Greece, Abstract

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1 Optimum design of deployable structures using genetic algorithms CJ. Gantes, P.O. Georgiou and V.K. Koumousis Department of Civil Engineering, National Technical Greece, Abstract The prefabricated deployable space frames investigated in this work are "smart" structures in the sense that they have the ability to adapt to changing needs by using the, usually destructive, snap-through phenomenon as a form of prestressing. Their design is very challenging due to the need to balance the conflicting requirements of desired flexibility during deployment and desired stiffness under service loads. This paper describes a first effort to escape from a heuristically driven, manual design by employing optimization via a genetic algorithm. 1 Introduction Deployable structures are prefabricated space frames consisting of straight bars linked together in the factory as a compact bundle, which can then be unfolded into large-span, load bearing structural shapes by simple articulation. Because of this feature they offer significant advantages in comparison to conventional, non-deployable structures for a wide spectrum of applications ranging from temporary structures to the aerospace industry, and they can be classified as "smart" structures due to their capability of transforming and adapting to changing needs. A fundamental design requirement is that these structures are self standing and stress free when fully closed or fully deployed. However, at intermediate geometric configurations during the deployment process

2 incompatibilities between the member lengths lead to the occurrence or second order strains and stresses resulting in a snap-through phenomenon that "locks" the structure in its deployed configuration. The structural response during deployment is, hence, characterized by geometric nonlinearities, and simulation of the deployment process is, therefore, a very important problem requiring sophisticated finite element modeling. The material behavior, on the other hand, must remain linearly elastic, so that no residual stresses reduce the load bearing capacity under service loads. Geometric design is performed according to a set of geometric constraints resulting from the requirement of zero stresses at the two extreme configurations. From a structural point of view, deployable structures have to be designed for two completely different loading conditions, under service loads in the deployed configuration, and during deployment. The structural design process is very complicated and requires successive iterations to achieve some balance between desired flexibility during deployment and desired stiffness in the deployed configuration. In this work genetic algorithms are employed to optimally assign the member properties of the two groups of elements that comprise the structure, assuming that geometric design has already been performed. The objective is to find optimal compromizing solutions that minimize the deployment effort and maximize the load carrying capacity of the deployed structure. For a set of nodes located at appropriate positions two distinct problems are solved. The one is a geometrically nonlinear problem that simulates the deployment action. This corresponds to a specific set of monotonically increasing imposed displacements. From this solution, the maximum stresses are traced in all steps. Then, by changing the boundary conditions and the set of loads, linear analyses of the structure are performed from which member stresses and nodal displacements are deduced. Based on the results of both analyses an augmented fitness function is formulated which corresponds to the multi-objective criterion as the weighted sum of the competing criteria and penalty terms that encompass the stress and displacement constraints of both solutions. Complete binary coding of the design variables is used to cover a wide range of properties. 2 Basic features of deployable structures The deployable, prefabricated space frames investigated here consist of socalled scissor-like-elements (SLEs), pairs of bars connected to each other at intermediate points with a pivotal connection and hinged at their end 256

3 /. ocvcicu arc to each other to form units with plan view of regular polygons which, in turn, are assembled into larger structures, either flat or curved (Figure 2). Each side of the polygons is a symmetrical SLE about an axis going though the pivotal connection, and it is called "outer" SLE. Each halfdiagonal of the polygons is a non-symmetrical SLE, consisting of bars of unequal length, and it is called "inner" SLE If lines connecting corresponding upper and lower nodes are parallel to each other, the units are called "flat", and are used to assemble flat deployable structures (Figure 2a). If these lines go through a pole, the units are called "curved", and are used to assemble spherical deployable structures (Figure 2b). Pivot Figure 1: A Typical Scissor-Like-Element (SLE) (a) Flat Structure (b) Curved Structure Figure 2: Deployable Structures in their Deployed Configurations Most deployable structures behave as mechanisms during deployment, and need, accordingly, external stabilization in order to develop a load bearing capacity in their deployed configuration^. This is undesirable because it increases the time and effort needed for erection and dismantling. To overcome this disadvantage geometric incompatibilities have to be introduced during deployment which cause the development of strains and stresses, culminating in a "snap-through" phenomenon, that "locks" the structure in its deployed configuration^, so that external 257

4 stabilization is no longer necessary. It, however, some ot these incompatibilities are still maintained in the open form, some members remain bent and have residual stresses. As a result, these members are more susceptible to buckling, leading to a reduction of the overall load bearing capacity of the complete structure*'*. It is possible to design deployable structures avoiding both those drawbacks^. A fundamental design requirement of such an improved structure is that it is self-standing and stress-free when fully closed or fully deployed. However, at intermediate stages during the deployment process geometric incompatibilities between the member lengths do exist, that also lead to the occurrence of strains and stresses which result in a snapthrough phenomenon that "locks" the structure in its deployed configuration. In order to achieve this goal a careful geometric design is required. From a structural point of view, two design situations have to be considered, one for the final configuration under external loads, and one during deployment. The structural behavior in the first situation is linear. Response during the second situation, the deployment process, is geometrically nonlinear, therefore, numerical modeling is needed, by means of a large displacement-small strain finite element formulation, in our case implemented in MSC-Nastran^. A displacement controlled Newton Raphson method or an automatic load incrementation algorithm is used to trace the load-displacement curve beyond the limit point. The effects of discrete joint sizes have been included in the numerical model. Computational savings for the deployment analysis are achieved by taking advantage of symmetry within each unit. Figure 3 shows successive deployment stages of this model. Figure 4 exhibits qualitatively the snap-through nature of the dismantling behavior for a curved square unit, as it resulted from a finite element analysis. The lower center node of the unit is hinged and additional appropriate boundary conditions are employed to prevent rigid body rotation. A concentrated load is applied on the upper center node, in order to achieve dismantling of the structure. Figure 4 shows the variation of this dismantling load as a function of the change in distance between the two center nodes. The deployment response is exactly the reverse, however, dismantling is usually simulated instead of deployment, as it is then more straightforward to define the initial geometry of the structure. Finite element analyses have indicated that during deployment and dismantling members of inner SLEs are in compression and bending, while members of outer SLEs are in tension. This has been verified by experimental measurements, and is in agreement with physical intuition. 258

5 auui/iuicu is vciy aiiu inquires Transactions the Built Environment vol 35, 1998 WIT Press, ISSN successive iterations to achieve some balance between desired flexibility during deployment and desired stiffness in the deployed configuration. In connection to the behavior mentioned in the previous paragraph, heuristic rules developed for manually driven design iterations indicate that symmetrical SLEs are preferably made of stiffer materials such as aluminum, as they are the principal load bearing members when the structure has to carry external loads. On the other hand, diagonal, nonsymmetrical SLEs attract much higher compression at intermediate stages during deployment and dismantling and are mainly responsible for the level of intensity of the snap-through phenomenon. Consequently, they are usually made of more flexible materials, such as plastics, so that the material remains elastic during deployment and residual deformations and stresses are avoided. Furthermore, hollow rectangular cross-sections have been found to be suitable for both inner and outer SLEs. Figure 3: Successive Dismantling Stages of Deployable Unit Changing distance between center nodes Figure 4: Structural Response During Dismantling In the present paper an effort is made to escape from this process of trial-and-error iterations, and use instead an optimization process employing genetic algorithms. 259

6 3 Basic features of genetic algorithms Genetic algorithms are local search algorithms that belong to the class of stochastic or random search algorithms. They are based on the concepts of natural selection and natural genetics^^, and rely on the principles of Darwinian theory of survival of the fittest. Although genetic algorithms are randomized, these algorithms are not a simple random walk in the space of solutions. In fact, they efficiently incorporate information from previous stages to create new search points in the design space, resulting in improved performance. Genetic algorithms are different from traditional optimization methods. They usually work with a coded set of the variables and not with the variables themselves; they search from a population of points rather than by improving a single point; they use objective function information without any gradient information, and their transition scheme is probabilistic compared to traditional methods using gradient information. An essential characteristic of a genetic algorithm is the coding of the variables that describe the problem. The most common coding method consists of transforming the variables to a binary string of specific length. This string represents the chromosome of the problem the length of which is defined by the number of zeros and ones in the binary string. For a specific problem that depends on more than one variables, a multivariable coding is constructed by simply concatenating as many single variable codings as the number of the variables of the problem. Each variable may have its own length corresponding to the number of its possible values specified for the particular application. By decoding the individuals of the initial population, the solution for each specific instance is determined and the value of the objective function that corresponds to each individual is evaluated. A simple genetic algorithm proceeds byfirstrandomly generating a population of a specific size. A pseudo-random generator, based on a specific seed that varies between zero and one, is used to generate the initial population. From this population, the next generation is evolved by performing three distinct operations for the reproduction; namely, selection, crossover and mutation. Based on the statistics of this population the next generation is reproduced with different opportunities, i.e. by following a bias law which assigns new probabilities to the members analogous to the statistics of the generation. This means that weak members will be assigned small probabilities and strong members will be assigned high probabilities of existence in the next generation. In this way, the next generation evolves so that thefittestsurvive and increase 260

7 uiwiiuwa uic uui ui uisci[jjjcai iiuiii uic population. In addition, several alternative elitist schemes that pass the best solutions of the current generation to the next one are also employed. In the process of the reproduction of the new generation, the operations of crossover and mutation are performed. With a specified probability of crossover, two members of the population are selected randomly and exchange part of their chromosomal information, by exchanging the parts of their string at randomly selected positions. The probability of crossover determines whether this exchange will occur for the selected pair of parents or not. Finally, certain digits of the chromosome may be altered with a probability of mutation. If the digits are found to be zero, they are changed to ones, and vice versa, with a specific probability. After crossover and mutation the population takes its final form in the current generation. Again, by decoding the strings and solving the sizing problem, new objective function values are determined which express the fitness of the population members for this generation. Simple statistics, i.e. the best, the worse and the average members, are deduced for this generation and the process goes on by performing reproduction, crossover and mutation. After several generations, the best member of the population evolves to represent a very satisfactory solution of the problem. Different schemes have been suggested for crossover and mutation, among which a two-point crossover or uniform crossover seem to perform better for a wide range of parameters'^. Also different mutation schemes were introduced to improve the convergence of the genetic algorithm. The basic parameters of a simple genetic algorithm are the population size of the generation, the probability of crossover and the probability of mutation. By varying these parameters, the behaviour of the algorithm is altered. The performance of a genetic algorithm can be envisaged as a balanced combination of exploration of new regions in the search space and exploitation of already sampled regions. The selection of optimal control parameters for a particular problem largely remains an open issue. 4 Application Consider the deployable structure shown in Figure 5 (perspective view and plan view), to be used as a tent for temporary shelter of victims of natural disasters. The structure consists of two intersecting, semi-circular arches. Each arch has five square units and the two arches share the central unit. 261

8 Scissor-like bracings are added to the deployed structure, as initial design efforts indicated that without them the structure is too flexible and deforms excessively when subjected to snow and wind loads. Figure 5: Deployable tent structure in its deployed configuration For the optimization process, the geometric layout of the structure was considered to be known, to reduce the number of design variables and simplify the problem. Besides the reduction of computational effort, this simplification was mostly needed in order to facilitate the communication between the genetic algorithm code and MSC-Nastran, which performed the linear and nonlinear finite element analysis. Thus, only material and cross-sectional properties had to be updated in the Nastran input file between iterations, and not the topology of the structure. It is envisaged that the more complex optimization process, involving geometric quantities as design variables will be carried out in future work. Each iteration of the genetic algorithm optimization involves four analyses. One is nonlinear and simulates the deployment process of one quarter of a unit, taking advantage of symmetry. The other three are linear predicting the response of the structure under three different service load combinations: (i) dead load and snow, (ii) dead load and wind, (iii) dead load, snow and wind. The constraints impose that the maximum stresses for inner and outer SLEs from linear and nonlinear analysis as well as the maximum deflection from linear analysis do not exceed the respective allowable limits. As for the design variables, they are related to material and crosssection properties for inner and outer elements. Materials are selected from a pool of available candidate materials. The attributes of each one used in the optimization process are the modulus of elasticity, Poisson's ratio, yield stress and cost per unit volume. Cross-sections are assumed to be hollow rectangular and are defined by their width, height and thickness. 262

9 me pruoiem nas s design variables, material, width, height and thickness for the two groups of inner and outer SLEs. Figure 6: Cross-sections of members of deployable tent As for the fitness function, it is defined as the total cost of members of inner and outer SLEs supplemented by penalty terms to account for violated constraints: fmin =^0^060 WjAjCj + (penalty terms for violated constra int s) (1) Thus, the steps of the optimization process can be summarized as follows: (l)initialize genetics (2)Run linear analyses for the 3 load cases and obtain maximum stresses and displacements (3)Run nonlinear analysis to simulate deployment and obtain maximum stresses (4)Create fitness function (5)Check for convergence and stop or return to step 2 This algorithm has been employed for the design of the deployable tent shown in Figure 5 with internal diameter equal to 5.80m. Optimization resulted in outer SLEs made of high-density polyethylenene and inner SLEs of the softer low-density polyethylenene, with the following crosssectional characteristics: bo=35mm, h<,=72mm, to=8mm, b,=26mm, hi=64mm, tj=7mm. 5 Summary and conclusions Deployable structures that exhibit snap-through type of behavior during deployment have been investigated. The design of such structures is very difficult due to the conflicting requirements of flexibility during deployment and stiffness under service loads. To deal with this task and minimize the cost, genetic algorithm optimization has been employed. The proposed algorithm has been applied successfully for the design of a deployable tent to be used in emergency situations. 263

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