A new hybrid CG-GAs approach for high sensitive optimization problems: With application for parameters estimation of FG nanobeams
Graphical abstract
Introduction
Optimization methods are widely used in different fields of sciences. The aims of using these methods are to find optimum design variables or to estimate parameters of a system. More recently, to increase accuracy and decrease computational costs, presentation of new or developed optimization techniques are of interest of researchers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Chuanwen and Bompard [7] combined particle swarm optimization with linear interior point. They used the method to optimize the reactive power in order to improve voltage level and decrease network losses. Xia and Wu [8] introduced a mix of simulated annealing and particle swarm optimization methods for scheduling the flexible job-shop problem. Kelnera et al. [9] presented a hybrid genetic algorithms and the interior point method for optimizing continuous functions. Vosoughi and Gerist [10] presented a new hybrid genetic algorithms and particle swarm optimization sensitivity base technique. They coupled the method with finite element method to detect damage of laminated composite beams via the modal analysis. Vosoughi [11] and Vosoughi et al. [12], [13] used hybrid genetic algorithms and particle swarm optimization methods for crack identification of beams and maximizing natural frequency and buckling load of laminated composite thick plates, respectively. Nguyen et al. [14] and Li et al. [15] presented a hybrid particle swarm and chemical reaction evolutionary optimization methods to introduce an efficient optimization solver. They applied the method to obtain optimum value of continuous functions. Li et al. [16] mixed particle swarm optimization and artificial bee colony methods to introduce an efficient optimization solver for high-dimensional optimization problems. Lopez-Garcia et al. [17] mixed genetic algorithms and cross entropy method to solve continuous optimization functions. They divided the population into two sub-populations and applied each of the optimization techniques on a sub-population, separately. Brewick and Smyth [18] combined pattern-search optimization with clustering algorithm to estimate modal damping ratio of a simulated bridge model. The model was subjected to moving traffic loads.
According to the above reviewed articles, mixed optimization techniques for solving complex optimization problems can be divided into two categories. First, mixing global optimization techniques and second, mixing local and global optimization methods. Among these two categories, the second category can yield to obtain more efficient results and to decrease the computational costs of the optimization solution procedure [9]. But sometimes, employing local optimization methods, which require calculating derivatives of the objective functions are not simple and cannot give global optimum points of the problem. In this study as a first attempt, an approach for solving high sensitive optimization problems is introduced. In this approach, the genetic algorithms and conjugate gradient method as two simple and well-known local and global optimization techniques respectively are used. These two methods are selected and used in this approach to decrease computational costs, to increase accuracy and to find reliable results.
Usefulness and applicability of the presented new approach is investigated by solving parameters estimation of nanobeams. Small scale parameter plays an important rule for analyzing nanostructure problems which has been yield to introduce different methods in literature for obtaining the parameter. A comprehensive literature review for small scale parameter estimation of nanostructures can be found in recently published article by Darabi and Vosoughi [19]. To apply the method for parameters estimation of nanobeams the governing equations of the beam, which are obtained using the first-order shear deformation theory and Eringen nonlocal elasticity theory, are discretized by employing the differential quadrature method (DQM). Then, the frequencies of the beam are obtained. These frequencies are named as measured frequencies and are used as input data. The DQM is coupled with conjugate gradient method to produce a mathematical solver to obtain the volume fraction coefficient and small scale parameter of the FG beams. To find these parameters, objective function of the problem is defined as a root mean square error between the measured frequencies and the frequencies obtained from the mathematical solver. For a specified design variable, which is the volume fraction coefficient in this study, the whole domain of the problem is divided into a number of layers or intervals. At each of these layers the best parameters are chosen using the DQ-CG. Then, a curve is fitted to the obtained optimum points at each layer or interval. Then, the whole domain is divided into a number of subdomains and by using equation of the fitted curve values of the objective function at each subdomain are calculated. After sorting the values of the objective function, minimum value is stored and the GAs is employed to find the best optimum solution of the problem in a circular domain limited to the nearest neighborhoods of the minimum point. If the convergence criterion has not been satisfied, the next minimum point is selected and the coupled DQ-GAs is used to estimate the parameters in a new limited circular domain. This procedure would be continued, till the convergence occurs. More details for the solution procedure is presented in the following sections and applicability, accuracy, usefulness and effectiveness of the presented approach are demonstrated.
Section snippets
The governing equations and solution procedure
Consider a functionally graded nanobeam with length L and thickness h (see Fig. 1). Employing the first-order shear deformation and Eringen’s nonlocal elasticity theories the free vibration equations of the beam can be written as follows [19], [20].
The presented hybrid CG-GAs optimization approach
The details of the presented new hybrid approach are discussed in this section. This approach is used to solve the following constrained optimization problem for estimating the small scale parameter and the volume fraction coefficient of the nanobeam. The objective function and the related constraints arewhere Si and Ei with i = 1,2 are the lower and upper bonds of the design variables.
For minimizing the above objective function, according to the upper and lower
Numerical results
Convergence and accuracy of the differential quadrature method for solving free vibration of the FG nanobeams were performed in our previous study [18]. So, without loss of time, influences of volume fraction coefficient and small scale parameter on free vibration analysis of fully clamped nanobeams are investigated and the results are presented in Table 1, Table 2. From the obtained results, it is obvious that changing volume fraction coefficient of the nanobeam from 0.5 to 10 does not affect
Conclusions
As a first attempt, a new hybrid optimization approach for solving optimization problems with high sensitive objective function is introduced. In this approach the conjugate gradient and genetic algorithms are used to introduce accurate and efficient numerical solvers. To employ these two mixed methods, according to the lower and upper bonds of a design variable, a number of layers are chosen and the conjugate gradient method is implemented to find optimum points of the objective function over
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