Optimal design of planar and space structures with genetic algorithms
Introduction
This paper is concerned with a computer-based systematic approach for optimum design of planar and space structures composed of one-dimensional elements. In structural optimization it is well-known that optimality criteria (OC) techniques and mathematical programming (MP) based methods have received great interest and application during the last several decades. Recently, genetic algorithms (GA), simulated annealing and evolutionary programming have attracted attention amongst the engineering design optimization community [1]. These new approaches show certain advantages over the more classical optimization procedures [2], e.g. they can successfully be applied to a broad range of diverse problem areas. In this study design optimization approach using a GA is employed. Genetic algorithms are global search methods which have found application in a wide spectrum of problem areas, including optimum structural design [3], [4], [5], [6], [7]. Here, discrete optimal designs of bar structures including beams, plane and space trusses, plane and space frames and grids is covered using a GA based procedure. Discrete variable optimization refers to the case where the design variables are to be selected from an available list. Integer and 0–1 optimizations are easily interpreted as two special cases of the general discrete variable optimization approach. The need for more research on discrete variable optimization is pointed out in Thanedar and Vanderplaats [8], where the three principal methods for discrete variable optimization are stated as the classical branch and bound methods, mixed linearization or approximation methods and ad-hoc methods. Included in the ad-hoc methods are simulated annealing and genetic algorithms.
The integrated GA approach transforms the constrained structural design problem into an unconstrained problem through the use of penalty functions. Recent penalty functions for GA applications are discussed in Michalewicz [9]. In this paper, a slightly modified version of the Joines and Houck’s method is used. The GA is a robust optimizer. A proper treatment of GA tools [11] allows both a thorough exploration of the design space and a satisfactory exploitation of the already obtained possible solutions in the search of the optima. A global optimum is not guaranteed, although, near-optimal solutions are found easily. In the weight optimization problems presented in this paper, discrete optimization is performed by selecting discrete sections of structural elements from available profile lists. Although only size optimization of bar structures is considered, GA integrated structural optimization can conceivably handle other structural types and more complex structures. For carrying out the necessary computations, a computer program GAOS (Genetic Algorithm Based Optimum Structural Design) is introduced. The general flowchart of GAOS is given in Fig. 1. The main components of the system are: (1) structural modeling (displacement finite element structural analysis algorithm); (2) modeling for optimum design problem (objective function, design variables and constraints involved in size optimization); and (3) the optimization algorithm (GA).
The paper first discusses the basic working scheme of a GA and the GA integrated structural optimization. Then the implementation of the GAOS program is discussed in detail with respect to discrete size optimization of trusses and frames with applications to steel structures composed of ready hot-rolled sections. Six example problems are given and fully discussed.
Section snippets
Structural optimization and GA
The main three components in the operation of a GA are: (1) the creation of an initial pool of designs; (2) combination of the designs in a pool in order to produce better designs; and (3) obtaining new generations of designs. These steps are explained below in the context of structural design optimization problems.
Structural optimization problem
A general discrete-sizing structural optimization problem is posed as,
minimizesatisfying whereIn , , , , Nc and Nd represent, respectively, the number of constraints and independent design variables, is the vector of design variables of dimension , is the objective function, gj is the jth constraint on structural response and the inequalities in Eq. (3) are side constraints on the design variables. In discrete structural
Genetic algorithm based structural optimization (GAOS)
The integrated structural optimization explained above is implemented as the program GAOS. The GAOS program is conceived to handle the optimal design of structures made up of 1D elements. In its present form it handles size optimization of plane and space trusses, plane and space frames, beams and grids with given topology and geometry. The general characteristics of the program are:
- 1.
Constraint violations are taken care of by penalty functions.
- 2.
A structural analysis program is included in the
Use of GAOS
For the operation of GAOS, one has to prepare an input file where in addition to typical structural analysis input options, data related to genetic operations are also included. These are: population size, maximum number of generations, mutation probability, and crossover probability. Additionally, two more data are also considered. One is a penalty coefficient, which is needed in tackling constraint violations and the other one is to indicate the element group to which individual elements
A drawback of a GA
Optimization of a structural system comprising numerous design variables is a challenging problem due to its huge size of search space. In fact, it is not easy for any optimization algorithm to carry out an effective exploration without locating a local optimum. The dimension of the search space even grows exponentially either with the addition of extra design variables or with the enlargement of profile lists. The tests performed on various structures show that a GA is generally quite
Numerical examples
Several standard test problems are used to verify the correctness and efficiency of the GAOS program in optimization of plane and space steel structures. In all problems considered, structures are designed for the minimum weight with the cross-sectional areas of structural members being the design variables. The multilevel optimization approach discussed above is implemented in all the problems. Ten thousand structural analyses are performed in each optimization level of the problems tested
Discussion of results and conclusion
In this paper, the program GAOS was implemented for a class of typical structural design problems to confirm the ability of the GA on structural optimization. These problems, except for the statically determinate truss, have been used as test problems in the literature. Throughout the problems, the GA is set to compete with various continuous and discrete optimization algorithms. While the test problems 1, 3, 5 and 6 are in the category of discrete variable optimization, the test problems 2 and
Acknowledgements
The authors are grateful to Prof. D. E. Goldberg, for providing his Turbo Pascal SGA source code, on which the program developed in this paper was based.
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