Harmony Search Heuristic Algorithm에 벌칙함수를 도입한 구조물의 이산최적화

Abstract

구조물의 단면 최적설계를 위하여 다양한 방법들이 개발되었으며, 또한 실제 구조물 설계에도 응용되어지고 있다. 그러나 개발된 대다수의 최적설계법은 Gradient-Based Mathematical Programming Method (GMPM)가 중심이며, 일반적으로 부재단면을 연속적인 변수로 가정을 한다. 대부분 구조공학의 설계에서는 경제성 및 실용성을 고려하여, 부재단면은 공장 생산된 표준단면의 사용성 및 시공현장의 제한성 등을 고려하여 이산단면 리스트로부터 선택되어져야 한다(이산최적화 문제). 한편 90년대 중반부터 GMPM 방법과는 달리 자기발견론적 방법, 즉 Genetic Algorithms, Simulated Annealing 등을 이용한 구조물의 이산 최적설계법이 개발되어 GMPM 기법이 내재하는 몇몇 문제점을 해결하고 있으나, 보다 실용적·효율적 이산최적기법은 여전히 연구대상이다. 최근 완벽한 하모니를 창출한다는 음악적 개념을 도입시켜 개념화한 Harmony Search Heuristic Algorithm(HSHA)을 이용한 최적화기법이 개발되어 그 유용성을 인정받고 있다. HSHA는 기존의 자기발견적 알고리즘(Heuristic Algorithm) 대비, 비교적 간단한 수학적 조건이 요구되며, 다양한 유형의 구조물의 최적화 문제에 보다 쉽게 적용할 수가 있는 것이 큰 특징이다. 또한 HSHA는 변수의 초기조건을 요구하는 경사도 탐색법과 달리 하모니 메모리 기여율(Harmony Memory Considering Rate)과 피치 조정률(Pitch Adjusting Rate)을 연산자로 한 확률론적 랜덤 탐색법을 사용한다. 본 연구의 주목적은 HSHA를 바탕으로 보다 실용적이고 효과적인 이산최적화설계법을 개발하고자 하는 것이다. HSHA는 지금까지 벌칙함수를 사용하지 않은 Pure HSHA를 이용한 구조물의 이산단면 최적화설계에 성공적으로 적용이 되었으며, 그 유용성 또한 증명이 되었다. 본 연구에서는 HSHA에 벌칙함수를 적용한 이산최적화설계법을 제안함과 동시에, 트러스 구조물에 이 기법을 적용하여 그 유용성을 기존 Pure HSHA 이산최적설계법과 비교·검토하였다. 그 결과 수렴과정에서 벌칙함수를 적용한 HSHA를 이용한 방법이 Pure HSHA를 이용한 방법보다 수렴률이 우수함을 증명했다. 따라서 벌칙함수를 적용한 HSHA를 이용한 새로운 방법인 이산단면변수의 구조최적화 문제에 효과적으로 적용가능하다고 판단된다. 또한 본 연구에서 제안한 벌칙함수를 도입한 HSHA 이산단면최적화기법은 트러스 구조물의 최적화 문제에만 한정되지 않으며, 골조, 막구조, 판구조 등의 구조물에도 용이하게 적용가능하다고 판단된다.; Traditionally, many gradient-based mathematical programming methods have beendeveloped and frequently used to solve structural optimization problems. Themajority of these methods assume that the cross-sectional areas called sizingvariables are continuous. In most practical design problems in structural engineering,however, the sizing variables have to be chosen from a list of discrete valuesbecause this is due to the availability of components in standard sizes andconstraints caused by construction and manufacturing practices. Although themathematical methods can consider the discreteness employing the round-offtechniques based on continuous solutions, the rounded-off solutions may result inthose far from optimum, or even result in infeasible values when the number ofvariables increases. Because most of the available optimization methods treat thedesign variables as continuous, they are very inadequate in the presence of discretedesign variables. On the other hand, a few methods based on mathematicalprogramming techniques were developed in order to handle the discrete nature ofdesign variables (Liebman et al., 1981; Hua, 1983; Zhu, 1986; John et al., 1987).They provide a useful strategy in solving a limited problem, but every method hasits drawbacks, which include low efficiency, limited reliability, and readily beingtrapped at local optimum. More detailed literature surveys were given by Templeman(1988). Over the last decade, in order to overcome the computational drawbacks ofmathematical methods, new optimization strategies based on heuristic algorithms suchas simulated annealing and genetic algorithms (GAs) have been devised for optimaldesign of discrete structural system. Especially, the GA-based discrete optimizationmethods have been vigorously studied by many researchers including Rajeev andKrishnamoorthy (1992, 1997), Lin and Hajela (1992), Wu and Chow (1995a, 1995b),Camp et al. (1998) and Pezeshk et al. (2000). The GA was originally proposed byHolland (1975) and further developed by Goldberg (1989) and others, which is aglobal search algorithm based on concepts from natural genetics and Darwiniansurvival-of-the-fittest. The heuristic algorithms including the GA-based discretesizing optimization methods for structures have occasionally overcome severaldeficiencies of mathematical methods. Seeking a more powerful, effective and robustmethod for discrete structural optimization is still a major concern to structuralengineer. The main purpose of this paper is to propose a more powerful and efficientoptimization method for structures with discrete sizing variables. In our earlierresearch (Lee et al., 2004), a new optimization method for structures with continuousvariables was proposed based on the harmony search (HS) heuristic algorithm andgood results were obtained. The recently developed HS algorithm was conceptualizedusing the musical process of searching for a perfect state of harmony (Geem et al.,2001). Compared to mathematical optimization algorithms, the HS algorithm imposesfewer mathematical requirements to solve optimization problems and the probabilityof being entrapped in a local optimum is reduced because this algorithm is not hillclimbing algorithm. Since the HS algorithm uses a stochastic random search, thederivative information is unnecessary. This new algorithm also considers severalsolution vectors simultaneously in a manner similar to the GAs. However, the majordifference between the GAs and the HS algorithm is that the latter generates a newvector from all the existing vectors, while the former generates a new vector fromonly two of the existing vectors (parents). In addition, the HS algorithm canindependently consider each component variable in a vector when it generates a newvector; the GAs cannot, because they have to maintain the gene structure. In this paper, a discrete search strategy using the HS algorithm with penaltyfunction is presented in detail and its applicability using several standard trussexamples is discussed. Several standard test examples from the literature waspresented to demonstrate the effectiveness and robustness of the proposed approachcompared to other optimization methods. The numerical results reveal that the HSalgorithm with penalty function proposed in this study is a powerful search anddesign optimization technique for structures with discrete-sized members. Theillustrative example revealed that the optimal results were better than those obtainedfrom all previous investigations. Also, the convergence capability results obtainedusing the proposed HS method outperformed those obtained using the GA-basedmethods. Finally a discrete search strategy using the HS algorithm with penaltyfunction is compared to optimization method using the pure HSHA. The optimizationmethod using the HS algorithm with penalty function revealed that the optimalresults were better than those obtained from all previous investigations. In conclusion, our study suggests that the new HS-based method is potentially apowerful search and optimization technique for solving structural optimizationproblems with discrete sizing variables. The HS algorithm-based method is simpleand mathematically less complex. The method is not limited to truss structuraloptimization problems. Besides trusses, this method can be applied to other types ofstructural optimization problems including frame structures, plates and shells.