tal and A. Nemirovskii, 'Potential reduction polynomial time method for truss topology design', SIAM J. Optim., 4, 596-612 (1994).
37. E. S. Levitin and B. T. Polyak, 'Constrained minimization methods', U.S.S.R. Comput. Math. Math. Phys, 6, 1-50 (1966).
38. V. Rezek, Implementation of a subgradient algorithm for topology optimization in the FE-program TRINITAS', Report LiTH-IKP-Ex-1320, Department of Mechanical Engineering, Linköping Institute of Technology, Linköping, Sweden, 1996.
39. J. Petersson, 'Optimization of Structures in Unilateral Contact', Dissertation No. 397, Department of Mechanical Engineering, Linköping Institute of Technology, Linköping, Sweden, 1995.
40. A. Klarbring and M. Rönnqvist, 'Nested approach to structural optimization in nonsmooth mechanics', Struct. Optim., 10, 79-86 (1995).