A hybrid power series and artificial bee colony algorithm (PS-ABC) method is applied to solve a system of nonlinear differential equations arising from the distributed parameter model of multiwalled carbon nanotube (MWCNT) cantilevers in the vicinity of thin and thick graphite sheets subject to intermolecular forces. The intermolecular forces are modeled using van der Waals forces. A trial solution of the differential equation is defined as sum of two polynomial parts. The first part satisfies the boundary conditions and does contain two adjustable parameters. The second part is constructed as not to affect the boundary conditions, which involves adjustable parameters. The ABC method is applied to find adjustable parameters of trial solution (in first and second part). The obtained results are compared with numerical results as well as analytical solutions those reported in the literature. The results of the presented method represent a remarkable accuracy in comparison with numerical results. The minimum initial gap and the detachment length of the actuator that does not stick to the substrate due to the intermolecular forces, as important parameters in pull-in instability of MWCNT actuator, are evaluated by obtained power series.
1. Introduction
Multiwalled carbon nanotubes (MWCNTs) have attracted considerable attention among other nanomaterials because of the potential advantages on markedly improved stiffness, strength, and elimination of main failure mechanism [1]. These novel materials can usually be visualized as nanoscale concentric cylinders rolled up by graphene sheets. MWCNTs are produced by different techniques such as chemical vapor deposition, laser ablation, and arc discharge [1, 2]. The nanotubes can provide various ranges of conductive properties depending on their atomic and geometrical structure [3]. The unusual properties of MWCNTs have motivated worldwide engineers to explore their applications in different fields of science [4].
Experimental investigations show that the conductance of CNTs is strongly influenced by the occurrence of buckling [5]. The repeatable transformation between the buckled state and normal state of CNTs produces good potential applications to create devices such as nanotransistors [5], nano-valve, and so forth, [6]. With recent growth in nanotechnology, MWCNTs are increasingly used in developing atomic force microscope (AFM) probes [1, 3, 7, 8] and nanoelectromechanical system (NEMS) switches [9–11].
In the nanoscale, the surface forces play an important role in the design and operation of the MEMS and NEMS devices. The van der Waals force and Casimir force are two significant forces in the scale of nanometers [12]. These forces are basically electromagnetic in nature, and they are important when the separation space between objects is very small [12].
In general, there are two basic approaches to understand the behaviors of CNTs: one is atomistic molecular dynamics simulation and the second is continuum mechanics. However, the molecular dynamic method is very time consuming and computationally expensive for a large-scale system.
In a recent work, Koochi et al. [13] applied a hybrid continuum model to investigate the molecular force-induced buckling of a freestanding MWCNT probes/actuators suspended over graphite sheets. They carried out a fourth order nonlinear ordinary differential equation for buckling of multiwalled carbon nanotube (MWCNT) probes/actuators. Koochi et al. [13] used Adomian decomposition method to obtain a solution for buckling and pull-in in stability of MWCNTs. Although, the results of Adomian decomposition method in comparison with numerical results are acceptable, but the results show that the accuracy of Adomian method near the pull-in instability conditions is decreased.
Many different methods have been developed to solve differential equations. However, the solution of nonlinear differential equations still is a challenge [14, 15]. Recently, artificial intelligence techniques are used to solve nonlinear differential equations and modeling engineering problems [16–21]. Lee and Kang [22] used parallel processor computers to obtain a trial solution for a first order differential equation. Meade and Fernandez [23] applied feed forward neural networks to solve linear and nonlinear ordinary differential equations. Lagaris et al. [15] introduced a new method to solve First order linear ordinary and partial differential equations using artificial neural networks. Malek and Beidokhti [24] applied a hybrid artificial neural network—Nelder-Mead optimization technique to solve high order linear differential equations. A hybrid artificial neural network- swarm intelligence method was used by Khan et al. [14] to solve a nonlinear differential equation.
The goal of an optimization problem can be stated as finding the combination of parameters (independent variables) which maximizes or minimizes the value of one or more dependent variables. The value or function to be optimized is called objective function.
Artificial bee colony (ABC) is one of the optimization algorithms, which is introduced by Karaboga in 2005 [25]. The motivation of this algorithm is the intelligent behavior of honey bees. ABC is a simple method with a few main common control parameters such as colony size and maximum cycle number.
Karaboga and Basturk [26] introduced artificial bee colony as an efficient algorithm for numerical function optimization. Karaboga and Akay [27] performed a comparative study on the ABC. They used ABC for optimizing a large set of numerical test functions. They compared the produced results of ABC algorithm with the results obtained by genetic algorithm, particle swarm optimization algorithm, differential evolution algorithm, and evolution strategies. They reported that the performance of the ABC is better than or similar to those of other population-based algorithms with the advantage of employing fewer control parameters.
In the present study, a combination of power series and artificial bee colony optimization algorithm is applied to obtain a power series solution for the nonlinear ordinary differential equations of MWCNT cantilevers. A remarkable accuracy for the presented method is achieved when the obtained results are compared with numerical results.
2. Mathematical Model
Figure 1 shows the schematic of a typical freestanding MWCNT near a surface consisted of N graphene layers, with interlayer distance d=3.35Å. Consider a MWCNT with the mean radius of RW, the length of L, and multiwall nanotube layers of NW. The gap between MWCNT and the surface is D.
Schematic of an MWCNT suspended over graphite sheets.
2.1. Electrostatic Domain
Based on continuum mechanics, an MWCNT is modeled by concentric cylindrical tubes. Eeff is Young’s modulus of MWCNT and the cross-sectional moment of inertia I is equal to π(Ro4-Ri4)/4 [13]. By applying the Euler theory and neglecting the effect of large displacement (finite kinematics) for L/De>10 [28], the governing equation of a cantilever MWCNT can be defined in the form of following boundary value differential equation [13]:(1a)EeffId4UdX4=qvdW(D-U)
subject to geometrical and natural boundary conditions as
(1b)U(0)=dUdX(0)=0,d2UdX2(L)=d3UdX3(L)=0,
where X is the position along MWCNT measured from the bended end, U is the deflection of MWCNT, and qvdw denote the intermolecular force per unit length of MWCNT. The intermolecular force of qWdv base on double-volume integral of Lennard-Jones potential and some simplification can be represented as
(2)qWdv=4C6σ2π2NNWRWd(D-U+Nd/2)4
for small number of graphene layers [13].
In the above equation, C6=15.2eVÅ6 is the attractive constants for the carbon-carbon interaction [29], and σ≈38nm-2 is the graphene surface density [30]. By substituting (2) in (1a), (1b) and using the following substitutions
(3)x=XL,u=UD+Nd/2,fn=4C6σ2π2NNWRWL4EeffI(D+Nd/2)6,
the dimensionless form of (1a), (1b) can be obtained as follows:
(4)d4udx4=f(1-u(x))5u(0)=u′(0)=0,atx=0,u′′(1)=u′′′(1)=0,atx=1,
where in all equations prime denotes differentiation with respect to x.
3. Artificial Bee Colony Optimization Algorithm
The artificial bee colony (ABC) algorithm was proposed by Karaboga in 2005. This method is a gradient free optimization algorithm for real parameter optimization, which inspired from intelligent foraging behavior of a bee colony [25].
In a natural bee colony, some tasks are performed by specialized individuals. In the ABC algorithm, the colony of artificial bees contains three groups of bees as the employed bees, the onlookers, and the scouts [26]. The onlooker bee is a bee waiting for making decision to choose a nectar source. A bee which is going to the nectar source which it previously visited is an employed bee. A scout bee carries out a random search around the hive. In the ABC algorithm, first half of the colony consists of employed artificial bees and the second half constitutes the onlooker bees. In the method proposed by Karaboga, the number of employed bees is equal to the number of food sources around the hive. The employed bee whose food source is exhausted by the employed and onlooker bees becomes a scout bee. The main steps of the algorithm can be summarized as [26].
Initializing the population which can be strategic or random.
Repeat procedure:
place the employed bees on the nectar sources in the memory;
place the onlooker bees on the nectar sources in the memory;
send the scouts to the search area for discovering new nectar sources.
Until requirements are met.
These specialized bees try to maximize the nectar amount stored in the hive by performing efficient division of labor and self-organization. The basic idea and details about ABC algorithm are explained in [25–27, 31]. In the ABC algorithm, proposed by Karaboga, the position of a nectar source represents a possible solution in the search space for the optimization problem, and the nectar amount of a food source represents the profitability (fitness) of the associated solution.
4. Solution Method
The governing equation of a nanotube cantilever was expressed by (1a), (1b). In order to solve (1a), (1b), assume a discretization of the domain D with m arbitrary points. Here, the problem can be written as the following set of equations:
(5)d4u(xi)dx4-(f(1-u(xi))5)=0,xi∈D,i=1,2,…,m
subject to given boundary conditions in (2).
Let us assume yT(x,v→) as an approximate solution to (1a), (1b) in which v→ is a vector containing adjustable parameters. These parameters (i.e., the adjustable parameters) can be evaluated by minimizing the following sum of squared errors, subject to given boundary conditions in (2)
(6)E(a→)=∑i=1m(d4yT(xi,v→)dx4-f(1-yT(xi,v→))5)2,xi∈[01].
In order to transform (5) to an unconstrained problem, yT(x,a→) can be written as
(7)YT(x,v→)=v1x5+v2x4-(10v1-4v2)x3+(20v1+6v2)x2+x2(x-1)4N(x,v→),
where v1 and v2 are adjustable parameters. N(x,a→) is a power series (N(x,v→)=∑i=0nvi+3xi) which involves adjustable parameters of v3 to vn. Here, (7) is in the form of a power series with adjustable coefficients, and it exactly satisfies the given boundary conditions of (1-b).
Now, an optimization technique can be applied in order to determine the optimal adjustable parameters of yT(x,v→) (i.e., v→) to minimize E(v→) in (6). Here, the ABC algorithm is used to evaluate the adjustable vector parameter of v to minimize (6).
For different sizes of the series (n =5,6,7,…,10), the domain of solution is divided to 21 collocation points with equal spices of 0.05 (xi∈{0,0.05,0.01…1}). This configuration was applied for all solutions in the following text. The control parameters of ABC are essential for obtaining an accurate solution. Increase of colony size increases the calculation time, and decrease of colony size decreases the accuracy of solution and may lead to trap in the local optimums. Table 1 shows the best obtained combination of user-specified parameters of ABC method, which are used for this problem in the following text. This combination of parameters is obtained by trial and error.
Control parameters of ABC algorithm.
Control parameter
Value
Type of initial population
Random
The number of colony size (employed bees + onlooker bees)
150 bees
The food source limit which will be abandoned if no improvement was observed
100 try
The maximum foraging try
450 try
In order to verify the convergence and accuracy of the present method, buckling of a typical nanotube-actuator with f=0.5 is computed for different size of series using PS-ABC method. The obtained results are compared with the numerical data as well as Adomian decomposition results reported by Koochi et al. [13]. Numerical results are obtained using Maple commercial software, which uses a combination of trapezoid as base scheme and Richardson extrapolation as enhancement scheme [32, 33]. In the case of fn=0.5, the tip deflection was numerically obtained as ut=0.08323. The variation of the nanotube cantilever tip deflection (utip), using different selected terms of series is shown in Table 2. This table ensures the convergence and accuracy of the results. As seen, higher accuracy can be obtained by evaluating more terms of the solution u(x). The relative error is computed from
(8)Error=|uAnalytical-uNumericaluNumerical|,
where uAnalytical and uNumerical are the cantilever MWCNT tip deflection computed from analytical method and the tip deflection computed using numerical method, respectively. Here, error indicates the relative error.
A comparison between the evaluated tip deflection of a typical MWCNT cantilever using different terms of PS-ABC method and Adomian decomposition method for f=0.5.
Series size
Tip deflection adomian [13]
Error
Tip deflection PS-ABC
Error
5
0.06250
2.491E-01
0.08278
5.50E-03
6
0.09533
1.453E-01
0.08319
5.13E-04
7
0.07516
9.700E-02
0.08324
6.41E-05
8
0.09350
1.233E-01
0.08323
0
9
0.07737
7.046E-02
0.08323
0
10
0.08842
6.228E-02
0.08323
0
The results of Table 2 show that the PS-ABC with series size of eight almost converged to the numerical result. Thus, the power series size of eight has been selected in the following text for convenience. Comparing this error with the same series size of Adomian method (i.e., eight terms and relative error of 0.12%) shows that the PS-ABC method could compute deflection of cantilever MWCNTs with more accuracy than Adomian method. The obtained power series with eight term for f=0.5 is as follows:
(9)u(x)=0.160624628x2-0.097464959x3+0.018038276x4+0.000022094x5+0.003837749x6-0.002404155x7+0.000580025x8.
5. Results and Discussion
Figure 2 shows the centerline deflection of nanotubes for selected values of parameter f.
Buckling of MWCNT cantilever in the vicinity of graphite sheets for different values of f.
This figure reveals that when the parameter of f is small, the buckling can be neglected, and by increase of f, the beam deflects into the substrate. As seen, the power series solution in comparison with the same size of Adomian series solution (i.e., [13]) is more powerful to simulate the deflection and instability of nanocantilever beams. Figure 3 shows the variation of tip deflection as a function of f. Figures 2 and 3 reveal that the Adomian method underestimated the bucking of nanotubes and Greens method overestimated it while the presented method obtained buckling of nanotubes with very good accuracy in comparison with numerical results.
Comparison between the tip buckling of MWCNT cantilevers obtained by different methods.
6. Conclusions
In this paper, an integration of power series and artificial bee colony optimization method has been utilized in order to obtain a solution for buckling of MWCNT cantilevers subject to small number of graphite layers. The governing differential equation is forth order and highly nonlinear due to the inherent of the van der Waals and electrostatic interactions. A trial solution which exactly satisfies the boundary conditions in the form of a power series with adjustable parameters was introduced. The artificial bee colony optimization algorithm was successfully applied to justify the adjustable parameters of the trial solution. The results of present method are compared with numerical results as well as Adomian decomposition method and Green’s method reported in the literature. It is found that the accuracy of present method remarkably is better than the same size of Adomian power series. Therefore, the PS-ABC method can provide an accurate and stable solution for study of MWCNT cantilevers. The present method can be easily extended to solve other nonlinear boundary value differential equations. The future work can be focused on the comparison between the efficiently of the ABC method and other available optimization methods to tackle the present boundary value differential equation.
Acknowledgments
The authors are grateful to Shahid Chamran University of Ahvaz for its support through this paper.
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