Keywords: evolutionary algorithms, optimization, finite element method, plane stress, bending plates, shells, three-dimensional structures.

Shape and topology optimization has been an active research area for some time. Recently, several innovative approaches for topology optimization have been developed. The present work is based on the application of the evolutionary algorithm and the finite element method to the optimization problems of two and three-dimensional structures. The optimization method concerns the simultaneous optimization of topology, shape, and material or thickness for two-dimensional structures. This paper is an extension of previous work elaborated by Burczynski, Poteralski and Szczepanik concerning such optimization problems [1]. Recently, evolutionary methods have found various applications in mechanics, especially in structural optimization [2].

The main feature of those methods is to simulate biological processes based on heredity principles (genetics) and the natural selection (the theory of evolution) to create optimal individuals (solutions) presented by single chromosomes. The main advantage of the evolutionary algorithm is the fact that this approach does not need any information about the gradient of the fitness function and ensures a strong probability of finding the global optimum. The fitness function is calculated for each chromosome in each generation by solving a boundary-value problem by means of the finite element method (FEM). In order to solve the optimization problem the fitness function, design variables and constraints should be formulated.

Parameterization is the key stage in the structural optimization. The great number of design variables results in the optimization process being ineffective. A connection between design variables (genes) and number of finite element leads to poor results. The better results can be obtained when the surface (or hyper surface) of mass density distribution (or thickness) is interpolated by suitable number of values given in control points. This number should provide the good interpolation and the number of design variables should be small.

Two different additional procedures are introduced:

• the additional procedure of topology optimization,
• the smoothing procedure.

An additional procedure of topology optimization is based on material elimination in the regions with low stresses. The final structure obtained after the optimization process has a rough boundary. In order to smooth this boundary, a special smoothing procedure is proposed.

Several numerical examples of evolutionary optimization of topology of two and three-dimensional structures for the various criteria are presented.

1

Burczynski T., Poteralski A., Szczepanik M., "Genetic generation of 2-D and 3-D structures", In: Computational Fluid and Solid Mechanics 2003, ed. K.J. Bathe, Vol. 2, pp. 2221-2225, Elsevier, 2003. doi:10.1016/B978-008044046-0.50545-5

2

Burczynski T., Osyczka A., (eds), Evolutionary Methods in Mechanics, Kluwer, Dordrecht, 2003.

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