Topology optimization of truss structures using genetic algorithms ; Santvarų topologijos optimizavimas genetiniais algoritmais
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VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Dmitrij ŠEŠOK TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES USING GENETIC ALGORITHMS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) Vilnius 2008 Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2004–2008. Scientific Supervisor Prof Dr Habil Rimantas BELEVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T). The dissertation is being defended at the Council of Scientific Field of Mechanical Engineering at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Rimantas KAČIANAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T). Members: Prof Dr Habil Raimondas ČIEGIS (Vilnius Gediminas Technical University, Technological Sciences, Informatics Engineering – 07T), Prof Dr Rymantas Tadas TOLOČKA (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T), Prof Dr Habil Vladas VEKTERIS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T), Dr Julius ŽILINSKAS (Institute of Mathematics and Informatics, Technological Sciences, Informatics Engineering – 07T).
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| Publié par | vilnius_gediminas_technical_university |
| Publié le | 01 janvier 2008 |
| Nombre de lectures | 64 |
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VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Dmitrij ŠEŠOK TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES USING GENETIC ALGORITHMS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)
Vilnius 2008
Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2004–2008. Scientific Supervisor Prof Dr Habil Rimantas BELEVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T). The dissertation is being defended at the Council of Scientific Field of Mechanical Engineering at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Rimantas KAČIANAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T). Members: Prof Dr Habil Raimondas ČIEGIS Gediminas Technical (Vilnius University, Technological Sciences, Informatics Engineering – 07T), Prof Dr Rymantas Tadas TOLOČKA University of (Kaunas Technology, Technological Sciences, Mechanical Engineering – 09T), Prof Dr Habil Vladas VEKTERIS Gediminas Technical (Vilnius University, Technological Sciences, Mechanical Engineering – 09T), Dr Julius ŽILINSKAS (Institute of Mathematics and Informatics, Technological Sciences, Informatics Engineering – 07T). Opponents: Prof Dr Habil Juozas ATKOČIŪNAS Gediminas Technical (Vilnius University, Technological Sciences, Mechanical Engineering – 09T), Prof Dr Habil Rimantas BARAUSKAS(Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T). The dissertation will be defended at the public meeting of the Council of Scientific Field of Mechanical Engineering in the Senate Hall of Vilnius Gediminas Technical University at 2 p. m. on 16 June 2008. Address: Saul5tekio al. 11, LT-10223 Vilnius, Lithuania. Tel.: +370 274 492, +370 274 496; fax +370 270 0112; e-mail:doktor@adm.vgtu.lt The summary of the doctoral dissertation was distributed on 16 May 2008. A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saul5tekio al. 14, LT-10223 Vilnius, Lithuania). © Dmitrij Šešok, 2008
VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS Dmitrij ŠEŠOK SANTVARŲ TOPOLOGIJOS OPTIMIZAVIMAS GENETINIAIS ALGORITMAIS Daktaro disertacijos santrauka Technologijos mokslai, mechanikos inžinerija (09T)
Vilnius 2008
Disertacija rengta 2004–2008 metais Vilniaus Gedimino technikos universitete. Mokslinis vadovas prof. habil. dr. Rimantas BELEVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Disertacija ginama Vilniaus Gedimino technikos universiteto Mechanikos inžinerijos mokslo krypties taryboje: Pirmininkas prof. habil. dr. Rimantas KAČIANAUSKAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Nariai: prof. habil. dr. Raimondas ČIEGIS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, informatikos inžinerija – 07T), prof. dr. Rymantas Tadas TOLOČKA technologijos (Kauno universitetas, technologijos mokslai, mechanikos inžinerija – 09T), prof. habil. dr. Vladas VEKTERIS Gedimino technikos (Vilniaus universitetas, technologijos mokslai, mechanikos inžinerija – 09T), dr. Julius ŽILINSKAS (Matematikos ir informatikos institutas, technologijos mokslai, informatikos inžinerija – 07T). Oponentai: prof. habil. dr. Juozas ATKOČIŪNAS(Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos inžinerija – 09T), prof. habil. dr. Rimantas BARAUSKAS technologijos (Kauno universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Disertacija bus ginama viešame Mechanikos inžinerijos mokslo krypties tarybos pos5dyje 2008 m. birželio 16 d. 14 val. Vilniaus Gedimino technikos universiteto senato pos5džių sal5je. Adresas: Saul5tekio al. 11, LT-10223 Vilnius, Lietuva. Tel.: (8 ) 274 492, (8 ) 274 496; faksas (8 ) 270 0112; el. paštasdoktor@adm.vgtu.lt Disertacijos santrauka išsiuntin5ta 2008 m. geguž5s 16 d. Disertaciją galima peržiūr5ti Vilniaus Gedimino technikos universiteto bibliotekoje (Saul5tekio al. 14, LT-10223 Vilnius, Lietuva). VGTU leidyklos „Technika“ 1481-M mokslo literatūros knyga. © Dmitrij Šešok, 2008
General characteristic of the dissertation Topicality of the problem. Truss systems are widely used in engineering practice. These systems form the framework of such constructions as bridges, towers, roof supporting structures, etc. The ability to design the rational structure in short terms is clear economical demand therefore the engineer must have at his disposal the methodology of optimization of such structures. Nowadays the good-performing optimization algorithms for sizing optimization of truss systems’ trusses or for shape optimization of truss systems are elaborated therefore the main attention of this work is devoted to the topology optimization of three-dimensional truss systems, which is obviously insufficiently explored so far. In this work the so-called “ground structure approach” is used for the topology optimization, where the optimal connection scheme between the given set of immovable nodes is sought. The search begins from the connection layout “all-to-all-nodes”; the number of trusses remaining in the final system is not known in advance and is determined by the constraints of problem. In mathematical terms, this is a highly non-convex, global optimization problem. The objective function of the problem is usually the total mass of the system or the total length of all truss elements (in case of uniform cross-section of all trusses this is an equivalent form of the function). The obtained optimal truss system still can be improved by allowing the nodes to wander in the system space, i.e. performing the successive shape optimization of truss system. Also, other step-by-step topology/shape optimization algorithms are potential. However, in this case the question whether the global solution is obtained remains unacknowledged. In this work the topology optimization and shape optimization will be integrated into one step of algorithm, thus increasing the possibility to obtain the better solution. Similar problems of practically significant size still are not solved to date due to the large number of design parameters: the typical truss system includes hundreds of trusses, and each truss may have different cross-section, material, etc. Thus, the solution time may be very long and, moreover, in case of improper solution technique, even unacceptable for engineering practice. Therefore, it is important to choose appropriate optimization algorithm and effective application technique of this algorithm. Review of papers in the field of topology optimization of truss systems showed up that the genetic algorithms (GA) may be promising here. In this work the GA were adapted for the topology and shape optimization of truss systems; these modified GA render better results than the classical GA. The investigation is done in step-by-step manner: starting from the simplest two-
dimensional problems and transiting to the three-dimensional truss systems only after investigation of all algorithmic matters. Aim and tasks of the work.The main objective of the present work is to create a technology enabling to perform the topology and shape optimization of three-dimensional truss systems of relevant for the engineering practice size employing the GA concept. The following tasks have to be performed: 1. Create the technology for simultaneous topology and shape optimization of truss systems using GA. 2. Create the algorithms implementing suggested technology and experimentally investigate their capabilities. 3. solution results depend on the concreteInvestigate, how the parameters of GA: the population size, the crossover and mutation probabilities, and the number of generations; propose the empirical formulas for optimal genetic parameters. 4. Analyse the possibility of application of suggested methods in engineering practice: solve typical three-dimensional truss system optimization problem of relevant for engineering practice size. Scientific novelty 1. The original technology for application of GA concept to the topology optimization of truss systems has been suggested and implemented in original software package. The classical GA has been complemented with an additional operation, purification of the genotype; this algorithm proved to be more effective than the classical GA. 2. The original technology for simultaneous topology and shape optimization of truss systems has been suggested and implemented in software; the mixed GA has been suggested and investigated for these purposes. Methodology of research. The computational solution methods are employed in the present work. The total truss system mass is taken as the objective function for the minimization subjected to the constraints of stability, statical equilibrium and maximum allowed stresses. The magnitude of the objective function is evaluated as well as the control of constraints is performed using the finite element method. The genetic algorithms are employed for the optimization. All the software is created by author of the present work. The solution results were tested using finite element package ANSYS.
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Practical value. The suggested methods can be used for optimization of three-dimensional truss systems of relevant for the engineering practice size. Defended propositions 1. The suggested modification of GA, the purification of genotype enables for more efficient topology optimization of truss systems. 2. The suggested technology of simultaneous topology and shape optimization of truss systems renders better solution than the successive topology and shape optimizations. 3. The suggested empirical formula for dependence of optimal population size on the maximum possible number of nodes in the truss system to be optimized renders the proper population size in advance. The scope of the scientific work.The thesis is presented in Lithuanian. It consists of eight chapters, the first of which is an introduction, and the last one presents the generalization of achieved results and summary. The thesis comprises 108 pages, 44 tables, including 66 illustrations. The list of references comprises 101 items. 1. Introduction The introductory chapter presents the relevance of the problem as well as the objective and the main goals of investigation, scientific novelty and originality. 2. Review of methods for global optimization of truss systems In this chapter the main types of optimization of truss systems are discussed: the sizing, shape and topology optimizations. The main global optimization methods used in engineering practice for optimization of truss systems optimization are described. Also the historical review of optimization of truss systems and the main traits of state-of-the-art research in this field are surveyed. Usually, the global optimization problems require huge computer resources therefore the stochastic (probabilistic) global optimization methods seem to be promising. The main attention is given to the genetic algorithms, which simulate the evolution laws of nature (Goldberg, 1989). Engineers employ the GA for optimization of truss systems for more than 20 years. Among the first works in the field the papers of Rajeev and Krishnamurthy (1992) and Jenkins (1992) should be mentioned.
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The present research is mainly connected to the various modifications of GA which allow for better results than the classical GA. As an example, the papers of I-Cheng Yeh (1999), Guan-Chun Luh and Chung-Huei Chueh (2004), Agarwal and Raich (2006) can serve. After review of literature it is concluded, the problems of global optimization of truss systems of relevant for the engineering practice size are still not solved and the methodology for them has to be developed. Also, so far only a few scientific papers dealt with a simultaneous topology and shape optimization of truss systems. 3. Comparison of classical and modified genetic algorithms for topology optimization of truss systems This chapter describes how to apply the GA concept to the topology optimization of truss systems. At the beginning the finite element method technology and software which obtain all necessary characteristics of truss system (stresses and internal forces in truss elements, displacements of nodes, total mass of system) are described. In the second part of chapter the coding of the truss system into the string of bits and reversal transformation are presented. Later on the solutions of topology optimization of particular truss systems are given. First of all, using the exhaustive, full-search algorithms the global solutions of small-scale problems are obtained; these solutions serve as a benchmark for the classical and modified GAs. The main attention is paid to the modified GA (Fig 1), which introduces one additional phase into classical GA: purification or repair of genotype (Fig 2). Here, besides the selection, crossover and mutation processes, each individual is additionally analyzed. Provided that there are particular trusses with stresses below some threshold values, they are eliminated from the truss system retaining this “improved” individual in the population. Evidently, as an alternative for purification of the genotype, a corresponding constraint can be led into the mathematical model:
σe≥σmin, (1) whereσeis stress in theethtruss, andσminis minimum allowable stress. However, experience gained in optimization of numerous truss systems clearly states, that such a constraint impedes the optimization process, because a number of individuals must be eliminated from each generated population. It
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is obvious from an engineering point of view: at the beginning of the optimization process, the truss system contains usually a fairly large number of elements and the probability to obtain the under-stressed truss is high. The suggested heuristics proved to retain more possibilities for obtaining the global solution, because it supposes only exclusion of certain truss elements, while the whole truss system is kept in optimization process. Technically, the purification of genotype supposes reversion of the values of genes that correspond to the under-stressed trusses.
Fig 1.Scheme of modified GA
Fig 2.Purification of genotype
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The purification of the genotype, i.e. introducing of all in advance known information about the problem into the algorithm, always assists in finding a better solution: better on average results are obtained in shorter on average solution time. 4. Improving of topology optimization results via additional shape optimization In this chapter the technology improving the obtained topology optimization results is described. That can be achieved via additional shape optimization of obtained truss system. In the first part of chapter the technology of truss system shape optimization using GA concept is presented. In the second part the topology optimization problem of 12-node truss system is solved. All the possibilities improving the obtained solution via shape optimization are described. The topology optimization employs the modified GA, while the shape optimization – classical GA. Fig 3 shows the obtained solution results: the best truss system after the topology optimization at the center, and the best one after the additional shape optimization on the right.
Fig 3.12-nodes truss system: initial system, results of topology optimization, results of topology and successive shape optimization The results of research show, the successive shape optimization can slightly improve (usually until %) the topology optimization results. 5. Customizing GA parameters Here the research is undertaken in order to ascertain the recommended population size yielding stable topology optimization results. In previous chapters the 8- and 12-node truss systems were optimized. These numerical examples clearly show that small population sizes do not ensure good and
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stable optimization results; the results stabilize starting from certain “optimal population size”popt. In order to generalize the conclusions on the population size, number different truss systems possessing 16, 20 and 3 possible nodes were optimized. Results on population size of all solved problems were approximated by second-order curve using the least square method. Thus, the recommended population sizeypossible nodes in the truss systemsubject to the number of x is as follows: y= −19.78+4.97x+0.03x2. (2) Graphically the obtained dependency is depicted in Fig 4. It should be noted, the formula (2) is of recommendatory character as it relays on a number of numerical experiments only.
Fig 4.population size on the number of possible nodes in the trussDependency of system 6. Merging together topology and shape optimization This chapter describes the technology for simultaneous topology and shape optimization of truss systems using GA concept. For that the mixed GA (Fig ) is invented.
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