Sizing and topology optimization of truss structures using genetic programming
Introduction
Structural optimization is the act of design and developing structures to take the maximum profit of available resources. This topic has attracted a great deal of interest amongst scholars and has become a challenging and critical research topic in the last decades.
Topological optimum design (TOD) aims to find the optimum layout of a structure for both types of continuous and discrete structures. For the first category, TOD discretizes the design space into small rectangular grids and each rectangle may contain material or void. However, for discrete structures (second category) such as trusses, this approach considers the number of elements and discretizes the design space into joints and the rigid elements which connect the joints and form a structure [1]. Truss optimal design tries to find the optimum distribution of stresses in the truss elements subjected to static or dynamic constraints in the design space.
Truss optimization problems classify into three perspectives: topology optimization seeks to find the optimal connectivity between joints, shape optimization tries to find the optimal coordinates of the joints in the design space and sizing optimization deals with the selection of the optimal cross-section areas for the elements of the structure. The combination of sizing and topology optimization tries to optimize the size of each element while incorporates new elements or eliminate the existing ones.
In each case, truss optimization problems aim to minimize an objective function, commonly the structural weight. These optimization problems are subjected to some constraints such as deflections of joints, element stresses, critical buckling load and natural frequencies.
Two major approaches to consider the design space are ground structure and open domain design. In the first approach, the boundaries of the design problem are well-defined; the ground structure represents the heaviest structure (the worst structure) including all potential connections between joints and all possible joints (with their preliminary position in design space) [2]. However, in the latter approach, the boundaries of the open design domain are ill-defined and no prior knowledge about the connectivity between nodes is known.
Many classical optimization methods such as sensitivity analysis or approximation concepts have been developed for structural optimization [3], [4], [5], [6], [7], [8], [9] which provide efficient solutions for solving truss optimization problems. However, they showed deficient capability to deal with non-small and complex problems [1], [10]; these mathematical methods lack efficacy to represent connectivity of truss elements [11].
Non-gradient and metaheuristic methods of solving structural optimization problems such as evolutionary algorithms have proved their capability in seeking the global optima for these problems [12]. These methods compensate the drawback in the conventional methods in solving non-convex problems; there has been a rapid growth in using evolutionary algorithms for structural optimization in the last two decades [1].
Evolutionary algorithms (EAs) provide a mathematical platform inspired by the natural selection principle of Darwin to solve optimization problems. Many studies have applied different EAs in structural optimization [1]. EAs are computationally expensive in comparison with conventional methods, but this is a slight issue because of the rapid development of computer systems and parallel processing. It led to development of powerful algorithms to handle large scale optimization problems [13], [14], [15].
Hajela and Lin [16] implemented genetic search methods and considered integer, discrete and continuous design variables for optimization of trusses. Hajela et al. [17] used a stochastic search strategy for optimization of load-bearing truss structures. Adeli and Kumar [18] applied a distributed genetic algorithm for optimization of large truss structures using multiprocessor systems. Yang and Soh applied different methods for truss optimization: they combined genetic algorithm and fuzzy rule-based system [19]; they later incorporated tournament selection in genetic algorithm [20] and presented a variable string length genetic algorithm for topology, shape and size optimization of trusses [21]. Deb and Gulati [11] applied a real-coded genetic algorithm for truss optimization and proposed a penalty-based approach for handling constrained truss optimization problem. Kaveh and Kalatjari [22] used the genetic algorithm, force method and some concepts of graph theory for topology and sizing optimization of truss structures. Lee and Geem [23] employed harmony search algorithm [24] for truss optimization; this algorithm inspired by the musical process of searching a perfect state of harmony for the composer. Li et al. [25] presented a hybrid heuristic algorithm composed of harmony search and particle swarm optimization algorithms to find an optimal truss design [26]. Camp [27] applied big bang-big crunch algorithm [28] for truss optimization; this algorithm in the first stage distributes population in the design space and the second stage draws the scattered particles into an order. Luh and Lin [29] used a two-stage ant algorithm [30] for the design of optimal trusses; this algorithm inspired by the foraging behavior of real ant colonies. Li et al. [31] applied their proposed algorithm for optimization of trusses with discrete design variables. Hadidi et al. [32] used a modified artificial bee colony algorithm, this algorithm inspired by the foraging behavior of honeybee swarm. Wu and Tseng [10] applied an adaptive multi-population differential evolution algorithm [33] to find the optimal design of truss structures. Luh and Lin [34] employed a two-stage mixed particle swarm optimization algorithm; first, they found the optimum topology and in the second stage, the sizing of the elements for the obtained topology optimized. Camp and Farshchin [35] used a modified teaching-learning based optimization algorithm for truss optimization; this population-based algorithm mimics a class and the interaction between teacher and students to provide better students and raise the average performance of the class [36]. Azad and Hasançebi [37] presented a refined self-adaptive step size search algorithm for truss optimization. Kaveh et al. [38] proposed a hybrid algorithm composed of swallow swarm optimization [39] and particle swarm optimization for truss optimization. Kaveh and Bakhshpoori [40] proposed water evaporation optimization algorithm and employed it on truss optimization problems. Mortazavi and Toğan [41] used a modified particle swarm optimization algorithm for simultaneous topology, sizing and shape optimization of trusses. Cheng et al. [42] applied a variant harmony search algorithm for truss optimization with discrete design variables.
Genetic Programming (GP) is a prominent evolutionary algorithm which employs computer programs as an individual in its population [43]. GP has been used widely as a tool to solve optimization problems in a variety of domains such as artificial intelligence, machine learning, pattern recognition, neural networks, controller design and circuit design [43]. GP employs tree-structures and evolves more efficiently than EAs which employ typical representation forms such as binary encoding [44].
In structural design and truss optimization, genetic programming has been used sparsely. Soh and Yang proposed a tree-based GP method for simultaneous topology, shape and size optimization of trusses [45]. They incorporated ground structure design space and performed this approach with open domain design space later [46]. They also integrated fuzzy logic based decision approach to control iteration process of proposed algorithm for both design space approaches [47]. Zheng et al. suggested a linear GP structure employing ground structure design space for simultaneous topology, shape and size optimization of trusses, [48].
Recently, Fenton et al. [49], [50] has used Grammatical Evolution which is a type of GP for sizing and topology optimization of structures with ground structure design space and developed their method for ill-defined boundary problems later.
Because GP has the capability of simultaneously structural synthesis and parameter tuning of its individuals, it explores the search space more efficiently than other EAs. and Its unique type of genetic operators avoid the local optima traps in complex problems.
There exists a gap in the literature about the efficacy of simultaneous optimization of sizing and topology of structures using genetic programming. This apparent gap in the literature prompted us to investigate whether GP might be more efficient in sizing and topology optimization of structures than other methods.
This study presents a simultaneous structural sizing and topology optimization by genetic programming (SOGP) for trusses. The proposed method is validated by some truss benchmark examples. This approach is a comprehensive process and can be developed for other types of structures like plates, shells, and frames.
Section snippets
Genetic programming
Genetic programming (GP) is a prominent evolutionary algorithm which adopts computer programs for forming its population, binary trees represent these computer programs. These trees are generated by some branches and leaves which are represented by functions and terminals, respectively [43].
Functions include arithmetic unary or binary operations, standard programming operations, logic functions or custom functions. Terminals include variables, constants, random variables or specific quantities.
Problem description
Variable and problem-specified constraints bound search space in a constrained optimization problem. Fig. 3 depicts that the search space can be split into four regions for a tree (an individual) in GP population.
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Variable boundary constraints: in this region, the tree satisfies the variable boundary constraints, but violates the problem-specified constraints boundary.
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Problem-specified constraints boundary: in this region, the tree satisfies the problem-specified constraints boundary, but
Structural optimization by genetic programming
This section presents the approach for structural optimization by genetic programming (SOGP). The goal is to create a computer program whose output is a truss structure. This truss has the optimal connectivities between joints and optimal cross-section areas of elements to reach the minimum weight in the feasible region of search space.
Results and discussion
We found that genetic programming provides better solutions for the problem of sizing and topology optimization problem. The proposed approach is applied to three problems; the results are compared with methods in the literature such: Conventional methods [4], [5], [9], Genetic algorithm (GA) [11], Harmony Search algorithm (HS) [23], Heuristic Particle Swarm Optimization (HPSO) [25], Ant Algorithm [29], Adaptive Multi-Population Differential Evolution (AMPDE) [10], Grammatical Evolution (GE)
Conclusion
In this paper, a genetic programming approach has been used to solve the optimization problem of sizing and topology optimization of truss structures simultaneously. Every genetic programming individual represented a truss structure and the blind-searched initial population of genetic programming evolved along the evolution process to the potential candidate solution. Genetic programming has the capability of structural synthesis and parameter tuning of its individuals; this feature allows GP
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