CCDs employ the methodology of response surfaces; their construction consists of adding points (star points) to a full factorial design [11] . However, they are very greedy in terms of the number of experiments compared to other types of designs but it is conceivable when the number of factors studied is low (between 2 and 4). The number of experiments in a central composite design is given by the following equation:

$N=2^k+2k+n_0$ (4)

where k is the number of factors. A central composite design is the sum of three terms:

The three components, namely axial distance, factorial points, and the number of center runs, play important and somewhat different roles. The factorial points contribute to the estimation of linear terms and two-factor interactions. The axial points contribute in a large way to estimation of pure quadratic terms. Axial distance greatly depends on the region of operability and region of interest while the rotatability, robustness of extrapolation, estimation precision of parameters, and location of optimized points all reside in the selection of α. The center runs provide an internal estimate of error (pure error) and estimation of uniform precision. Significant differences in centre points play an important and different role in different designs of interest.

Central composite designs can be used in two domains: spherical and cubic. In the first case, two designs are used: the Circumscribed Central Composite (CCC) and the Central Composite Inscribed (CCI), in the second case, it’s called Central Composite Face centered (CCF).

In this type of design, each factor takes 5 levels and the experimental field is spherical. In this case, if k = 2 factors, the experimental field is within a circle of radius α. The two axes represent the variation of two coded factors X₁ and X₂. Coded values of each factor are given in the experiment’s matrix reported in Table 1 .

The first four experiments represent the full factorial design (possible combinations of −1 and +1 levels of each factor). Experiments 5 to 8 represent the star points; each factor has its largest value in the design. It is shown that for two factors, and to maintain rotatability, the value of α depends on the number of experimental runs in the factorial portion of the central composite design [12] - [15] :

$\alpha =\sqrt[4]{2^k}$ (5)

The last point of the matrix represents the center of the domain (0, 0). After building the experiments matrix with coded values, the next step consists of building the design of experiments using relation (3).

The experimental domain occupied by a CCI design is a spherical domain, the value of α is equal to 1, and the extreme coded values are then −1 and 1. This design is composed of a point at the center of the domain (0, 0), the vertices of the domain which are combinations between the levels −a and a of each factor and star points which are located at a unit distance from the center of the domain.

The CCI designs have the same functionality as the CCC designs described above. The advantage here comes from the fact that the extreme values of the coded factors are −1 and +1, which facilitates the transition from the matrix of experiments to the design of experiments. The value of a is given by:

$a=\frac{\sqrt{k}}{k}$ (6)

In the case of two factors, a is equal to 0.7071, the vertices of the design correspond to the first four experiments in the matrix as shown in Table 1 , experiments 5 to 8 represent the star points, considering a single point in the center of the domain (experiment 9).

In this type of design, the experimental domain is a cubic domain; each factor requires 3 levels, which are −1, 0, and 1. These designs are used in the case where no operating point of the device is known, the experimental domain is limited by a square, which gives the possibility to find the optima that are next to the extreme values of the factors (which is not possible in a spherical domain). The value of α is equal to 1; the vertices of the domain are combinations with the values −1 and 1 of the input factors, which explains the number of levels of each factor, which is limited to 3.

The matrix of experiments for a CCF design also consists of 3 parts as shown in Table 1 .

The use of a design in a spherical area is recommended since each factor requires five levels, which gives high accuracy for the mathematical models. A spherical field is generally used whenever we know an operating point of the device to optimize; this is usually the center of the spherical field. When the location of the optimum is totally unknown, a design in a cubic field is recommended. The optimum can be found for extreme levels of factors. Figure 1 shows the graphical representation of the three types of CCDs for two input factors.

To define the quality of models for each response, the ANalysis of VAriance (ANOVA) is studied; for this, several terms must be defined [16] [17] :

SCT is the sum of squares, that is the sum of squared deviations between the results of simulations ( $Y_i$) and the average ( $\bar{Y}$); it is given by:

$SCT={\displaystyle \underset{i=1}{\overset{N}{\sum }}{\left(Y_i-\bar{Y}\right)}^2}$ (7)

The total sum of squares (SCT) can be given by the following equation, also called the equation of analysis of variance or regression equation:

$SCT=SCM+SCE$ (8)

The first term reflects the variation of responses calculated by the model ( ${\hat{Y}}_i$) around the mean value, given by:

$SCM={\displaystyle \underset{i=1}{\overset{N}{\sum }}{\left({\hat{Y}}_i-\bar{Y}\right)}^2}$ (9)

The second term reflects the sum of square of the residuals:

$SCE={\displaystyle \underset{i=1}{\overset{N}{\sum }}{\left(Y_i-{\hat{Y}}_i\right)}^2}$ (10)

A statistical test to reject the hypothesis (H₀) that the model does not describe the variation of the experiments will then be defined. When this hypothesis is verified, it can be shown that the statistic F_c follows respectively (p − 1) and (N − p) degrees of freedom with p the number of coefficients to be estimated and N the number of experiments:

$F_c=\frac{SCM\left(p-1\right)}{SCE\left(N-p\right)}$ (11)

Thus, the hypothesis (H₀) is rejected with probability α if:

$F_c>F_{\left(\alpha ,p-1,N-p\right)}$ (12)

In this equation, $F_{\left(\alpha ,p-1,N-p\right)}$ is the (1 − α) quantile of a Fisher’s law with (p − 1) and (N − p) degrees of freedom.

The model can describe the variation of the experimental results if the probability of rejecting the null hypothesis is low (Prob (F_c) < 5%).

All these data lead to the construction of the ANOVA table, summarizing the different results as shown in Table 2 . ANOVA table could be used to calculate the coefficient of determination R² from the following equation:

$R^2=\frac{SCM}{SCT}=1-\frac{SCE}{SCT}$ (13)

This coefficient reflects the model’s contribution to the observed response variation. The R² value is always between 0 and 1. Values close to 1 indicate a very good model.

In the presence of several explicative variables, which is usually the case, it is imperative to avoid using the coefficient of determination R² to estimate the descriptive quality of the model. The adjusted coefficient of determination is used; it is given by the following formula:

$R_{adj}^2=1-\frac{N-p}{SCT}$ (14)

Another coefficient to describe the predictive ability of the model, called $R^2{}_{pred}$ is given by the formula:

$R_{pred}^2=1-\frac{PRESS}{SCT}$ (15)

where PRESS (Prediction Residual Error Sum of Square), the sum of squared residuals, is given by the following formula:

$PRESS={\displaystyle \underset{i=1}{\overset{N}{\sum }}e{\left(i\right)}^2}$ (16)

The regression has been done without the experience i and for each (N − 1) points we calculate ${\hat{Y}}_i$ at point i: $e\left(i\right)=Y_i-{\hat{Y}}_i$, the procedure has been repeated for each point ( $i=1,2,\cdots ,N$) where ${\hat{Y}}_i$ is the value of the response calculated by model and $Y_i$ the value of the response obtained by simulation at point i. The value of $R^2{}_{pred}$ ranges between 0 and 1. Large $R^2{}_{pred}$ (>0.7) indicates that the model has good predictive ability and will have small prediction errors.

When there are several responses to be optimized, the notion of desirability will be used. This concept was introduced by E.C. Harrington [18] in 1965, and was later developed, notably by G. Derringer [19] . It is based on the transformation of all responses obtained from different measurement scales into an identical dimensionless desirability scale (individual desirability). The values of the individual desirability functions (d_i) are between 0 and 1. Then, the set of individual desirabilities is gathered into a single global desirability D [20] , which is their geometric mean. The particularity of the geometric mean is that the nullity of at least one of the individual desirabilities leads to the nullity of the global desirability. The highest D value is obtained under conditions where the combination of the different responses is globally optimal.

In order to use the global desirability function, the problem could be formulated as:

$MaxD={\left[{\displaystyle \prod d_i^{wi}}\right]}^{1/{\displaystyle \sum wi}}$ (17)

where D represents overall desirability, d_i is the individual desirability for the ith response and wi is the weight of the ith response.

The traditional desirability function method proposed by Derringer is employed to calculate individual desirability for each response. We use Y_i, Y_c, Y_{i, min}, Y_{i, max} and w_i to denote, respectively, the predicted value, the target value, the lowest acceptable value, the highest acceptable value and the weight of desirability function of the ith output response.

To look for a maximum value for a response, the individual desirability is calculated as:

$d_i=\{\begin{array}{ccc}0& \iff & Y_i\le Y_{i\text{,min}}\\ {\left[\frac{Y_i-Y_{i\text{,min}}}{Y_{i\text{,max}}-Y_{i\text{,min}}}\right]}^{w_i}& \iff & Y_{i\text{,min}}\le Y_i\le Y_{i\text{,max}}\\ 1& \iff & Y_i\ge Y_{i\text{,max}}\end{array}$ (18)

To look for a minimum value for a response, the individual desirability is calculated as:

$d_i=\{\begin{array}{ccc}1& \iff & Y_i\le Y_{i\text{,min}}\\ {\left[\frac{Y_i-Y_{i\text{,max}}}{Y_{i\text{,min}}-Y_{i\text{,max}}}\right]}^{w_i}& \iff & Y_{i\text{,min}}\le Y_i\le Y_{i\text{,max}}\\ 0& \iff & Y_i\ge Y_{i\text{,max}}\end{array}$ (19)

In the case where we wish to have a target value of a response, the desirability function is calculated as:

$d_i=\{\begin{array}{ccc}0& \iff & Y_i\le Y_{i,min},Y_i\ge Y_{i,max}\\ {\left[\frac{Y_i-Y_{i\text{,min}}}{Y_c-Y_{i\text{,min}}}\right]}^{r_i}& \iff & Y_{i,min}\le Y_i\le Y_c\\ {\left[\frac{Y_i-Y_{i\text{,max}}}{Y_c-Y_{i\text{,max}}}\right]}^{r_i}& \iff & Y_c\le Y_i\le Y_{i,max}\\ 1& \iff & Y_i=Y_c\end{array}$ (20)

To maximize the overall desirability function, Nemrodw software uses the simulated annealing method. This method is a general probabilistic algorithm for optimization problems [21] [22] . It uses a process of searching for a global optimal solution in the solution space that is analogous to the physical process of annealing.

In a transmission system, a multichannel output multiplexer filter (OMUX) is behind the power amplifier module. Its role is to select the channel narrow-band signal and thus, eliminate the frequency noise created by amplifiers. This filter is characterized by low insertion losses and high quality factor due to its location between the power amplifier and antenna. The selectivity has to be high because the different channels to be multiplexed can have very similar bands. An OMUX filter is made with a dual-mode cylindrical cavity and filled with air [7] [23] .

The system studied consists of a parallelepiped-shaped resonator ( Figure 2 ) truncated at the 4 corners and short-circuited in a cylindrical cavity. It has four contacts with the metal walls, which ensure its maintenance.

The structure was analyzed by the finite element method (FEM) [24] ; FEM is a popular technique for analyzing microwave components [25] since both planar and waveguide structures (including antennas) can be treated. Modeling a microwave component with FEM requires discretizing the structure into small mesh elements before solving.

The dimensions of reference are its diameter denoted by Dc = 39.86 mm and its height denoted by Hc = 28 mm. The resonator thickness E is equal to 3 mm.

The relative permittivity of the dielectric material is ε_r = 12.6 with a loss tangent tanδ = 5.5 × 10⁻⁵, the metallic conductivity σ of the cavity is equal to 4.76 × 10⁷ S∙m⁻¹.

The electromagnetic simulation given in Figure 3 shows that the filter presents two excited modes (at the frequencies F₀ and F₂) and a non-excited mode at the frequency F₁ (between F₀ and F₂).

The quality factor Q₀ of the resonator is equal to 10,145. The aim of the optimization of this resonator is to hold off the two frequencies F₁ and F₂ (F₁ ≥ 4.5 GHz and F₂ ≥ 6 GHz) from the fundamental frequency F₀ and have the largest value of the quality factor Q₀.

In order to optimize the characteristics of resonator, a first study was done according to the parameters describing the cylindrical cavity ( Figure 2 ). The two factors describing the cavity are its height Hc (which varies between 21 and 35 mm) and its diameter Dc (which varies between 30 and 50 mm). The thickness of the resonator E is not involved in the optimization; it has been calculated to have a frequency of the first excited mode F₀ always equal to 4 GHz.

$\hat{Y}=b_0+b_1{X}_1+b_2{X}_2+b_{11}{X}_1^2+b_{22}{X}_2^2+b_{12}{X}_1{X}_2+\epsilon$ (21)

We are interested in this part of the design of experiments in which factors have many levels; there is no difficulty in changing the levels of these two factors, so we chose a CCC design that allows five levels per factor. In our case, the number of input factor is equal to two, resulting in 9 experiments to be performed. The model used is a polynomial of degree 2 written as follow.

The design of experiments has been constructed from the experiment matrix containing the coded values of input factors. In this study, the experiment is a simulation using the software EMXD [26] that is based on FEM. The design of experiments contains 9 lines; the thickness of the resonator is calculated for each line of the design, which gives a frequency of the first excited mode F₀ equal to 4 GHz. The value of α in our case is 1.4; the experimental range and levels of independent variables are given in Table 3 .

After building the experimental design and performing the simulations, the next step is to validate the mathematical models for each response. The statistical software ‘NemrodW) [27] has been used to study the regression analysis of experimental data and to draw the response surface plot. The standard analysis of variance (ANOVA) and model coefficients for the response F₁ are presented in Table 4 . The ANOVA confirms adequacy of the quadratic model since its Prob (F_c) value is equal to 0.955.

The fit of the model was checked by the determination coefficient (R²). For response F₁, the value of R² is 0.98, thus ensuring a satisfactory adjustment of the model to the experimental data and indicating that approximately 98% of the variability in the dependent variable (response F₁) could be explained by this model. The value of the adjusted determination coefficient is 0.946, attesting once more, the reliability of the model.

The ANOVA tables are given for each response; the Prob (F_c) values and the R², $R^2{}_{adj}$ coefficients are given in Table 5 .

It could be seen from Table 5 that the mathematical models for all responses are validated: Prob (F_c) < 5% and R², $R^2{}_{adj}$ are close to 1.

The experimental domain is defined by the variation of two factors; graphical analysis can help us study their effects on each response. The effects of input factors Dc and Hc on all responses are given by three-dimensional graphics called response surfaces ( Figure 4 ). The horizontal plane of the figure represents the range of variation of the two factors Dc and Hc; the vertical axis materializes the variation of responses.

By observing these figures, we see that: the frequency of the first excited mode F₁ increases when Hc increases and Dc decreases; the frequency of the second excited mode F₂ increases when Hc increases and Dc decreases; the quality factor Q₀ increases by increasing Dc and Hc.

The good quality of the models could be seen from the model adequacy graph. This graph is used to compare the simulated responses and the responses estimated by the model. The measured (experimental) responses are placed on the x-axis and the estimated (model-calculated) responses on the y-axis. If the scatterplot is aligned on the equation line y = x, the descriptive quality of the model is excellent (we consider that the values calculated by the model are very close to the measured values). The adequacy graph for each response is given in Figure 5 and shows a very good quality of the mathematical model; indeed, the experimental points, are placed perfectly on the line y = x.

The search for a multi-criteria optimum consists of finding the level of factors that maximizes the value of the global desirability function. In our case, we are looking for a maximum value for all the responses.

Figure 6 shows the individual desirability functions d1, d2 and d3 of the responses F₁, F₂ and Q₀. A value of 4.5 GHz has been specified as the minimum accepted value for the response F₁. We are also looking for a value of F₂ higher than 6 GHz and a maximum value of the answer Q₀ (above 10,000). The same weight is assigned to the three responses, which gives an overall desirability function:

$D={\left(d1\times d2\times d3\right)}^{1/3}$ (22)

A high percentage of individual desirability for both responses F₁ (d1 = 100%) and F₂ (d2 = 100%) is obtained. The values of these two responses calculated by the model are 4.57 GHz for F₁ and 6.7 GHz for F₂.

The value of the individual desirability function for the response Q₀ is 11.7%. This value is considered low since the value searched for Q₀ is 30,000 and the model finds 12,341. All these values of the individual desirability functions lead to an overall desirability of 48.91%.

Optimization results obtained by the CCC design are given in Table 6 . In this table, the optimal values of the input factors, as well as the calculated responses and the simulated responses, are given. The good quality of models could be seen from the insignificant difference between calculated and simulated responses. A quality factor of 12,291 has been found, leading to an improvement of 21.1% compared to the reference quality factor (10,145). Frequency isolation is well ensured since the frequency F₁ is greater than 4.5 GHz and the frequency F₂ is greater than 6 GHz.

The optimal value found using a CCC design is given in Figure 7 ; it can be seen that this value is located at the limit of the experimental region. These results lead us to create a new CCF design with only 6 new experiments instead of 9, as shown in Figure 7 .

After performing the simulations, the mathematical model has been validated and a multicriteria optimization has been conducted leading to results in Table 7 .

The good quality of models could be seen from the insignificant difference between calculated and simulated responses. A quality factor of 12,680 has been found, leading to an improvement of 24.9% compared to the reference quality factor (10,145). Frequency isolation is well ensured since the frequency F₁ is greater than 4.5 GHz and the frequency F₂ is greater than 6 GHz.

In this section, the RSM is used, taking into account three factors: the dimensions of the cavity (Dc and Hc) and the radius (R) of the plate. The design of experiments used in this section is CCF, in this case, the field is cubic which gives the possibility to find optima which is next to the extreme values of the factors. For k = 3 factors, the number of experiments to be carried out is 15 and the corresponding experimental design is given in Table 8 .

It has been shown previously that the quality factor increases with the cavity dimensions Dc and Hc. We have therefore reduced the experimental ranges of these two factors where the quality factor is at its maximum. The new experimental ranges of the input factors are: Dc between 44 and 48 mm, Hc between 28 and 36 mm and R between 8 and 14.8 mm.

The statistical parameter obtained from ANOVA was shown in Table 9 .

It can be seen that the model of the response Q₀ is of high quality since the values of R², $R^2{}_{adj}$, and $R^2{}_{pred}$ are close to 1. On the other hand, the predictive quality of the mathematical models of the responses F₁ and F₂ is poor ( $R^2{}_{pred}$ = 0.361 for the response F₁ and 0.215 for F₂). These results will be verified by estimating the difference between the calculated and simulated responses on the k + 1 test points proposed by the Nemrodw software. In this case, 4 test points are necessary to validate the model at any point in the experimental domain. The calculated responses on these points as well as the values obtained by simulations are given in Figure 9 .

The bad predictive quality of the models for the responses F₁ and F₂ could be clearly seen in Figure 9 ; indeed, the experimental test points are not closely distributed around the line y = x.

In the present case, we are mainly interested in the quality of the model of the Q₀ response. Indeed, according to the experimental design ( Table 8 ), frequency isolation could be easily ensured. The goal is then to find the maximum of the quality factor Q₀.

Multi-criteria optimization results are given in Table 10. The best solution has been found for a radius R = 8 mm, a diameter Dc of 43.6 mm and a height Hc of 32.7 mm. The results obtained can show the relevance of the coupling the level set and design of experiments method. Indeed, although the independent use of each of these two methods improves the quality factor (respectively 21.48% and 25.36% compared to the reference value), their coupling as described in this work leads to an improvement of 25.77% as shown in Figure 10 . In addition, we have demonstrated a solution whose volume is 34.5% lower than that identified by using only designs of experiments method, and therefore more economical to manufacture.

For validating the optimization techniques, the initial and optimized resonators have been fabricated using a stereolithography process [34] ; the photograph of the fabricated resonators is given in Figure 11 .

The left part of Figure 12 shows the simulated responses of both the initial and optimized resonators, while the right part presents the measured responses of the fabricated resonators. It can be observed that the optimized resonator operates exactly at 4 GHz. The isolation frequency is well ensured, and the measured quality factor is 12,760, resulting in an improvement of 25.77% compared to the reference structure.