Fernando Bastos *
Adelaide Cerveira †
Joaquim Gromicho ‡
† Departamento de Matemática, UTAD, Vila Real, Portugal
‡ Vrije Universiteit, Amsterdam & ORTEC International, Gouda, The Netherlands
]]> jgromicho@ortec.nl
Abstract:
The design of truss structures is an important engineering activity which has traditionally been done without optimization support. Nowadays we witness an increasing concern for efficiency and therefore engineers seek aid on Mathematical Programming to optimize a design. In this article, we consider a mathematical model where we maximize the stiffness with a volume constraint and bounds in the cross sectional area of the bars, [2]. The basic model is a large-scale non-convex constrained optimization problem but two equivalent problems are considered. One of them is a minimization of a convex non-smooth function in several variables (much less than in the basic model), being only one non-negative. The other is a semidefinite programming problem. We solve some instances using both alternatives and we present and compare the results.
Keywords: truss topology design, stiffness, non-smooth convex programming, descent method, semidefinite programming, duality, interior point methods
Texto completo apenas disponível em PDF.
Full text only in PDF.
]]> Bibliography
1 M. S. BAZARAA, H. D. SHERALI, AND C. M. SHETTY, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, 1993. [ Links ]
2 A. BEN-TAL AND M. P. BENDSØE, A new method for optimal truss topology design, SIAM Journal Optimization, 3 (1993), pp. 322-358. [ Links ]
3 A. BEN-TAL AND A. NEMIROVSKI, Potential reduction polynomial time method for truss topology design, SIAM Journal Optimization, 4 (1994), pp. 596-612. [ Links ]
4 A. BEN-TAL AND A. NEMIROVSKI, Optimal design of engineering structures, Optima, 47 (1995), pp. 4-8. [ Links ]
5 B. BORCHERS, CSDP, 3.2 User's Guide, Optimization Methods and Software, 11 (1999), pp. 597-611 [ Links ]
6 B. BORCHERS, CSDP, A C library for Semidefinite Programming, Optimization Methods and Software, 11 (1999), pp. 613-623. [ Links ]
7 C. M. BRANCO, Mecânica dos Materiais, Fundação Calouste Gulbenkian, Lisboa, 1994. [ Links ]
8 M. A. C. CERVEIRA, Optimização do desenho de estruturas, master's thesis, Universidade de Lisboa, Portugal, 1997. [ Links ]
9 E. W. CHENEY AND A. A. GOLDSTEIN, A Newton's method for convex programming and Tchebycheff approximation, Numeric Mathematics, 1 (1959), pp. 253-268. [ Links ]
10 E. DE KLERK, C. ROOS, AND T. TERLAKY, Semi-definite problems in truss topology optimization, Tech. Report Nr. 95-128, Faculty of Technical Mathematics and Informatics, Delft University of Technology, November 1995. [ Links ]
11 C. HELMBERG, Semidefinite programming for combinatorial optimization, tech. report, Konrad-Zuse-Zentrum für Informationstecghnik Berlin, 2000. [ Links ]
12 J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algoritms I: Fundamentals, vol. 305 of A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1993. [ Links ]
13 J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algoritms II: Advanced Theory and Bundle Methods, vol. 305 of A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1993. [ Links ]
14 R. A. HORN AND C. R. JONHSON, Matrix Analysis, Cambridge University Press, Cambridge, 1985. [ Links ]
15 R. A. HORN AND C. R. JONHSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. [ Links ]
16 J. E. KELLEY, The cutting plane method for solving convex problems, Journal of the Society for the Industrial and Applied Mathematics, 8 (1960), pp. 703-712. [ Links ]
17 U. KIRSCH, Optimum Structural Design: Concepts, Methods and Applications, McGraw-Hill, New York, 1981. [ Links ]
18 D. G. LUENBERGER, Linear and Nonlinear Programming, Addison-Wesley, Reading Massachusetts, 1984. [ Links ]
19 J. M. MULVEY, R. J. VANDERBEI, AND S. A. ZENIOS, Robust optimization of large-scale systems, Operations Research, 43 (1995), pp. 264-281. [ Links ]
20 M. PATRIKSSON AND J. PETERSSON, A subgradient method for contact structural optimization, LiTH-MAT-R-1995-25, (1995). [ Links ]
21 R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. [ Links ]
22 N. Z. SHOR, Convergence rate of the gradient descent method with dilation of the space, Cambridge, 6 (1970), pp. 102-108. [ Links ]
23 N. Z. SHOR, Utilization of the operation of space dilation in the minimization of convex functions, Cambridge, 6 (1970), pp. 7-15. [ Links ]
24 N. Z. SHOR, Cut-off method with space extension in convex programming problems, Cambridge, 13 (1977), pp. 94-96. [ Links ]
25 N. Z. SHOR, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1985. [ Links ]
26 G. N. VANDERPLAATS, Numerical Optimization Techniques for Engineering Design: With Applications, Series in Mechanical Engineering, McGraw-Hill, New York, 1984. [ Links ]
]]>