Article

Searching for the superior solution to the population-based optimization problem: Processing of the wear resistant commercial AA6061 AMCs

International Journal of Damage Mechanics 2014, Vol. 23(7) 899–916 ! The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1056789513518951 ijd.sagepub.com

Ali Mazahery1,2, Mohsen Ostad Shabani3 and Ahmed Elrefaei2

Abstract In this present work, stir casting technique was used to fabricate aluminum matrix composites with varying weight percentages of SiC (5, 10 and 15) reinforcements. The proper selection of process parameter such as pouring temperature, stirring speed, stirring time and pre-heat temperature of reinforcement can all influence the quality of the fabricated composites. The porosity level of composite should be minimized and the chemical reaction between reinforcement and matrix should be avoided. Optimization is an applied science which explores the best values of the parameters of a problem that may take under specified conditions. The execution of an optimized stir casting technique yields relatively homogenous and fine microstructure which improves the addition of reinforcement material in the molten metal. The influence of SiC content, SiC size and secondary mechanical processing with different rolling reductions on the dry sliding wear characteristics of Al matrix composites has been assessed using a pin-on-disc wear test. The porosity and hardness of the resultant composites were also examined. Keywords SiC, optimization, particle swarm optimization

Introduction Particle swarm optimization (PSO) is a parallel population-based computation technique proposed by Kennedy and Eberhart (1995), which was motivated by the organisms behavior such as schooling of ﬁsh and ﬂocking of birds. PSO can solve a variety of diﬃcult optimization problems (Beasley 1

Light Metal Casting Research Centre (LMCRC), McMaster University, Ontario, Canada Leichtmetallkompetenzzentrum LKR, Austrian Institute of Technology, Austria 3 Karaj Branch, Islamic Azad University, Karaj, Iran 2

Corresponding author: Mohsen Ostad Shabani, Karaj Branch, Islamic Azad University, Karaj, Iran. Email: [email protected]

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et al., 1993; Engelbrecht et al., 2005; Krohling and Mendel, 2009; Mazahery and Shabani, 2012; Milani and Nazari, 2012a; Nazari and Riahi, 2012; Shabani and Mazahery, 2012a). PSO’s major diﬀerence from genetic algorithm (GA) is that PSO uses the physical movements of the individuals in the swarm and has a ﬂexible and well-balanced mechanism to enhance and adapt to the global and local exploration abilities, whereas GA uses genetic operators. Another advantage of PSO is its simplicity in coding and consistency in performance (Brits, 2002; Kennedy, 1999; Kennedy and Eberhart, 2001; Kennedy and Mendes, 2002; Mazahery and Shabani, 2012b; Shabani and Mazahery, 2012b; van den Bergh and Engelbrecht, 2006). GAs are a group of randomized methods used in function optimization (Shabani and Mazahery, 2013a). Early developments in the ﬁeld of GAs are generally credited to Holland and since then it has been successfully applied to various optimization problems. A GA population consists of several individuals, which work in parallel to identify a best solution. Individuals are potential solutions that gradually change or mature over time to converge to an optimal solution. In GA each individual is represented by a binary string of 0 s and 1 s of a ﬁxed length called chromosome, consisting of several genes. A GA starts oﬀ with an initial population of randomly generated chromosomes. During successive iterations, named generations, the initial chromosomes advance towards stronger chromosomes by reproduction among candidate solutions of the previous generation. In the next step, the binary strings are decoded and converted into optimization variable values, using a linear scaling. The objective function is evaluated from the established optimization variable values and a measure of worth or ﬁtness is evaluated. A high ﬁtness value would indicate a better solution than a low ﬁtness value. In the natural world, this would be an individual’s ability to survive in its present environment (Kennedy, 1999; Kennedy and Eberhart, 2001; Kennedy and Mendes, 2002; Mazahery and Shabani, 2012b; Shabani and Mazahery, 2013a; van den Bergh and Engelbrecht, 2006). During the last decade novel computational methods have been introduced in some ﬁelds of engineering sciences, including the materials science. Metal matrix composites (MMCs) are gaining wide spread popularity in several technological ﬁelds owing to its improved mechanical properties when compared with conventional metals/alloys (Bauri and Surappa, 2009; Blumenthal et al., 1994; Mazahery and Shabani, 2012c, 2012d; Pyzik and Aksay, 1989; Rhee, 1970; Shabani and Mazahery, 2012c, 2012d, 2012e; Xiao, 2010). Among the several categories of MMCs, Al based composites are ﬁnding wide spread acceptance especially in applications where weight and strength are of prime concern (Karnezis et al., 1998; Mazahery and Shabani, 2012e; Razavi et al., 2008; Shabani and Mazahery, 2012f; Tian and Fu, 2011; Xu et al., 2012). Weight reduction of vehicles to reduce fuel consumption and CO2 emissions has driven developments to extend the use of aluminum alloys by the automobile industry. Several wrought AA6xxx alloys based on the Al-Mg-Si system are available for car body outer panels, but the use of Al-Mg-Si is still a challenge for castings. Their lack of ductility has resulted in cracking during riveting in production and has limited their use in industry. Increased ductility of die cast Al-Mg-Si based casting alloys is therefore of critical interest (Shabani and Mazahery, 2011a; Asthana, 1998). Another major limitation is proved to be the low wear resistance (Leon and Drew, 2000; Mazahery and Shabani, 2012f; Moustafa et al., 2002; Shabani and Mazahery, 2013b; Thakur and Dhinan, 2001; Tham et al., 2001). In recent years, investigators have successfully attempted to improve the tribological properties by reinforcing it with high modulus ceramic particulates such as SiC, thus making it an increasingly suitable candidate for even wider spectrum of applications (Cocen and Onel, 2002; Mazahery and Shabani, 2011, 2012g; McKimpson and Scott, 1989; Varma et al., 2000; Zhong, 2000). To fully utilize the potential of a A6061/ceramic composition, the judicious selection of primary and secondary processing and heat treatment procedure becomes important. It has been demonstrated, for example, that the diﬀerent primary processing techniques are

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capable of imparting totally diﬀerent microstructural features and mechanical properties to a material of given composition. The microstructural and properties variation for a ﬁxed metal/ceramic formulation can also be realized by varying the secondary processing parameters such as reduction percentage in the case of rolling. Although the eﬀects of various parameters have been critically investigated for monolithic and composites materials, no attempt has been made to model and optimize the microstructure and mechanical characteristics of the AA6061/SiC composites. Particular emphasis was given in this study to optimization of the grain morphology and SiC particulates redistribution within the matrix.

Experimental procedure Al6061 alloy matrix composites reinforced with SiC particulates were synthesized through a semisolid processing route. In order to achieve sound quality composite, the furnace is placed on the ﬂoor and the temperature of the furnace is precisely measured and controlled. The process involved melting the alloy in the graphite crucible (Quaak and Kool, 1994). Stirring was done via an impeller made of graphite and driven by a variable ac motor. The crucible was heated to about 680 C in a resistance furnace and after melting the aluminum. SiC particles with average sizes of 1, 5, 20 and 50 mm as reinforcing particles were added to the melt and stirred at 400/500/600/700 r/min. After the addition of SiC during the process, the stirring time were noted at 5, 10 and 15 min. As stirring continued, the furnace temperature was gradually lowered until the melt reached a temperature in the liquid–solid state. In order to minimize the oxidation of molten aluminum, nitrogen gas is used inside and outside of the crucible. Steel die was placed below the furnace. The casting was obtained by pouring composite slurry into the die. The obtained casts were annealed at 530 C for 2 h, machined to samples of 150 mm length, 50 mm width and 10 mm thickness and cold rolled to different ﬁnal reduction. Pin-on-disc type wear tester was used for dry sliding wear tests. The test was done at sliding velocity of 0.3 m/s within a load range of 10–40 N. Figure 1 shows schematic diagram of the abrasion wear test. The disk was made of the steel hardened up to 63 HRC with a diameter of 50 mm and a thickness of 10 mm. The pin test sample dimensions were 6 mm diameter and 25 mm length. At a given load level, weight loss from the worn surfaces was found to increase linearly with sliding distance, except during the transient period at the beginning of the test. The wear rates were calculated from the slope of weight loss versus sliding distance curves determined at several applied load levels within the range of 10–40 N. The wear rates measured in weight units were then converted to volumetric wear rates.

Figure 1. Schematic diagram of the abrasion wear test.

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The experimental density of the composites was obtained by the Archimedes method of weighing composites ﬁrst in air and then in water, while the theoretical density was calculated using the mixture rule according to the volume fraction of the SiC particles. Metallographic samples were prepared using standard metallographic techniques, etched with standard aluminum etching solutions and examined by a optical microscope to determine the distribution of the SiC particles. The volume fraction of SiC particles was measured by means of an image analyzer system attached to the microscope. The tensile test samples were machined according to the ASTM E8M standard and oriented along the rolling direction.

Genetic algorithm GA is a computational technique, which uses principles of Darwinian natural selection in order to solve a wide variety of problems (Shabani and Mazahery, 2013b). Based on the idea of survival of the ﬁttest and natural selection, GA is a class of parallel iterative algorithm with speciﬁc learning ability, which repeats evaluation, selection, crossover and mutation after initialization until the stopping condition is satisﬁed. As originally proposed, a simple GA usually consists of three processes selection, genetic operation and replacement (Shabani and Mazahery, 2012b). The population comprises a group of chromosomes that are the candidates for the solution. The ﬁtness values of all the chromosomes are evaluated using an objective function in a decoded form. A particular group of parents is selected from the population to generate oﬀspring by deﬁned genetic operations of crossover and mutation (Mazahery and Shabani, 2012a). The ﬁtness of all the oﬀsprings is then evaluated using the same criterion and the chromosomes in the current population are then replaced by their oﬀsprings, based on a certain replacement strategy. Such a GA cycle is repeated until a desired termination criterion is reached. If all go well throughout the process of simulated evolution, the best chromosome in the ﬁnal population can become a highly evolved and more superior solution to the problem (Mazahery and Shabani, 2012h; Petalas et al., 2009; Sathiya et al., 2009; Shabani et al., 2011; Shabani and Mazahery, 2011b; Wu et al., 2010; Zhao and Suganthan, 2011). According to the natural features, especially, the population-size-changing rule on human beings, this paper presents a novel improved GA with variable population-size which obtains a better balance between the variety and continuity. The main idea of present GA is described as follows. Of most feature species, parents are neither dead after their reproduction right away, nor living forever. In fact, the individual is apt to die when it is old. In present GA each individual is designated a dying probability according to its living generations. Deﬁne DIE_PROBABILITY [Q] as the dying probability of those individuals living Q generations. For example, considering the expected max living generations is 3, we could set the DIE_PROBABILITY [0, 1, 2, 3] to be 0, 0.3, 0.7 and 1, respectively. After the performance of the traditional genetic operations, the individual should undergo a dying process. In the dying process, each individual is determined whether to die according to its DIE_PROBABILITY. On the other hand, the size of features’ population always increases, until there is a contagious disease or a war breaking out among the population. In general, the disease or the war will reduce the size of the population sharply. In present GA, the size of the population is reduced to the initial size when it reaches the given limitation, of which the individuals with higher ﬁtness have more opportunities to survive (Mazahery and Shabani, 2012h).

Particle swarm optimization The PSO was originally designed by Kennedy and Mendes (2002) and Kennedy and Eberhart (1995, 2001) and has been compared to GAs for eﬃciently seeking optimal or near-optimal solutions in

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large search spaces. The PSO algorithm conducts search using a population of particles which correspond to individuals in a GA (Kennedy, 1999). A population of particles is initially randomly generated. Each particle represents a potential solution and has a position represented by a position vector. A swarm of particles moves through the problem space, with the moving velocity of each particle represented by a velocity vector. At each time step, a function representing a quality measure is calculated by using as input. Each particle keeps track of its own best position, which is associated with the best ﬁtness it has achieved so far in a vector (Kennedy and Eberhart, 2001). Furthermore, the best position among all the particles obtained so far in the population is kept track as output. In addition to this global version, another local version of PSO keeps track of the best position among all the topological neighbors of a particle. At each time step, by using the individual best position and global best position, a new velocity for particle is updated. The computation for PSO is easy and adds only a slight computational load when it is incorporated into the conventional GA (Kennedy and Eberhart, 1995). Furthermore, the ﬂexibility of PSO to control the balance between local and global exploration of the problem space helps to overcome premature convergence of elite strategy in GA, and also enhances search ability (Kennedy and Mendes, 2002). PSO ﬁrst initializes a group of random particles and then ﬁnds the optimal solution by multiple iterations. Suppose that the searching space is D-dimensional and m particles form the colony (Kennedy and Eberhart, 2001). The ith particle represents a D-dimensional vector Xi (i ¼ 1, 2, . . . ,m). It means that the ith particle locates at Xi ¼ (xi1, xi2, . . . ,xiD) (i ¼ 1, 2, . . . ,m) in the searching space. The position of each particle is a potential result. We could calculate the particle’s ﬁtness by putting its position into a designated objective function. When the ﬁtness is higher, the corresponding Xi is ‘‘better’’. The ith particle’s ‘‘ﬂying’’ velocity is also a D-dimensional vector, denoted as Vi ¼ (vi1, vi2, . . . ,viD) (i ¼ 1, 2, . . . , m). Denote the best position of the ith particle as Pi ¼ (pi1, pi2, . . . ,piD), and the best position of the colony as Pg(pg1, pg2, . . . ,pgD), respectively. The PSO algorithm could be performed by the following equations: Xikþ1 ¼ Vikþ1 þ Xik

ð1Þ

Vikþ1 ¼ wk Vik þ c1 r1 ð pik Xik Þ þ c2 r2 ð pgk Xik Þ

ð2Þ

Where k represents the iterative number, w is the inertia weight, c1 and c2 are learning rates, r1 and r2 are random numbers between 0 and 1, Vi 2 [Vmin,Vmax] where Vmin and Vmax are the designated vectors. The termination criterion for the iterations is determined according to whether the max generation or a designated value of the ﬁtness of Pg is reached (Kennedy and Mendes, 2002).

Modeling Compared with GA, PSO has a much more profound intelligent background and can be performed more easily. Given that particle optimization mainly evolves via the comparison of the self position, the surrounding positions and the global positions of all the particles as a singleton pattern, PSO may prematurely converge. A good diversity must have a strong ability to ﬁnd the best position; otherwise, the undesired local solution may appear when the population is not well diversiﬁed. Using diﬀerent evolution mechanisms, such as crossover and mutation, GA can enhance the diversity of results, but, to some extent, it leads to a large number of redundant iterations, resulting in long computing times and low problem-solving eﬃciency. Several researchers have combined PSO and GA to form diﬀerent hybrid algorithms. Shi combined a variable population-size genetic algorithm (VPGA) and PSO to form a hybrid algorithm

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that generates initial populations for VPGA and PSO according to a certain proportion. The new population is obtained by evolution according to the corresponding algorithm rules. Gandelli randomly divided the total population into two subpopulations and evolved the two subpopulations using GA and PSO operations, respectively. Kao proposed a GA–PSO hybrid algorithm for geometric proportional populations. Kuo integrated the mutation mechanism of GA with PSO. Valdez combined GA and PSO using fuzzy logic to integrate the results of both methods and for parameter tuning (Beasley et al., 1993; Brits, 2002; Engelbrecht et al., 2005; Kennedy, 1999; Kennedy and Eberhart, 1995, 2001; Kennedy and Mendes, 2002; Krohling and Mendel, 2009; Mazahery and Shabani, 2012a, 2012b; Milani and Nazari, 2012; Nazari and Riahi, 2012; Shabani and Mazahery, 2012a, 2012b; van den Bergh and Engelbrecht, 2006). The hybrid algorithms above have evolution rules for a certain proportion of the population and only implement information transfer for particles after one generation or just use one GA operator. In fact, this is a formal hybrid and cannot exchange information many times in the PSO or in the GA evolution process. These approaches cannot fully exploit a complement of two algorithms. Therefore, the present study proposes a new PSO–GA hybrid algorithm. The PSO–GA hybrid algorithm diﬀers from previous hybrids in the following ways: . The GA and the PSO approaches result in a hybrid hierarchy. First, all particles in the population are evolved by the PSO with smaller iterations, and then M-optimized particles are selected for the implementation of genetic operators . Two information transfers are accomplished. The optimized particles after the PSO form an initial GA population, and the PSO updates the velocity and the position of all particles after genetic operations. Better optimize the coeﬃcients, an eﬀective hybrid optimization algorithm based on PSO and GA is developed. First, the population is evolved over a certain number of generations by PSO and the best M particles are retained; the remaining pop_size_M particles are excluded. Second, pop_size_M new individuals are generated by GA operators for the remaining best M particles. Finally, pop_size_M new individuals are combined with the remaining best M particles to form a new population for the next generation. (1) Relevant parameters, such as the number of particles (pop-size), the number of particles retained after evolution by PSO (M), the PSO weight coeﬃcients (c1 and c2), the crossover and mutation probabilities for GA (pc and pm), the maximum particle velocity (Vmax), the number of PSO generations evaluated (kmax) and the number of generations for the hybrid algorithm (max-gen), are initialized. (2) The pop-size initial particles and their ﬁtness values are generated and calculated, respectively (3) Let gen ¼ 1. (4) If gen max-gen, step 5 is carried out; otherwise, step 4 is carried out. (5) Let k ¼ 1. (6) If k max-k, step 7 is carried out; otherwise, step 9 is carried out. (7) The position and the velocity of the particles are updated according to equations (1) and (2). (8) Let k ¼ k þ 1, then return to step 6. (9) The pop-size particles are ranked according to ﬁtness, and the M particles with the lowest ﬁtness values are selected. (10) The pop-size M individuals obtained by GA and M particles are combined to form new pop-size particles. (11) Let gen ¼ gen þ 1, then step 4 is carried out. (12) The best ﬁtness values and solutions, namely, the position, are outputted.

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Figure 2 shows the ﬂowchart of combined GA-PSO model which has been used in this investigation. PSO is an eﬀective algorithm which gives quality solutions in a reasonable computational time and consists of less numbers parameters as compared to the other evolutionary metaheuristics. Mutation, a commonly used operator in GA, has been introduced in PSO so that trapping of solutions at local minima or premature convergence can be avoided.

Figure 2. The flowchart of the model used in this study.

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It is well known that the unreinforced matrix alloy wear more rapidly than the reinforced composite materials. A number of studies have been engaged during the last 20 years on wear behavior of Al based particulate reinforced composite. Reinforcement of Al based alloy with SiC are usually found to improve the wear resistance under both abrasion and lubricated sliding conditions. Experimental results of the wear rate and hardness test are presented in Figure 3. It is noted that the addition of hard ceramic SiC particles increases the hardness of Al alloy. Figure 4 shows the wear rate and hardness results of cold rolled composites, respectively. It is observed that there is an improvement in wear of the cold rolled composites which might be explained by the increase in the hardness of them. Fabrication of these composites via compocasting technique plus reduction (90%) during cold rolling process lead to reasonably uniform distribution of particles in the matrix and minimum clustering or agglomeration of the reinforcing phase and thus higher hardness and wear resistance. Figure 5 shows the eﬀect of the rolling process on the porosity content of the composites. It is noted that the application of the heavy reduction (90%) during cold rolling process results in approximately 0% volume fraction porosity. The porosity level increased, since the contact surface

Wear Rate (10-12m3/m)

(a) 2.7 2.4

Unreiforced Al

5% SiC

2.1

10% SiC

15% SiC

1.8 1.5 1.2 0.9 0.6 0.3 0 1.0E+3 1.5E+3 2.0E+3 2.5E+3 3.0E+3 3.5E+3 4.0E+3

Sliding distance (m)

Hardness (HRC)

(b) 90 85 80 75 70 65 60 0

3

6

9

12

15

Volume fraction of SiC

Figure 3. a: Effect of the sliding distance on the wear rate of Al–SiC composites under an applied load of 10 N and b: hardness.

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Wear Rate (10-12m3/m)

1.2 1.1 1 0.9 0.8

%5 SiC

%10 SiC

%15 SiC

0.7 0

10

20

30

40

50

60

70

80

90

Reduction (%) (b) 2.9 5% SiC

2.7

10% SiC

15% SiC

Wear Rate (10-12m3/m)

2.5 2.3 2.1 1.9 1.7 0

10

20

30

40

50

60

70

80

90

Reduction (%)

Figure 4. The effect of the rolling process on the wear of the composites under the applied load of: (a) 10 N, (b) 20 N, (c) 40 N and hardness of the composites (d).

area was increased. This is attributed to pore nucleation at the SiC particulate sites and to hinder liquid metal ﬂow due to more particle clustering. The mechanical property results of the composites are shown in Figure 6. It is noted that there would be a great enhancement in tensile behavior of the composites while using rolling reduction in comparison to the as-cast composites. It can also be seen that tensile strength increases when the reduction quantity is increased. It is observed that there is a simultaneous improvement in the strength and ductility of the cold rolled composites which might be explained by the reduction in porosity content. A more detailed analysis of the eﬀect of c1 and c2 parameters is necessary to determine the sensitivity of the parameters in the overall optimization procedure. The eﬀect of the cognitive and social parameters on the general optimization history can be seen in Figure 7 where objective function values versus iteration number is shown.

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Volume % Porosity

5 4 3 2 1

5% SiC

10% SiC

15% SiC 0 10 20 30 40 50 60 70 80 90

0

Reduction (%)

Figure 5. The effect of the rolling process and SiC volume fraction on the porosity content of the composites. (a) 350

UTS (MPa)

300 5% SiC 10% SiC 15% SiC

250 200 150 100 50 0 0

10 20 30 40 50 60 70 80 90

Reduction (%)

Elongation (%)

(b)

8 7

5% SiC

6

10% SiC 15% SiC

5 4 3 2 1 0 0

10 20 30 40 50 60 70 80 90

Reduction (%)

Figure 6. The effect of the rolling process and SiC volume fraction on a: the tensile strength and b: the elongation of the composites.

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Objective Values

10

C1 1=0

C1 = 1

C1 1=3

C1 = 4

C1 = 2

8 6 4 2 0 +0 0.0E+

2.0E+3

44.0E+3

6.0E+3

Numbers Of Iteration

Figure 7. Objective function values vs. iteration number of cognitive and social parameters.

The parameters were varied according to the formula c2 ¼ 4 c1, where c1 was changed in the interval [4, 0]. It can be seen clearly that when only cognitive c1 ¼ 4 or only social values c2 ¼ 4 are used the resulting history converges very fast, within the ﬁrst iterations, but do not improve after the initial convergence, which indicate that the algorithm reaches a local suboptimal and is not able to escape from it, due to the lack of information exchange either from social or cognitive sources, respectively. When higher emphasis is placed on the social exchange of information (c1 ¼ 1, c2 ¼ 3) the algorithm is able to gradually converge to better regions of the design space, but again if a local optimum is found it cannot escape from it. This fact indicates an overshooting of the algorithm where particle positions are updated based mostly on social information not promoting the exchange of individual information which could be useful when an individual particle has found a better region of the design space. When the cognitive and social parameters are in balance or with slightly higher cognitive value, solutions converge near the global optimum solution; these two situations are shown by the gradual decrease of objective function values and subsequent improvements over time. Notice that a slightly higher value for the C2 results in better solutions. This is the result of the fact that promoting some global exchange of information causes the swarm to point toward the best regions, but individuals concentrate more in their own search regions and avoid overshooting the best design space region.

Inertia weight variation effect. The concept of linearly decreasing inertia weight was introduced by van den Bergh and Engelbrecht (2006) and is given by: w ¼ wmax

wmax wmin iter itermax

ð3Þ

where iter is the current iteration number, while itermax is the maximum number of allowable iteration. Usually the value of w is varied between wmax ¼ 0.9 and wmin ¼ (0.2, 0.3 or 0.4).

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Objective Values

10

w_min = 0.3 w

w w_min = 0.2

w_min = 0.4 w

w = 0.7

8 6 4 2 0 +0 1.0E+3 2.0E+3 3.0E+3 44.0E+3 5.0E+ +3 6.0E+3 7.00E+3 0.0E+

Numbers Of Iteration

Figure 8. Objective function values vs. iteration number of inertia weight.

Therefore, the particle is to use lager inertia weight during the initial exploration and gradual reduce its value as the search proceeds in further iterations. Results are given in Figure 8. Based on this ﬁgure, the best global optimum solution is reached by wmin ¼ 3. Notice that ﬁxed inertia weight allow for a fast convergence to sub-optimal regions of the design space while linearly decreasing inertia weight updates converge slowly, due to the initial global search nature force by higher cost function update values, and gradually converges to optimal regions of the design space approximately after latest iteration. Figure 9 illustrates the convergence behaviors of the three methods. It is noted that the GA–PSO converges faster than the conventional GA and PSO approach. In addition, the computation time for GA optimization method is longer compared with PSO methods. Several experiments were conducted to determine the best parameter values of the proposed algorithm. The total number of runs for each combination of parameter values is 30. Figure 10 shows performance of three algorithms. As can be seen, GA–PSO is more repeatable than PSO and GA algorithms. GA–PSO has more global search ability at the end of evolution, which is required to jump out of the local optima in some cases. Goldberg suggested that appropriate values of user speciﬁed GA parameters can be selected as crossover probability (Pc) (0.5–0.6) and mutation probability (Pm) (0.01–0.02). The proposed method uses a GA operator to improve the diversity of PSO particles. Therefore, larger probabilities of crossover and mutation are set, namely, Pc ¼ 0.7 and Pm ¼ 0.05. Although better particles may degenerate as probabilities Pc and Pm become larger, best M particles Q1, Q2, . . . ,Qm are retained without genetic implementation in the proposed algorithm. The eﬀects of momentum factor and learning rate on the performance of the developed model are shown in Figure 11. As can be seen, the optimum values of momentum factor and learning rate are 0.4 and 0.6, respectively. These parameters were chosen to ensure convergent trajectories. GAs and PSO are much similar in their inherent parallel characteristics, whereas experiments show that they have their speciﬁc advantages when solving diﬀerent problems. What we would like to do is to obtain both their excellent features by synthesizing the two algorithms. Final optimized parameters for the mechanical and tribological properties in applied load 10 N and 90% reduction are 668 s stirring time, 645 r/min speed of stirrer, 27.81 mm particle size of SiC,

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Objective Values

10

PSO-GA

PSO

8 6 4 2 0 0.0E+0

2.0E+3

4.0E+3

6.0E+3

Numbers Of Iteration

Figure 9. Objective function values vs. iteration number of the three methods.

6

Objective Values

PSO O-GA

PSO

GA

4

2

0 1

5

9

13

17

21

25

29

Number Figure 10. Objective function values vs. number of runs for each method.

11.03% volume percentage of SiC, 0.48% porosity, 83.8 HRC hardness, 303.9 MPa ultimate tensile strength (UTS), 6.66% elongation and 1.11% 1012 m3/m wear rate. Final optimized parameters for applied load 20 N and 90% reduction are 688 s stirring time, 679 r/min speed of stirrer, 34.11 mm particle size of SiC, 11.14% volume percentage of SiC, 0.42% porosity, 84.1 HRC hardness, 313.2 MPa UTS, 6.55% elongation and 2.2% 102 m3/m wear rate. Porosity formation in cast MMCs can be attributed to the air trapped in the clusters of particles as well as the hindered metal ﬂow inside them. Increasing amount of porosity is observed by

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MAPE

6 5 4 3 2 1 0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.9

1.0

Momentum Factor (b) 12

MAPE

10 8 6 4 2 0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Learning Rate Figure 11. The effects of (a) momentum factor and (b) learning rate on the performance of the developed model.

increasing the volume fraction of SiC particulates. The decreased porosity of MMCs during rolling is due to the ﬂow of the matrix alloy under the applied shear and compressive forces resulting in ﬁlling the voids. The increased rolling reduction provides easier ﬂow of the matrix alloy and hence results in decreased porosity. In fact, the major reason for rolling the particulate metal matrix composites (PMMCs) is to close the pores and attain improved mechanical properties. The wear loss is inversely proportional to the hardness of alloys. The wear rate of the unreinforced alloy is found to be higher than that of the composites. Shabani and Mazahery emphasized that higher hardness of composite could be achieved by ceramic reinforced particulate (B4C) because B4C particle acts as an obstacle to the motion of dislocation. In case of unreinforced Al alloy, the depth of penetration is governed by the hardness of the specimen surface and applied load. Iwai found that both mild wear in the initial sliding distance and severe wear rate decreased with increasing volume fraction. The lowest value of mass loss in wear test was distinct for Al–15 vol. % SiC and the highest mass loss in wear test was for bare Al alloy. Although the rate of change for the composites is smaller than that of the matrix, the wear rate of the matrix and the composites decrease with the sliding distance.

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In case of Al matrix composite, the depth of penetration of the harder asperities of hardened steel disk is primarily governed by the protruded hard ceramic reinforcement. Thus, the major portion of the applied load is carried by SiC particles. The role of the reinforcement particles is to support the contact stresses preventing high plastic deformations and abrasion between contact surfaces and hence reduce the amount of worn material. However, if the load exceeds a critical value, the particles will be fractured and comminuted, losing their role as load supporters (Thakur and Dhinan, 2001). It is also noted that the wear rate in all the samples increases marginally with applied load. The increase in the applied load leads to increase in the penetration of hard asperities of the counter surface to the softer pin surface, increase in micro cracking tendency of the subsurface and also increase in the deformation and fracture of asperities of the softer surface. Alpas and Zhang while investigating the wear of particle reinforced MMC under diﬀerent applied load conditions identiﬁed three diﬀerent wear regimes. At low load (regime I), the particles support the applied load in which the wear resistance of MMCs is better than Al-alloy. At regime II, wear rates of MMCs and Al-alloy were similar. At high load and transition to severe wear (regime III), the surface temperature exceeded the critical value. D. P. Mondal opinion was that the applied load aﬀects the wear rate of alloy and composites signiﬁcantly and is the most dominating factor controlling the wear behavior (Cocen and Onel, 2002; Leon and Drew, 2000; Mazahery and Shabani, 2011, 2012f; McKimpson and Scott, 1989; Moustafa et al., 2002; Shabani and Mazahery, 2013b; Thakur and Dhinan, 2001; Tham et al., 2001). The cumulative volume loss increases with increasing applied normal load and the contact surface temperature increases as the applied load increases. Karnezis et al. (1998) found that the wear rate varies with normal load, which is an indicative of Archard’s law and is signiﬁcantly low in the case of composites. Increasing the applied load induces changes in the optimum condition. In design variables, particle size and volume percentage of SiC are increased. Larger particle size and volume percentage of SiC induce more wear resistance. These changes are desirable and depend on the conditions (mechanical properties or tribological properties or both with percentage of importance). The results show that the novel technique implemented in this investigation has an acceptable performance. Therefore, this work shows the usefulness of an intelligent way to predict the performance of Al matrix composites using combined PSO and novel improved GA. The execution of an optimized stir casting technique yields relatively homogenous and ﬁne microstructure which improves the addition of reinforcement material in the molten metal.

Conclusion The aim of the present study was to provide preliminary information on the eﬀect of primary and secondary processing on the microstructure, tribological and mechanical properties of a SiC reinforced aluminum-based composite produced by the compocasting. The microstructural studies revealed the more uniform distribution of the particles in the matrix system of the cold rolled samples. In overall it can be concluded that Al6061-SiC exhibits better mechanical and tribological properties than the unreinforced alloy. Hardness, tensile strength, elongation and wear resistance of the composites are found to increase with the increase in rolling reduction. Simulation results show that novel improved GA is more eﬃcient than standard GAs. The standard gbest and lbest PSO approaches share information about a best solution found by the swarm or a neighborhood of particles. Sharing this information introduces a bias in the swarm’s search, forcing it to converge on a single solution. Based on the studies, GA the computation time increases more rapidly with the number of generations than that of PSO. Results show that GA-PSO is a promising method with good global convergence performance.

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Funding The author(s) received no ﬁnancial support for the research and/or authorship of this article.

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Searching for the superior solution to the population-based optimization problem: Processing of the wear resistant commercial AA6061 AMCs

International Journal of Damage Mechanics 2014, Vol. 23(7) 899–916 ! The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1056789513518951 ijd.sagepub.com

Ali Mazahery1,2, Mohsen Ostad Shabani3 and Ahmed Elrefaei2

Abstract In this present work, stir casting technique was used to fabricate aluminum matrix composites with varying weight percentages of SiC (5, 10 and 15) reinforcements. The proper selection of process parameter such as pouring temperature, stirring speed, stirring time and pre-heat temperature of reinforcement can all influence the quality of the fabricated composites. The porosity level of composite should be minimized and the chemical reaction between reinforcement and matrix should be avoided. Optimization is an applied science which explores the best values of the parameters of a problem that may take under specified conditions. The execution of an optimized stir casting technique yields relatively homogenous and fine microstructure which improves the addition of reinforcement material in the molten metal. The influence of SiC content, SiC size and secondary mechanical processing with different rolling reductions on the dry sliding wear characteristics of Al matrix composites has been assessed using a pin-on-disc wear test. The porosity and hardness of the resultant composites were also examined. Keywords SiC, optimization, particle swarm optimization

Introduction Particle swarm optimization (PSO) is a parallel population-based computation technique proposed by Kennedy and Eberhart (1995), which was motivated by the organisms behavior such as schooling of ﬁsh and ﬂocking of birds. PSO can solve a variety of diﬃcult optimization problems (Beasley 1

Light Metal Casting Research Centre (LMCRC), McMaster University, Ontario, Canada Leichtmetallkompetenzzentrum LKR, Austrian Institute of Technology, Austria 3 Karaj Branch, Islamic Azad University, Karaj, Iran 2

Corresponding author: Mohsen Ostad Shabani, Karaj Branch, Islamic Azad University, Karaj, Iran. Email: [email protected]

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et al., 1993; Engelbrecht et al., 2005; Krohling and Mendel, 2009; Mazahery and Shabani, 2012; Milani and Nazari, 2012a; Nazari and Riahi, 2012; Shabani and Mazahery, 2012a). PSO’s major diﬀerence from genetic algorithm (GA) is that PSO uses the physical movements of the individuals in the swarm and has a ﬂexible and well-balanced mechanism to enhance and adapt to the global and local exploration abilities, whereas GA uses genetic operators. Another advantage of PSO is its simplicity in coding and consistency in performance (Brits, 2002; Kennedy, 1999; Kennedy and Eberhart, 2001; Kennedy and Mendes, 2002; Mazahery and Shabani, 2012b; Shabani and Mazahery, 2012b; van den Bergh and Engelbrecht, 2006). GAs are a group of randomized methods used in function optimization (Shabani and Mazahery, 2013a). Early developments in the ﬁeld of GAs are generally credited to Holland and since then it has been successfully applied to various optimization problems. A GA population consists of several individuals, which work in parallel to identify a best solution. Individuals are potential solutions that gradually change or mature over time to converge to an optimal solution. In GA each individual is represented by a binary string of 0 s and 1 s of a ﬁxed length called chromosome, consisting of several genes. A GA starts oﬀ with an initial population of randomly generated chromosomes. During successive iterations, named generations, the initial chromosomes advance towards stronger chromosomes by reproduction among candidate solutions of the previous generation. In the next step, the binary strings are decoded and converted into optimization variable values, using a linear scaling. The objective function is evaluated from the established optimization variable values and a measure of worth or ﬁtness is evaluated. A high ﬁtness value would indicate a better solution than a low ﬁtness value. In the natural world, this would be an individual’s ability to survive in its present environment (Kennedy, 1999; Kennedy and Eberhart, 2001; Kennedy and Mendes, 2002; Mazahery and Shabani, 2012b; Shabani and Mazahery, 2013a; van den Bergh and Engelbrecht, 2006). During the last decade novel computational methods have been introduced in some ﬁelds of engineering sciences, including the materials science. Metal matrix composites (MMCs) are gaining wide spread popularity in several technological ﬁelds owing to its improved mechanical properties when compared with conventional metals/alloys (Bauri and Surappa, 2009; Blumenthal et al., 1994; Mazahery and Shabani, 2012c, 2012d; Pyzik and Aksay, 1989; Rhee, 1970; Shabani and Mazahery, 2012c, 2012d, 2012e; Xiao, 2010). Among the several categories of MMCs, Al based composites are ﬁnding wide spread acceptance especially in applications where weight and strength are of prime concern (Karnezis et al., 1998; Mazahery and Shabani, 2012e; Razavi et al., 2008; Shabani and Mazahery, 2012f; Tian and Fu, 2011; Xu et al., 2012). Weight reduction of vehicles to reduce fuel consumption and CO2 emissions has driven developments to extend the use of aluminum alloys by the automobile industry. Several wrought AA6xxx alloys based on the Al-Mg-Si system are available for car body outer panels, but the use of Al-Mg-Si is still a challenge for castings. Their lack of ductility has resulted in cracking during riveting in production and has limited their use in industry. Increased ductility of die cast Al-Mg-Si based casting alloys is therefore of critical interest (Shabani and Mazahery, 2011a; Asthana, 1998). Another major limitation is proved to be the low wear resistance (Leon and Drew, 2000; Mazahery and Shabani, 2012f; Moustafa et al., 2002; Shabani and Mazahery, 2013b; Thakur and Dhinan, 2001; Tham et al., 2001). In recent years, investigators have successfully attempted to improve the tribological properties by reinforcing it with high modulus ceramic particulates such as SiC, thus making it an increasingly suitable candidate for even wider spectrum of applications (Cocen and Onel, 2002; Mazahery and Shabani, 2011, 2012g; McKimpson and Scott, 1989; Varma et al., 2000; Zhong, 2000). To fully utilize the potential of a A6061/ceramic composition, the judicious selection of primary and secondary processing and heat treatment procedure becomes important. It has been demonstrated, for example, that the diﬀerent primary processing techniques are

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capable of imparting totally diﬀerent microstructural features and mechanical properties to a material of given composition. The microstructural and properties variation for a ﬁxed metal/ceramic formulation can also be realized by varying the secondary processing parameters such as reduction percentage in the case of rolling. Although the eﬀects of various parameters have been critically investigated for monolithic and composites materials, no attempt has been made to model and optimize the microstructure and mechanical characteristics of the AA6061/SiC composites. Particular emphasis was given in this study to optimization of the grain morphology and SiC particulates redistribution within the matrix.

Experimental procedure Al6061 alloy matrix composites reinforced with SiC particulates were synthesized through a semisolid processing route. In order to achieve sound quality composite, the furnace is placed on the ﬂoor and the temperature of the furnace is precisely measured and controlled. The process involved melting the alloy in the graphite crucible (Quaak and Kool, 1994). Stirring was done via an impeller made of graphite and driven by a variable ac motor. The crucible was heated to about 680 C in a resistance furnace and after melting the aluminum. SiC particles with average sizes of 1, 5, 20 and 50 mm as reinforcing particles were added to the melt and stirred at 400/500/600/700 r/min. After the addition of SiC during the process, the stirring time were noted at 5, 10 and 15 min. As stirring continued, the furnace temperature was gradually lowered until the melt reached a temperature in the liquid–solid state. In order to minimize the oxidation of molten aluminum, nitrogen gas is used inside and outside of the crucible. Steel die was placed below the furnace. The casting was obtained by pouring composite slurry into the die. The obtained casts were annealed at 530 C for 2 h, machined to samples of 150 mm length, 50 mm width and 10 mm thickness and cold rolled to different ﬁnal reduction. Pin-on-disc type wear tester was used for dry sliding wear tests. The test was done at sliding velocity of 0.3 m/s within a load range of 10–40 N. Figure 1 shows schematic diagram of the abrasion wear test. The disk was made of the steel hardened up to 63 HRC with a diameter of 50 mm and a thickness of 10 mm. The pin test sample dimensions were 6 mm diameter and 25 mm length. At a given load level, weight loss from the worn surfaces was found to increase linearly with sliding distance, except during the transient period at the beginning of the test. The wear rates were calculated from the slope of weight loss versus sliding distance curves determined at several applied load levels within the range of 10–40 N. The wear rates measured in weight units were then converted to volumetric wear rates.

Figure 1. Schematic diagram of the abrasion wear test.

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The experimental density of the composites was obtained by the Archimedes method of weighing composites ﬁrst in air and then in water, while the theoretical density was calculated using the mixture rule according to the volume fraction of the SiC particles. Metallographic samples were prepared using standard metallographic techniques, etched with standard aluminum etching solutions and examined by a optical microscope to determine the distribution of the SiC particles. The volume fraction of SiC particles was measured by means of an image analyzer system attached to the microscope. The tensile test samples were machined according to the ASTM E8M standard and oriented along the rolling direction.

Genetic algorithm GA is a computational technique, which uses principles of Darwinian natural selection in order to solve a wide variety of problems (Shabani and Mazahery, 2013b). Based on the idea of survival of the ﬁttest and natural selection, GA is a class of parallel iterative algorithm with speciﬁc learning ability, which repeats evaluation, selection, crossover and mutation after initialization until the stopping condition is satisﬁed. As originally proposed, a simple GA usually consists of three processes selection, genetic operation and replacement (Shabani and Mazahery, 2012b). The population comprises a group of chromosomes that are the candidates for the solution. The ﬁtness values of all the chromosomes are evaluated using an objective function in a decoded form. A particular group of parents is selected from the population to generate oﬀspring by deﬁned genetic operations of crossover and mutation (Mazahery and Shabani, 2012a). The ﬁtness of all the oﬀsprings is then evaluated using the same criterion and the chromosomes in the current population are then replaced by their oﬀsprings, based on a certain replacement strategy. Such a GA cycle is repeated until a desired termination criterion is reached. If all go well throughout the process of simulated evolution, the best chromosome in the ﬁnal population can become a highly evolved and more superior solution to the problem (Mazahery and Shabani, 2012h; Petalas et al., 2009; Sathiya et al., 2009; Shabani et al., 2011; Shabani and Mazahery, 2011b; Wu et al., 2010; Zhao and Suganthan, 2011). According to the natural features, especially, the population-size-changing rule on human beings, this paper presents a novel improved GA with variable population-size which obtains a better balance between the variety and continuity. The main idea of present GA is described as follows. Of most feature species, parents are neither dead after their reproduction right away, nor living forever. In fact, the individual is apt to die when it is old. In present GA each individual is designated a dying probability according to its living generations. Deﬁne DIE_PROBABILITY [Q] as the dying probability of those individuals living Q generations. For example, considering the expected max living generations is 3, we could set the DIE_PROBABILITY [0, 1, 2, 3] to be 0, 0.3, 0.7 and 1, respectively. After the performance of the traditional genetic operations, the individual should undergo a dying process. In the dying process, each individual is determined whether to die according to its DIE_PROBABILITY. On the other hand, the size of features’ population always increases, until there is a contagious disease or a war breaking out among the population. In general, the disease or the war will reduce the size of the population sharply. In present GA, the size of the population is reduced to the initial size when it reaches the given limitation, of which the individuals with higher ﬁtness have more opportunities to survive (Mazahery and Shabani, 2012h).

Particle swarm optimization The PSO was originally designed by Kennedy and Mendes (2002) and Kennedy and Eberhart (1995, 2001) and has been compared to GAs for eﬃciently seeking optimal or near-optimal solutions in

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large search spaces. The PSO algorithm conducts search using a population of particles which correspond to individuals in a GA (Kennedy, 1999). A population of particles is initially randomly generated. Each particle represents a potential solution and has a position represented by a position vector. A swarm of particles moves through the problem space, with the moving velocity of each particle represented by a velocity vector. At each time step, a function representing a quality measure is calculated by using as input. Each particle keeps track of its own best position, which is associated with the best ﬁtness it has achieved so far in a vector (Kennedy and Eberhart, 2001). Furthermore, the best position among all the particles obtained so far in the population is kept track as output. In addition to this global version, another local version of PSO keeps track of the best position among all the topological neighbors of a particle. At each time step, by using the individual best position and global best position, a new velocity for particle is updated. The computation for PSO is easy and adds only a slight computational load when it is incorporated into the conventional GA (Kennedy and Eberhart, 1995). Furthermore, the ﬂexibility of PSO to control the balance between local and global exploration of the problem space helps to overcome premature convergence of elite strategy in GA, and also enhances search ability (Kennedy and Mendes, 2002). PSO ﬁrst initializes a group of random particles and then ﬁnds the optimal solution by multiple iterations. Suppose that the searching space is D-dimensional and m particles form the colony (Kennedy and Eberhart, 2001). The ith particle represents a D-dimensional vector Xi (i ¼ 1, 2, . . . ,m). It means that the ith particle locates at Xi ¼ (xi1, xi2, . . . ,xiD) (i ¼ 1, 2, . . . ,m) in the searching space. The position of each particle is a potential result. We could calculate the particle’s ﬁtness by putting its position into a designated objective function. When the ﬁtness is higher, the corresponding Xi is ‘‘better’’. The ith particle’s ‘‘ﬂying’’ velocity is also a D-dimensional vector, denoted as Vi ¼ (vi1, vi2, . . . ,viD) (i ¼ 1, 2, . . . , m). Denote the best position of the ith particle as Pi ¼ (pi1, pi2, . . . ,piD), and the best position of the colony as Pg(pg1, pg2, . . . ,pgD), respectively. The PSO algorithm could be performed by the following equations: Xikþ1 ¼ Vikþ1 þ Xik

ð1Þ

Vikþ1 ¼ wk Vik þ c1 r1 ð pik Xik Þ þ c2 r2 ð pgk Xik Þ

ð2Þ

Where k represents the iterative number, w is the inertia weight, c1 and c2 are learning rates, r1 and r2 are random numbers between 0 and 1, Vi 2 [Vmin,Vmax] where Vmin and Vmax are the designated vectors. The termination criterion for the iterations is determined according to whether the max generation or a designated value of the ﬁtness of Pg is reached (Kennedy and Mendes, 2002).

Modeling Compared with GA, PSO has a much more profound intelligent background and can be performed more easily. Given that particle optimization mainly evolves via the comparison of the self position, the surrounding positions and the global positions of all the particles as a singleton pattern, PSO may prematurely converge. A good diversity must have a strong ability to ﬁnd the best position; otherwise, the undesired local solution may appear when the population is not well diversiﬁed. Using diﬀerent evolution mechanisms, such as crossover and mutation, GA can enhance the diversity of results, but, to some extent, it leads to a large number of redundant iterations, resulting in long computing times and low problem-solving eﬃciency. Several researchers have combined PSO and GA to form diﬀerent hybrid algorithms. Shi combined a variable population-size genetic algorithm (VPGA) and PSO to form a hybrid algorithm

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that generates initial populations for VPGA and PSO according to a certain proportion. The new population is obtained by evolution according to the corresponding algorithm rules. Gandelli randomly divided the total population into two subpopulations and evolved the two subpopulations using GA and PSO operations, respectively. Kao proposed a GA–PSO hybrid algorithm for geometric proportional populations. Kuo integrated the mutation mechanism of GA with PSO. Valdez combined GA and PSO using fuzzy logic to integrate the results of both methods and for parameter tuning (Beasley et al., 1993; Brits, 2002; Engelbrecht et al., 2005; Kennedy, 1999; Kennedy and Eberhart, 1995, 2001; Kennedy and Mendes, 2002; Krohling and Mendel, 2009; Mazahery and Shabani, 2012a, 2012b; Milani and Nazari, 2012; Nazari and Riahi, 2012; Shabani and Mazahery, 2012a, 2012b; van den Bergh and Engelbrecht, 2006). The hybrid algorithms above have evolution rules for a certain proportion of the population and only implement information transfer for particles after one generation or just use one GA operator. In fact, this is a formal hybrid and cannot exchange information many times in the PSO or in the GA evolution process. These approaches cannot fully exploit a complement of two algorithms. Therefore, the present study proposes a new PSO–GA hybrid algorithm. The PSO–GA hybrid algorithm diﬀers from previous hybrids in the following ways: . The GA and the PSO approaches result in a hybrid hierarchy. First, all particles in the population are evolved by the PSO with smaller iterations, and then M-optimized particles are selected for the implementation of genetic operators . Two information transfers are accomplished. The optimized particles after the PSO form an initial GA population, and the PSO updates the velocity and the position of all particles after genetic operations. Better optimize the coeﬃcients, an eﬀective hybrid optimization algorithm based on PSO and GA is developed. First, the population is evolved over a certain number of generations by PSO and the best M particles are retained; the remaining pop_size_M particles are excluded. Second, pop_size_M new individuals are generated by GA operators for the remaining best M particles. Finally, pop_size_M new individuals are combined with the remaining best M particles to form a new population for the next generation. (1) Relevant parameters, such as the number of particles (pop-size), the number of particles retained after evolution by PSO (M), the PSO weight coeﬃcients (c1 and c2), the crossover and mutation probabilities for GA (pc and pm), the maximum particle velocity (Vmax), the number of PSO generations evaluated (kmax) and the number of generations for the hybrid algorithm (max-gen), are initialized. (2) The pop-size initial particles and their ﬁtness values are generated and calculated, respectively (3) Let gen ¼ 1. (4) If gen max-gen, step 5 is carried out; otherwise, step 4 is carried out. (5) Let k ¼ 1. (6) If k max-k, step 7 is carried out; otherwise, step 9 is carried out. (7) The position and the velocity of the particles are updated according to equations (1) and (2). (8) Let k ¼ k þ 1, then return to step 6. (9) The pop-size particles are ranked according to ﬁtness, and the M particles with the lowest ﬁtness values are selected. (10) The pop-size M individuals obtained by GA and M particles are combined to form new pop-size particles. (11) Let gen ¼ gen þ 1, then step 4 is carried out. (12) The best ﬁtness values and solutions, namely, the position, are outputted.

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Figure 2 shows the ﬂowchart of combined GA-PSO model which has been used in this investigation. PSO is an eﬀective algorithm which gives quality solutions in a reasonable computational time and consists of less numbers parameters as compared to the other evolutionary metaheuristics. Mutation, a commonly used operator in GA, has been introduced in PSO so that trapping of solutions at local minima or premature convergence can be avoided.

Figure 2. The flowchart of the model used in this study.

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It is well known that the unreinforced matrix alloy wear more rapidly than the reinforced composite materials. A number of studies have been engaged during the last 20 years on wear behavior of Al based particulate reinforced composite. Reinforcement of Al based alloy with SiC are usually found to improve the wear resistance under both abrasion and lubricated sliding conditions. Experimental results of the wear rate and hardness test are presented in Figure 3. It is noted that the addition of hard ceramic SiC particles increases the hardness of Al alloy. Figure 4 shows the wear rate and hardness results of cold rolled composites, respectively. It is observed that there is an improvement in wear of the cold rolled composites which might be explained by the increase in the hardness of them. Fabrication of these composites via compocasting technique plus reduction (90%) during cold rolling process lead to reasonably uniform distribution of particles in the matrix and minimum clustering or agglomeration of the reinforcing phase and thus higher hardness and wear resistance. Figure 5 shows the eﬀect of the rolling process on the porosity content of the composites. It is noted that the application of the heavy reduction (90%) during cold rolling process results in approximately 0% volume fraction porosity. The porosity level increased, since the contact surface

Wear Rate (10-12m3/m)

(a) 2.7 2.4

Unreiforced Al

5% SiC

2.1

10% SiC

15% SiC

1.8 1.5 1.2 0.9 0.6 0.3 0 1.0E+3 1.5E+3 2.0E+3 2.5E+3 3.0E+3 3.5E+3 4.0E+3

Sliding distance (m)

Hardness (HRC)

(b) 90 85 80 75 70 65 60 0

3

6

9

12

15

Volume fraction of SiC

Figure 3. a: Effect of the sliding distance on the wear rate of Al–SiC composites under an applied load of 10 N and b: hardness.

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Wear Rate (10-12m3/m)

1.2 1.1 1 0.9 0.8

%5 SiC

%10 SiC

%15 SiC

0.7 0

10

20

30

40

50

60

70

80

90

Reduction (%) (b) 2.9 5% SiC

2.7

10% SiC

15% SiC

Wear Rate (10-12m3/m)

2.5 2.3 2.1 1.9 1.7 0

10

20

30

40

50

60

70

80

90

Reduction (%)

Figure 4. The effect of the rolling process on the wear of the composites under the applied load of: (a) 10 N, (b) 20 N, (c) 40 N and hardness of the composites (d).

area was increased. This is attributed to pore nucleation at the SiC particulate sites and to hinder liquid metal ﬂow due to more particle clustering. The mechanical property results of the composites are shown in Figure 6. It is noted that there would be a great enhancement in tensile behavior of the composites while using rolling reduction in comparison to the as-cast composites. It can also be seen that tensile strength increases when the reduction quantity is increased. It is observed that there is a simultaneous improvement in the strength and ductility of the cold rolled composites which might be explained by the reduction in porosity content. A more detailed analysis of the eﬀect of c1 and c2 parameters is necessary to determine the sensitivity of the parameters in the overall optimization procedure. The eﬀect of the cognitive and social parameters on the general optimization history can be seen in Figure 7 where objective function values versus iteration number is shown.

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International Journal of Damage Mechanics 23(7)

Volume % Porosity

5 4 3 2 1

5% SiC

10% SiC

15% SiC 0 10 20 30 40 50 60 70 80 90

0

Reduction (%)

Figure 5. The effect of the rolling process and SiC volume fraction on the porosity content of the composites. (a) 350

UTS (MPa)

300 5% SiC 10% SiC 15% SiC

250 200 150 100 50 0 0

10 20 30 40 50 60 70 80 90

Reduction (%)

Elongation (%)

(b)

8 7

5% SiC

6

10% SiC 15% SiC

5 4 3 2 1 0 0

10 20 30 40 50 60 70 80 90

Reduction (%)

Figure 6. The effect of the rolling process and SiC volume fraction on a: the tensile strength and b: the elongation of the composites.

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Objective Values

10

C1 1=0

C1 = 1

C1 1=3

C1 = 4

C1 = 2

8 6 4 2 0 +0 0.0E+

2.0E+3

44.0E+3

6.0E+3

Numbers Of Iteration

Figure 7. Objective function values vs. iteration number of cognitive and social parameters.

The parameters were varied according to the formula c2 ¼ 4 c1, where c1 was changed in the interval [4, 0]. It can be seen clearly that when only cognitive c1 ¼ 4 or only social values c2 ¼ 4 are used the resulting history converges very fast, within the ﬁrst iterations, but do not improve after the initial convergence, which indicate that the algorithm reaches a local suboptimal and is not able to escape from it, due to the lack of information exchange either from social or cognitive sources, respectively. When higher emphasis is placed on the social exchange of information (c1 ¼ 1, c2 ¼ 3) the algorithm is able to gradually converge to better regions of the design space, but again if a local optimum is found it cannot escape from it. This fact indicates an overshooting of the algorithm where particle positions are updated based mostly on social information not promoting the exchange of individual information which could be useful when an individual particle has found a better region of the design space. When the cognitive and social parameters are in balance or with slightly higher cognitive value, solutions converge near the global optimum solution; these two situations are shown by the gradual decrease of objective function values and subsequent improvements over time. Notice that a slightly higher value for the C2 results in better solutions. This is the result of the fact that promoting some global exchange of information causes the swarm to point toward the best regions, but individuals concentrate more in their own search regions and avoid overshooting the best design space region.

Inertia weight variation effect. The concept of linearly decreasing inertia weight was introduced by van den Bergh and Engelbrecht (2006) and is given by: w ¼ wmax

wmax wmin iter itermax

ð3Þ

where iter is the current iteration number, while itermax is the maximum number of allowable iteration. Usually the value of w is varied between wmax ¼ 0.9 and wmin ¼ (0.2, 0.3 or 0.4).

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Objective Values

10

w_min = 0.3 w

w w_min = 0.2

w_min = 0.4 w

w = 0.7

8 6 4 2 0 +0 1.0E+3 2.0E+3 3.0E+3 44.0E+3 5.0E+ +3 6.0E+3 7.00E+3 0.0E+

Numbers Of Iteration

Figure 8. Objective function values vs. iteration number of inertia weight.

Therefore, the particle is to use lager inertia weight during the initial exploration and gradual reduce its value as the search proceeds in further iterations. Results are given in Figure 8. Based on this ﬁgure, the best global optimum solution is reached by wmin ¼ 3. Notice that ﬁxed inertia weight allow for a fast convergence to sub-optimal regions of the design space while linearly decreasing inertia weight updates converge slowly, due to the initial global search nature force by higher cost function update values, and gradually converges to optimal regions of the design space approximately after latest iteration. Figure 9 illustrates the convergence behaviors of the three methods. It is noted that the GA–PSO converges faster than the conventional GA and PSO approach. In addition, the computation time for GA optimization method is longer compared with PSO methods. Several experiments were conducted to determine the best parameter values of the proposed algorithm. The total number of runs for each combination of parameter values is 30. Figure 10 shows performance of three algorithms. As can be seen, GA–PSO is more repeatable than PSO and GA algorithms. GA–PSO has more global search ability at the end of evolution, which is required to jump out of the local optima in some cases. Goldberg suggested that appropriate values of user speciﬁed GA parameters can be selected as crossover probability (Pc) (0.5–0.6) and mutation probability (Pm) (0.01–0.02). The proposed method uses a GA operator to improve the diversity of PSO particles. Therefore, larger probabilities of crossover and mutation are set, namely, Pc ¼ 0.7 and Pm ¼ 0.05. Although better particles may degenerate as probabilities Pc and Pm become larger, best M particles Q1, Q2, . . . ,Qm are retained without genetic implementation in the proposed algorithm. The eﬀects of momentum factor and learning rate on the performance of the developed model are shown in Figure 11. As can be seen, the optimum values of momentum factor and learning rate are 0.4 and 0.6, respectively. These parameters were chosen to ensure convergent trajectories. GAs and PSO are much similar in their inherent parallel characteristics, whereas experiments show that they have their speciﬁc advantages when solving diﬀerent problems. What we would like to do is to obtain both their excellent features by synthesizing the two algorithms. Final optimized parameters for the mechanical and tribological properties in applied load 10 N and 90% reduction are 668 s stirring time, 645 r/min speed of stirrer, 27.81 mm particle size of SiC,

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Objective Values

10

PSO-GA

PSO

8 6 4 2 0 0.0E+0

2.0E+3

4.0E+3

6.0E+3

Numbers Of Iteration

Figure 9. Objective function values vs. iteration number of the three methods.

6

Objective Values

PSO O-GA

PSO

GA

4

2

0 1

5

9

13

17

21

25

29

Number Figure 10. Objective function values vs. number of runs for each method.

11.03% volume percentage of SiC, 0.48% porosity, 83.8 HRC hardness, 303.9 MPa ultimate tensile strength (UTS), 6.66% elongation and 1.11% 1012 m3/m wear rate. Final optimized parameters for applied load 20 N and 90% reduction are 688 s stirring time, 679 r/min speed of stirrer, 34.11 mm particle size of SiC, 11.14% volume percentage of SiC, 0.42% porosity, 84.1 HRC hardness, 313.2 MPa UTS, 6.55% elongation and 2.2% 102 m3/m wear rate. Porosity formation in cast MMCs can be attributed to the air trapped in the clusters of particles as well as the hindered metal ﬂow inside them. Increasing amount of porosity is observed by

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International Journal of Damage Mechanics 23(7) (a) 8 7

MAPE

6 5 4 3 2 1 0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.9

1.0

Momentum Factor (b) 12

MAPE

10 8 6 4 2 0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Learning Rate Figure 11. The effects of (a) momentum factor and (b) learning rate on the performance of the developed model.

increasing the volume fraction of SiC particulates. The decreased porosity of MMCs during rolling is due to the ﬂow of the matrix alloy under the applied shear and compressive forces resulting in ﬁlling the voids. The increased rolling reduction provides easier ﬂow of the matrix alloy and hence results in decreased porosity. In fact, the major reason for rolling the particulate metal matrix composites (PMMCs) is to close the pores and attain improved mechanical properties. The wear loss is inversely proportional to the hardness of alloys. The wear rate of the unreinforced alloy is found to be higher than that of the composites. Shabani and Mazahery emphasized that higher hardness of composite could be achieved by ceramic reinforced particulate (B4C) because B4C particle acts as an obstacle to the motion of dislocation. In case of unreinforced Al alloy, the depth of penetration is governed by the hardness of the specimen surface and applied load. Iwai found that both mild wear in the initial sliding distance and severe wear rate decreased with increasing volume fraction. The lowest value of mass loss in wear test was distinct for Al–15 vol. % SiC and the highest mass loss in wear test was for bare Al alloy. Although the rate of change for the composites is smaller than that of the matrix, the wear rate of the matrix and the composites decrease with the sliding distance.

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In case of Al matrix composite, the depth of penetration of the harder asperities of hardened steel disk is primarily governed by the protruded hard ceramic reinforcement. Thus, the major portion of the applied load is carried by SiC particles. The role of the reinforcement particles is to support the contact stresses preventing high plastic deformations and abrasion between contact surfaces and hence reduce the amount of worn material. However, if the load exceeds a critical value, the particles will be fractured and comminuted, losing their role as load supporters (Thakur and Dhinan, 2001). It is also noted that the wear rate in all the samples increases marginally with applied load. The increase in the applied load leads to increase in the penetration of hard asperities of the counter surface to the softer pin surface, increase in micro cracking tendency of the subsurface and also increase in the deformation and fracture of asperities of the softer surface. Alpas and Zhang while investigating the wear of particle reinforced MMC under diﬀerent applied load conditions identiﬁed three diﬀerent wear regimes. At low load (regime I), the particles support the applied load in which the wear resistance of MMCs is better than Al-alloy. At regime II, wear rates of MMCs and Al-alloy were similar. At high load and transition to severe wear (regime III), the surface temperature exceeded the critical value. D. P. Mondal opinion was that the applied load aﬀects the wear rate of alloy and composites signiﬁcantly and is the most dominating factor controlling the wear behavior (Cocen and Onel, 2002; Leon and Drew, 2000; Mazahery and Shabani, 2011, 2012f; McKimpson and Scott, 1989; Moustafa et al., 2002; Shabani and Mazahery, 2013b; Thakur and Dhinan, 2001; Tham et al., 2001). The cumulative volume loss increases with increasing applied normal load and the contact surface temperature increases as the applied load increases. Karnezis et al. (1998) found that the wear rate varies with normal load, which is an indicative of Archard’s law and is signiﬁcantly low in the case of composites. Increasing the applied load induces changes in the optimum condition. In design variables, particle size and volume percentage of SiC are increased. Larger particle size and volume percentage of SiC induce more wear resistance. These changes are desirable and depend on the conditions (mechanical properties or tribological properties or both with percentage of importance). The results show that the novel technique implemented in this investigation has an acceptable performance. Therefore, this work shows the usefulness of an intelligent way to predict the performance of Al matrix composites using combined PSO and novel improved GA. The execution of an optimized stir casting technique yields relatively homogenous and ﬁne microstructure which improves the addition of reinforcement material in the molten metal.

Conclusion The aim of the present study was to provide preliminary information on the eﬀect of primary and secondary processing on the microstructure, tribological and mechanical properties of a SiC reinforced aluminum-based composite produced by the compocasting. The microstructural studies revealed the more uniform distribution of the particles in the matrix system of the cold rolled samples. In overall it can be concluded that Al6061-SiC exhibits better mechanical and tribological properties than the unreinforced alloy. Hardness, tensile strength, elongation and wear resistance of the composites are found to increase with the increase in rolling reduction. Simulation results show that novel improved GA is more eﬃcient than standard GAs. The standard gbest and lbest PSO approaches share information about a best solution found by the swarm or a neighborhood of particles. Sharing this information introduces a bias in the swarm’s search, forcing it to converge on a single solution. Based on the studies, GA the computation time increases more rapidly with the number of generations than that of PSO. Results show that GA-PSO is a promising method with good global convergence performance.

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Funding The author(s) received no ﬁnancial support for the research and/or authorship of this article.

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