Metaheuristic Optimization Techniques for an Electromagnetic Multilayer Radome Design

1. Ant Colony Optimization for a Continuous Domain

Originally, ACO was introduced in 1992 by Dorigo [10], and it has been used to solve many combinational optimization problems consisting of a set of discrete decision variables. The idea of applying ACO in solving the continuous optimization problem was presented in 2008 by Socha and Dorigo [6]. The flowchart of the ACO_R algorithm is illustrated in Fig. 1.

Fig. 1

Flowchart of ACO_R algorithm.

In ACO_R, the solutions are kept and sorted in a solution archive (Fig. 2), in which the associated weight of solution l defined in Eq. (1) represents the strength of the solution in producing new solutions.

Fig. 2

Solution archive of the ACO_R algorithm with sli as the value of parameter i in group l and the objective value f(l) with associated weight ω(l).

To generate new solutions, a kernel is selected by probability that is computed for each group as in Eq. (2).

Roulette wheel selection [11] is applied to select a solution kernel. Each probability p(l) is presented as a proportion of the wheel (Fig. 3), and a random selection process is made similarly to rotate the roulette wheel. An m new random number group according to the parameterized normal distribution is used with mean ( μli) and standard deviation ( σli) for ant i in group l.

Fig. 3

Roulette wheel with a probability of solution p(l) as the proportion of the wheel.

ξ>0 is a constant parameter that is similar to the pheromone evaporation rate in ACO [10]. Then, m new solutions are added to the solution archive and re-ordered. Non-fit solutions are removed, and only the k best ones are kept after each iteration.

2. Transmission Coefficient Characteristic of the C-Sandwich Radome

A C-sandwich radome, which is a multilayer radome (style d) [7], is considered in this work. The sandwich wall has five layers that were developed to cover an antenna array operating in a wide-band frequency range. The C-sandwich construction consists of three skins that are interleaved by two cores. Typically, the relative permittivity of skin is greater than that of the core. To guarantee that the input and output of the propagation wave are equal, a C-sandwich wall is designed with the first and last skins having the same thickness, and the widths of the two cores are the same as well. To fabricate a radome, the thickness of each layer is designed as the multiple of plies. The propagation wave on the radome with layer i has thickness d_i, relative permittivity ɛ_i, loss tangent tanδ_i, and refractive index n_i. The F_{j (j ∈[1,7])}/B_k(k∈[1,6]) are the forward and backward propagation waves, respectively. The forward and backward fields at the first left interface are related to Eq. (5).

τ_i =1+r_i and r_i is the intrinsic reflection coefficient of each interface and is calculated by the Fresnel equation. If one uses the formula in Eq. (5) and simplifies it to Eq. (6), the transmission coefficient can be obtained by Eq. (7).

1. Fitness Function

The main objective of this research is to maximize the transmission coefficient of a multilayer structure radome, which is calculated by Eq. (7) in a specific frequency band with various angles of incidence in both the TE and TM modes. To ensure the stability of the transmission coefficient, the following objective function is considered for optimization:

where T(f,θ) is the transmission coefficient at frequency f and incident angle θ.

2. Optimization Design Result

The optimized thickness design is compared with an effective solid radome type with a half-wavelength thickness. The transmitted power is 80% greater in both polarizations, and the wide angle of incidence is the target of optimization. The design of the multilayer radome structure for optimization is shown in Table 1.

Table 1

C-sandwich radome layer properties

According to the EMA method, a multilayer radome can be approximated by its effective material property. The effective relative permittivity (ɛ_ff) of the radome after optimization is calculated by Eq. (9), where n is the total number of layers, and d_i, ɛ_i are the thickness and permittivity of layer i, respectively. By using a half-wavelength equation to assume the thickness of the effective medium, Eq. (10) can be used to compute the thickness of the effective half-wavelength solid-type radome at a resonant frequency of 10 GHz for the X-band and 15 GHz for the Ku-band with an angle of incidence θ= 70°.

The first optimized radome of this study is the design for the X-band (8–12 GHz) frequency. According to a comparative analysis with 30 execution times (Table 2), ACO_R requires less time (56.69 seconds) to reach the convergence than PSO (70.18 seconds). Fig. 4 presents the comparison of transmitted power stability between the optimized multilayer radome and the effective solid radome with the stability of a multilayer radome in various angles, with both TE and TM modes being under 10% change. The transmission coefficient characteristic of the C-sandwich radome after optimization is presented in Fig. 5 in the X-band.

Table 2

Comparison of the 30 execution times between ACO_R and PSO for the X-band frequency

Fig. 4

Stability of the transmitted power of an optimized multilayer structure with an effective solid-type radome in the X-band.

Fig. 5

Optimized transmission coefficient of the multilayer structure for the X-band.

To give a more effective decision, the same design properties of the multilayer radome are applied in the Ku-band (12–18 GHz). Table 3 shows the same result to demonstrate that ACO_R has better performance than PSO. Fig. 6 illustrates that the stability in the transmitted power of the optimized multilayer structure is less than 10%, which is outstanding compared with that of the effective solid radome. Fig. 7 shows the transmission coefficient of the optimized radome for the TE and TM modes in the Ku-band.

Table 3

Comparison of the 30 execution times between ACO_R and PSO for the Ku-band frequency

Fig. 6

Stability of the transmitted power of the optimized multilayer structure with an effective solid-type radome in the Ku-band.

Fig. 7

Optimized transmission coefficient of the multilayer structure for the Ku-band.

According to the computed result, ACO_R gives a promising value in optimizing multi-objective problems applied in the electromagnetic multilayer radome design.