1 Introduction

Nowadays, both Copper (Cu) and Aluminum (Al) are important metals in the electrical industry due to their proper electrical and thermal conductivity as well as high corrosion resistance and also chemical and mechanical properties [1, 2]. There have been many studies on different welding methods in order to joint these two materials. However, it is difficult to join them using current welding methods due to their tremendous differences in chemical and mechanical properties and tendency to form some brittle Inter-Metallic Compounds (IMCs) [3]. Thus, different techniques of joining copper and aluminum, such as Friction Stir Welding (FSW), have turned into research subjects. The versatility and easy reproducibility of the FSW process, the improvement of weld quality and mechanical properties of the final joint make this an important study [4]. That’s why this is a challenging task and a possible breakthrough in joining Cu/Al sheets.

The FSW method uses a dedicated tool to stir and generate frictional heat [5]. Due to the rotating, plunging and retreating a non-consumable tool into the workpiece, the FSW is distinguished from other welding methods [6, 7]. The FSW process is widely used in various industries including pipelines, petrochemicals, constructions, automotives and shipbuildings [8]. The cost is very important in large industries and this factor is entirely based on the determination of the FSW and the relevance between existing parameters and how to achieve the desired state. Achieving this demand, on the other hand, depends on frequent and onerous experiments, so modeling methods, which relied on mathematical relevance to the model of industrial systems in order to make an optimal point between input and output parameters, are expressed. In the FSW, the relation between mechanical properties and the process’ input parameters should be established by a model. The dependence of mechanical properties on parameters of the FSW has been meaningfully approved through various studies [9]. Parameters of the process are entirely based on experiences or extracted from the database [10]. It should be kept in mind that the desired measures of mechanical properties may not achieved via the input parameters considered by this method. So, the production of models that provide a connection between mechanical properties and process variables is seemed to be a basic task [11].

Therefore, solid-state joining methods such as Friction Stir Welding (FSW), Rolled Welding (RW) and Explosion Welding (EW) have received much attention [12]. These methods, however, have a few drawbacks. For example, both friction stir welding and rolled welding have no versatility and there are various safety problems involved in explosion welding as well [13].

Since emerged, the Algorithm Method has been used in several areas; and, numerous publications, from medicine to robotics passing by food science or zoology, can be found in this field [14, 15]. In the current study, working on FSW processes of Al5083 and Cu has been only addressed. The relationship between input and output parameters is quite evident in the subject of Design of Experiments (DoE) [16] and algorithms [17, 18]. Algorithms provide no acceptable results in experiments with low data count and can be only implemented by at least 30 experimental data and above [19]. The welding operation requires an operator with a considerable skill and experience to designate all desired welding parameters together [20]. To this end in modern industries, there has been a lot of research to replace DoE and algorithm systems in order to eliminate the need for a highly-skilled operator [21]. Algorithm systems in industrial processes are of a great importance for several reasons such as increasing efficiency, improving work conditions and ensuring high quality of manufactured goods [22].

A numerical model has been recommended by Wahid et al. [23] to enhance the tensile strength of the FSW process on a Al6082-T6 sheet. The tool’s shoulder diameter, rotation speed and welding speed were all considered as process parameters and their influence on the joint’s tensile strength was probed. In addition, the post impacts of variance in temperature, the macro and micro-structural examinations, the micro-hardness and also the fracture behavior of the submerged FSW joint were prospected. Some experimental studies on FSW, Al/Cu and Al/Brass sheets have been worked out by Ismaili et al. [24]. Studying their mechanical and micro-structural properties, it was found that the highest strength of 80% belongs to the strength of aluminium-based metal. In their opinion, the good flow of materials and the creation of narrow multilayer inter-metallic compounds in the inter-metallic joint was recognized as the main reason of this proper coupling. The process parameters in lap joint FSW of Al5456 alloy have been optimized by Salari et al. [25]. This study was executed on various thicknesses of the sheet and the micro-structure was estimated at the weld region. In aforesaid research, four-pin geometric varieties namely conical threaded type, cylindrical-conical threaded type, stepped conical threaded type and flared-triflute pin tool as well as the rotation speeds of 600 ~ 800 rpm were asigned as the input welding factors to create the joints. The output parameters of the study, on the other hand, were included the micro-hardness and micro-structure of both the workpiece and the heat-affected zone (HAZ). After investigation, their relationship to the selected processing environments were diagnosed. It was witnessed that the stepped conical threaded pin tool at the rate of 600 rpm, as compared with other combinations, has produced the weld joints with a more homogeneous micro-structure, finer grain size, and improved micro-hardness level. Pitchipoo et al. [26] have conducted various experimental studies with a multi-objective optimization technique based on the Dragonfly Algorithm (DFA) which was proposed for optimizing the weld strength in FSW of the Al6082-T6 alloy. It determines the combination of optimal parameters namely the tool’s rotation speed, welding speed, and tilt angle presented to achieve the optimal tensile strength and impact strength of the welded workpiece.

Another numerical model has been established by Gopi et al. [27] to envisage the strength of welding joint in two-sided FSW of Al6082-T6 alloy. The tool’s axle speed/rotation speed, welding speed, and geometric features of the tool were considered as process parameters. The workpiece’s strength with the confidential level of 95% was anticipated by means of the developed model.

Using the Hybrid Fuzzy-Taguchi & Gra (HFTGRA) optimization method, a dissimilar FSW of Al/Cu metal as well as the optimization of parameters have been accomplished by Sahu et al. [28]. The trial performance was prepared by Taguchi's L16 symmetrical cluster through four process parameters, videlicet the tool’s rotation, the welding speed, the tool’s pin offset, and the plunging depth. The tensile strength, the compressive strength, the elongation percentage, the bending angle, the weld nut thickness and the average micro-hardness at the weld zone were all considered as output responses. An optimal combination of input parameters was attained by using a hybrid method in which fuzzy concepts had been integrated with grey relational analysis.

Plenty of studies have been carried out on the effect of dissimilar welding parameters of aluminium (Al) and copper (Cu) on the micro-structure and mechanical properties of the weld zone as well as the detection of inter-metallic phases that occurs at the weld zone. In fact, several works have already addressed the dissimilar FSW of these materials, in both butt and lap joint configurations. However, Al/Cu lap joining has been explored much more than FSW, on which only a few number of studies have been conducted so far [29]. These studies have concluded with different results and could not achieve high advantages, yet very few studies possess the mechanical properties of the prediction model.

Previous researches have indicated that the mechanical properties of the FSW welds are affected by different process parameters. However, a comprehensive study on the effect of these parameters on dissimilar metal welding is still a challenging issue and needs further study. Therefore, in this scholarship paper, it is attempted to comprehensively investigate the interactions and main effects of the process parameters on the mechanical properties of Al5083 and pure copper dissimilar welds using the ANN, ICA and PSO methods.

2 Materials and methods

The sheets used as the base metals in FSW were made up of Al5083 and pure copper with dimensions of 100 × 20 × 5 mm3. The mechanical properties of both Al5083 and pure copper are respectively depicted in Table 1. In rectangle shapes with the same dimensions, the samples were cut by guillotine.

Table 1 Mechanical properties of Al5083 and pure Cu sheets
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To accurately adjust the welding speed in this study, an FP4M milling machine has been used in Tabriz. The parameters of the FSW process are presented in Table 2. Many parameters have been reported in friction stir welding, but the tool’s rotation speed, traverse speed and angle deviation can be mentioned as its most important parameters. The tool’s pin geometry, circular or quadrangular, has the most important influence on mechanical properties.

Table 2 The parameters of FSW process
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To remove the oxide layers at the bonding surface, the sanding operation was performed and the sheets were finally cleaned with acetone. For welding, two plates in the form of butt weld were utilized. Using two H13 tools with the micro-hardness of 52 HRC (Hardness Rockwell-C), the welding was done at the ambient temperature. The FSW was primarily studied for Al5083 and pure copper. Few studies was found regarding the joint of thin sheets even for the stated alloys and none of them had used an annealed alloy. The detail of used experimental tests are as portrayed in Fig. 1.

Fig. 1
figure 1

Experimental details; a FP4M milling welding machine, b Placement of samples in clamping and c Pin geometry with manufactured dimension

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The Al/Cu welding has been our most challenging combination yet. Regardless of the system restrictions, the applied load or the generated heat, the softness of Al5083 is too high to support the load of FSW in Al/Cu welding. However, the FSW process has some limitations in butt-joining of thin sheets. The thickness reduction, resulting from the forging effect of the shoulder, can significantly reduce the mechanical resistance in thin plates. Creating serious problems in thin plate welds, the existence of minor defects is usually resolved in thick plate welds instead.

During the tentative test, many experiments were not satisfactorily performed. Thanks to a more rigorous study hypothesis, proper samples were obtained by changing the geometry of different tools. In order to overcome the softness of aluminum sheets and the rise of shoulder diameter, and also to avoid the vertical instability of the sheet, shortening the pin’s length was suggested (Fig. 2).

Fig. 2
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image of the; a tensile test specimens, b tensile test dimensions and c micro-hardness

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In the FSW, the tool’s geometry is one of the most important parameters of the process. Therefore, not only the geometry of both shoulder and pin, which are in contact with the parts to be welded, but also how the tool’s body resists to the applied mechanical tensions and dissipates the generated heat during the process are essential.

Moreover, during all FSW joining processes, the pure copper was placed at the tool’s Retreating Side (RS), while the Al5083 was kept at the Advancing Side (AS) of the welding tool. Many trials were primarily performed on choosing the optimum welding process parameters.

According to various articles, mechanical properties gain the highest level in the 4th pass, and since then, increasing the number of passes has no effect on mechanical properties. While the tool’s pin is passing through the weld piece, recrystallization occurs at the same time. By passing the pin across the junction, the heat releases. The heat release subtracts the energy required for recrystallization, thus fine-graining does not occur. Activation of the persistent recrystallization mechanism causes the grain sizes to become finer as the pin passes; and after the pin passes through the generated heat, the grain growth is barricaded by the mechanism.

In the current research, the mechanical properties, for two reasons, were at variance in each pass. First, due to the better overlap with the communication machine, three algorithmic models—namely Artificial Neural Networks (ANN), Imperialist Competitive Algorithm (ICA), Particle Swarm Optimization (PSO)—were used. In the ANN, the interaction between the complexity and the error rate is clearly controlled for high-dimensional data. In the ICA, by increasing the number of experiments, the relationship between the data and the interaction of effects is measured. Second, the dynamic recrystallization has the highest level in the 4th pass and accordingly leads to the best result. To do so, three algorithmic models (ANN, ICA and PSO) were applied in each pass and the most compatible model was observed in the 4th pass. Therefore, the presented model in this study is of the 4th pass and has the highest generalization over empirical experiments.

3 Results

3.1 Experimental details

In this paper, various empirical investigations have been carried out on Al5083 and pure copper sheets. The result of empirical experiments is illustrated in Table 3. It has been attempted to gradually apply the process of designing a full factorial experiment in order to work out the Design of Experiments and obtain an acceptable algorithm.

Table 3 Experimental test using Full Factorial Method
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As observed, the mechanical properties, as functions of input parameters, are modified by the four input parameters—including the rotation speed, the traverse speed, the pin’s geometry and the pin’s angle deviation. In this area, input variables have been surveyed and the minimum and maximum levels of both theoretical and practical constraints have been determined. Using the full factorial method, 72 experiments are presented in Table 3.

According to the ASTM E8-04 standard, the trial pieces were prepared for the tensile test and the test at a constant speed of 1 mm/min and a strain rate of 0.003 at the ambient temperature. All the samples were welded by the SANTAM STM-150 in Semnan University. The welded sample’s metallography (detecting the micro-hardness) was implemented according to the ASTM-E384 by the Vickers’ hardness test via Bohler machine at the force of 250 gr in 10 s. The micro-hardness was measured at five spots with the distance of 2 mm from each other on the cross-section of the welded sample. The standard deviations are also represented in Table 3. According to the results, the standard deviations are below 10% which it seems desirable.

The maximum tensile force at the rotation speed of 1150 rpm, as compared to rotation speeds of 1000 rpm and 850 rpm, was respectively improved by 76.8% and 227.7%. “The higher rotation speed, the more generated heat”; because there are more friction and stirring phenomena at higher rotation speeds. Therefore, “The higher rotation speed, the higher mechanical properties of the welded samples”.

The tool’s geometry is the most important factor in the friction stir welding process and plays the most important role in the material flow. The friction stir welding tools consist of a pin and a shoulder. Generating the local heat and material flow are the function of these tools. The first function is to generate the heat at the beginning of the pin's contact with the material due to the friction. Some extent of the heat arises from deformation of the plastic materials as well. The pin sinks as far as the shoulder sits on the workpiece surface. At this step, a great deal of the process heat is produced by the friction between the shoulder and the workpiece. In terms of the heat output, the pin and shoulder’s size ratio is important too. However, other design parameters have a little effect on the generated heat. the workpiece’ warming range is also determined by the shoulder. The second function of the tool is to rotate and move the material. The formation of micro-structure and consequent properties depend on the tool’s geometry. A concave shoulder and a threaded cylindrical pin are commonly used. The material's displacement volume in circular tool pin, as compared to the quadrangular tool pin, is respectively reduced up to 60% and 70%. The reduction of frictional force, the possibility of moving that part of the material which has deformed the plastic, facilitating the tool’s collapsing motion, and increasing the intersection joint between the pin and the material which has deformed the plastic are some advantages of these designs. The dynamic-to-static volume ratio, which is important for creating a proper path for material flow, is the main advantage of these pins. Given the tool’s important geometrical effect on the metal flow, the resulting micro-structure, which is directly related to the flow pattern, will be different for each tool.

3.2 Artificial neural network

The ANN is one of the most important and well-known models in predicting and approximating functions, which can obtain a good estimation for modeling and achieving effective parameters in welding processes [30]. The ANN is widely used in research and problem solving in which an approximate function is required for non-linear mapping between input and output parameters [31]. Classic optimization methods are used for relentless and regular functions and can be confined to minimum local points for them.

A Multi-Layer Perceptron (MLP) is a type of feed-forward artificial neural network that generates a set of outputs from a set of inputs. The MLP is a neural network which connects multiple layers in a directed graph; it means that the signal path through the nodes only goes to one direction. The MLP network consists of input, output, and hidden layers [32].

In this study, the ICA is utilized to evaluate the mechanical properties and the micro-hardness in the FSW. The reason of using this algorithm is that less ICA has been used to optimize welding processes—despite of the ability to optimize the objective function, the ability to find an appropriate global optimal point and the ability not to get stuck in local optimal points and eventually high convergence speed.

Investigating the efficiency of this algorithm in optimizing the FSW processes is one of the objectives of this research. The structure of a two-layer ANN with a hidden layer, in which there are 7 neurons, and a sigmoid tangent conversion function in the first and second layers, in which the ICA optimization algorithm is investigated and the parameters’ influences and interferences are determined, is demonstrated in Fig. 3.

Fig. 3
figure 3

Structure of neural network

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3.3 Imperialist competitive algorithm

It was suggested in the ICA that the evolutionary computing should be inspired not by a natural phenomenon, but also a socio-political phenomenon [33]. The ICA, along with other optimization methods, has been proposed as a new tool for evolutionary calculations and solving optimization problems and successfully applied to a great deal of problems in this field. This algorithm is based on the modeling of colonial phenomenon process and is competitive with other evolutionary algorithms [34]. In the next Nvar optimization problem, a country is a 1 × Nvar array. This array is defined as Eq. (1):

$$Country \, = \, \left[ {P_{1} , \, P_{2} , \, P_{3} , \ldots , \, P_{Nvar} } \right]$$
(1)

The cost of a country is obtained by evaluating the Function (f) in the design variables according to Eq. (2):

$$Cost_{i} = \, f \, \left( {Country_{i} } \right) \, = \, f \, \left( {P_{i1} , \, P_{i2} , \, P_{i3} , \ldots , \, P_{iNvar} } \right)$$
(2)

Like other optimization algorithms, this algorithm forms a country which a number of Ncoun have randomly formed the primary population. The Countryi is divided into two categories; colonial and imperialist. In the ICA, some of the best elements of Nimp population are known as Imperialists and the rest of Ncol population as Colonies. With the formation of the early empires, the imperialistic rivalry between them begins. Any empire who fails to increase its power in the colonial rivalry will be gradually removed from the arena of colonial rivalry and replaced by stronger empires [35]. The new imperialist country begins to attract colonial countries with a policy of assimilation and change in their cultures and customs. The colonial countries’ movement towards the imperialist countries is shown in Eq. (2).

$$X \, \sim \, U \, \left( {0, \, \beta , \, d} \right), \beta \, > \, 1; \, \theta \, \sim \, U \, \left( { - \gamma , \, \gamma } \right)$$
(3)

where X and θ are random numbers. U stands for the displacement function and d for the distance between the colony and the imperialist. Also γ and β are optional numbers in the search range of the colonies.

During this movement, if the colonial countries are in better positions than the imperialists, they will possess the imperialists’ position. The colonial countries’ movement towards the imperialist countries and also the steps of the colonial competition algorithm are respectively depicted in Figs. 4 and 5. Due to their power, the imperialist countries attract these colonies according to Eq. (4):

$$T.C_{.n} = \, Cost \, \left( {imp} \right) \, + \, \varepsilon \, mean\left\{ {Cost \, \left( {col} \right)} \right\}$$
(4)

where T.C.n is the total power of the nth empire and Ɛ is considered as a positive number between 0 and 1.

Fig. 4
figure 4

Moving toward the imperialist competitive algorithm

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Fig. 5
figure 5

Steps of the imperialist competitive algorithm

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In addition, the Cost (imp) and [Cost (col)] nth empire is and average total cost of empire colonies. By calculating the power of each imperialist, the weaker imperialist loses his colonies and gradually disappears, so other imperialists take over these colonies. Meeting the optimal point and remaining only one empire, the algorithm stops. The probability of a colony being taken over by any empire is calculated through Eq. (5):

$$P_{Pn} = \left| {\frac{{N.T.C_{n} }}{{\mathop \sum \nolimits_{i = 0}^{{N_{imp} }} N.T.C_{i} }}} \right|$$
(5)

where N.T.Cn is the total normalized cost of that empire [36]. Escaping from local minimum points as well as having no dependence on any certain network structure are two important advantages of the ICA. Due to the pins’ irregular changes and variations in all FSW parameters via different inputs, not being trapped in local minimum points is very important to optimize the processes in this type of welding. In order to model, the combination of ANN and ICA has been used in this research. The pin’s geometric shape, the traverse speed, the rotation speed, and the pin’s angle are all considered as the input and the micro-hardness as the output. It is assumed, in the optimization process, that the input parameters are persistently constant. The variation range in ANN of the input parameters includes the pin’s shape (cylindrical or cubic), the traverse speed in three levels (27, 32 and 40 mm/min), the rotation speed in four levels (700, 850, 1000 and 1150 rpm), and also the pin’s angle in three levels (2°, 3° and 4°).

Several levels of FSW parameters are provided in Table 4 that the process designer can achieve any of the mentioned mechanical properties and micro-hardness factors in the experiment according to the pin’s angle and geometry, traverse speed and rotation speed. The optimal micro-hardness levels, obtained by the colonial competitive algorithm, should be experimentally tested and measured after determining the optimal welding parameters. In this regard, a friction stir welding step was performed with the optimal parameters, which previously attained, and the experimentally obtained micro-hardness was compared with the levels calculated by the neural network (Fig. 6).

Table 4 Parameters of ICA algorithms
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Fig. 6
figure 6

Comparison between the neural network data and the real data in the tensile test

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The appropriate parameters of the ICA algorithm are as specified in Table 4. The average difference between the micro-hardness levels—which estimated by the neural network in the optimal states obtained via the ICA algorithm—and the micro-hardness levels—which obtained from the experimental details performed by the optimal states—was about 18 ~ 22.

3.4 Particle swarm optimization

The PSO algorithm is a population-based evolutionary algorithm. Like many meta-exploration algorithms, the PSO algorithm begins by creating a “random” population of individuals, which is now referred to as Swarm of Particles (Fig. 7).

Fig. 7
figure 7

Comparison between the neural network data and the real data in the hardness

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As with all evolutionary algorithms, each particle in Swarm is a different set of indeterminate parameters that their optimal levels must be determined; and in fact, each particle represents a spot in a solution space. Like other evolutionary algorithms, these parameters can be defined as real or binary levels, depending on the problem conditions.

Adaptability, convergent responses, proximity, quality, and high stability of responses are some of the advantages of this method [37]. The basis of the algorithm is concerned with searching for a solution space in a convenient way to move a flock of particles to the best position they have encountered in the past, in the hope of reaching a better position in the process, and finally, converging all particles at the optimal point.

In the PSO algorithm, each population’s member has an adaptive speed (relocation) according to which it moves in the search space. In addition, each one has a memory too, namely the best position it can find in the search space. They remember the receipt; therefore, each member moves in two directions. First, towards the best situation they have met. Second, towards the best situation that the best member in their neighborhood has met. The random generation of particles is the first step in the optimization using the PSO algorithm. Each particle has both of position and velocity vectors. To evaluate the performance of each particle, its suitability must be determined. This fitness is determined by the objective function [38]. Calculating the micro-hardness of two pieces of Al5083 and pure copper in FSW was the objective function used in this research. in the next step, the particles’ velocity is updated according to the best position that each particle has met so far and also the best position determined by the whole population. (Eq. (1))

$$V_{i} \left( t \right) \, = \, W.V_{i} \left( {t - 1} \right) + C_{1} \times R_{1} \left( t \right) \times \left( {P_{besti} {-} \, X_{i} \left( t \right)} \right) + \, C_{2} \times R_{2} \left( t \right) \times \left( {G_{best}^{t} {-} \, X_{i} \left( t \right)} \right)$$
(6)

Regardingly, Vi and Xi stand for the velocity and the position of each particle. As well, C1 and C2 are the learning coefficients; R1 and R2 are random levels; Pbest is the best position each particle has ever achieved; Gbest is the best position determined by the whole population; W is the Inertia Weight obtained from Eq. (3); and the parameter t represents that performing PSO calculations is in the step t.

$$(X_{i} \left( t \right) = \, X_{i} \left( {t - a} \right) + V_{i} \left( t \right)$$
(7)

Inertia Weight

$$W = \, \left( {W_{1} - W_{2} } \right)\frac{{max_{iter} - iter}}{{max_{iter} }} + w_{2}$$
(8)

In these equations, W1 and W2 are the minimum and maximum levels of the inertia weight, which are obtained by analyzing the sensitivity of their levels. In this study, their levels were respectively choosed as 0.1 and 0.9. By using the Peripheral Neighborhood PSO algorithm, once a new Gbest is found in the search process, all particles start moving towards the new Gbest, and the search space begins to shrink as the new Pbest is found. When all the particles in the Swarm converged to the Gbest, the optimal answer found by the algorithm is indicating the Gbest position. The particles’ dislocation, in two dimensions, is dramatically shown in Fig. 8 [39, 40].

Fig. 8
figure 8

Dislocation of the particles

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The using ANN is a two-layer neural network with a hidden layer in which there are 5 neurons and a sigmoid tangent conversion function in the first and second layers in which the PSO algorithm is investigated and the parameters’ influences and interferences are determined.

In this research, the PSO algorithm has been used to optimize the ANN for modeling the micro-hardness in FSW. The traverse speed, rotation speed, and pin’s angle are considered as the input and the micro-hardness considered as the output. The variation range in ANN of the input parameters includes the pin’s shape (cylindrical or cubic), the traverse speed in three levels (27, 32 and 40 mm/min), the rotation speed in four levels (700, 850, 1000 and 1150 rpm), and also the pin’s angle in three levels (2°, 3° and 4°). The linear regression analysis between the experimental data (train), in predicting the neural network to calculate the micro-hardness, is shown in Fig. 9. It is observed that the errors are small, compared to the experimental data, and the neural network is reasonably trained.

Fig. 9
figure 9

Comparison between the imperialist competitive algorithm data and the real data in the tensile test

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Several levels of friction stir welding parameters are provided in Table 5 that the process designer can achieve any of the mentioned mechanical properties and micro-hardness factors in the experiment according to the pin’s angle and geometry, traverse speed and rotation speed. In the next step, the PSO algorithm generates decision variables based on Eq. (1) ~ (3) by comparing the results obtained from the micro-hardness simulation model and examining the objective functions (Table 6). As the input data, these levels are inserted into the neural network by the PSO algorithm. This process is continued as far as the answer converges to the optimal level. Utilizing the PSO optimizer function, the levels ​​of objective functions for different program implementations in the neural network are presented in Table 5. As witnessed, the minimum level of the objective function is equal to 0.094426, which obtained using 300 particles (Fig. 10).

Table 5 Objective function levels for different program executions in the ANN by the PSO optimizer function
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Table 6 Appropriate parameters of PSO algorithm
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Fig. 10
figure 10

Comparison between the imperialist competitive algorithm data and the real data in the hardness

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The appropriate parameters of the PSO algorithm were as presented in Table 7. The optimal micro-hardness levels, obtained by the PSO algorithm, should be experimentally tested and measured upon determining the optimal welding parameters. For this goal, an FSW step was performed with the optimal parameters, which already attained, and the experimentally obtained micro-hardness was compared with the levels calculated by the ANN. The PSO algorithm is applied and the actual data is compared to (Figs. 11 and 12).

Table 7 The result of algorithm performance for different number of layers
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Fig. 11
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Comparison between the particle swarm optimization data and the real data in the tensile test

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Fig. 12
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Comparison between the particle swarm optimization data and the real data in the hardness

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The average difference between the micro-hardness levels—which estimated by the neural network in the optimal states obtained via the PSO algorithm—and the micro-hardness levels—which obtained from the experimental details performed by the optimal states—was about 14 ~ 18%.

4 Conclusion

Analyzing different geometries and optimizing the process parameters, this essay has been established to extend knowledge about mechanical properties and micro-hardness of FSW in Al5083 and pure copper (Cu). This was accomplished through the full factorial analysis of FSW welds in Al5083 and pure copper. The parameters and tools’ geometries have been also investigated for Al/Cu dissimilar welds. Supported by mechanical properties and micro-hardness test results, the full factorial method has been used for FSW welding. Overlapped FSW samples have been welded with satisfactory results and the tool’s geometry and welding parameters have been successfully defined. Moreover, copper welds were much more influenced by external errors. The multilayer Artificial Neural Networks (ANN), the Imperialist Competitive Algorithm (ICA) and the Full Factorial Method have been used to analyze. The obtained results indicate that the effects of process parameters involved in the FSW process of square section samples can be investigated by the mathematical models, especially the ANN model, with no need of costly and time-consuming tests and experiments. A slight 14% decrease of mechanical properties in the base material was revealed by confirmatory trials of Al5083 welds. The ICA model, on the other hand, was perfectly fitted with the small error influence. Using the evaluation factors of mechanical properties, the micro-hardness test, and the R2 ~ 0.943 analysis, it was possible to conclude the optimum parameters of joined material. The average difference between analyses proved that the rotation speed of 1150 rpm, the traverse speed of 40 mm/min, and the pin angle of 2° were the optimum conditions. The optimum conditions led to improve the maximum tensile force up to 15 ~ 21%.