Abstract
Reliability-based structural optimization methods use mostly the following basic design criteria: I) Minimum weight (volume or costs) and II) high strength of the structure. Since several parameters of the structure, e.g. material parameters, loads, manufacturing errors, are not given, fixed quantities, but random variables having a certain probability distribution P,stochastic optimization problems result from criteria (I), (II), which can be represented by
Here,f=f(ω,x) is a function on ℛr depending on a random element ω, “E” denotes the expectation operator andD is a given closed, convex subset of ℛr. Stochastic approximation methods are considered for solving (1), where gradient estimators are obtained by means of the response surface methodology (RSM). Moreover, improvements of the RSM-gradient estimator by using “intermediate” or “intervening” variables are examined.
Similar content being viewed by others
References
Barthelemy JF, Haftka R (1993) Recent advances in approximation concepts for optimum structural design. In: Rozvany G (ed.) Optimization of large structural design, Kluwer Acad. Publ., Dordrecht
Biles WE, Swain JJ (1979) Mathematical programming and the optimization of computer simulations. Mathematical Programming Study 11:189–207
Bouleau N et al. (1986) Une méthode numérique adaptée au calcul probabiliste des structures. J. Theoretical and Applied Mechanics 5 (5):781–801
Box GE, Draper NR (1987) Empirical model-building and response surfaces. J. Wiley, New York-Chichester-Brisbane-Toronto-Singapore
Bucher CG, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Structural Safety 7:57–66
Cornell CA (1967) Bounds on the reliability of structural systems. J. Struct. Div., ASCE 93:171–200
Ditlevsen O (1979) Narrow reliability bounds for structural systems. J. Struct. Mech. 7:453–472
Duysinx P, Zhang WH, Fleury C (1995) A new separable approximation scheme for topological problems and optimization problems characterized by a large number of design variables. In: Olhoff N, Rozvany GIN (eds.) WCSMO-1, Structural and multidisciplinary optimization, Pergamon, Elsevier Science, Oxford-New York-Tokyo, pp. 1–8
Ermoliev Yu (1988) Stochastic quasigradient methods. In: Ermoliev Yu, Wets J-B (eds.) Numerical techniques for stochastic optimization, Springer-Verlag, Berlin-Heidelberg-New York, pp. 141–185
Eschenauer HA et al. (1991) Engineering optimization in design processes. Lecture Notes in Engineering, Vol. 63, Springer-Verlag, Berlin
Fleury C (1989) First and second order convex approximation strategies in structural optimization. Structural Optimization 1:3–10
Fleury C, Braibant V (1986) Structural optimization: A new dual method using mixed variables. Int. J. Num. Meth. Eng. 23:409–428
Fleury C (1989) Efficient approximation concepts using second order information. Int. J. Num. Meth. Eng. 28:2041–2058
Galambos J (1977) Bonferroni inequalities. The Annals of Probability 5:577–581
Jacobsen Sh H, Schruben Lee W (1989) Techniques for simulation response optimization. Operations Research Letters 8:1–9
Kounias EG (1968) Bounds for the probability of a union, with applications. The Annals of Math. Stat. 39:2154–2158
Marti K (1990) Stochastic optimization methods in structural mechanics. ZAMM 70:T742-T745
Marti K (1992) Semi-stochastic approximation by the response surface methodology (RSM). Optimization 25:209–230
Marti K (1992) Stochastic optimization in structural design. ZAMM 72:T452-T464
Marti K (1996) Stochastic optimization methods in engineering. In: Dolezal J, Fidler J (eds.) System modelling and optimization, Chapman and Hall, London, pp. 75–87
Marti K (1997) Approximation and derivatives of probabilities of survival in structural analysis and design. Structural Optimization 13(4):230–243
Myers RH (1971) Response surface methodology. Allyn and Bacon, Boston
Nguyen VH, Strodiot JJ, Fleury C (1987) A mathematical convergence analysis of the convex linearization method for engineering design problems. Engineering Optim. 11:195–216
Prased B (1983) Explizit constraint approximation forms in structural optimization. Part 1: analysis and projections. Computer Methods in Appl. Mech. and Eng. 40:1–26
Prasad B (1984) Novel concepts for constraint treatments and approximations in efficient structural synthesis. AIAA Journal 22 (7):957–966
Rackwitz R, Thoft-Christensen P (eds.) (1992) Reliability and optimization of structural systems '91. Springer-Verlag, Berlin-Heidelberg-New York
Reinhart J (1997) Stochastische Optimierung von Faserkunststoffverbundplatten; Adaptive stochastische Approximationsverfahren auf der Basis der Response-Surface-Methode. VDI-Verlag, Düsseldorf
Schittkowski K, Zillober C (1995) Sequential convex programming methods. In: Marti K, Kall P (eds.) Stochastic programming: Numerical techniques and engineering applications, LNEMS Vol. 423, Springer-Verlag, Berlin-Heidelberg-New York-London, pp. 123–141
Schmit LA, Miura H (1976) Approximation concepts for efficient structural synhesis. NASA CR-2552
Schmit LA, Farshi B (1974) Some approximation concepts for structural synthesis. AIAA J. 12 (5):692–699
Schoofs AJG (1987) Experimental design and structural optimization. Dissertation, TU Eindhoven, Netherlands
Svanberg K (1987) The method of moving asymptotes — a new method for structural optimization Int. J. for Num. Math. In Eng. 24:359–373
Svanberg K (1993) The method of moving asymptotes (MMA) with some extensions. In: Rozvany G (ed.) Optimization of large structural systems, Vol. 1, Kluwer Acad. Publ., Dordrecht, pp. 555–566
Svanberg K (1995) A globally convergent version of MMA without linesearch. In: Olhoff N, Rozvany GIN (eds.) WCSMO-1, Structural and multidisciplinary optimization, Pergamon, Elsevier Science, Oxford-New York-Tokyo, pp. 9–16
Vanderplaats GN, Thomas HL (1991) The state of the art of approximation concepts in structural optimization. In: Rozvany G (ed.) Optimization of large structural systems, Berchtesgaden, Sept. 23–Oct. 4, 1991, Essen University, Essen, Vol. 1, pp. 109–121
Wasan MT (1969) Stochastic approximation. Cambridge University Press, Cambridge
Zhang WH, Fleury C (1994) Recent advancs in convex approximation methods for structural Optimization. In: Toppings BHV, Papadrakakis M (eds.) Advances in structural optimization. Proceed. 2nd Int. Conf. on Comput. Structures Techn., Athens, Aug. 30–Sept. 1, 1994, CIVIL- COMP Press, Edinburgh, pp. 83–90
Zhang WH, Fleury C, Duysinx P (1995) A generalized method of moving asymptotes (GMMA) including equality constraints. In: Olhoff N, Rozvany GIN (eds.) WCSMO-1, Structural and multidisciplinary optimization, Pergamon, Elsevier Science, Oxford-New York-Tokyo, pp. 53–58
Zillober C (1993) A globally convergent version of the method of moving asymptotes. Structural optimization 6:166–174
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Marti, K. Solving stochastic structural optimization problems by RSM-based stochastic approximation methods — gradient estimation in case of intermediate variables. Mathematical Methods of Operations Research 46, 409–434 (1997). https://doi.org/10.1007/BF01194863
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01194863