A new structural optimization method based on the harmony search algorithm
Introduction
Structural design optimization is a critical and challenging activity that has received considerable attention in the last two decades. Designers are able to produce better designs while saving time and money through optimization. Traditionally, various mathematical methods such as linear, nonlinear, and dynamic programming have been developed to solve engineering optimization problems. However, these methods represent a limited approach, and no single method is completely efficient and robust for all types of optimization problems.
Some techniques, including the penalty-function, augmented Lagrangian, and conjugate gradient methods, search for a local optimum by moving in a direction related to the local gradient. Other methods apply the first- and second-order necessary conditions to seek a local minimum by solving a set of nonlinear equations. These methods become inefficient when searching for the optimum design of large structures due to the large amount of gradient calculations that are required. Usually, these techniques seek a solution in the neighborhood of the starting point, similar to local hill climbing. If there is more than one local optimum in the problem, the result will depend on the selection of the initial point, and the solution will not necessarily correspond to the global optimum. Furthermore, when the objective function and constraints have multiple or sharp peaks, the gradient search becomes difficult and unstable [1].
The computational drawbacks of mathematical methods (i.e., complex derivatives, sensitivity to initial values, and the large amount of enumeration memory required) have forced researchers to rely on meta-heuristic algorithms based on simulations to solve optimization problems. The common factor in meta-heuristic algorithms is that they combine rules and randomness to imitate natural phenomena. These phenomena include the biological evolutionary process (e.g., the evolutionary algorithm proposed by Fogel et al. [2], De Jong [3], and Koza [4] and the genetic algorithm proposed by Holland [5] and Goldberg [6]), animal behavior (e.g., tabu search proposed by Glover [7] and the ant algorithm proposed by Dorigo et al. [8]), and the physical annealing process (e.g., simulated annealing proposed by Kirkpatrick et al. [9]). In the last decade, these meta-heuristic algorithms, especially the genetic algorithm have been broadly applied to solve various structural optimization problems, and have occasionally overcome several deficiencies of conventional mathematical methods. These include researches by Adeli and Cheng [1], Rajeev and Krishnamoorthy [10], [11], Koumousis and Georgious [12], Hajela and Lee [13], Adeli and Kumar [14], Wu and Chow [15], [16], Soh and Yang [17], Camp et al. [18], Shrestha and Ghaboussi [19], Erbatur et al. [20], and Sarma and Adeli [21]). To solve complicated optimization problems, however, new heuristic and more powerful algorithms based on analogies with natural or artificial phenomena remain to be explored.
Recently, Geem et al. [22] developed a new harmony search (HS) meta-heuristic algorithm that was conceptualized using the musical process of searching for a perfect state of harmony. Compared to mathematical optimization algorithms, the HS algorithm imposes fewer mathematical requirements and does not require initial values for the decision variables. Furthermore, the HS algorithm uses a random search, which is based on the harmony memory considering rate and the pitch adjusting rate (these are defined in the following section), instead of a gradient search, so derivative information is unnecessary. Although the HS algorithm is a comparatively simple method, it has been successfully applied to various optimization problems including the traveling salesperson problem, the layout of pipe networks, pipe capacity design in water supply networks, hydrologic model parameter calibrations, cofferdam drainage pipe design, and optimal school bus routings [22], [23], [24], [25], [26].
This paper proposes a new structural optimization method based on the HS algorithm. Various truss examples, including large-scale trusses under multiple loading conditions with continuous sizing variables (fixed geometry), are presented to demonstrate the effectiveness and robustness of the new method. Although the method proposed in this paper is applied to truss structures, it is a general optimization procedure that can be easily adapted to other types of structures, such as frame structures, plates, and shells.
Section snippets
Harmony search algorithm
The new HS meta-heuristic algorithm was derived by adopting the idea that existing meta-heuristic algorithms are found in the paradigm of natural phenomena. The algorithm was based on natural musical performance processes that occur when a musician searches for a better state of harmony, such as during jazz improvization [22]. Jazz improvization seeks to find musically pleasing harmony (a perfect state) as determined by an aesthetic standard, just as the optimization process seeks to find a
Statement of the optimization design problem
Design objectives that can be used to measure design quality include minimum construction cost, minimum life cycle cost, minimum weight, and maximum stiffness, as well as many others. Typically, the design is limited by constraints such as the choice of material, feasible strength, displacements, eigen-frequencies, load cases, support conditions, and technical constraints (e.g., type and size of available structural members and cross-sections, etc.). Hence, one must decide which parameters can
HS algorithm-based structural optimization and design procedure
Solutions to the size optimization problems described by Eqs. , can be obtained using the HS algorithm process by (a) defining value bounds for the design variables, (b) generating the initial harmony memory (HM), (c) improvising a new harmony, (d) evaluating the objective function under the constraint functions using structural analysis, and (e) updating the initialized HM. Here, we will focus on the HS algorithm mechanism to arrive at optimum values.
Fig. 3 shows the detailed procedure of the
Numerical examples
Standard test cases that have been used in previous truss size optimization papers were considered in this study using a FORTRAN computer program developed to demonstrate the efficiency and robustness of the HS algorithm. These cases include a 10-bar planar truss subjected to a single load condition, a 17-bar planar truss subjected to a single load condition, an 18-bar planar truss subjected to a single load condition, a 22-bar space truss subjected to three load conditions, a 25-bar space
Conclusions
The recently developed HS meta-heuristic algorithm was conceptualized using the musical process of searching for a perfect state of harmony. Compared to gradient-based mathematical optimization algorithms, the HS algorithm imposes fewer mathematical requirements to solve optimization problems and does not require initial starting values for the decision variables. The HS algorithm uses a stochastic random search based on the harmony memory considering rate (HMCR) and pitch adjusting rate (PAR),
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