Circular Antenna Array Synthesis Using Multiverse Optimizer

Antenna array synthesis is one of the most popular topics in the electromagnetic field. Since achieving a desired antenna radiation pattern is a mathematical problem, in the literature, there are various optimization algorithms applied to the synthesis process of different kinds of antenna arrays. In this study, Multiverse Optimizer (MVO) and modified MVO (MMVO) are used to perform circular antenna array (CAA) synthesis. During the exploration, exploitation, and local search phases of calculation, MVO uses three concepts in cosmology; white hole, black hole, and wormhole. Convergence capability of this nature-inspired algorithm is employed for finding optimum amplitude and position values of CAA elements in order to achieve an array pattern with low maximum sidelobe level (MSL) and minimum circumference. 'e performance of MVO and MMVO was tested on five design examples of pattern synthesis, and the obtained results were compared with ten different algorithms. 'e simulation results show that MVO and MMVO provide low MSLs with small circumferences.


Introduction
In recent days, with the developing technology, devices have started to host more wireless components. e application of wireless technology in all areas of data transfer and communication has accelerated the spread of its use. However, increment in use of this technology brought some drawbacks such as interference and electromagnetic pollution. To overcome this issue and use frequency spectrum much more efficiently, RF components are improved year by year. Antennas, which constitute one of the most important elements of wireless technology, also took part in this improvement. As a result, different antenna types appeared such as wire, horn, dish, and array. e antenna arrays consist of a combination of multiple antenna elements and they are widely used in communication. Electromagnetic radiation of each element can be constructively or destructively added. us, whole array can radiate towards desired direction while minimizing the interference on undesired angles [1]. anks to their useful and flexible structure, antenna arrays can provide high gain, high directivity, and high beam steering capability with less interfering effect. ese features of antenna arrays make them popular both in military and civil applications. ey are widely used in sonar, radar, navigation systems, cellular systems, and satellite communications.
Antenna arrays take their names according to the geometric shape of overall array structure. Array elements can be positioned on a single line, called linear antenna arrays, or distributed over a plane, called planar antenna arrays. Array structures can be in form of circle and ellipse too. Moreover, by creating nested structures from circular or elliptical arrays, concentric circular and concentric elliptical antenna arrays are achieved.
In this paper, circular antenna array (CAA) is considered for the array synthesis. Unlike the linear antenna arrays, the main lobe of CAA can be directed between 0 and 360 degrees. According to the changes in position, amplitude, and phase of array elements, or changes in circle radius, overall radiation pattern is shaped. However, achieving a desired antenna pattern is a challenging task since the mentioned array parameters have to be calculated. It is seen from the literature that the optimization algorithms can be used to overcome this kind of pattern synthesis problems. e following can be given as examples of these studies: linear antenna array synthesis using Tabu Search Algorithm (TSA) [2], Harmony Search Algorithm (HSA) [3], and Touring Ant Colony Optimization (TACO) [4]; planar antenna array synthesis using Taguchi Method (TM) [5] and Genetic Algorithm (GA) [6]; elliptical antenna array synthesis using Backtracking Search Optimization Algorithm (BSA) [7]; circular antenna array synthesis using Differential Evolution (DE) Algorithm [8]; and concentric CAA synthesis using BSA [9]. Multiverse Optimizer (MVO), which is a nature-inspired algorithm, was proposed in 2016 [10] and used to solve practical optimization problems in engineering such as welded beam design, gear train design, three-bar truss design, pressure vessel design, and cantilever beam design. In [10], the performance of MVO was compared with GA, Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), and Gravitational Search Algorithm (GSA). e majority of test results show that MVO outperforms the popular algorithms in the literature. MVO was also used for multiobjective optimal antenna synthesis of microwave wireless power transmission [11].
In the following section, MVO and MMVO are explained in detail. Section 3 includes the CAA geometry and equation of array factor. Section 4 presents results and discussions. Section 5 covers the conclusion part.

Multiverse Optimizer
Multiverse theory was proposed by physicists, and MVO is inspired by this theory. It is believed that more than one big bang occurred, and each one of them is resulted with the birth of a universe [19,20]. e idea behind this theory is the opposite of universe, which states the presence of other universes in addition to ours. Besides, according to this model, these universes may interact and/or collide with each other.
MVO is based on three concepts in the cosmology: white holes, black holes, and wormholes [10]. Physicists state that the white hole, which is also considered as big bang, is the main component of the birth of a universe [21]. It appeared after the collision of two parallel universes [22]. Black holes are the white holes in reverse and, because of their extremely high gravitational force, they attract everything including light beams [23,24]. Wormholes can be described as time/ space highways where objects are able to travel instantly between two different points of a universe or two different universes too [25].
Basically, MVO converges the target by sending and receiving the objects (variables) between the universes (solutions). is sending/receiving process depends on the inflation rates (fitness values) of the universes calculated over iterations. If a universe has high inflation rate, it is highly probable to have white holes and tends to send object towards other universes. If a universe has low inflation rate, most probably it has a black hole and tends to receive objects. is mentioned process is the exploration step of the algorithm and is represented mathematically as follows [10]: where Ui is the ith universe, p is the number of parameters (variables), n is the number of universes (solutions), x j i is the jth parameter of ith universe, r1 is a random number in [0, 1], NI (Ui) is normalized inflation rate of the ith universe, and x j k is the jth parameter of kth universe selected by a roulette wheel selection mechanism.
In the exploitation phase of the algorithm, it is assumed that each universe is using wormholes to transfer the objects through space randomly regardless of inflation rate. Besides, wormholes are used to perform local updates in each universe by establishing a connection to the best universe formed so far. is mechanism is formulated as follows [10]: where X j is the jth parameter of best universe formed so far; r2, r3, and r4 are the random numbers in [0, 1]; and ub j and lb j are used to define upper and lower bound of jth variable, respectively. WEP and TDR are coefficients which are explained below.
WEP stands for the wormhole's existence probability in the universe. In the algorithm, this coefficient is increased linearly over repetitions to emphasize exploitation. WEP is calculated as follows [10]: 2 International Journal of Antennas and Propagation where, by default, min is defined as 0.2 and max is defined as 1. L indicates the maximum iteration value and l represents the current step of the iteration. Traveling distance rate (TDR) is also a coefficient. It is used while sending objects through a wormhole to the best universe obtained so far. e formula for TDR is as follows [10]: where p indicates the exploitation accuracy and is equal to 6 as a default value. e pseudocode of the MVO is shown in Figure 1.

Modified MVO (MMVO). As is stated in No Free
Lunch (NFL) eorem, there is no single algorithm to solve every problem set [26]. Besides, it is realized that the performance of an algorithm can be improved by modifying it and thus the algorithm can be used to solve problems in different fields. Even though the performance of original MVO is good enough to optimize the problems, structure of the algorithm itself is suitable for modifications to improve the convergence. erefore, in this paper, the exploitation phase of the MVO is changed as follows: and this modification aims to perform the solution updates locally using the best universe and current universe instead of upper and lower bounds of problem set. us, by shortening the exploitation steps, the algorithm is improved and forced to find better amplitude and position values for array elements.

Circular Antenna Array
In Figure 2, geometry of a nonuniform CAA is depicted. N isotropic radiators (antenna elements) are placed nonuniformly on a circle of radius a in the x-y plane. e array factor of such structure can be mathematically described as follows [1]: In order to focus the main beam towards the (θ o , ϕ o ) direction, the excitation phase is written as follows [1]: In the above, I n and α n indicate the excitation amplitude and excitation phase of the nth antenna element, respectively. d is used to define the arc distance between the adjacent antenna elements. ϕ n is the angular position of the nth radiator on x-y 1. Initialize the universe population U 2. Initialize l, WEP and TDR 3. while l < L 4.
Calculate the fitness for all universes 5.
Sort the universe population and get SU 6. NI = Normalized fitness values of the universes 7.
for each universe indexed by i 8.
Update WEP and TDR 9.
for each object indexed by j 11.
end for 27. end for 28.

Results and Discussion
In this section, the overall performance of MVO and MMVO on CAA synthesis is evaluated in detail. e purpose of performing array synthesis is to achieve a desired antenna pattern. As mentioned in the Introduction, different synthesis processes were applied to different kinds of antenna arrays with different algorithms. e main aim of those works is to get low sidelobe levels, place nulls in desired directions, and increase the gain or directivity of the antenna.
In this paper, MVO and MMVO algorithms were used to find optimum amplitude and position values of CAA elements in order to achieve low MSL and minimum circumference. e obtained results from both MVO and MMVO were comprehensively compared with the results of ten different algorithms which are GA [14], PSO [15], FA [16], SQP [13], SADE [13], HA [12], ALO [12], BBO [13], SOA [17], and SA [18]. During the optimization, the following fitness function was used: where ϕ MSL is the angular interval of the sidelobe region that is minimized, AF max is the peak of main beam, and C is the desired circumference. In each iteration step, the difference between the desired circumference and the arc distance summation is taken into account and the algorithm reduces it while keeping the sidelobe level to a minimum. e fitness function is used for CAA with 8, 10, 12, and 20 elements. In order to observe the real performance of MVO and MMVO and to make a fair comparison, 20 independent runs were performed under the default parameters.  Figure 3. Table 1 illustrates MSL and circumference values of the radiation patterns obtained using MVO and MMVO. Table 1 also shows the comparison results of GA [14], PSO [15], FA [16], SQP [13], SADE [13], HA [12], ALO [12], BBO [13], SOA [17], and SA [18]. It can be clearly observed from Table 1 that MSL and circumference values of MVO and MMVO are better than those of GA [14], PSO [15], FA [16], SQP [13], SADE [13], HA [12], ALO [12], BBO [13], SOA [17], and SA [18]. e convergence curves of this example can be seen in Figure 4.
In the fourth example, the number of array elements is set to 20 and the sidelobe suppression region is expanded to ϕ � [− 180°, − 14°] and ϕ � [14°, 180°]. Result comparison of this synthesis example is made in Table 2, and array patterns are shown in Figure 7. According to Table 2, it is evidently seen that the MSL and circumference values obtained by MVO and MMVO are better than those of BBO [13], SADE [13], SQP [13], HA [12], and ALO [12].
In the last example, while suppressing the MSL, the circumference value is fixed to the circumference value of 20-element uniform CAA which is 10 λ (d i � 0.5 λ). After 20 independent runs, − 12.51 dB and − 12.62 dB MSL are achieved for MVO and MMVO, respectively. e obtained patterns are shown in Figure 8. In Table 2, the MSL results of patterns produced by MVO and MMVO are compared with the results of BBO [13], SADE [13], SQP [13], HA [12], and ALO [12]. It is apparent from Table 2 that MSL values of MVO and MMVO are better than the MSL values of the other compared algorithms. e amplitude and position values for all design examples are shown in Table 3. Besides, box-and-whisker plots are depicted in Figure 9 to exhibit the overall statistical view of 20 independent runs for each design case. From this figure, improvement in results can be seen as an outcome of modification in the algorithm.

Conclusions
In this paper, MVO and MMVO algorithms were presented for the synthesis of CAA. ese algorithms were used to calculate the optimum amplitude and position values of the isotropic radiators in the CAA in order to achieve low MSL and minimum circumference. Five different design examples were used to see the overall performance of the MVO and MMVO against ten different optimization algorithms. It was observed that MVO and MMVO have offered minimum sidelobe levels with competitive circumference values. MVO and MMVO can be a good alternative while solving the optimization problems in antenna and electromagnetics.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.   International Journal of Antennas and Propagation 9