A hybrid approach based on an improved gravitational search algorithm and orthogonal crossover for optimal shape design of concrete gravity dams
Introduction
Concrete gravity dams have been well-known as an important engineering structure that the economy and safety of these structures depend on an appropriate shape design. Hence, finding a proper shape design of concrete gravity dams is considered as an important problem in design approach of dams. To achieve this purpose, several alternative schemes with various patterns should be selected and modified to obtain a number of feasible shapes. Therefore, the proper shape of dam considering the economy and safety of design, structural considerations, etc. is selected as the final shape [1]. In order to reliably achieve an optimal shape for dams instead of this try and error procedure, optimization techniques have been effectively utilized.
Several interesting researches on the design optimization of arch dams under static and dynamic loads have been carried out and reported in [2], [3], [4], [5], [6], [7], [8], [9]. Although the conventional mathematical models have been utilized for analysis approximation and optimization task in these studies, the effects of arch dam–water–foundation rock interaction have been neglected. In recent years, the optimal shape design of arch dams including dam–water–foundation rock interaction has been developed by few researchers [1], [10], [11], [12], [13], [14]. Recently, a study has also been introduced by Salajegheh et al. [15] so that the shape optimal design of concrete gravity dams including hydrodynamic effects is achieved using hybrid of gravitational search algorithm (GSA) and particle swarm optimization (PSO). A study as updated and revised version of the conference paper [15] has also been introduced by Slajegheh and Khosravi [16] so that shape optimal design of concrete gravity dams including hydrodynamic effects is achieved using the hybrid of GSA and PSO.
Recently, gravitational search algorithm (GSA) as a new meta-heuristic optimization method has been proposed by Rashedi et al. [17]. In GSA, the agents as a collection of masses interact with each other based on the Newtonian gravity and the laws of motion. To find the best solution in the search space, the agents share information using the gravitational force. The high performance and the global search ability of GSA in solving various nonlinear functions have been demonstrated by Rashedi et al. [17]. The GSA method suffers from slow searching speed in the last iterations. Due to this fact, the convergence rate of GSA decreases. To eliminate this drawback of GSA, an improved gravitational search algorithm (IGSA) was introduced by Li and Zhou [18]. The IGSA method is based on the idea of memory and social information of PSO. Although IGSA has the characteristics of high accuracy and stability, it has the shortcomings of premature convergence, low searching accuracy and iterative inefficiency. When an agent in the population finds its current optimal position, the other agents will gather close to it rapidly. Thus, the IGSA algorithm traps into local optimum.
This study proposes a hybrid approach based on IGSA and an orthogonal crossover (OC) operator [19], which is called IGSA-OC. This hybrid approach is introduced to eliminate drawbacks of IGSA, and improve the performance of IGSA. The optimal shape of gravity dams with consideration of dam–water–foundation rock effects is investigated using the proposed IGSA-OC. To find the optimal shape of concrete gravity dams, the interaction effects of dam–water–foundation rock subjected to earthquake loading are considered.
In order to reduce the computational cost of optimization process subjected to earthquake loading, many methods of forecasting equipped with the techniques have been utilized as metamodel. Support vector machine (SVM) proposed by Vapnik [20] as an efficient metamodel has been used for modeling the high non-linear system based on small sample. This metamodel has been applied in many classification and regression problems successfully. SVM based on the structural risk minimization (SRM) rules is superior to artificial neural networks (ANNs), which have been developed the traditional empirical risk minimization (ERM) inductive principle [21]. Also, the problems as over learning, dimension disaster and local minimum are eliminated in SVM [21]. Weighted least squares support vector machine (WLS-SVM) regression was introduced by Suykens et al. [22] to decrease the training computational effort of SVM in the large-scale problem. According to the practice, WLS-SVM is more robust and precise than that of SVM and least squares version of SVM (LSSVM) [21], [22]. In this study, the WLS-SVM regression is utilized to predict the time history response of gravity dams. WLS-SVM can significantly reduce the computing effort of optimization procedure.
To demonstrate the high performance and the global search ability of the proposed IGSA-CO, first, four well-known benchmark functions in literatures are optimized using IGSA-CO. The results of IGSA-CO are compared with the standard GSA and the other modified GSA methods. The numerical results demonstrate the efficiency and robustness of the proposed IGSA-CO. Then, the optimal shape of concrete gravity dams is found using IGSA-CO. Furthermore, the optimal solutions obtained by the proposed IGSA-OC are compared with those of the standard GSA, IGSA and PSO. The optimal results revealed that the proposed IGSA-OC can create a robust tool for effectively optimizing concrete gravity dams.
Section snippets
Optimization problem of concrete gravity dam
The optimization problem of concrete gravity dam subjected to earthquake loading is expressed as follows:where f, gi and t are the objective function, ith constraint from m inequality constraints and the time, respectively. XL and XU are the lower bound and the upper bound of the design variables, X, respectively. T is the earthquake duration.
Constraint handling approaches in conjunction with meta-heuristic optimization methods
Finite element model of dam–water–foundation rock system
In order to consider the effects of fluid-structure interaction in the optimum design, FEM is utilized in this study. In the FEM framework, the discretized dynamic equations of the fluid and structure are simultaneously considered to obtain the coupled fluid-structure equation.
Gravitational search algorithm
Gravitational search algorithm (GSA) was introduced by Rashedi et al. [17] as a new stochastic population based search algorithm based on the law of gravity and mass interactions. In GSA, each agent of the population represents a potential solution of the optimization problem. The ith agent in tth iteration is associated with a position vector, , and a velocity vector, . D is dimension of the solution space. Based on Rashedi et al. [17], the mass of
Orthogonal crossover (OC)
Orthogonal array based orthogonal crossover (OC) as efficient means has been incorporated into evolutionary algorithms (EAs) to solve global optimization problems [19], [32], [33], multi-objective optimization problems [32], [34], and constrained optimization problems [35]. The researches have shown that OC outperform standard and existing crossovers in EAs. In OC, an orthogonal array is integrated into the classical crossover operator so that two parents can be used to generate a set of
The proposed hybrid approach of IGSA-OC
However, IGSA shows significant performance in the initial iterations, drawbacks still exit in IGSA. When an agent in the population finds its current optimal position, the other agents will gather close to it rapidly. Due to locating the position in a local optimum, the agents will not be able to search over again in the solution space. Hence, the algorithm traps into local optimum. Accordingly, this study proposes a hybrid of IGSA and OC, called IGSA-OC, to resolve the aforementioned drawback.
WLS-SVM regression
WLS-SVM is described as the following optimization problem in primal weight space [21]:Subject to the following equality constraints:where is a training data set of n points; xi ∈ Rn and yi ∈ R are input and output data, respectively. Operator φ(.): Rn → Rd is a function which maps the input space into a higher dimensional space, also named feature space, where they can be separated by a hyper-plane with low training error. The
Verification of the proposed IGSA-OC
In order to demonstrate the effectiveness and performance of the proposed IGSA-OC, this method is tested using the four representative standard benchmark functions reported in [38]. These functions have been extensively utilized to evaluate the performance of optimization algorithms. The functions, dimension, the admissible range of the variable and the optimum are summarized in Table 1. All the functions are to be minimized. The first three functions are unimodal functions whereas the last
The shape optimization of concrete gravity dam
In order to investigate the competence of the proposed IGSA-OC for the shape optimization of concrete gravity dams, Pine flat dam located on King's River near Fresno, California is considered as a real-world structure. The properties of the dam structure are 400 ft height with a crest length of 1840 ft and its construction about 9491.94 kip concrete [39], [40]. The S69E component of Taft Lincoln School Tunnel during Kern country, California, earthquake (July 21, 1952) is selected as the
Conclusion
A novel hybrid optimization approach is introduced to find the optimal shape of concrete gravity dams including dam–water–foundation rock interaction subjected to earthquake loading. The proposed optimization approach is based on hybrid of an improved gravitational search algorithm (IGSA) and orthogonal crossover (OC), called IGSA-OC. The purpose of the proposed IGSA-OC is to utilize the advantages of both IGSA and OC methods and increase the probability of finding the global optimum in IGSA.
References (41)
- et al.
Advances in concrete arch dams shape optimization
Appl. Math. Model.
(2011) - et al.
Optimal design of arch dams subjected to earthquake loading by a combination of simultaneous perturbation stochastic approximation and particle swarm algorithms
Appl. Soft. Comput.
(2011) - et al.
GSA: a gravitational search algorithm
Inform. Sci.
(2009) - et al.
Parameters identification of hydraulic turbine governing system using improved gravitational search algorithm
Energy Convers. Manage.
(2011) - et al.
Weighted least squares support vector machine local region method for nonlinear time series prediction
Appl. Soft. Comput.
(2010) - et al.
Weighted least squares support vector machines: robustness and sparse approximation
Neurocomputing
(2002) Theoretical and numerical constraint-handling techniques used with evaluation algorithms: a survey of the state of art
Comput. Method Appl. M
(2002)- et al.
Transient analysis of dam–reservoir interaction including the reservoir bottom effect
J. Fluid Struct.
(2005) - et al.
A modified gravitational search algorithm for slope stability analysis
Eng. Appl. Artif. Intell.
(2012) - et al.
Constitutive model for the triaxial behavior of concrete