Keywords: probabilistic algorithms, laminate topology, shape optimization, fuzzy sets.

Recently, in order to use anisotropic properties of composite materials in an appropriate manner a lot of efforts have been put into introduction and application of effective optimization algorithms in the design of 2-D laminated plated and shell structures. The second still open problem in this area is associated with the effective definition of design variables describing the analyzed structures. Both problems are discussed in the present problems and various numerical examples are demonstrated. They are connected with the material, shape and topology optimization.

In the present work the topology optimization is understood in the sense of the stacking sequence optimization only. Shape optimization is understood in the sense of variations of boundary contours of meshes meaning also the structural boundaries in the broad sense. The material optimization is understood in the sense of the elimination from an initial structure parts in order to minimize the total volume and satisfy the inequality constraints in the form of allowable displacements and/or allowable stresses in the structures. The fundamental concept is based on the discrete procedure that generates and controls the elimination process. Two methods have been proposed and successfully adopted:

• A random generation of the base points in the initial FE mesh; then they are randomly joined by finite elements to crate an element of a population,
• An elimination of areas with the use of spline method – it is an extension of the first methods in order to accelerate and improve the computational effectiveness of the numerical procedures.

After generation of a new FE mesh the distributions of displacements and stresses are evaluated with the use of the FE NISA II code to check the constraint conditions.

The topology optimization is a classical one and is associated with the appropriate choice of discretised fibre orientations in order to maximize the required function. In our problem we intend to maximize the allowable strains in a rectangular plate loaded by an external uniform pressure and compressive forces in both directions. However, in order to take into account uncertainties in the experimental evaluation of allowable strains they are treated as fuzzy numbers and described with the use of assumed membership functions. Thus the discussed problem belongs to the class of fuzzy optimization problems. The optimal configurations are determined for different values of the a-cuts. The optimum design in a fuzzy environment is a new approach although some attempts have been made in this area starting from the pioneering works of Zimmermann [1].

For both optimization problems the optimization strategy is based on the use of probabilistic algorithms, i.e. variants of genetic algorithms and simulated annealing algorithm. The effectiveness and convergence of optimization procedures are demonstrated and discussed herein.

The present work is an extension of the methods introduced in Refs [2,3]. However, now, the optimization methodology is used to completely new numerical examples dealing with stacking sequence optimization of cylindrical shells and reinforcement of the junction of cylindrical shells.

1

Zimmermann H.J. (1976), "Description and optimization of fuzzy systems", International Journal of General Systems, 2, 1976, pp. 209-215. doi:10.1080/03081077508960870

2

Gurba W., Probabilistic Methods of Composite Structures Optimization, Ph.D. Thesis, Kracow University of Technology, Krakow 2001 (in Polish).

3

Muc A., "Transverse Shear Effects in Discrete Optimization of Laminated Compressed Cylindrical Shells", Composite Structures, 1997, 38, 489-497. doi:10.1016/S0263-8223(97)00084-6

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