World Journal of Engineering Review of algorithms of structural optimization with discrete variables 1 1 1 1 Yancang Li, , Beibei Heng, , Lingren Kong, , Weijuan Yang, Article information: Downloaded by California State University Fresno At 09:57 25 October 2017 (PT) To cite this document: 1 1 1 1 Yancang Li, , Beibei Heng, , Lingren Kong, , Weijuan Yang, (2011) "Review of algorithms of structural optimization with discrete variables", World Journal of Engineering, Vol. 8 Issue: 3, pp.231-236, https://doi.org/10.1260/1708-5284.8.3.231 Permanent link to this document: https://doi.org/10.1260/1708-5284.8.3.231 Downloaded on: 25 October 2017, At: 09:57 (PT) References: this document contains references to 34 other documents. To copy this document: permissions@emeraldinsight.com The fulltext of this document has been downloaded 19 times since 2011* Access to this document was granted through an Emerald subscription provided by emerald-srm:409465 [] For Authors If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.com Emerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services. Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. *Related content and download information correct at time of download. World Journal of World Journal of Engineering 8(3) (2011) 231-236 Engineering Review of algorithms of structural optimization with discrete variables Downloaded by California State University Fresno At 09:57 25 October 2017 (PT) Yancang Li, Beibei Heng, Lingren Kong and Weijuan Yang College of Civil Engineering, Hebei University of Engineering, Handan 056038, China liyancang@163.com (Received 22 July 2010; accepted 17 December 2010) Abstract Much work has been done on the optimization of the discrete variable structure design. In order to handle the optimization problem effectively, the related theories, methods, and other beneficial results were summarized. On the basis of analyzing the predecessors’ research, the development direction was introduced. Then, some practical methods, including their improvements, for discrete structural optimization were analyzed. Finally, the Ant Colony Optimization algorithms were shown as promising methods. This work has significance in theory and practice for the development of structural optimization. Key words: Discrete variable structure, Optimization, Review, Ant colony algorithms 1. Introduction In the actual structural design, not all of the design variables can be handled as limited discrete values (Guo, 2000). The usual optimization design methods must deal with the results to meet the engineering design specifications and various technical standards, but the discrete results treated by continuous variable optimization method is not necessarily feasible or optimal (Zhang et al., 2003). This situation has signified the study of related optimization algorithms. This paper presented the new results of the structural discrete variable optimal design theory including its advantages and disadvantages, and pointed out its direction of development. This work will give rise to the study of structural design optimization. The paper is organized as follows. In section 2, the characteristics of discrete structural optimization were introduced. In the following part, attention ISSN:1708-5284 was paid to the methods of discrete structural optimization. Then a prediction of discrete variable structural design optimization was proposed. Finally, the advantages of ant colony algorithm were illustrated. 2. Characteristics of discrete structural optimization In the structural optimization, the objective function and constraints of the mathematical model of optimum design are discontinuous. As a result, the ordinary methods, such as various gradient algorithms of sensitivity analysis, K-T conditions, and so on, are not valid. The difficulties in structural optimization design of discrete variables are as follows. (1) Traditional mathematical analysis is not valid. In traditional mathematics, the mixeddiscrete variable optimization problem belongs to combinational optimization 232 (2) Downloaded by California State University Fresno At 09:57 25 October 2017 (PT) (3) (4) (5) Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236 problem. With the increase of design variables, the number of combination increases exponentially, namely, called as the “combination explosion” with the NPhard problems. The difficulty of NP-hard problem is the number of combination increasing rapidly with the increasing of design variables (Li et al., 2010). Shape optimization mainly includes section variable optimization and shape (coordinates) variable optimization. It is difficult to process the two kinds of variable optimization simultaneously. Topological optimization is similar to the shape optimization. They both have a topology between nodes and no-bar connections. In dealing with different types of optimization problems, optimal topological solutions can’t be obtained without the rods recovery strategy. It is difficult to handle the layout optimization, which deals with three nature variables simultaneously: cross-section, shape and topology optimization. Little work has been done on the optimization of structure type, which is a high level of structural optimization. 3. Discrete structural optimization algorithms Algorithms for discrete variable structural optimization design can be divided into three types: accurate algorithms, approximate algorithms and heuristic algorithms. Sun introduced the traditional methods of optimization design such as the enumeration method, implicit enumeration method, rounded method, branch and bound method, cut plane method, the penalty function method, Bala’s method, integer gradient method, etc. (Sun et al., 2002). Here we mainly deal with the newly developed heuristic algorithms. 3.1. Genetic algorithm (GA) As a randomized, parallel and adaptive optimization algorithm, GA is based on the idea of “the survival of the fittest”. It includes selection, crossover and mutation processes. The basic idea is that the problem will be expressed by encoded chromosomes, chromosome groups through reproduction, crossover and mutation and other operations, continuously evolved and converged to solve the optimal problem. Chromosome length is positively correlated with coding accuracy. When selecting parameters the number of aspects should be considered. GA is a recent rise heuristic algorithm, especially for the complex and nonlinear problems. Therefore, it is more suitable for discrete optimization. Burczynski optimized the continuous structures on the topology optimization (Burczynski and Koko, 1998). Zhang and Wang employed it to optimize the shape of the structure (Zhang and Wang, 2001). The application results show the unique advantage of GA in the structural optimization. But, GA has the defects of too many times re-analysis of structure, lower search efficiency and prematurity. Many experts and scholars have extensively studied and proposed various improved genetic algorithms, such as layered GA, a restructuring big foreign across generation mutant algorithm, self-adaptive genetic algorithm, parallel genetic algorithm based on niche genetic algorithm, and so on. These improvements have been used successfully in the optimization. 3.2. Simulated annealing method Simulated annealing is based on the principle of annealing process of solids, with Monte Carlo iterative solution strategy and an optimal algorithm. The law comes from statistical mechanics. First, a high temperature system is determined for faster search, to define low-energy area. Declining with temperature, accuracy of search for the optimal solutions is also rising. Using the probability jump to avoid the local optimal solution, the chance of getting the global optimal solution is increased. The temperature attenuation parameters and step of optimization are both applied to the optimal speed and precision (Shung et al., 2011). Because it does not need any gradient information and Hessian matrix, the algorithm is more suitable for optimization design of discrete variables. Much effort has been made to improve the performance of the classic simulated annealing algorithm. Du proposed mixed simulated annealing algorithm, which can improve the speed of the algorithm’s convergence (Du et al., 2001). Gao put forward an improved and simulated annealing algorithm (Gao et al., 2002), Gao combined the simplex method with the simulated annealing algorithm and proposed improvement measures, giving the algorithm a better solution efficiency and global optimization ability(Gao, 2007); Wu Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236 Downloaded by California State University Fresno At 09:57 25 October 2017 (PT) proposed a hybrid discrete variable optimization method of simulated annealing (Wu, 1997). 3.3. Artificial neural network As a heuristic search method, the Hopfield Artificial Neural Network was successfully used in structural optimization design (Hopfield, 1982). Dhingra used Hopfield to solve the three rod nonstatically structural optimization and welding structural optimization problem; Lu employed Hopfield neural network model to solve the discrete structural optimization problem (Lu, 1997). Now, the chaotic cellular neural networks model, fuzzy neural network model, and other improvements have had been employed to the structural optimization design. 3.4. Tabu search Local search in the optimization process easily falls into local optimal solution, and can’t get global optimal solutions. The Tabu search algorithm was put forward on the problem. Its main idea is that in the search process, the searched local optimal solutions are marked and will not be searched again in the next search process, so as to ensure the effective search path is different (Marcos et al., 2008). Bland applied it to the optimization of truss design (Bland, 1995). The method of Tabu search, neural network and genetic algorithm were commonly used in combination in order to overcome the weakness of the basic algorithm (Luo and Huo, 2005). 3.5. Ant colony algorithm Biologists found that the ants in the foraging process will release some special secretionspheromones. The ants within the scope can feel the pheromones and tend to move toward the intensity route. This behavior of ants is a kind of information feedback phenomenon. Inspired by such biological phenomena, Marco Dorigo, and other Italian scholars, put forward the basic ACO (Ant Colony Optimization) (Blum, 2005; Colorm et al., 1991). The algorithm was successfully used to solve the famous TSP (Traveling Salesman Problem) (Lao, 2009). After ten years of development, some improvements were proposed (Yang et al., 2009; Hong et al., 2010; Dorigo et al., 2002; Zhou et al., 2004). Numerical simulation results demonstrated that the algorithm has many advantages, and it can be used as a new heuristic method in solving the combinatorial optimization problems (Balseiro et al., 2011; Kaji, 2001; Karaboga et al., 2004). 233 In recent years, as stochastic optimization methods, the ACO algorithms have been widely used in single-objective optimization problems and multiobjective optimization problems. In the combinational optimization it has gradually show its unique advantages (Li and Li, 2007). ACO algorithms have become very promising heuristic algorithms. 3.6. Other methods At present, many scholars are engaged in the structural optimization design of discrete variables research. Guo proposed a full stress design method based on GA (Guo, 2004). Zhang proposed unilateral search algorithms (Zhang et al., 2005); Li proposed a two-stage algorithm for structural optimization (Li and Zhang, 2006). In 2004, the self-organization behavior of search algorithm was applied to discrete structural optimization (Li and Gong, 2004). In 2006, the particle swarm algorithm was employed in the discrete structural optimization (Wang and Liu, 2006). In 2007, the artificial fish colony optimization algorithm was introduced into the application of discrete structural optimization (Zhang et al., 2007). Many scholars are trying to apply these intelligent algorithms to the structural optimization design, but we still have a long way to go. 4. Prediction of discrete variables in structural optimization design The direction of the discrete variables structure optimization design can be shown as follows. (1) Specific problem. How to reduce the difference between the mathematic model and the practical engineering structure is a big problem for the structure optimization. (2) Multi-objective problem. In reality, many optimization problems have multiple targets and contradictions. How to deal with the interaction of the constraint is another direction we should pay attention to. (3) Structural optimization based on reliability. Structural reliability has gradually become a criterion of modern structure optimization design. The structural optimization based on the reliability will be an important research direction. (4) The study on the high-efficiency optimization methods of handling the shape (geometric), topology, layout optimization simultaneity will be a promising direction too (Jiang et al., 2007) 234 Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236 5. Conclusions Structural optimization with discrete variables has become a promising direction in civil engineering. At present, much work has been done, but further study is still needed. In order to deal with this NP-hard problem more efficiently, we summarized the common methods of the structural optimization, especially the heuristic algorithms. As a novel heuristic algorithm, the ant colony algorithm will gradually show its advantage in this field. Downloaded by California State University Fresno At 09:57 25 October 2017 (PT) Acknowledgements The work was supported by the Fund of Building Technology of Hebei Province (2009–128), Foundation of Hebei Educational Committee and Hebei University of Engineering Funds for Distinguished Young Scholar. References Blum C., 2005. Ant colony optimization: introduction and recent tends. Physics of Life Reviews 2(4), 353–373. Bland J A., 1995. Discrete variable optimal structural design using Tabu search. Structural Optimum 10, 87–93. Burczynski Tadeusz and Koko Grzegorz, 1998. Topology optimization using boundary elements and genetic algorithms. Baecelona: IMN E 5, 1–12. Colorm A., Dorigo M. and Mlniezzo V., 1991. Distributed optimization by ant colonies. Proceeding of the First European Conference on Artificial Life, Elsevier Publishing, Paris France, 134–142. Colorni A. and Dorigo M., Maniezzo V., 1994. Ant system for job–shop scheduling. Belgian Oper. Res. Statist. Computer. Sci. 34, 39–53. Dorigom L., Gambardeila L.M. and Middendore M., 2002. Guest editorial: special section on ant colony optimization. IEEE Transactions on Evolutionary Computation 6(4), 317–319. Du Z.H., Li S.L., Wu M.Y., Li S.Y. and Zhu J., 2001. Mixed SPMD simulated annealing algorithm and its application. Journal of Computers 24(1), 335–342. Gao Q.S., Zhang S.D., Pan D.H. and Liu X.H., 2002. Parametric design of the Simulated Annealing approach. Systems engineering theory and practice 8, 41–44. Gao Z.Y., 2007. Improved simulated annealing algorithm for the simple. Journal of Nanyang Normal University 6(3), 30–32. Guo P.F., Han Y.S and Wei Y.Z., 2000. Discrete variable structural optimization design of full stress design method. Engineering Mechanics 17(1), 94–98. Guo P.F., 2004. Discrete structural optimization Fibonacci algorithm. Journal of Liaoning Institute of Technology 23(4), 1–4. Hong T.S., Yu C.D., Liang H.Y., Dong Y.Z., Hong J and Wei D.L., 2010. Study on the route optimization of military logistics distribution in wartime based on the Ant Colony Algorithm. Computer and Information Science 3(1), 1913–1918. Hopfield J., 1982. Neural Networks and Physical Systems with Emergent Collective Computer Abilities. Proc Natl Acad Sci 79(6), 2554–2558. Jiang X. and Deng Z., 2007. Optimization of concrete mix based on artificial neural network and simulated annealing. World Journal of Engineering 4(1), 38–44. Kaji T., 2001. Approach by ant tabu agents for traveling salesman problem. Proceedings of the IEEE International Conference on Systems 5, 3429–3434. Karaboga N., Kalinli A. and Karaboga D., 2004. Designing digital filters using ant colony optimization algorithm. Engineering Applications of Artificial Intelligence 17, 301–309. Lao J., 2009. Ant colony algorithm for TSP problem of a number of improvement strategies. Science Technology and Engineering 9(9), 2459–2462. Li Q.Y. and Gong Y.B., 2004. Discrete structural optimization in an effective bionic algorithm. Modern Manufacturing Engineering 5, 32–34. Li Y.C. and Li W. Q., 2007. Adaptive ant colony optimization algorithm based on information entropy: foundation and application. Fundam. Inform 77(3), 229–242. Li Y.M. and Zhang Y.G., 2006. Consider the structure of the overall stability of the monolayer shells optimal design. Building Structures 36(4), 77–80. Li Y.Y., Tan T. X. and Li X. S., 2010. A gradientbased approach for discrete optimum design. Structural and Multidisciplinary Optimization 41(6), 881–892. Downloaded by California State University Fresno At 09:57 25 October 2017 (PT) Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236 Lu J.G., 1997. Artificial neural networks and multilevel optimization techniques and practices. Wuhan, Press of Huazhong University of Science. Luo H.L and Huo D., 2005. Topology optimization of truss structures with discrete variables of genetic tabu search algorithm. Henan Science 23(6), 909–911. Ramonet B. L., 2011. An Ant Colony algorithm hybridized with insertion heuristics for the time dependent vehicle routing problem with Time Windows. Computers & Operations Research 38(6), 305–310. Sun H.C., Chai S. and Wang Y.F., 2002. Discrete structural optimization, Dalian, Press of Dalian University of Technology. Wang X.Y and Liu R.F., 2006. Chaotic particle swarm optimization algorithm in truss design. Journal of Taiyuan University of Technology 27(6), 478–480, 485. Wu J.G., 1997. Works mixed discrete variable structural optimization simulated annealing method. Engineering Mechanics 3, 139–143. Yang J., Yang S. and Zeng Q.G., 2009. Pheromonebased ant colony algorithm strength. Computer Applications 29(3), 865–869. 235 Yeh S.F., Chu C.W., Chang Y.J. and Lin M.D., 2011. Applying tabu search and simulated annealing to the optimal design of sewer networks. Engineering Optimization 43(2), 159–174. Zhang M.F., Shao C and Gan Y., 2007. Based on simulated annealing mutation operator with a mixture of artificial fish school algorithm. Electronics 34(8), 1381–1385. Zhang M.H and Wang S.J., 2001. Genetic algorithm to structural shape optimization. Mechanical Science and Technology 20(6), 824–826. Zhang Y.N., Liu B. and Guo P.F., 2003. Hybrid genetic algorithm for optimal design of building structures. Journal of NortheasternUniversity 24(10), 985–988. Zhang Y.N., Liu J.P., Liu B., Dong J.K. and Zhu Z.Y., 2005. Improved one-way search genetic algorithm optimization of engineering design. Mechanics Quarterly 26(2), 293–298. Zhou S.J., Li Y.C and Cui H.L., 2004. Improved ant colony algorithm based on information entropy and its application. Quantitative and Technical Economics Research 10, 104–109. Downloaded by California State University Fresno At 09:57 25 October 2017 (PT)

1/--страниц