Designing Thinner Broadband Multilayer Radar Absorbing Material Through Novel Formulation of Cost Function

Modern military and civic applications require broadband microwave multilayer radar absorbing material (MRAM), which typically consists of multiple layers that increase the thickness of platforms like aircraft, ships, and vehicles, thereby adding to their weight. To address this, MRAM design methods strive to optimize layer parameters to reduce thickness. In this article, we emphasize the importance of the cost function used in the optimization process, which can significantly affect the outcomes. This article investigates the impact of common cost function formulations on achieving global optima and proposes a new cost function that produces thinner MRAM designs without increasing algorithmic complexity. The new formulation works as a two-step optimizer and yields thinner results. The article compares the results for twelve different MRAMs with those of previous literature, and achieves maximum and minimum thickness reductions of 22.9562% and 0.009%, respectively. The findings highlight that the cost function formulation significantly impacts the design outcome and can be used to obtain thinner designs. Additionally, the proposed approach is more effective than other commonly used formulations for finding the optimal layer sequence and minimizing thickness using a given search algorithm.


I. INTRODUCTION
Radar absorbing materials (RAMs) have a wide range of uses in today's military and civic industries [1], [2], [3], [4]. RAMs by reducing the electromagnetic (EM) reflection of the platforms help to reduce the detectability of the target. In view of this, multi-layer RAMs (MRAMs) have piqued the interest of researchers in recent decades due to the high likelihood of achieving very low reflectivity for broad operating bandwidth (BW) and angle of incidence with both transverse electric (TE) and transverse magnetic (TM) polarization. For this purpose, the surface of the platform is coated with MRAM. This coating is often referred to as radar absorbing paint. However, The associate editor coordinating the review of this manuscript and approving it for publication was Mohamed Kheir . this increases the thickness of the platform's surface and, as a result the weight and volume of the platform. Despite extensive research to improve RAM performance and reduce thickness, designs of thin MRAMs that operate in the desired frequency range are highly demanded [5], [6]. In the design problem of an N-layer MRAM with a given material database, although there are 2N variables that need to be optimized, the optimization becomes difficult due to an increase in convergence time with the increase in the number of layers and other design requirements. In this regard, various single-objective meta-heuristic algorithms [7], [8], [9], [10], [11], [12] as well as multi-objective algorithms [5], [13], [14] have been used.
The design of a MRAM is generally a multi-objective problem. The commonly used objectives are to increase the operational BW and to reduce the thickness of the MRAM [15]. There are different approaches to solving this problem. First, the problem is solved using multiobjective algorithms [16]. However, multiobjective solutions require a lot of resources and processing. Second, a single objective optimization algorithm is used by combining the two objectives using the weighted sum method. In this method, the multiple objectives are combined using a weight factor that decides the priorities amongst the objective functions. However, the value of the weight factor needs to be carefully chosen, since it plays an important role in obtaining the final solution. Therefore, the weight factor should be considered as a variable for the optimization problem.
For any optimization algorithm, the formulation of the cost function plays a very important role in finding the optimal solution for the problem. However, to the best of our knowledge, the effect of the formulation of the cost function on the final design obtained using an optimization algorithm has not been studied for MRAM design problem. In literature, two formulations are most commonly used. Both these formulations focus on the maximization of reflection loss (RL) in the BW under consideration. However, for practical purposes, the best solution will be the thinnest MRAM with the required level of performance (RL) in the desired BW. This can be achieved by designing a cost function that focuses on minimizing thickness while satisfying the performance requirement.
This article investigates the impact of different cost function formulations on the design of MRAM and introduces a new cost function formulation that can determine the optimal layer sequence and thicknesses for MRAM design based on a given RL requirement, without relying on the weight factor. To evaluate the proposed formulation, we utilized the Particle Swarm Optimization (PSO) [8], [17], [18] algorithm with a range of weight factor values and compared the outcomes with those obtained using commonly used formulations (discussed in section III). Furthermore, we compared our designs with the findings reported in [8] to demonstrate the effectiveness and superiority of the proposed formulation. In addition to this, we also test the applicability of the proposed formulation to wide-angle incidence MRAM design. We also discuss the absorption mechanisms in MRAM and analyze the loss per layer in a MRAM structure.
The remainder of this article is structured as follows: Section II presents the absorption mechanism and physical model of MRAM, while Section III outlines the problem formulation, including commonly used and new cost functions. In Section IV, we analyze the cost functions and present comparative results. Lastly, Section V provides a concluding summary of the paper.

II. PHYSICAL MODEL OF MRAM
As the name suggests, a MRAM works through absorption of EM wave. In this section, we explain the various absorption mechanisms in a MRAM, the physical model and RL calculation for MRAM.

A. ABSORPTION MECHANISMS OF MRAM
When an EM wave encounters a material, it interacts with its constituent atoms and molecules. The behavior of the EM wave depends on the frequency of the wave and the nature of the material's molecular structure. The commonly used materials in MRAM to absorb EM waves are lossy dielectric materials, lossy magnetic materials, and conducting materials. All these materials dissipate the energy of the EM waves by conversion of electrical energy in the form of heat [3], [4]. The primary mechanism for dielectric loss at microwave frequencies is dipole relaxation, whereas for magnetic loss is magnetization rotation within the domains. The conducting materials dissipate EM energy due to finite conductivity of the material. These mechanisms can be combined by creating composites that incorporate multiple materials with different absorption mechanisms [14]. Another way to combine these effects is to use a multilayer structure, where each layer may have a different absorption mechanism [3]. Multi-layering also helps to increase losses of an incoming EM wave mechanisms, such as internal reflection and phase cancellation [15].
The performance of MRAM is evaluated using RL (Reflection coefficient in dB). RL can be defined as the ratio of reflected power from the MRAM to the incident power [4], [15]. This implies that RL represents the absorbed power in relation to the incident power, without taking into account the actual magnitude of the incident power. Consequently, RL remains unaffected by changes in incident power levels. The optimization process aims to improve RL, and therefore, the optimization results are independent of power of incident signal. It is worth highlighting that the reliability and safety of MRAM structures face significant challenges when dealing with high peak and average microwave power sources. Design failures can occur due to EM breakdown and MRAM overheating. EM breakdown is particularly concerning with very short pulse durations (in microseconds) of high peak power. Conversely, sustained continuous wave operations with high average power can lead to system overheating [19]. The dissipated energy within the system, resulting from timeaveraged losses, transforms into heat, which may cause issues depending on the chosen design materials. Therefore, the maximum incident power on the MRAM must be considered, and the selection of materials for MRAM should be tailored to the power of the incident signal in a given application. Typically, objects coated with MRAM are located far from the source, resulting in low incident power and minimal heat build-up [3], [4].

B. PHYSICAL MODEL AND REFLECTION LOSS CALCULATION
The physical configuration of a MRAM with N layers on a Perfect Electric Conductor (PEC) substrate is illustrated in Fig. 1. Each planar layer 'i' is assumed to be infinite along the x-and y-axes and parallel to the PEC. The layers have different thicknesses d i , frequency-dependent complex VOLUME 11, 2023 permeabilities µ i , and permittivities ϵ i . An EM wave of a specific frequency is incident from the air (layer 0) to the first interface of the N-layer MRAM, where it is partially absorbed by each layer, partially reflected at each interface, and partially transmitted to the next layer until it is reflected back by the PEC (layer N + 1), which acts as a perfect reflector. The reflection coefficient of the MRAM is calculated using various formulations that express the reflection coefficient of an EM wave impinging upon a planar layered media as a function of constitutive parameters, the thickness of each layer, the angle of incidence, and polarization [8], [20], [21].
In Fig. 1, if angle of incidence is θ o , then the reflection coefficient R TE/TM i at i th layer for both polarization is defined as follows: where r TE i and r TM i are given by and where k i is the wavenumber of the i th layer and is expressed as follows: ϵ i (f ) and µ i (f ) are frequency depended complex permittivity and permeability of i t h layer respectively and are expressed as: In the above equations, ω (= 2πf) is the radian frequency, ϵ 0 and µ 0 are permittivity and permeability of free space respectively and are given by As the last layer of MRAM is PEC, the reflection coefficient of the last layer is taken as 1 for TE polarization and -1 for TM polarization. Thus, the total reflection coefficients R TE 0 and R TM 0 for the MRAM are recursively calculated using (1).

III. PROBLEM FORMULATION
The optimization problem for designing MRAM involves multiple objectives, including maximizing operational BW and minimizing MRAM thickness. Operational BW refers to the frequency range where the MRAM's RL is below a specified threshold. The literature offers several approaches to formulate the cost function for this multi-objective optimization problem. In this article, we focus on two of the most frequently used formulations.
A. FORMULATION 1 In this formulation, the thickness of the MRAM is considered by summing the thickness of all the layers while the RL is taken into account by considering the sum of RL at all frequency points [13], [15], [21]. The cost function is formulated by combining two conflicting objectives (OF a and OF b ) using a weighted sum approach.
where A and C are normalization factors and given as Here, OF a focuses on the maximization of the RL in the desired BW and the OF b focuses on the minimization of the thickness. The final value of cost function CF 1 depends on OF a and OF b , as well as the weighting factor α which decides the priority between OF a and OF b .

B. FORMULATION 2
In this formulation, the thickness of the MRAM is again considered by summing the thickness of all the layers while the RL is taken into account by considering the sum of maximum RL at a given angle for all the frequencies [7], [8], [10]. Similar to CF 1 , the cost function CF 2 is formulated by combining the two aforementioned objective functions OF c and OF d .
OF c = 20 log 10 (max(|R 91018 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
here, OF c focuses on the minimization of maximum RL provided by the MRAM in desired frequency range while OF d focuses on the reduction of the total thickness.

C. NEW FORMULATION
In the above formulations, OF a and OF c focus on the maximization of RL in the desired BW. For MRAM applications, generally, in the desired BW, the RL must be below a required threshold and the thickness should be as low as possible. In practice, the actual value of the RL may not be important as long as it is below a predefined threshold. Therefore, based on this information we propose a new formulation as follows: where, RC dB is the RL (R TE/TM 0 ) in dB, RL des is the desired level of RL (dB) in the BW under consideration. OF 2 is defined as: The cost function CF new looks same as (10) and (17), however, unlike (10) and (17) the α is fixed at 0.9. Here, OF 1 focuses on minimizing the number of frequency points at which the RL is greater than the predefined threshold. As the RL criteria is satisfied, the summation in (18) goes to zero, and therefore contribution from OF 1 goes to zero. After this point, only OF 2 will contribute to CF new . Equation (20) becomes independent of α and the algorithm then focuses only on minimizing the thickness of the MRAM. With CF new , the algorithm effectively works as a two-step optimization problem because once the RL criteria is fulfilled, the algorithm automatically switches to the optimizing thickness of MRAM.

D. HEURISTIC OPTIMIZATION ALGORITHM FOR DESIGNED MRAM
The MRAM optimization process utilizes the PSO algorithm in conjunction with the cost functions. PSO is an evolutionary algorithm inspired by the collective behavior of swarms, flocks, herds, and schools of animals searching for food [18]. In the PSO algorithm, each swarm member, or particle, adjusts its search strategy using its own experiences and those of other particles in the swarm. Mathematically, each particle represents a potential solution or a point in the D-dimensional search space. The global optimum is represented as the location of the food source in the model. The PSO algorithm uses a particle's fitness value and velocity, which are determined based on the best experiences of the entire swarm and the individual particle, to guide its flight direction. This algorithm has demonstrated success in optimizing a wide range of problems and is straightforward to implement [17]. With regard to MRAM design utilizing PSO from a given material database, we consider each particle is characterized by a set of selected material from the database and its thickness for each layer. For this purpose, PSO uses integer number codes for the materials in the database [15]. Therefore, in Table 1, Table 2, and Table 3, we use integer numbers 1, 2, 3, . . . etc. to identify different materials.

E. VERIFICATION SOFTWARE FOR DESIGNED MRAM
The Computer Simulation Technology (CST) Microwave Studio Suite (MWS) is a widely used EM simulation tool for designing microwave structures and validating computed results. In this article, Finite Element Method (FEM) was used with the Frequency Domain Solver (FDS) module to simulate all designs. The FDS module has shown excellent simulation performance in array structures, while CST is capable of accurately simulating skin effects, resulting in precise results. These outstanding features make CST an effective validation tool for many studies on MRAM design. For this specific study, the optimized MRAM layers were modeled as a unit cell in CST, as demonstrated in Fig. 2.  To model the MRAM structure, an infinite periodic repetition (unit cell) is used along both the x and y axes. Each layer is assigned material from the CST database and for user-defined materials, the material characteristics can be assigned. For externally supplied material characteristic parameters, the CST studio internally fits the supplied data for future use. The fitting error between originally supplied VOLUME 11, 2023 data and fitted data will lead to a deviation of the results simulated in CST from the calculated results. Table 2 presents parameters of 16 different types of materials. Among all these materials, Material-8 and Material-12 give higher fitting error. For sake of comparison, Fig. 3 (a), Fig. 3 (b), Fig. 3 (c), and Fig. 3 (d) illustrate the fitting result for Material 8, Material 12, Material 6, and Material 1 respectively. The structure is simulated with floquet ports which simulate planar wave incidence. This approach enables obtaining reflection coefficients that are close to actual measurements for the designed MRAMs.

IV. RESULTS AND ANALYSIS A. COMPARISON OF FORMULATIONS OF COST FUNCTION
To demonstrate the effectiveness of the proposed formulation, we optimized three different databases using formulation 1, formulation 2, and the proposed formulation for the same set of design requirements. The databases are listed in Tables -1, 2 and 3 along with the integer number code for optimization algorithm (PSO [17], [18]). For all designs, RL des was set to -10 dB. For Table 4 an operational BW of 8-12.5 GHz is considered for all the databases. The first database (Table 1) is of epoxy and carbon black (CB) composite. The Epoxy-CB composites are fabricated by mixing 1, 2, and 4 weight percentage of CB with the epoxy resin. The material parameters are then measured using [22]. The measured permittivity of the composites is shown in Fig. 4. As there are no magnetic inclusions, the permeability of the composites is assumed to be 1.  GA [7]. These materials include frequency-dependent lossless dielectric, lossy dielectric, lossy magnetic, and relaxation magnetic materials.
The third database (Table 3) is of composites whose parameters are published in the literature [15], [21]. The database is composed of materials fabricated using epoxy as a matrix and multiwalled carbon nanotubes (MWCNT) and Carbon nanofiber (CNF) as filler.  Table 4 presents the optimized MRAM designs obtained from three databases, where only normal incidence is considered for all models. The PSO algorithm implementation remains constant for a given database, except for the cost function formulation. The algorithm is run for 20 independent trials, and the best solution is selected. Table 4 includes the thickness obtained for various values of α, as well as the BW obtained for each model. To maintain the table's conciseness, the individual designs are included in Appendix. Additionally, Fig. 5, Fig. 6, and Fig. 7 display the reflection coefficient versus frequency curves for cost function CF 1 , cost function CF 2 , and cost function CF new respectively. As can be observed from Table 4, for all the databases, when α = 1 all the three objective functions provide the thickest designs. This is because, for α = 1, all three objective functions do not take thickness into consideration. For other values of α significant decrease in the thickness can be observed for all the cases.
For cost function, CF 1 , with Database I, as the value of α decreases from 0.9 to 0.5, the thickness of the MRAM decreases from 39 mm to 21.4 mm. With the decrease in value of α, the priority of the cost function shifts towards the minimization of the thickness. However, the obtained solution does not satisfy the BW requirement when α is equal to 0.5 (shown in Fig. 5(a)). In a similar way, we carried out the optimization for Database II and Database III using the cost function CF 1 . For Database II as α decreases from 0.9 to 0.5 the optimized thickness decreases from 7.2 mm to 0.5 mm while satisfying BW requirement (shown in Fig. 5(b)). Whereas, for Database III, as α decreases from 0.9 to 0.5, the thickness decreases from 15.2 mm to 8 mm while satisfying BW requirement (shown in Fig. 5(c)).
For cost function CF 2 , with Database I, as the value of α decreases from 0.9 to 0.5, the thickness of the MRAM decreases from 47.8 mm to 3.7 mm. However, the obtained solution does not satisfy the BW requirement when α is equal to 0.5 (shown in Fig. 6(a)). In a similar way, we carried out the optimization for Database II and Database III using cost function CF 2 . For Database II as α decreases from 0.9 to 0.5 the optimized thickness decreases from 13.1 mm to 0.5 mm while satisfying BW requirement (shown in Fig. 6(b)). Whereas, for Database III, as α decreases from 0.9 to 0.5, the thickness decreases from 12.9 mm to 7.1 mm while satisfying BW requirement (shown in Fig. 6(c)). Therefore, for CF 1 and CF 2 , it is observed that the obtained optimized results depend on the value of weight factor α. In addition to that, BW requirements may not be satisfied for all the scenarios.
For cost function CF new , with Database I, as the value of α decreases from 0.9 to 0.5, the thickness of the MRAM of 28.6 mm is obtained for all values of α. In addition to this, the obtained solution satisfies the BW requirement for all the values of α (shown in Fig. 7(a)). In a similar way, we carried out the optimization for Database II and Database III using cost function CF new . For Database II as α decreases from 0.9 to 0.5, a constant optimized thickness of 0.4 mm is obtained while satisfying BW requirement for all the values of α (shown in Fig. 7(b)). For Database III, as α decreases from 0.9 to 0.5, a constant thickness of 6.9 mm is obtained while satisfying the BW requirement for all the values of α (shown in Fig. 7(c)). Therefore, for CF new , it is observed that the obtained optimized results do not depend on the value of weight factor α. In addition to this, the BW requirement is satisfied for all the scenarios.
From the above comparison, it can be concluded that the proposed cost function CF new provides constant thickness for all the values of α under consideration. Thus the cost function formulation can be considered independent of the value of α. In addition to this, for all the databases, the proposed cost function provides the thinnest MRAM design which can satisfy the BW requirement.

B. DESIGNED MRAMs AND COMPARATIVE RESULTS
To further test the effectiveness of the proposed formulation, we compare the results obtained for the optimization of MRAM design using the proposed cost function with the results reported in [8] for the same set of parameters. For the PSO implementation, the values of different search parameters such as the number of iterations, initial population, acceleration coefficient, inertial weight, the maximum number of layers, and maximum particle velocity are taken as reported in [8]. The designs are obtained at different broadband frequency ranges (viz., 2-8, 8-12, 12-18, and 2-18 GHz) for normal incidence and oblique incidence (viz .  30 o , 45 o , 60 o and 75 o ). The predefined threshold RL des required for proposed cost function CF new is set to the maximum RL reported in [8] for each design. For further verification, each design is also simulated in CST. It is observed that the CST simulated results match with the computed results. The results obtained for the optimization of MRAM design with normal and oblique incidence are discussed as follows.

1) NORMAL INCIDENCE
We consider four different MRAM designs labeled as Model 1, Model 2, Model 3, and Model 4 corresponding to 2-8, 8-12, 12-18, and 2-18 GHz bands, respectively. The VOLUME 11, 2023 FIGURE 5. Calculated RL at normal incidence for designed structures described in Table 4 optimized with cost function CF 1 with various values of α (= 1, 0.9, 0.7, 0.5) for (a) Database I (b) Database II (c) Database III. FIGURE 6. Calculated RL at normal incidence for designed structures described in Table 4   thicknesses of each design obtained at the end of the optimizations are given in Table 5. For comparison, the results of MRAM designs in [8] are also presented in Table 5. From Table 5, it is observed that MRAM designs with the proposed cost function have the least total thickness.
The reflection coefficient versus frequency curves for Model 1 and Model 2 are illustrated in Fig. 8. As can be seen from Fig.8.(a) the RL for Model 1 is below the desired RL des of −21.6397 dB in the entire frequency range of 2-8 GHz and in Fig. 8.(b) the RL for Model 2 is below the desired RL des of -26.1052 dB in the entire frequency range 8-12 GHz.
The reflection coefficient versus frequency curves for Model 3 and Model 4 are illustrated in Fig. 9. As can be seen from Fig.9.(a) the RL for Model 3 is below the desired RL des of -23.9903 dB in the entire frequency range of 12-18 GHz and in Fig. 9.(b) the RL for Model 2 is below the desired RL des of -16.2544 dB in the entire frequency range 2-18 GHz. In addition to this, the reduction in thickness for Model 1 is 12.4918%; for Model 2 is 22.9562%; for Model 3 is 2.3547%; for Model 4 is 1.0067%. Therefore, the proposed cost function formulation can satisfy the BW requirement in the desired frequency range while providing thinner structures for normal incidence.

2) OBLIQUE INCIDENCE
In order to evaluate the performance of the proposed algorithm for oblique angles of incidence, 8 different MRAM structures operating at the frequency range 2-18 GHz (Model 5 -Model 12) are designed for both TE and TM polarization. For a fair comparison, the RL des is set to the maximum value reported in [8]. The designs obtained with the proposed cost function are shown in Table 6 and Table 7. From Table 6 and Table 7, it is observed that MRAM designs with the proposed cost function have the least total thickness. The reflection coefficient versus frequency curves for Models 5-12 are illustrated in Fig. 10-Fig. 12. As can be seen from Fig. 10 the RL for Model 5 is below the desired RL des of -19.3096 dB in the entire frequency     In addition to this, the reduction in thickness for Model 5 is 16.8587% and for Model 6 is 2.3001%.
As can be seen from Fig. 11 the RL for Model 7 is below the desired RL des of -26.5193 dB in the entire frequency range 2-18 GHz and the RL for Model 8 is below the desired RL des of -12.2444 dB in the entire frequency range 2-18 GHz. In addition to this, the reduction in thickness for Model 7 is 12.6104% and for Model 8 is 1.9368%.
As can be seen from Fig. 12 the RL for Model 9 is below the desired RL des of -29.1497 dB in the entire frequency   As can be seen from Fig. 13 the RL for Model 11 is below the desired RL des of -25.3794 dB in the entire frequency range 2-18 GHz and the RL for Model 12 is below the desired RL des of -3.2196 dB in the entire frequency range 2-18 GHz. In addition to this, the reduction in thickness for Model 9 is 12.1425%; for Model 10 is 0.009%; for Model 11 is 7.2023%; for Model 12 is 1.6987%. Therefore, the proposed cost function formulation can satisfy the BW requirement in the desired frequency range while providing thinner structures for oblique incidence.

C. OBLIQUE INCIDENCE AT MULTIPLE ANGLES
In the earlier subsection IV-A and IV-B, the MRAMs are optimized for single incidence angle (i.e., either normal incidence or oblique incidence). However, in practice, the EM wave can be incident at any angle. Therefore, MRAM is generally designed for wide-angle of incidence. To demonstrate the effectiveness of the proposed formulation, we optimize MRAM using all three formulations (discussed in subsections III-A, III-B, and III-C) considering simultaneous multiple angle of incidences (0 • , 20 • , 40 • , and 50 • ). In this regard, we consider Database II (Table 2) to optimize MRAM that provides RL less than -10 dB up to 50 • wide-angle incidence in the frequency range of 8-12.5 GHz.
For CF1 and CF2, we carry out the optimization for multiple values of α in order to obtain the thinnest MRAM to satisfy the BW (8)-12.5 GHz) and RL (less than −10 dB) requirements. We obtained the optimized solutions for CF1 (Model 13), CF2 (Model14) and CFnew (Model 15) for α = 0.6, α = 0.7, and α = 0.9 respectively. The corresponding optimized solutions are presented in Table-8 Fig. 14 it can be observed that, for all three models, we obtain the RL less than -10 dB. Further, from Table 8 it can be observed that Model 15, obtained utilizing proposed formulation CFnew, provides thinnest (6.7092 mm) MRAM while satisfying BW and RL requirements.

D. LOSS ANALYSIS
We investigated the loss mechanism of Model 4 with the help of CST software. For this purpose, we use the time domain simulation tool in CST to show energy dissipation of propagating wave travelling through different layers of MRAM. The time domain tool in CST studio helps to measure the signal strength at various probe locations in the structure as a function of time. Therefore, it aids in understanding the behavior of structure with respect to time [23]. In the case of MRAM we measure the electric field intensity of the incident signal at various probe locations inside the MRAM. In this regard, we select Model 4 (the details of the Model 4 are presented in Table 2) and simulate it using time domain tool in CST. In order to record the evolution of electric signals, we set several ideal probes along the z-axis which are parallel to the orientation of the strongest components of electric vectors in a MRAM as shown in Fig. 15(a). The probes are placed just above the end of each layer. The source of excitation (at probe location P 0 ) is the default Gaussian signal in CST simulation software. Fig. 15(b) illustrated the obtained electric field intensities at various probe locations (P 1 , P 2 , P 3 , and P 4 ). From Fig. 15(b), it can be observed that, as the wave propagates through the MRAM, the electric field intensity decreases. This is because the energy is absorbed into the lossy materials used in the MRAM. Fig. 15(c) illustrates the loss percentage of incident energy in various materials (used in Model 4) with respect to frequency. From Fig. 15(c) it can be observed that, below 5.3 GHz, Material 9 (layer 4) is more lossy compared to Material 6 and Material 16. Whereas, above 5.3 GHz, Material 16 (layer 1) is more lossy material. Material 6 (layer 2 and 3) is least lossy among all three materials in the considered frequency range (i.e., 2 -18 GHz). Further, it can be observed that almost all the energy is absorbed into the MRAM.

V. CONCLUSION
In this article, we study the effect of the formulation of the cost function on a given search algorithm and propose a new formulation of the cost function for the optimum design of MRAM. We optimized three separate databases using two commonly used and the proposed formulation for different values of weight factor α. We found that the formulation of cost function indeed affects the obtained optimal solution and therefore can be used to obtain better solutions. In addition to this, the proposed formulation provides a thinner design solution and is independent of weight factor. Moreover, any algorithm with the proposed formulation effectively works as a two-step optimizer. Therefore, the proposed formulation is useful in obtaining thinner design with fever optimization trials compared to the other considered formulations for a given algorithm without increasing the complexity of the algorithm.
Further, we compare the results from the proposed formulation with twelve designs from the existing literature. Our proposed formulation provides better results for all the twelve models from existing literature. A maximum thickness reduction of 22.9562% is obtained for Model 2 and a minimum reduction of 0.009% is obtained for Model 10. The proposed formulation is also applicable for designing MRAM for wide-angle of incidence and provides thinner solutions compared to other considered formulations. Therefore, the proposed modification of the cost function is straightforward to implement in any optimization technique, making it a useful tool for producing thinner MRAM designs. In the future, we can enhance the cost function by considering specific design goals like maximizing absorption frequency or reducing layer count. This improvement would lead to better solutions for MRAM design.   Table 9 to Table 17 presents the details of the MRAM designs considered in Table 4. For each layer, the selected material VOLUME 11, 2023    number and the thickness of each layer are listed for various values of α. The details of the selected material for each layer can be found from its material number in Table 1, Table 2    and Table 3 for Database 1, Database 2, and Database 3 respectively. A maximum of 10 layers are considered for each absorber design. The maximum allowed thickness for each layer is set to 30 mm. For each case, the algorithm is run for 500 iterations and 20 independent trials. The best solution obtained is considered for each case.