Full Length ArticleSizing, layout and topology design optimization of truss structures using the Jaya algorithm
Graphical abstract
Introduction
Metaheuristic algorithms try to achieve a dynamic balance between exploration of search space (i.e. “diversification” phase) and exploitation of accumulated search experience (i.e. “intensification” phase). This balance allows to quickly identify regions of design space containing high quality solutions as well as to bypass regions that either were already explored or are far from global optimum [1], [2], [3].
Metaheuristic optimization methods such as, for example, genetic algorithms (GA) [4], [5], simulated annealing (SA) [6], [7], evolution strategies (ES) [8], [9], ant colony optimization (ACO) [10], [11], particle swarm optimization (PSO) [12], [13], harmony search optimization (HS) [14], [15], artificial bee colony algorithm (ABC) [16], big bang-big crunch optimization (BB-BC) [17], charged system search (CSS) [18], firefly algorithm (FFA) [2], teaching-learning-based optimization (TLBO) [19], flower pollination algorithm (FPA) [20], swallow swarm optimization algorithm (SSO) [21] and water evaporation optimization (WEO) [22], have been successfully used in every field of science and engineering.
Truss structures are very often selected as benchmark design problems to test the efficiency of metaheuristic algorithms. Just to cite a few examples from the extensive optimization literature, GA using different search operators and re-analysis strategies [23], [24], [25], [26], [27], [28], [29], [30], differential evolution with various search schemes [31], [32], [33], [34], [35], SA based on single or multi-level search [36], [37], [38], [39], particle swarm optimization with different modelling of social/individual behavior, combination of global/local best and center of mass particles [40], [41], [42], [43], [44], harmony search optimization with different search and parameter adaptation strategies [45], [46], [47], [48], [49], big bang big crunch big bang-big crunch optimization with different definition of trial designs and infrequent explosions [48], [50], [51], [52], teaching-learning based optimization [53], [54], [55], [56], artificial bee colony algorithm with adaptive penalty approach [57], firefly algorithm [58], [59], cultural algorithm [60], flower pollination algorithm [61], water evaporation algorithm [62], hybrid methods combining two or more metaheuristic algorithms, as well as metaheuristic algorithms and gradient-based optimization [63], [64], [65], [66]. Further information can be found in classical textbooks [2], review papers [67], [68] and studies comparing the relative efficiency of several metaheuristic algorithms in static and dynamic truss optimization problems [69], [70].
The continuously increasing computational power has favored the blooming of new metaheuristic algorithms that are often claimed by authors to be very competitive with the most popular state-of-the-art optimizers. However, finding the global optimum at a reasonably low computational cost for all problems with a limited sensitivity to the selection (i.e. size and composition) of initial population and the setting/adaptation of internal parameters that drive the search process remains an unresolved issue in metaheuristic optimization.
An interesting metaheuristic algorithm that has a very simple formulation and does not require internal parameters is the JAYA algorithm (JA) developed by Rao in 2016 [71]. The JA algorithm was successfully tested on several benchmark functions. Rao and Waghmare [72] later utilized the JA for solving constrained mechanical design problems such as robot gripper, multiple disc clutch brake, hydrostatic thrust bearing and rolling element bearing. The efficiency of the JA with respect to other metaheuristic algorithms was demonstrated also for these test problems. The JA was used also for sizing optimization of a micro-channel heat sink [73] by taking thermal resistance and pumping power as objective functions and micro-channel width, depth and fin width as design variables. Once again JA resulted very competitive or even better than those obtained for TLBO and multi-objective evolutionary algorithms.
The main objective of this study is to evaluate the suitability of the JA algorithm for weight minimization of truss structures. This test suite is selected because of the tremendously large amount of data available in literature, which allows comprehensive and detailed comparisons to be carried out. Sizing optimization problems of trusses with 200, 942 and 1938 elements, and combined sizing-layout optimization problems of trusses with 25, 45 and 47 elements are solved for that purpose. Sizing optimization problems include up to 204 design variables while combined sizing-layout optimization problems include up to 81 design variables. Discrete sizing/layout variables and simplified topology optimization also are considered. The original JA formulation is modified in order to improve convergence speed thus reducing the number of structural analyses required in the optimization.
The results obtained by the JA are compared with those of other state-of-the-art metaheuristic optimization methods including variants of genetic algorithms, differential evolution, simulated annealing, particle swarm, harmony search, big bang big crunch, artificial bee colony, teaching-learning based optimization, cultural algorithm, firefly algorithm, flower pollination algorithm, water evaporation optimization, hybrid particle and swallow swarm optimization, hybrid particle swarm, ant colony and harmony search optimization. Comparisons with gradient-based optimizers also are presented in the article. The performance of JA is evaluated in terms of minimum weight, standard deviation on optimized weight and number of structural analyses required in the optimization process. In all test problems, JA is compared with the best solutions available in metaheuristic optimization literature as well as with commercial software. A statistical analysis of the best, average and worst optimized weights and corresponding standard deviations obtained over independent optimization runs is performed. Results prove that the proposed algorithm is very competitive with the other metaheuristic methods and its convergence speed is similar to gradient-based optimizers.
The paper is structured as follows. The formulation of the design optimization problem for truss structures is recalled in Section 2. The JAYA algorithm is described in Section 3 along with its implementation for truss structure problems. Section 4 describes the test problems and discusses optimization results. Section 5 summarizes the main findings of this study. Sensitivity of JA convergence behavior to population size is analyzed in detail in the Appendix.
Section snippets
Optimization of truss structures
The objective of truss optimization is to minimize the weight of the structure under design constraints such as element stresses and nodal displacements. The sizing optimization problem of a truss structure can be stated as:where the A vector contains the sizing variables (i.e. cross-sectional areas of bars), W(A) is the weight of the truss structure. γi and Li,
The JAYA algorithm
The JAYA algorithm (JA) is a new metaheuristic optimization method proposed in 2016 by Rao [71]. The word “Jaya” means “victory” in Sanskrit. This population based algorithm is based on the concept that the search process should always move toward the best design and avoid the worst design. The search engine continuously tries to get closer to success (i.e. to reach the best design) trying at the same time to avoid failure (i.e. by moving away from the worst design) [73]. A definite strength
Test problems, optimization results and discussion
Six trusses previously optimized with various metaheuristic methods were considered in this study in order to demonstrate the efficiency of the JA algorithm in sizing-layout optimization of truss structures. The design examples regarded the sizing optimization of a planar 200-bar truss, a spatial 942-bar tower and a spatial 1938-bar tower; these test cases included up to 204 design variables. Combined sizing-layout optimization was performed for a spatial 25-bar transmission tower, a planar
Conclusions
A very recently developed metaheuristic optimization method called Jaya algorithm was applied in this study to sizing, layout and topology design of truss structures for the first time ever. Jaya is a non-parametric algorithm that attempts to approach the best design included in the population and simultaneously escape from the worst design. Here, the original formulation of JA was modified in order to improve the convergence speed of the optimization process by reducing the number of required
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