Evolutionary Algorithm For Structural Optimization

This poster paper summarizes ongoing dissertation research (under the direction of Dr. Christopher M. Foley) concerned with the utilization of genetic algorithms for the optimization of steel structures that have semi-rigid connections using advanced inelastic analysis and performance based engineering design. An evolutionary algorithm has been developed that extends the standard genetic algorithm thru the incorporation of problem specific information and allows the user to impose subjective preferences in an intuitive fashion via graphical monitoring of the rank based fitness statement components. Recently, work was completed on a new bilinear selection mechanism that will be incorporated into the aforementioned evolutionary algorithm. The bilinear selection mechanism will allow a more intuitive application of dynamic selection pressures in a way that is very efficient with respect to coding and implementation. 1 EVOLUTIONARY ALGORITHM A hybrid rank-based evolutionary algorithm that takes advantage of a-priori problem specific information and operates on a high cardinality heuristic genetic representation was developed (Voss and Foley 1999(b), Voss and Foley 1999(a)). A rank based fitness statement combined with generationally dependant penalty exponents was proposed to condition the seven components of the fitness statement so they participate fairly during the evolutionary process. Translocation crossover and intelligent mutation were utilized to maintain genetic diversity. A graphical method was proposed to monitor the progress of the components of the fitness function, allowing the user to interact with the evolutionary process. Generationally dependant non-linear rank based selection was used to orchestrate a soft landing near the global optimum for an example problem with 20 discrete design variables. 2 BILINEAR SELECTION SCHEME FOR RANKED POPULATIONS (μ, 8, ", $) The development of a new selection scheme was motivated by the desire for a generationally dependant selection mechanism that could be adjusted in an intuitive manner. The selection mechanism (Voss and Foley 1999(c)) is defined by a four parameter (μ, 8, ", $) bilinear cumulative distribution function. The selection mechanism provides an intuitive feel for how the selection pressure and takeover time are affected by changes in the model parameters. Expressions for the mean and variance with respect to the bilinear distribution were developed along with an expression for takeover time. It was shown that the mean is a function of only one of the parameters ($). The mean and variance of the proposed selection scheme were correlated with those found in tournament selection for various tournament sizes and equivalent values of (",$) were tabulated. Figure 1, illustrates the correlated takeover times for the bilinear distribution and compares them with experimental and theoretical results(Goldberg and Deb 1991). Figure 1: Comparison of Bilinear Selection with Tournament Selection. In Figure 2, it is shown that superposition of the proposed bilinear distribution allows one to approximate more complex cumulative distributions in an efficient manner. Expressions for the mean and variance for two superimposed bilinear distributions were derived. Figure 2: Superposition of Two Bilinear Distributions to Form a Multilinear Distribution (μ i,8i,"i,$i,fi). The following code was the primary motivation for the development of the bilinear distribution: do i = 1, n1 "1 = 0.8 0.6 * ( current_gen / max_gen) $1 = 0.3 + 0.5 * ( current_gen / max_gen) individual(i) = CalcRank( μ1,"1, $1, 8 ) end do do i = n1+1, n1 + n2 "2 = 0.4 0.2 * ( current_gen / max_gen) $2 = 0.3 + 0.5 * ( current_gen / max_gen) individual(i) = CalcRank( μ2,"2, $2, 8 ) end do where: CalcRank is a function that returns a rank value from the bilinear distribution, n1 and n2 are the number of individuals that are to be chosen from the first and second distributions respectively, current_gen is the current generation, max_gen is the number of generations in the evolutionary simulation, and rateExp is an exponent that controls how fast the selection parameters are modified. The example code illustrates the implementation of the bilinear selection strategy. Selection during early generations is engineered so that exploration is around large areas of promising individuals and transitions into an elitist selection strategy that has a large selection variance. This would allow the algorithm to converge while combining any beneficial remaining genetic material from nearly identical individuals. The implementation of a fairly sophisticated selection mechanism that alters both the takeover time and the distribution variance generationally in an intuitive manner is implemented in just a few lines of code. The code also implies the ease in which more than two bilinear distributions could be superimposed to engineer selection distributions to any degree of accuracy advocated by a particular evolutionary algorithm. . AcknowledgmentsThe authors would like to acknowledge the support of theNational Science Foundation (USA) Grant Number CMS9813216 under the direction of Dr. Priscilla P. Nelson.The views expressed in the paper are those of the authorand not necessarily the sponsor. ReferencesGoldberg, D. E. and Deb, K. (1991). A ComparativeAnalysis of Selection Schemes Used in Genetic AlgorithmsIn G. J. E. Rawlins (ed.), Foundations of Genetic Algo-rithms, 69-93. San Mateo, CA: Morgan Kaufmann. Voss, M. S. and Foley, C. M. (1999(c)). Bilinear SelectionScheme For Ranked Populations. Proceedings of Geneticand Evolutionary Computation Conference Late BreakingPapers. Orlando, FL, (submitted for publication). Voss, M. S. and Foley, C. M. (1999(b)). EvolutionaryAlgorithm for Structural Optimization. Proceedings ofGenetic and Evolutionary Computation Conference.Orlando, FL, Voss, M. S. and Foley, C. M. (1999(a)), Rank BasedEvolutionary Algorithm for Structural Optimization.Computers & Structures (submitted for publication).