Keywords: topology optimisation, probability, reliability, optimal design, SIMP method, stochastic.

Topology optimisation has more than one hundred years of history and is still an expanding field in optimal design. The first two milestones are the works by Michell [1] in 1904, and seventy years later in 1973, the work by Rossow and Taylor [2]. The "modern" topology optimisation of continuum type structures was first published independently by Rozvany and Zhou [3,4], and Bendsoe and Kikuchi [5] at the end of the eighties.

It can be stated that the achievements of structural optimisation and reliability based design optimisation are considerable and one can see that topology optimisation as a research field has achieved significant results, especially in the last two decades. In spite of a clear concept, the solution of topology optimisation problems poses significant technical challenges. Problems are typically large-scale and discrete, and often exhibit some numerical difficulty associated with underlying mechanics (such as instability of members, checkerboard patterns in continua). For these reasons, the majority of topology optimisation research has focused on deterministic design problems, neglecting the uncertainty that arises in most engineering applications. Until the end of the last century one could hardly find any publications on topology optimisation that considered uncertainties. Fortunately, this trend has changed and a great number of works have been published over the last twelve years. Generally the publications are classified according to the object of the topology optimisation (trusses, continuum type structures, grids, plates, etc.), the type of numerical procedures (e.g. homogenisation method, the solid isotropic material with penalisation (SIMP) approach, the evolutionary structural optimisation (ESO) method, the level set-based topology optimisation method, meta-heuristic methods, etc.), and the formation of the uncertainties (loading, geometry, stiffness, production tolerance). The aspect common to many of these different approaches is the need to reformulate the uncertainties and create an alternate deterministic formulation. The uncertain objective function, or the uncertain constraint, can be replaced by statistical averages. An alternative is to minimise the influence of stochastic variability on the mean design by including higher order statistics such as variance. These approaches are commonly referred to as robust design optimisation (RDO). The description of the reliable and robust tool for structural shape optimisation is the object of the paper by Sienz and Hinton [6] in 1997. Reliability based design optimisation (RBDO) seeks to constrain or minimise (or maximise) a measure of the probability of failure, such as the reliability index. Other methods also use the upper bound theorems of the stochastic optimisation to provide an equivalent deterministic expression which bounds the probabilistic ones.

After a relatively detailed review of the probability based topology optimisation achievements, three types of stochastic topology optimisation developments are discussed [7-9]. Finally, selected numerical examples close this chapter.

[1]

A.G.M. Michell, "The Limits of Economy of Material in Frames Structures", Philosophical Magazine, 8, 589-597, 1904. doi:10.1080/14786440409463229

[2]

M.P. Rossow, J.E. Taylor, "A Finite Element Method for the Optimal Design of Variable Thickness Sheets", J. AIAA, 11, 1566-1569, 1973. doi:10.2514/3.50631

[3]

M. Zhou, G.I.N. Rozvany, "The COC algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization", Comp. Meth. Appl. Mech. Eng., 89, 309-336, 1991. doi:10.1016/0045-7825(91)90046-9

[4]

M. Zhou, G.I.N. Rozvany, "DCOC: An Optimality Criterion Method for Large Systems. Part I: Theory. Part II: Algorithm", Structural Optimization, 5(1-2), 12-25, 1992. doi:10.1007/BF01744690 & 6(4), 250-262, 1993. doi:10.1007/BF01743384

[51]

M.P. Bendsoe, N. Kikuchi, "Generating Optimal Topologies in Structural Design Using a Homogenization Method", Comp. Meth. Appl. Mech. Eng., 71, 197-224, 1988. doi:10.1016/0045-7825(88)90086-2

[6]

J. Sienz, E. Hinton, "Reliable structural optimization with error estimation, adaptivity and robust sensitivity analysis", Computers and Structures, 64(1-4), 31-63, 1997. doi:10.1016/S0045-7949(96)00170-8

[7]

J. Lógó "New Type of Optimality Criteria Method in Case of Probabilistic Loading Conditions", Mechanics Based Design of Structures and Machines, 35(2), 147-162, 2007. doi:10.1080/15397730701243066

[8]

J. Lógó, M. Ghaemi and A. Vásárhelyi "Stochastic compliance constrained topology optimization based on optimality criteria method", Periodica Polytechnica-Civil Engineering, 51(2), 5-10, 2007. doi:10.3311/pp.ci.2007-2.02

[9]

J. Lógó, M. Ghaemi and M. Movahedi Rad "Optimal topologies in case of probabilistic loading: The influence of load correlation", Mechanics Based Design of Structures and Machines, 37(3), 327-348, 2009. doi:10.1080/15397730902936328

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