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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 210

Topology Optimization Using Probabilistic Compliance Constraints

J. Lógó1, S. Kaliszky12 and M. Ghaemi1

1Department of Structural Mechanics, Budapest University of Technology and Economics, Hungary
2Research Group of the Computational Structural Mechanics, Hungarian Academy of Science, Budapest, Hungary

Full Bibliographic Reference for this paper
J. Lógó, S. Kaliszky, M. Ghaemi, "Topology Optimization Using Probabilistic Compliance Constraints", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 210, 2006. doi:10.4203/ccp.83.210
Keywords: topology optimization, stochastic programming, optimality criteria, compliance, optimal design, minimum volume design.

Recently topology optimization has become one of the most "popular" topics in the expanding field of optimal design but the majority of the papers deal with deterministic problems. The reason for the introduction stochastic programming theory, (more generally probabilistic notation) is to attempt to consider in a more rational way the fact that the precise strength of a structure is not known, among the constraints there are probabilistic inequalities and perhaps even more importantly, that the loadings applied to the structure are not known with any degree of precision. There is an extensive and expanding literature in this area. Marti made significant achievements in this expanding field [1,2]. Melchers used a significant simplification in his optimality criteria based reliability design [3]. Recently Kharmanda et al. [4] have integrated the reliability analysis into a deterministic topology optimization problem by the introduction of the reliability constraint into the standard SIMP procedure. In the field of topology design the stochastic mathematical programming theory is missing.

The aim of this research is to introduce a new type of probability based topology design procedure and to compare the obtained results with optimal topologies calculated on deterministic way. The paper is divided into three parts: the first part deals with the deterministic topology optimization briefly, the second part presents the probabilistic based design and the third part compares the topologies obtained by the use of the stochastic and deterministic approaches.

Introducing the deterministic problem an iterative technique (SIMP) and the connected numerical examples will be briefly discussed briefly. The object of the design (ground structure) is a rectangular disk with a given loading and support conditions. The material is linearly elastic. The design variables are the thicknesses of the finite elements. To obtain the correct optimal topology some filtering method (the ground elements are subdivided into further elements) has to be applied to avoid the so-called "checker-board pattern" [5]. The optimization problem is to minimize the penalized weight of the structure subjected to a given compliance and side constraints.

In the proposed probabilistic topology optimization method: the minimum penalized weight design of the structure is subjected to compliance constraint which has uncertainties and side constraints. If the compliance value is given by the standard normal distribution function, the mean value and variance, then the deterministic topology problem is modified and the compliance constraint is substituted by a probability constraint. By the use of the recommendation of Prekopa [6] this probabilistic expression can be used as a constraint for the original problem. By the use of the first order optimality criteria a redesign formula of the stochastically constrained topology optimization problem can be derived.

The new classes of optimal topologies with their analytical and numerical confirmation are presented. A standard finite element computer program with quadrilateral membrane elements is applied in the numerical calculation. Through the numerical examples the paper compares the deterministic optimal topologies and optimal topologies obtained in case of uncertain situations.

K. Marti "Stochastic Optimization in Structural Mechanics", ZAMM 70, T742-T745, 1990.
K. Marti "Stochastic Optimization in Structural Mechanics", ZAMM 12, 795-811, 2003. doi:10.1002/zamm.200310072
R.E. Melchers "Optimality-critera-based Probabilistic Structural Design", Structural and Multidisplinary Optimization, 23, 34-39, 2001. doi:10.1007/s00158-001-0163-1
G. Kharmada, N. Olhoff, A. Mohamed, M. Lemaire "Reliability-based Topology Optimization", Structural and Multidisplinary Optimization, 26, 295-307, 2004. doi:10.1007/s00158-003-0322-7
J. Lógó "New Type of Optimal Topologies by Iterative Method", Mechanics Based Design of Structures and Machines, 33(2), 149-172, 2005. doi:10.1081/SME-200067035
A. Prékopa "Stochastic Programming", Akadz'emia Kiadz'o and Kluwer, Budapest, Dordrecht, 1995.

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