Genetic Algorithms for Preliminary 2-D Structural Design

Michael Leung' Gale E. Nevill, Jr."'

The University of Florida Gainesville, Florida

Abstract Preliminary structural design, in which the basic

configuration or shape of a structure is determined, plays a crucial role in the early stage of the design process. Various computer-aided design (CAD) techniques to support the search for preliminary designs have been studied and applied. These include calculus-based gradient search, knowledge-based (expert) systems, and connectionist (neural) networks. Unfortunately, however, all these techniques have serious limitations. This paper presents a promising new technique, genetic algorithms (GAS), for generating good preliminary designs by using the mechanics of natural selection. Genetic algorithms coupled with a binary 2-D array representation scheme are used to solve a preliminary 2-D structural design problem and the results are presented. Optimal and nearly optimal preliminary design solutions obtained empirically demonstrate the efficacy of applying GAS to problems involving 2-D geometric features.

I. Introduction Preliminary structural design, a starting point in the

design process, provides an initial rough structural configuration which may or may not satisfy the problem goals or constraints. However, an effective preliminary design eliminates poor alternatives and r edm redesign costs, and thus leads the rest of the design process to refine more efficiently the initial state to a complete, good design, including detailed dimensions of the structure. Various computational techniques to ayist designers in the search for the optimal preliminary designs have been studied. The most important of these are discussed briefly below.

A. Calculus-based Gradient Search Calculus-based gradient search13J6 is used extensively

in engineering. This method requires at least the first derivative of the objective function with respect to each independent variable. Then, starting with a randomly chosen point in the domain, the search iteratively

Wraduate Student, Department of Mechanical Engineering "Professor, Department of Aerospace Engineering, Mechanics, and

Engineering Science

generates more points converging to the nearest minimum or maximum. The method locates a relative optimum quickly; however, it is often a local rather than global optimum in a search space of multiple local optima, and depends on the starting point selected. In addition, the first derivative is not easily determined for most complex problems.

B. Knowledge-based (Exuert) Svstems Knowledge-based ~ e a r c h , ~ ~ ' ~ another alternative to

deduce a promising design, uses a set of design rules containing heuristic knowledge. In general, a knowledge- based system is a computer program which incorporates techniques of artificial intelligence to perform a specific task that normally would have to be performed by a human expert. MOSAIC1' is a knowledge-based system designed for creating light-weight structural configurations to support a given set of forces with available supports in a 2-D domain. To create a successful knowledge-based system, precise rules or heuristics should be coded. However, in design processes designers often use large quantities of experience-based knowledge which is difficult to identify, acquire, and represent explicitly as rules.

C. Connectionist (Neural) Networks Connectionist such as the Bol tzma~

machine and harmony theory, often using simulated annealing,S are other engineering optimization techniques. In these techniques, a design problem is modeled by a set of soft constraints, which ought to be satisfied. The role of the networks is to globally maximize the degree of constraint satisfaction. Comectionist networks are effective at locating the global optimum in problems that deal with soil constraints. However, some design problems cannot avoid the consideration of hard constraints, which must be satisfied. For example, a structural design system must be kinematically stable. Further, the knowledge is incorporated in a network by assigning proper comection weights between the nodes which represent important features of the problem domain. For a structural design problem, the correct comection weights are difficult to determine explicitly and teaching the network all the knowledge involved in even very simple structural design is very difficult.

Copyright American Institute Aeronautics and Astronautics, h., 1994. All righta reaemed.

D. The Promise of Genetic Algorithms Since all three previously described techniques have

serious deficiencies, a more robust search method for automatically and efficiently finding good preliminary designs is clearly needed. The potential of GAS as promising function optimizers has recently received considerable interest from researchers in different areas.

The ideas of using GAS as global optimizers were initiated by Holland6 at the University of Michigan about 1975. The standard mechanics of GAS, including the three major artificial genetic operators (reproduction, crossover, and mutation) are clearly defined by G~ldberg.~ Extensive research has proven that GAS seek the global or nearly global optimum effectively and quickly in a broad range of complex search spaces. In addition to effective performance, GAS are easy to implement.

Engineers frequently apply GAS as function optimizers to solve complex problems. For instance, Hajela5 applies GAS to search for the optimal cross-sectional area for each member of the 10 member truss such that the minimum weight of the truss is achieved while the displacement constraints are satisfied. More applications of GAS to engineering optimization problems can be

Moreover, GAS have the ability to learn and induce control rules. Odetayo and M~Gregor'~ present a GA-based method for automatically inducing control rules for a dynamic physical system: a polecart system. A wheeled cart has a rigid pole hinged to its top. The induced control rules determine whether a force should be applied to the left or right to keep the pole balanced as the cart is moving along a straight bounded track. Compared to other learning algorithms solving this problem, GAS are found both effective and robust.

In addition to engineering application, GAS are broadly applied in other diverse areas. Caldwell and Johnston1 use GAS in an interesting 2-D application to search for criminal suspects' facial composites. The results reveal that a likeness of a criminal suspect evolves after a witness, who serves as the fitness function (analogous to an objective function in most other common search techniques), evaluates about 120 GA generated facial composites out of 34 billion possibilities. This evidence shows the enormous potential of GAS applications; therefore, further explorations are expected to be beneficial.

The focus of this paper is exploration of the application of GAS to preliminary structural design tasks involving 2- D geometry. To evaluate this application, GAS are used to solve a truss design problem of multiple local optima. The goal in this problem is to automatically generate a kinematically stable, light-weight, simple truss configuration which supports all the given forces with the available supports in a 2-D space. Reliable structural design solutions obtained consistently reveal the power of GAS in preliminary design tasks. The problem domain, specific objective, methodology, and experimental results will be presented in more detail.

Stable ......... .................

Stable . . . . . . . . Unstable . . . . Structure

Unstable .... ...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .................................................................................

Fig. 1. Illustration of the structural design problem domain including examples of stable and unstable forces and structures.

11. Preliminary 2-D Structural Desirm Problem Domain and Obiective

The problem domain is defined using two general principles: 1) all structural members of a truss should have similar length and 2) triangular structures are desirable shapes in structural designs. Figure 1 illustrates this domain composed of equilateral triangles. The inputs to this problem are the presence (or absence) of a force and support at each node. Each dotted line segment represents a possible structural member. In Figure 1, the bold solid lines indicate the presence of structural members which create a simple truss configuration. Each member is assumed to be attached at the ends by pin joints. The objective in this problem is to search for a truss configuration which will stabilize this force-support system and also achieves the minimum weight (the least number of structural members required). Once the preliminary shape of a stable structure is obtained by using GAS, further engineering analyses can be applied to determine the exact length and cross-sectional area of each member.

In this design problem, a kinematically stable truss is defined as a number of triangular structures which are contiguously connected together and pin-jointed to at least two supports. Likewise, a force is supported if it is touching either a kinematically stable truss or a support. Some examples are given in Figure 1 to clarify the ideas of stability in this 2-D structural design problem.

111. Binan 2-D Array Remesentation Scheme and Crossinn Over

Binary string representation scheme is traditionally chosen as a direct analogy to chromosomes as GAS resemble the mechanics of natural selection and biological genetics. The string repmntation is well suited for many problems. However, when a pair of strings which represents 2-D design configurations is crossing over, some geometric features may be lost due to the linearity of the representation scheme. A 2-D array is preferred because it can better enable the patterns of related 2-D geometric featurea to stay close together during crossing

Truss Configuration . . ... . . ...,,,,,, . . . . to be Represented: . . 1 :"..

:. 0 ". . . ." 2 '.'.

2-D Ar ray 0 0 0 Representation1 0 1 1 1 0

0 0 0 0 0

Fig. 2. Illustration of the technique using binary 2-D array representation in a simplified problem domain.

over, thus improving the convergence rate and robustness of the sea r~h .~ Figure 2 illustrates how a binary 2-D array interprets a 2-D truss configuration in the preliminary structural design problem domain previously presented. Each bit in an array represents a single triangular panel. An activated bit (equal to 1) represents structural members being present on the three sides of the corresponding triangle. The corresponding crossover operator swaps portions of a pair of arrays. One of the three methods as illustrated in Figure 3 is randomly chosen every time a pair of arrays crosses over.

IV. Fitness Function Used to Evaluate Structural Designs

This section describes how the fibless function is formulated and explains how this function properly evaluates both complete and incomplete solutions for this structural design problem to guide the GA search to the optimum. The fitness function which measures the structural design performance is developed by using a penalty method concept." This method subtracts a penalized weight from a large constant and therefore always results in a nonnegative nhber , which is a requirement for GAS. In the present structural design problem, the penalized weight is the combination of the total weight and the degree of instability of a candidate structural configuration. This function is defined below.

Fitness = C,, - (C, x AAM) - (C, x TM)

where

C,, = A positive constant to maintain nonnegative fitness value

AAM = &proximate number of Additional Structural Members to support all forces -

TM = Total number of structural Members present C, = Weight factor for instability related to AAM C, = Weight factor for system weight related to TM

The second term, -(C, x AAM), penalizes candidate truss designs in which unstable forces exist. The penalty is proportional to the number of additional members needed to balance a force-support system. If all forces in a system are supported by a kinematically stable truss, this term becomes zero. The third term, -(C, x TM), allows the fitness function to reward lighter truss systems. The weight factor C, is higher than the weight factor C, because a truss being stable is more important than being light-weight. However, the ratio of C, to C, should be carefully selected to make the fitness function a soft penalty function. If C, is much higher than q, the function becomes a harsh penalty function. This harsh penalty function tends to reward stable designs exceedingly and, hence, the GA search more likely converges to a heavy stable structure. A set of effective constants are C, = 300, C, = 1.5, C, = 1.

The variable TM in a candidate design can be determined easily by counting the number of structural members present. However, it is difficult to determine an exact value for the variable AAM, which refers to the number of additional members needed to complete an unstable structural design. Therefore, AAM has been estimated heuri~tically.~

V. Ex~erimental Results A constant population of 100 arrays in each generation

was used in the GA search to solve this structural design problem. This implementation was tested thoroughly with many different inputs. The results revealed that the GA approach consistently produced promising solutions. The solutions of a selected set of input forces and supports, as illustrated in Figure 4, were used to perform statistical analysis. In 30 runs of the program, 10 and 14 searches converged to global and nearly global optimums, respectively. The other six searches converged to stable but bulky structures. The probability of finding the best design in each run was thus 0.33. Typically, it took 110

Fig. 3. Three different models of selecting crossing sites and crossing over.

to 140 generations for a search to converge. The evolution of one of the optimum structural configurations from the randomly created first generation is graphically presented in Figure 5. The entire search performance, which includes the maximum fitness and average fitness in each generation, is plotted in Figure 6. This figure shows the progrw of an initially poor design being manipulated by GA operators causing it to improve rapidly and gradually converge to the limit.

VI. Conclusions and Recommendations The overall goals in this research were accomplished.

Applying GAS to generate effective truss design configurations demonstrates the feasibility of application of GAS to preliminary 2-D structural design tasks. In

Fig. 4. Input forces and supports in the 2-D problem Domain.

a. Generation 1

c. Generation 30

addition to their effectiveness, GAS are easy to implement as is described in this paper. However, creation of a good fitness function can be quite complex if accurate evaluation is difficult.

Standard GAS can be applied, in general, to a wide range of optimization problems of continuous and discrete variables. However, minor modification may enhance the search performance in specific problems. For example, in the structural design problem introduced, the linearity of the conventional binary string representation makes the GA search sensitive to the design orientation in a 2-D domain, thus causing inconsistent search performance. This weakness can be overcome by using binary 2-D arrays and a corresponding crossover operator. This works because arrays better represent 2-D features of alternative structural designs and the resultant GA search is more efficient and robust. Correspondingly, when a 3- D design problem is to be solved by GAS, it is expected that the first priority for a representation scheme will be using 3-D arrays. The crossover operator used in this design problem to select and exchange partial structures between a pair of potential designs is one alternative. Further investigations of different crossing over techniques may be worthwhile.

The design problem solved in this research project can be made more practical by expanding the evaluation criteria coded in the fitness function. For example, if obstacles are allowed to be placed in the 2-D domain where structural members are prohibited, more sophisticated heuristics are needed to estimate the number

b

& c. Generation 60

P d. Generation 118

Fig. 5. The best c o n f i i t i o n obtained in specified generation.

2290

Generation

Fig. 6. Maximum fitness and average fitness versus number of generation of GA search.

of extra members for constructing a truss which can avoid obstacles. The function can also be made to punish a design (partial or complete) in proportion to the estimated maximum stress to encourage low stress structural design solutions.

In conclusion, genetic algorithms have the potential to be a powerful tool to assist in the prelimhry stages of structural design processes. Further research is necessary to explore and improve these promising algorithms.

REFERENCES 'Caldwell, C. and Johnston, V.S., "Tracking a

Criminal Suspect through "Face-Space" with a Genetic Algorithms," Proceedings of the Fourth International Conference on Genetics Algorithms, San Diego, CA, pp. 416-421, 1991.

%arcelon, J.H. and Nevill, G.E., Jr., "Qualitative Analysis of Preliminary Designs Using Artificial Neural Networks," Proceedings of ASME Computers in Engineering Conference, Boston, Massachusetts, pp. 171-180, August, 1990.

3Goldberg, D.E., "Computer-Aidad Pipeline Operation Using Genetic Algorithms and Rule Learning. Part I: Genetic Algorithms in Pipeline Optimization," Engineering with Computers, Vol. 3, pp. 35-45, 1987.

'Goldberg, D.E., Genetic Algorithm in Search, Optimization, and Machine Learning, Addison Wesley, Reading, MA, 1989.

'Hajela, P., "Genetic Search Strategies in Multicriterion Optimal Design," A M Journal, Vol. 31, No. 12, pp. 354-363, June 1991.

6Holland, J.H., Adaptation in Natural and Artificial Systems, The University of Michigan Press, Ann Arbor, 1975.

'Jackson, P., Introduction to Expert Systems, 2nd ed., Addison-Weslzy, Reading, Mass., 1990.

'Jain, P. and Agogino, A.M., "Optimal Design of Mechanisms Using Simulated Annealing: Theory and

Application,'Design Automation Confernce 04th: 1988: Kissimmee, IZ), Institute of Electrical and Electronics Engineers, New York, pp. 233-240, 1988.

b g , M., 'Genetic Algorithms for Prelimiuary 2-D Structural Design,' Master's Thesis, University of Florida, 1992.

"'Nevill, G.E., Jr. and Paul, G.H., Jr., "Knowledge- based Spatial Reasoning for Designing Structural Configuratibns,' Proceedings of the ASME Computers in Engineering Conference, Vol. 1, pp. 155-160, August 1987.

"Nordvik, J.P. and Renders, J.M., 'Genetic Algorithms and Their Potential For Use In Process Control: A Case Study,' Proceedings of the Fomh International ConJmence on Generic Algorithms, Morgan Kaufmana Publishers, San Diego, CA, pp. 416-421, 1991.

l20detay0, M.O. and McGregor, D.R., 'Genetic Algorithm For Inducing Control Rules For A Dynamic System," Proceeding of the Zhird Inrernarional Conference on Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, pp. 177-182, 1989.

13Pspalambros, P.Y. and Wilde, D.J., Principles of Optimal Design: M&ling and Computation, Cambridge University Press, Melbourne, Australia, 1988.

"Richardson, J.T., Palmer, M.R., Liepins, G. and Hilliard, M., 'Some Guidelines for Genetic Algorithms with Penalty Functions,' Proceedings of the 3hird International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, pp. 191-197, 1989.

"Rumelhart, D.E. and McClelland, J.L., Parallel DttributedProcasing: Explorations in the Microstructure of Cognition, MIT Press, Cambridge, MA, Vol. 1, 1986.

16Russell, D., Optimization Theory, W. A. Benjamin, Inc., New Yo*, 1970.

"Tanimoto, L.S., The Elements of Artificial Intelligence, Computer Science Press, New York, NY, pp. 491- 504, 1990.