Optimal design of multilayer radar absorbing materials: a simulation-optimization approach

Multilayer radar absorbing materials with light weight, strong absorption, and wide absorption bandwidth are urgently demanded with the increase of electromagnetic pollution. However, the current design methods with only simulation operation or optimization strategy are not comprehensive. Here, a simulation-optimization approach including electromagnetic simulation and numerical calculation is proposed based on the interaction between different software, in which the homogeneous medium substitution method is presented to simplify the complicated structures. Besides, return loss and impedance matching of different layer structures are investigated. From the simulated results, it can be found that the structures with better impedance matching have superior absorbing performance. The optimal method shows the advantages of fast and efficient, which has tremendous potential for various applications, such as military stealth and electromagnetic wave elimination.


Introduction
With the rapid development of wireless communication technologies and the wide use of electronic devices, electromagnetic pollution continuously increases and causes many negative impacts, such as interferences with the accuracy of electronic equipment, and even damages to human health [1][2][3]. Moreover, the requirement for radar cross-section (RCS) reduction in aircraft and warships has become more urgent due to the demand for military stealth technology [4]. To eliminate undesirable electromagnetic radiation and improve the stealth performance of military weapons, radar absorbing materials (RAMs) with strong absorption, thin thickness, and broad absorption bandwidth have attracted considerable research interest [5][6][7][8]. According to the transmission line theory, a well-matched impedance between the free space and materials is desired, so that the incident waves can gradually enter the RAMs and be attenuated through absorption and energy conversion mechanisms [9][10][11].
Compared to single-layer structures, multilayer RAMs with more design freedom are preferred to obtain high absorbing capability [12][13][14]. However, the design of such multilayer structures with various parameters presents a real challenge, resulting in a multi-objective optimization problem [15]. More recently, the research on multilayer RAMs mainly focuses on materials optimization and structure design. In materials, tremendous efforts have been dedicated to the preparation of lightweight and strong loss materials, including magnetic materials [16][17][18], carbonbased materials [19,20], and conductive polymer composites (CPCs) [21][22][23]. Distinguished from the aforementioned materials, metamaterials also contribute to improving the absorbing properties [24,25]. The ideal electromagnetic parameters can be obtained by the arrangement of artificial functional unit cells, which paves a new way for the design of wideband strong absorption RAMs [26][27][28]. In addition, another structural design method is to adjust the order and thickness of the different materials for the improvements of impedance matching [29]. To achieve a satisfactory compromise between the absorbing properties and the total thickness, several optimization approaches have been presented, including genetic algorithm (GA) [30][31][32], deep neural network (DNN) [33,34], particle swarm optimization (PSO) [35,36], and artificial bee colony (ABC) [37]. For instance, a DNN has been proposed to forecast the reflection coefficients based on the structural parameters, which can obtain the optimal configuration efficiently [33]. Simultaneously, the PSO method is also applied to the design of RAMs, so that a three-layer absorber with ultrawide absorption band, light weight, and superior performance is realized [35]. Although the above strategies have the potential for predicting the optimal structures, most of them are theoretical and numerical. To this end, massive simulation results are required, making the whole optimization process more complicated. Therefore, it is of great significance to investigate the simulation-optimization methods for RAMs design.
Herein, a simulation-optimization approach to the multilayer RAMs design is proposed and step-impedance structures with different layers at Ka-band are acquired. To simplify the construction process of complex structures, the effective electromagnetic parameters are extracted by the S-parameters retrieval method. Then, the synthesis simulation method based on CST and MATLAB is adopted to realize the automatic adjustment of material and thickness in each layer, and the optimal structure can be determined quickly at last. Furthermore, it is found that the calculated results based on transmission line theory are in good agreement with the simulated results. Hence, the simple genetic algorithm (SGA) is utilized for searching the impedance tapered structures with broader absorption band and superior absorption performance. Finally, impedance matching of the proposed ten-layer optimal structure with the increase of layers at different frequencies is demonstrated. It is shown that the impedance increases from 0 to 377 Ω, indicating the effectiveness of the gradient structure for electromagnetic energy dissipation. The proposed simulation-optimization method integrates electromagnetic simulation and numerical calculation, exhibiting great efficiency in RAMs design, which will provide more application potential in the field of military stealth and electromagnetic wave elimination.
2 Structure and theory of multilayer radar absorbing materials

Electromagnetic wave absorption mechanism
Electromagnetic waves are reflected and transmitted during the propagation between different media, so that the diffuse mechanism of multilayer absorbing materials is tedious. Traditionally, the ray tracing algorithm is used to compute the return loss of RAMs by calculating the sum of the electromagnetic waves emitted from the surface of the topmost media. Undoubtedly, the absorbing properties of the multilayer RAMs can be obtained, but the direct demonstration of impedance matching is lacking. Transmission line theory is considered a promising alternative, which can calculate the impedance through the thickness and electromagnetic parameters of each layer. Hence, the transmission line theory is determined as the basis of the subsequent numerical calculation. The schematic diagram of the transmission model is shown in Fig. 1.
According to the transmission line theory, when the incident electromagnetic wave is along the normal direction, the total return loss of the RAMs can be expressed as Eq. (1).
In Eq. (1), Z 0 and Z n are the characteristic impedance of free space and input impedance of the absorber, respectively. Eventually, the return loss calculation formula is obtained using the normalization method, as shown in Eq. (2). And the input impedance Z n is depicted in Eq. (3).
Thereinto, ε r and μ r refer to the relative complex permittivity and permeability. Thus, the return loss can be calculated by the recursive method when the related parameters are informed.

Numerical computation analysis and electromagnetic simulation method
Simulation method without manual calculation has the advantages of effective and intuitive, which modeling, optimization, and solution processes are included. CST Microwave Studio has been extensively applied to solve the complex electromagnetic simulation, due to its simple operation and high calculation accuracy. By adding field monitors and post-processing template, return loss of the multilayer RAMs can be obtained. Then, the optimization can be realized by parameters sweeping with the given material parameters and structural thickness.
In simulation, the RCS method is used to obtain the return loss of RAMs. According to the relevant theory, the RCS of metal ground plane and RAMs are required, and then the return loss can be calculated by Eq. (6).
In Eq. (6), σ RAM and σ m are the RCS of RAMs and ground plane with the unit m 2 . The military standard for the measurement methods for reflectivity of RAMs is also taken into consideration, and the metal ground plane is set as 180 mm × 180 mm × 4 mm. In addition, the simulation software has strong robustness in structures, so the uniform medium modeling is proposed to save computing resources and shorten simulation time. Through the S-parameters retrieval method, the relative complex permittivity and permeability of different layers are extracted. Figure 2 illustrates the simplified schematic of the complex structures.

Automatic simulation method based on CST and MATLAB
A great amount of simulation with different materials and thicknesses is necessary for obtaining a well-matched structure. Furthermore, it is time-consuming and laborious to determine the optimal parameters in massive simulation results. In order to overcome the abovementioned (5) Z n = n Z n−1 + n th k n d n n + Z n−1 th k n d n (6) Γ = 10 lg RAM m limitations, the synthesis simulation towards multilayer RAMs design is proposed. Because both MATLAB and CST compile with the COM interface, VBA code can be implemented to achieve the data interaction. Consequently, the automatic simulation process and results saving are realized. In addition, combined with the powerful data processing ability of MATLAB, the absorbing performance of each structure is evaluated efficiently.
The procedures of the proposed co-simulation method are illustrated in Fig. 3, where the stages surrounded by a red dotted line are the key to automation. By setting up loops in MATLAB, the update of structures and export of simulation data are realized. The proposed co-simulation method integrates the merits of parametric modeling of CST and data processing of MATLAB, which improves the efficiency of the structural design and provides a new method for the calculation of multilayer RAMs.

Genetic algorithm for searching the ultrathin, broadband, and high-performance structures
Genetic algorithm (GA) is developed to search for the global optimal solution by simulating the process of biological evolution through natural selection. To further improve the optimize efficiency, the simple genetic algorithm (SGA) is designed to predict the return loss based on the transmission line theory. Furthermore, the predicted results are compared with the simulated results, which are shown in Fig. 4. As seen in Fig. 4a, the return loss of RAM is obtained by subtracting the RCS of the metal ground plane from the RCS of RAM. From Fig. 4b, it is observed that the simulated results and predicted results are in good agreement, which demonstrates the great potential of the proposed method in realizing rapid design. The optimization process of multilayer RAMs design based on SGA is provided in Fig. 5. Normally, relevant parameters should be set in advance, including physical model parameters such as the number of structural layers, frequency range and thickness range, and calculation parameters such as initial population size, maximum iterations, crossover probability, and mutation probability. In addition, materials with excellent absorbing performance at Ka-band are chosen from other literature to form the material database. Table 1 shows the relevant references of these materials and the naming scheme in this article. There are sixteen materials and the thickness is also divided into sixteen according to its values range, so the final code used to denote the structure is composed of 8-bit binary encoding. And the corresponding coding scheme changes when crossover and mutation occur, resulting in the update of the multilayer structure. Ultimately, the program terminates when the maximum iteration is reached or the adaptive evaluation of the results is eligible. In conclusion, the optimization process of this method takes less than 1 min, manifesting the advantages of high convenience and accuracy.

Results and discussion
The proposed simulation-optimization approach is very effective for the multilayer RAMs design, which consists of co-simulation and SGA optimization algorithm. To further explain the necessity of the impedance matching design, the three-layer structures with different thicknesses and order are simulated. Table 2 displays the design parameters of the three-layer RAMs at Ka-band. Figure 6 describes the absorbing properties of the three-layer RAMs with the thickness t varies from 0.3 to 0.5 mm with a step of 0.1 mm. It can be found that different simulation results are presented as the thickness of each layer changes. According to Fig. 6a, when t = 0.4 mm, the bandwidth with return loss less than −10 dB reaches the maximum. It is observed that the absorbing properties do not improve with the increase of the structural thickness, indicating that there is no linear correlation between them. From Fig. 6b, the impedance between the topmost material and free space varies with frequency due to the dispersion of materials. And the return loss increase as the input impedance deviates from the intrinsic impedance of the free space. In addition, it is seen that the impedance for the thickness of 4 mm is 377 Ω at 29.38 GHz, which is exactly the intrinsic impedance of free space. Accordingly, the return loss with this thickness reaches nearly −30 dB at 29.38 GHz. Thus, it is important to optimize the thickness for achieving a well-matched impedance in the RAMs design.
Based on the above analysis, the structural thickness of each layer is set as 0.4 mm, and the influence of materials ordering on the absorbing performance is researched, as shown in Fig. 7. It can be observed that the return loss varies with the materials ordering and the order 4/16/11 presents the optimal absorbing properties. Therefore, the materials selection of each layer is also very notable to attain considerable absorbing performance and operating bandwidth.
In order to validate the availability of the proposed SGA method, RAMs with different layers are designed and optimized. The return loss results of these optimal structures are described in Fig. 8 and the related parameters are listed in Table 3. As displayed in Fig. 8, the return loss values of these structures with different layers are all below −10 dB, and the return loss of the ten-layer structure 3# is even less than −20 dB. Then, the impedance matching of the proposed ten-layer optimal structure 3# with the increase of the layers is investigated, as depicted in Fig. 9. From  Fig. 9a, the impedance increases from 0 Ω of the metal to 377 Ω of the free space with the growth of layers, implying that the impedance matching is becoming better. Figure 9b intuitively depicts the impedance changes with the increase of layers at different frequencies. According to Fig. 9b, the variation of the impedance in the middle layers is not continuously rising. Hence, it can be inferred that the better absorbing performance of structures will be achieved, if the ascending state of impedance matching is maintained constantly. Fig. 6 Absorbing properties of the three-layer RAMs with different thicknesses. a Return loss. b Impedance between the topmost material and free space Fig. 7 Absorbing properties of the three-layer RAMs in different orders. a Return loss. b Impedance between the topmost material and free space To further clarify the electromagnetic energy dispassion mechanism of multilayer RAMs, the power loss density of the proposed three-layer RAMs and four-layer RAMs are researched, respectively. Figure 10 shows the three-layer RAMs with different thicknesses at 32 GHz and 36 GHz. It is noticed that the structure with t = 0.4 mm possesses stronger energy loss ability and the main electromagnetic energy attenuation occurs in the area of the top layer. However, the same structure with t = 0.5 mm has no significant energy consumption layer, which corresponds to the return loss in Fig. 6. It indicates that the absorbing properties are largely determined by the materials thickness. Ideally, electromagnetic waves are expected to gradually enter the multilayer RAMs and be dissipated in the non-surface layer, so the direct reflection at the material-air interface can be avoided. Obviously, for the above three-layer structures, the electromagnetic waves are dissipated at the material-air interface, resulting in surface reflection and poor absorbing performance. It can be deduced that the limited number of layers is difficult to obtain perfect impedance matching.
As a comparison, the power loss density of the optimal four-layer RAMs with different structures is simulated, as displayed in Fig. 11. According to Fig. 11, the main electromagnetic energy attenuation occurs in the second layer, while the energy loss in the topmost layer is relatively week. Combined with the return loss results in Figs. 6a and 8a, the absorbing properties of the four-layer structures are better than the three-layer structures, indicating that the four-layer design makes the electromagnetic waves enter the RAMs more easily and then be consumed. In addition, it is found that the material of the second layer used in 3# structure is the same as the top layer of the mentioned three-layer structures. Even though this material is served to lose electromagnetic energy, it leads to different absorbing effects, which proves that the thickness and impedance matching between materials have a great influence on the absorbing performance. Thus, it can be reasonably inferred that the impedance matching will be better and the absorbing properties will be further improved, by increasing the number of structural layers and optimizing the multilayer design. It is worth noting that the proposed optimal method is suitable for the multilayer RAMs design at any band, which will greatly promote the rapid application of radar absorbing materials.

Conclusion
Well-matched multilayer RAMs with broader absorption band and superior absorption performance are designed by the proposed simulation-optimization approach. Thereinto, the uniform medium is utilized to replace the modeling of complex structures, so that the simulation period can be shortened on the premise of ensuring the simulation accuracy. Then, three-layer RAMs with different thicknesses and arrangements are simulated. In terms of simulated results, these structures display great differences in return loss, indicating the necessity of the structural design. Additionally, several RAMs with return loss less than −10 dB at Ka-band are optimized via the SGA method, and the impedance matching results of the proposed ten-layer optimal structure are investigated. It is implied that the absorbing performance becomes better with the improvement of impedance matching. The presented simulation-optimization approach can realize automatic simulation and rapid optimization, which widens the way to solve the comprehensive optimization of the multilayer RAMs design.
Author contribution For this work, Huiming Yao and Jiapeng Yang researched the co-simulation method based on CST and MATLAB. Huiming Yao and Han Li applied the simple genetic algorithm to the optimization of multilayer radar absorbing materials. Jianchun Xu studied the simulation method of multilayer absorbing materials based on electromagnetic simulation software. Huiming Yao and Ke Bi wrote the manuscript with contribution from all the other authors. All authors participated in the discussion of the results and are in agreement with the content of manuscript.