Layout optimization of trusses using improved GA ... - Springer Link

Acta Mechanica 146, 87 107 (2001)

ACTA MECHANICA 9 Springer-Verlag 2001

Layout optimization of trusses using improved GA methodologies O. Hasangebi and F. Erbatur, Ankara, Turkey (Received May 31, 1999; revised September 7, 1999)

Summary. The main concern of the paper is the simultaneous treatment of size, shape and topology variables in the optimum design of space trusses. As compared to only size optimization, this is a challenging, more difficult and complex problem. The paper discusses a solution algorithm which is based on the use of GAs. Two new methodologies, "annealing perturbation" and "adaptive reduction of the design space", are introduced in conjunction with GAs, bringing additional increase in computational efficiency. Some common problems in handling shape and topology design considerations are eliminated, which in turn provides a large and a flexible design environment. A numerical problem is presented to test the performance of the proposed methodologies and to compare the results with those existing in the literature. Furthermore, the paper studies a second problem designed to observe the efficiency of GAs in a considerably large and complex design space.

1 Introduction One of the interesting and challenging areas of structural optimization emerges to be optimum design with respect to size, shape and topology considerations. Obviously, in most cases such a design and optimization problem brings additional complexity which the traditional techniques may fail to treat. The complexity of the problem is mainly due to the simultaneous consideration of variables of different natures. Additionally, especially with the inclusion of topology variables the design space is exposed to discontinuity and singularity. Previous studies utilizing well known classical techniques are summarized in some good review papers available in the literature [1], [2]. Recently, genetic algorithms (GAs) along with other evolutionary algorithms, such as evolutionary strategies (ESs) and evolutionary programming (EP), proved to be robust and assertive techniques, yielding practical and improved solutions to problems difficult to be solved with traditional techniques [3]- [6]. The efficiency of GAs was tested by a few researchers on either size and shape optimization with a fixed topology [7], [8], or size and topology optimization with fixed shape approaches [9], [10]. A simultaneous size, shape and topology optimization with GAs was first attempted by Grierson and Pack [11]. A frame structure was considered as the test problem, and its weight is minimized under the imposed displacement constraints. Later, Rajan [12] addressed the simultaneous size, shape and topology optimization of truss structures under stress and displacement constraints. In this study, not only the member connectivities but also the support conditions are taken as topology design variables. Rajeev and Krishnamoorty [13] performed a similar study, introducing two new ideas. The first one is a two-phase method integrated to increase the computational efficiency, and the

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O. Hasanqebi and F. Erbatur

second one is the implementation of a variable length GA (VGA) in conjunction with topology optimization. A recent study on the same issue is by Shrestha and Ghaboussi [14]. The study proposes a new string representation sctaeme along with a physical design space concept through which the evolved structures are free to assume any shape and topology variation. This paper discusses a solution algorithm, which is automated to concurrently handle the size, shape and topology optimum design of 2-D and 3-D truss structures. It is compiled in Borland Delphi source code, and has no limitations in the number of nodes, members and loading conditions. It uses a high level programming coding scheme, through which the common problems encountered in previous applications are easily avoided, bringing a wide, flexible and practical design environment to the problems considered. The present study also proposes two methodologies in conjunction with GAs, which bring an additional increase in the efficiency of the numerical solution. The first methodology brings the use of annealing perturbation in a GA to display a more exploitative search. In the second methodology, at a certain stage of the process the design space is reduced adaptively to condense on a smaller sub-space after a general search is adequately achieved. The standard test problem, a 47-bar truss tower (Fig. 1), taken from the literature, is used to compare the current study with the previous studies, and also to test the performance of the proposed methodologies. Additionally, a second problem, a 224-bar space truss pyramid structure (Fig. 4), is designed and solved to test the ability of GAs in large and complex structural problems.

2 Design problem model 2.1 Problem formulation A general size, shape and topology problem in the context of structural weight optimization is stated as follows. Find a vector of cross-sectional areas, AN, xl = [A~I ATe1 = [A1,A2,...,A~,...,AN~] C ( S u e )

i = 1,...,N~

(1)

and a matrix of nodal point coordinates, PNpxd = [Pj,~], in the d-dimensional Cartesian coordinate system

Pg~d =

Pj,~ ,1

j = 1,...,Np

and

k= 1,...,d

(2)

PNp,d

such that the weight of a given structure, W(A~, Pj,~), is minimized, Ne W(A~,Pj,k) = E p~AiL~

where

L~

=

Li(Pj,~),

(3)

i=1

controlled by a set of imposed limitations on stress, stability and displacement:

gin(Ai,Pj,k) = I~?l- I(~d~zzl- C~ (i) will be observed, respectively. Such an adaptation strategy allows to satisfy the tendency of a joint to move in a certain direction, and thus better design solutions are searched in a region which is not previously covered. The proposed methodology also suggests a reduction scheme in the topology design set. The idea is to limit the inclusion of unstable topologies to the designs in the following generations. In this way, the search is restricted to those parts of the design space which are dominated by stable topologies. This is achieved by recognizing the essential members of a given structure, the absence of which may lead to an unstable topology. A maximum of ten of the best feasible designs obtained so far is used for this process. If a member appears to be present in all the best ten designs, it is considered as an essential member, and is not allowed to vanish in the coming generations. In this way, the design space is sufficiently reduced by directing the search towards seeking for the absence/presence of non-essential members. Besides, since all further designs necessarily include the essential members, a higher stability is observed in the following generations. At this point adaptation of the population to the reduced design space is needed. Otherwise, individuals in the current population would point out some haphazard points in the reduced design space due to the new definition of the bounds for design variables. Note that Eqs. (7) and (8) yield different values for the same string when different bounds are imposed for the variables. Such a case leads to a loss of design information obtained during the former generations as observed in a restart process. Therefore, the next population has to be created in such a way that the previous search experience will be retained, and at the same time new design points will be incorporated into the population for a thorough search. For this, firstly the adaptation of the best design is accomplished. This is done by altering its binary string representation in such a way that it maps the same solution in the new design space definition. Later, the next population (the first population of the new design space) is created. For each individual of the current population, a random number (r) is created between 0 and 1, and the following criteria are used to initiate the next population: if r >_ 0.60, the current individual (Jc) is replaced with a neighbour of the best design (db) obtained by annealing perturbation; if 0.10 < r < 0.60, it is directly copied into the next population, and otherwise (if r _< 0.10) a new individual (J~) is randomly created and replaced with the current individual in the next population. Accordingly, on average the proportions of Jb, Jc and d~ to appear in the next population happens to be 40%, 50% and 10%, respectively. One point deserves to be mentioned here. At a first glance, it might be thought that the population would be dominated with the neighbours of the best designs in the remaining generations. However, this is not observed. A meaningful explanation lies in the following two considerations: (i) since the design space is adequately confined to a region favoured by the best design, the other individuals also have acceptably good fitness scores; (ii) the use of annealing perturbation in every child design contributes a judicious genetic diversity.

5 Numerical examples The efficiency of the solution algorithm is investigated through two test problems. In both problems, the structures are designed for minimum weight with respect to size, shape and/or topology considerations, and constraints are defined to cover stress, stability and displacement limitations. The first problem, a 47-bar truss (Fig. 1), is chosen from the literature to

Layout optimization of trusses

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allow possible c o m p a r i s o n with previous continuous and discrete methods. The second problem, a 224-bar space truss (Fig. 4), on the other hand, is newly designed a n d introduced to observe the characteristics o f G A s in large and complex structural systems. The following control p a r a m e t e r s are selected in b o t h problems: p o p u l a t i o n size = 100, generation number = 1000, m u t a t i o n p r o b a b i l i t y = 0.005, crossover probability = 0.90. The c o m p u t a t i o n times for the first and second test p r o b l e m are 12 and 54 seconds per generation with a C P U 300 micro computer, respectively. A full list o f design data, final designs a n d comparison with other methods are presented for each problem.

5.! 47-bar truss tower The first p r o b l e m is a 47-bar truss tower shown in Fig. 1, which is often used as a test p r o b l e m in structural size and shape optimization literature [22], [23]. The truss has 47 members and 23 nodes, and is subjected to three different loading conditions. The u p p e r limits o f stresses are chosen as 20 ksi and 15 ksi for members in tension and compression, respectively. Euler buckling constraint is considered on compression members, and no displacement constraint is imposed. Symmetry of the truss is used to link the members, and thus the n u m b e r of independent size variables is reduced to 27. The position of nodes 15, 16, 17 and 22 remains unchanged, the nodes 1 and 2 are allowed to move only in the x direction, and the other Table 1. Design data for a 47-bar truss tower Constraint data Stress constraints

(~rt)~: _< 20ksi, i = 1,...,47 (~ro)i_