The RSM method can be readily adapted to develop an analytical model for a complex problem. With this analytical model, an objective function with constraints can be easily created and evaluated, and computation time can be saved. A polynomial approximation model is commonly used for a second-order fitted response (u) and can be written as follows: (2) u = β 0 + ∑ j = l k β j x j j + ∑ j = 1 k β j j x j 2 + ∑ j = l k β i j x i x j + ε β : regression coefficients ,       x : design variables ε : random error ,     k : number of design variables

The least squares method is used to estimate unknown coefficients. Matrix notations of the fitted coefficients and the fitted response model should be as shown in equation (3): (3) β ^ = ( X ′ X ) − 1 X ′ u   ,   u ^ = X β ^where β̂ is a vector of the unknown coefficients which are usually estimated to minimize the sum of the squares of the error term. It should be evaluated at the data points. RSM can be applied in connection with FEM and the response actually represents FEM result.

Kriging is a method of interpolation named after a South African mining engineer D. G. Krige, who developed the technique while trying to increase accuracy in predicting ore reserves. In the kriging model, the global approximation model for a response y(x) is represented as: (4) y ( x ) = β + ν ( x )where x is the design variable vector, β is a constant, and ν(x) is the realization of a stochastic process. In Equation (4), ν(x) has the mean zero, variance σ², and non-zero covariance. The weight of electromagnet Weight_total is replaced by y(x) to make a surrogate approximation model. Let y ( x ) ^ be an approximation model. Hereafter, ^ means the estimator. When the mean squared error between y(x) and y ( x ) ^ is minimized, y ( x ) ^ becomes: (5) y ( x ) ^ = β ^ + r T ( x ) R − 1 ( y − β ^ q )where r is the correlation vector, R is the correlation matrix, y is the observed data and q is the unit vector. The definitions of R and r are well explained in Refs [8,9].

The unknown correlation parameters of θ₁, θ₂,…,θ_n defined in R are calculated from the formulation according to: (6) maximize − [ n s ln ( σ 2 ^ ) + ln | R | ] 2where θ_i (i = 1,2,…,n) > 0. In this study, a GRG (generalized reduced gradient) algorithm built in the Excel program was utilized to determine the optimum parameters.

Optimization formulation: (7) Minimize : Weight total ( dv 1 , … , dv 7 ) (8) Subject to       F normal ( dv 1 ,   … ,   dv 7 ) ≥ 611.689

Figure 6 shows the prototype electromagnet of the reference model which is analyzed. The comparison of the static force obtained from FEM simulation and experimental test is shown in Figure 7. From the results, the use of FEM is validated, as it can be observed, since the comparison of normal force by simulation and experiment test shown in Figure 7 displays good agreement. Selection of the design variables is very important setup in optimization procedure.

Seven dimensions are selected as the design variables as shown in Figure 8, which shows the flux density vector in electromagnet of 2D and 3D FEM. The magnetization characteristics of the S20C, SM490A material is as shown in Figure 9. Table 1 shows the design variables and levels.

Table 2 shows the table of mixed orthogonal array and simulation results by 2D FEM. The mixed orthogonal array table is determined by considering the number of the design variables and each level of them. After getting experimental data by FEM, the function to draw a response surface is extracted. In order to determine equations of the response surface for each response value (weight, normal force), several experimental designs have been developed to establish the approximate equation using the smallest number of experiments. The purpose of this paper is to minimize the objective function (Weight_total) with constraints of normal force (F_normal). The two fitted second order polynomial of objective functions for the seven design variables are as follows: (9) F normal = 148.75 + 26.1 dv 1 − 2.41 dv 2 − 10.02 dv 3 − 32.18 dv 4 − 1.63 dv 5 + 35.16 dv 6 + 3.01 dv 7 − 7.14 E − 2 dv 1 2 + 1.3 E − 3 dv 2 2 + 0.23 dv 3 2 + 0.35 dv 4 2 + 5.56 E − 2 dv 5 2 − 0.9 dv 6 2 − 3.72 E − 4 dv 7 2 Weight total = 8.765 + 0.212 dv 1 − 0.248 dv 2 − 0.154 dv 4 − 0.488 dv 4 − 0.113 dv 5 + 0.264 dv 6 + 0.065 dv 7 + 1.84 E − 3 dv 1 2 + 2.8 E − 3 dv 2 2 + 6.54 E − 3 dv 3 2 + 5.95 E − 3 dv 4 2 + 7.58 E − 3 dv 5 2 − 4.05 E − 3 dv 6 2 + 2.64 E − 6 dv 7 2

The adjusted coefficients of the multiple determination R²_adj for normal force and weight are weight_total (99%) and F_normal (100%). Normal force and weight of experiment result according to change of the design variables are shown in Table 2. As many parameters are defined as design variables, a large simulation time is required due to the large number of required experiments. Therefore, it is necessary to establish the significant parameters to investigate the influence on the design result. The Pareto chart of normal force and weight shows the magnitude and importance of an effect. This chart displays the absolute value of the effects. The geometries of electromagnet can be defined by seven parameters, as shown in Table 1 and Figure 8. The most effective design variables of normal force and weight are dv₇, dv₁ according to the Pareto chart in Figure 10.

The values of ineffective design variables are determined by RSM. Figure 11 shows response optimization to find optimal solution according to response curves. The slope of the response function in Figure 11 shows the sensitivity of design variable. According to the response optimization the most sensitive design variables of normal force and weight are dv₇, dv₁.

The purpose of this paper is to minimize the objective function (Weight_total) with the constraint of normal force (F_normal). The kriging interpolation method, the optimum parameters of θ₁, …, θ₇ are determined by solving Equation (6). Then, the estimator β is calculated. In Tables 3 and 4, the optimal point is searched to find the point of less than 11.412% of the weight and greater than 7.754% of the normal force of the initially designed electromagnet in the magnetic levitation system. The optimum estimators are shown as Table 4. The optimum solution is the same in the RSM and kriging interpolation method. The reason is that the optimum solution is near the boundary value. However, response values, Weight_total and F_normal by RSM and kriging method, respectively, are different. The simulation result of the predicted optimum set is shown Table 5, displaying good agreement.