A new design approach based on differential evolution algorithm for geometric optimization of magnetorheological brakes

First, an initial population, contains NP individuals, is created by randomly sampling from the search space. Each individual is a vector containing D design variables ${{\bf{x}}}_{i}=({x}_{1},{x}_{2},\mathrm{...},{x}_{D})$ and is created by

$$\begin{eqnarray}&&{x}_{i,j}={x}_{j}^{l}+{\rm{rand}}\,[\mathrm{0,1}]\times ({x}_{j}^{u}-{x}_{j}^{l})\\ &&\,i=1,\,2,\ldots ,NP;\,j=\mathrm{1,2},\ldots ,D\end{eqnarray} \tag{ 1 }$$

where ${x}_{j}^{l}$ and ${x}_{j}^{u}$ are the lower and upper bounds of ${x}_{j},$ respectively; ${\rm{rand}}\,[\mathrm{0,1}]$ is a uniformly distributed random number in [0, 1]; NP is the size of population; D is the number of design variables.

Second, each individual in the population called the target vector x_i is used to generate a mutant vector v_i by means of mutation operations. Several popular mutation operations are usually used in the DE as follows

$$\begin{eqnarray}&&-\mathrm{rand}/1:{{\bf{v}}}_{i}={{\bf{x}}}_{{r}_{{\rm{1}}}}+F\times ({{\bf{x}}}_{{r}_{{\rm{2}}}}-{{\bf{x}}}_{{r}_{{\rm{3}}}})\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&-\mathrm{rand}/2:{{\bf{v}}}_{i}={{\bf{x}}}_{{r}_{{\rm{1}}}}+F\times ({{\bf{x}}}_{{r}_{{\rm{2}}}}-{{\bf{x}}}_{{r}_{{\rm{3}}}})+F\times ({{\bf{x}}}_{{r}_{4}}-{{\bf{x}}}_{{r}_{5}})\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&-\mathrm{best}/1:{{\bf{v}}}_{i}={{\bf{x}}}_{{\rm{best}}}+F\times ({{\bf{x}}}_{{r}_{1}}-{{\bf{x}}}_{{r}_{2}})\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&-\mathrm{best}/2:{{\bf{v}}}_{i}={{\bf{x}}}_{{\rm{best}}}+F\times ({{\bf{x}}}_{{r}_{{\rm{1}}}}-{{\bf{x}}}_{{r}_{2}})+F\times ({{\bf{x}}}_{{r}_{3}}-{{\bf{x}}}_{{r}_{{\rm{4}}}})\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray*}&&-\mathrm{current}\mbox{--}\mathrm{to}\mbox{--}\mathrm{best}/1:{{\bf{v}}}_{i}={{\bf{x}}}_{i}+F\\ &&\,\times ({{\bf{x}}}_{{\rm{best}}}-{{\bf{x}}}_{i})+F\times ({{\bf{x}}}_{{r}_{1}}-{{\bf{x}}}_{{r}_{2}})\end{eqnarray*}$$

where integers ${r}_{1},{r}_{2},{r}_{3},{r}_{4},{r}_{5}$ are randomly chosen from $\{\mathrm{1,2,...},\,NP\}$ such that ${r}_{1}\ne {r}_{2}\ne {r}_{3}\ne {r}_{4}\ne {r}_{5}\ne i;$ the mutation factor F is randomly selected within [0, 1]; and ${{\bf{x}}}_{{\rm{best}}}$ is the best individual in the current population.

After mutation, the jth components ${v}_{ij}$ of mutant vector ${{\bf{v}}}_{i}$ are reflected back to the allowable region if their boundary constraints are violated. This procedure is carried out by

$$\begin{eqnarray}&&{v}_{ij}=\left\{\begin{array}{c}2{x}_{j}^{l}-{v}_{ij}\,{\rm{if}}\,{v}_{ij}\lt {x}_{j}^{l}\\ 2{x}_{j}^{u}-{v}_{ij}\,{\rm{if}}\,{v}_{ij}\gt {x}_{j}^{u}\\ {v}_{ij}\,\,\,\,{\rm{otherwise}}\end{array}\right.\end{eqnarray} \tag{ 6 }$$

Third, to increase the diversity of the population, each target vector ${{\bf{x}}}_{i}$ will create a trial vector ${{\bf{u}}}_{i}$ by means of replacing some elements of the mutant vector ${{\bf{v}}}_{i}$ by some elements of the target vector ${{\bf{x}}}_{i}.$ This is implemented as follows

$$\begin{eqnarray}&&{u}_{ij}=\left\{\begin{array}{c}{v}_{ij}\,{\rm{if}}\,{\rm{rand}}\,[\mathrm{0,1}]\leqslant CR\,{\rm{or}}\,j={j}_{{\rm{r}}{\rm{a}}{\rm{n}}{\rm{d}}}\\ {x}_{ij}\,{\rm{otherwise}}\end{array}\right.\end{eqnarray} \tag{ 7 }$$

where $i\in \{\mathrm{1,\; 2,...},NP\};j\in \{\mathrm{1,\; 2,...},D\};$ ${j}_{{\rm{r}}{\rm{a}}{\rm{n}}{\rm{d}}}$ is an integer selected from 1 to D; CR is the crossover control parameter $\in \,[\mathrm{0,1}]$.

Finally, based on the value of the objective function, the trial vector ${{\bf{u}}}_{i}$ is compared to the target vector ${{\bf{x}}}_{i}.$ The better one with the lower objective function value will be selected for the next generation

$$\begin{eqnarray}&&{{\bf{x}}}_{i}=\left\{\begin{array}{c}{{\bf{u}}}_{i}\,{\rm{if}}\,f({{\bf{u}}}_{i})\leqslant f({{\bf{x}}}_{i})\\ {{\bf{x}}}_{i}\,{\rm{otherwise}}\end{array}\right.\end{eqnarray} \tag{ 8 }$$

If the stopping criteria are met after finishing the selection phase, the searching process will stop and optimal results will be obtained. Otherwise, the algorithm will be repeated from step 2 until the stopping criteria are satisfied.

In population-based search methods, the balance between global exploration and local exploitation significantly influences their success [27]. In the DE, the mutation scheme plays a crucial role in its searching ability and convergence rate. There are at least five mutation operators proposed for the DE with different purposes such as 'rand/1', 'rand/2', 'best/1', 'best/2', 'current-to-best/1'. For instance, with the mutation operator 'rand/1', the DE is good at global search, but bad at local search and hence leads to a slow convergence to the global optimal solution [28]. In contrast, with the mutation operator 'current-to-best/1', the DE is good at local search, but bad at global search and easy to be trapped at the local optimal solutions [29]. Thus, to balance the global and local search ability and increase the convergence speed of the DE, a novel adaptive mutation scheme based on the absolute deviation of the objective function between the best individual and the whole population in the previous generation is proposed. This scheme uses two mutation operators. The first is 'rand/1' which aims to ensure diversity of the population and prohibits the population from getting stuck in a local optimum, and the other is 'current-to-best/1' which aims to accelerate the convergence speed of the population by means of guiding the population toward the best individual. These two mutation operators will be adaptively chosen based on a delta value, which is the absolute deviation of the objective function between the best individual and the entire population in the previous generation. The modified mutation scheme is described as follows

$$\begin{eqnarray}&&{{\bf{v}}}_{{\rm{i}}}=\left\{\begin{array}{l}{{\bf{x}}}_{{\rm{r}}1}+F\times ({{\bf{x}}}_{{\rm{r}}2}-{{\bf{x}}}_{{\rm{r}}3})\,\,\,\mathrm{if}\,delta\gt threshold\\ {{\bf{x}}}_{{\rm{i}}}+F\times ({{\bf{x}}}_{{\rm{best}}}-{{\bf{x}}}_{{\rm{i}}})+F\times ({{\bf{x}}}_{{\rm{r}}2}-{{\bf{x}}}_{{\rm{r}}3})\,\mathrm{otherwise}\end{array}\right.\end{eqnarray} \tag{ 9 }$$

where F is a mutation factor, which is randomly created in [0.4, 0.85]; threshold is a criterion value, which is set by the user (e.g. threshold = 10⁻³).

It should be noted that the value of delta will reduce gradually during the searching process. In the first generation, the delta is often bigger than the threshold. This leads to the mutation operator 'rand/1' which will be used to search, and helps assure the global search ability of the DE. However, after the global search process, the diversity of the population is gradually stable and the delta becomes smaller than the threshold. The deviation of the objective function between the best individual and the entire population becomes small, and then the mutation operator 'current-to-best/1' will be utilized for the searching process. This helps enhance the local search ability and the convergence rate of the DE. Moreover, the mutation factor F is randomly created in the interval [0.4, 1.0] instead of being fixed as in the original DE. This helps increase the variety of searching directions for both cases of the delta (delta > threshold and delta ≤ threshold).

From the selection phase in section 2.4, it can be seen that each trial vector u_i created after the crossover phase will be compared with the target vector x_i to choose the best individual for the next generation. This technique may ignore some useful information of unselected individuals. Although an individual is not good compared to its target individual in the pair, it can be still be better than other individuals in the whole population. Hence, to hold good information for the next generation, in this paper, the elitist selection technique introduced in [30] will be utilized in order to replace the basic selection in the DE. This procedure is described as follows: first, the children population C containing trial vectors is mixed with the parent population P including target vectors for creating a combined population Q. Next, NP best individuals in Q are selected to construct the new population for the next generation. In this way, the best individual of the whole population is always stored for the next generation and hence the algorithm will obtain a better convergence rate.

As mentioned in section 1, the objectives of MRB optimal design are diverse. In this paper, the aim of our research is to reduce the mass of MRBs in order to decrease the MRB size and cost. Hence, the objective of the optimization is to find the lightest structure of the MRBs that can provide the required braking torque. The optimal design problem can be expressed as [18]

$$\begin{eqnarray}&&\begin{array}{c}{\rm{Minimize}}\,{m}_{b}={V}_{{\rm{d}}}{\rho }_{{\rm{d}}}+{V}_{{\rm{h}}}{\rho }_{{\rm{h}}}+{V}_{{\rm{s}}}{\rho }_{{\rm{s}}}+{V}_{{\rm{M}}{\rm{R}}}{\rho }_{{\rm{M}}{\rm{R}}}\\ \,\,\,\,\,\,\,+{V}_{{\rm{b}}{\rm{o}}{\rm{b}}}{\rho }_{{\rm{b}}{\rm{o}}{\rm{b}}}+{V}_{{\rm{c}}}{\rho }_{{\rm{c}}}\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\rm{Subject}}\,{\rm{to}}\,{{x}_{i}}^{L}\leqslant {x}_{i}\leqslant {{x}_{i}}^{U},i=1,\,2\ldots n\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&[{{T}}_{0}]\leqslant {T}_{{b}}\end{eqnarray} \tag{ 12 }$$

where m_b is the total mass of MRB, V_d, V_h, V_s, V_MR, V_bob and V_c are respectively the geometric volume of the disc, the housing, the shaft, the MRF, the bobbin and the coil of the brake; ρ_d, ρ_h, ρ_s, ρ_MR, ρ_bob and ρ_c are density of the discs, the housing, the shaft, the MRF, the bobbin and the coil material, respectively; x_i^L, x_i^U are the lower and upper geometric dimension limits of each design variable x_i ; n is the number of design variables; [T₀] is the required braking torque of the MRB; T_b is the induced braking torque.

Two optimization methods: the original DE and improved DE are applied to the above problem via the integrated procedure. The flowchart of the ANSYS-MATLAB integrated optimization using the DE and improved DE algorithm is shown in figure 1. First, modeling data are generated in MATLAB by an optimization method. These data are then used to create the CAD model in ANSYS. In order to solve the magnetic circuits of the MRBs, a 2D-axisymmetric couple element (PLANE13) is utilized to solve the magnetic circuits of the MRBs. It is noted that the rheological parameters of the MRF such as yield stress and post yield stress are dependent on the magnetic density across the MRF ducts and calculated by [31]

$$\begin{eqnarray}&&Y={Y}_{\infty }+({Y}_{0}-{Y}_{\infty })(2{e}^{-B{\alpha }_{SY}}-{e}^{-2B{\alpha }_{SY}})\end{eqnarray} \tag{ 13 }$$

where Y stands for a rheological parameter of the MRF. The value of Y tends from the zero-applied field value Y₀ to the saturation value Y_∞. α_SY is the saturation moment index of the Y parameter. B is the applied magnetic density.

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In this study, it is assumed that the shape of the MRB coils is rectangular and its geometric dimensions are calculated as follows

$$\begin{eqnarray}&&\begin{array}{c}{w}_{{\rm{c}}}={n}_{{\rm{w}}{\rm{c}}}.{d}_{{\rm{c}}}\\ {h}_{{\rm{c}}}={n}_{{\rm{h}}{\rm{c}}}.{d}_{{\rm{c}}}\end{array}\end{eqnarray} \tag{ 14 }$$

where w_c and h_c are correspondingly the width and the height of coil; n_wc and n_hc are respectively the number of wire layers along the width and the height of the coil, which are the positive discrete integer variables; d_c is the diameter of wire. It is observed from equation (14) that w_c and h_c are discrete variables. This is a benefit of the proposed method compared to gradient optimization methods [18].

In the finite element analysis (FEA) implemented to investigate the MRB characteristics, a current density is applied to the coils of the MRBs. Assuming that the coil is fully wound and the insulation layer on the wire is thin, the applied electric current can be considered uniformly distributed over the whole cross-section of the winding with a density j,

$$\begin{eqnarray}&&j=\displaystyle \frac{NI}{{S}_{{\rm{c}}{\rm{o}}{\rm{i}}{\rm{l}}}}=\displaystyle \frac{{n}_{{\rm{w}}{\rm{c}}}.{n}_{{\rm{h}}{\rm{c}}}I}{{w}_{{\rm{c}}}\,.{h}_{{\rm{c}}}}=\displaystyle \frac{{n}_{{\rm{w}}{\rm{c}}}.{n}_{{\rm{h}}{\rm{c}}}.I}{{n}_{{\rm{w}}{\rm{c}}}.{n}_{{\rm{h}}{\rm{c}}}\,.{d}_{{\rm{c}}}^{2}}=\displaystyle \frac{I}{{d}_{{\rm{c}}}^{2}}\end{eqnarray} \tag{ 15 }$$

where N and I are correspondingly the number of turns in the winding and the applied current.

After obtaining the magnetic density across the MRF ducts, the braking torque and off-state force is estimated by [18]. The value of braking torque is put into MATLAB in order to check the constraint. The mass of the MRBs (the objective function of the MRB optimal design problem) is calculated by equation (10) and the convergence criteria are evaluated. If the convergence criteria are satisfied, the program stops and the best design is output to the MATLAB screen.

The conventional MRB surveyed in this study includes two types: the MRB with a rectangular-shaped envelope and the MRB with a 7-segment polygonal envelope (in this paper, it is referred to as polygonal-shaped conventional MRB). First, the conventional MRB with a rectangular-shaped envelope, as shown in figure 3, is considered. In this case, significant geometric dimensions of the MRB such as the coil height h_c, the coil width w_c, the disk thickness t_d, the inner radius of the disk R₁, the outer radius of the disk R_d, the outer radius of the brake R and the housing thickness t_h are considered as design variables.

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The optimum results obtained by this study and those obtained from the ANSYS optimization tool using first-order algorithm [18] are shown in table 3. From the results, it is observed that the DE-based methods and the first-order ANSYS optimization tool offer a solution very close to each other. However, the number of wire layers along the height and the width of the coil obtained from the ANSYS optimization tool are, respectively, 6.82 and 11.2, which are not integers. This is one of the inherent disadvantages of the ANSYS optimization tool. In order to solve this problem, a further optimal solution of the MRB is performed in which the number of wire layers along the height and width of the coil is predetermined based on the previous optimal solution. The optimal solution, in which the number of wire layers along the height and width of the coil is kept constant, respectively, at 7 and 11, shown in brackets in the 'first order' column in table 3. It is noteworthy that the number of wire layers along the height and width of the coil should be chosen such that the number of coil turns is as close as that in the previous optimal solution (In this case, the number of coil turns is 77 which is almost equal to that in the previous optimal solution, 76.38). The results showed that the optimal solution obtained by the ANSYS optimization tool is almost coincident with that obtained from the DE and improved DE methods. Considering the optimal solution from the DE and improved DE method, it is found that the improved DE requires fewer number of generations than the original DE (97 generations for the improved DE, 235 generations for the original DE). Therefore, the computation cost of the improved DE is also reduced significantly (1.159 h for the improved DE, 2.807 for the original DE). It is also noted that the first-order method in the ANSYS optimization tool required 30 iterations in this case to achieve the convergence criteria and in each iteration 25 generations are required for the direction search of the design variables. Therefore, a total of 750 generations has been used in this case. The convergence history of the design variables, and the mass and braking torque are shown in figures 4(a)–(c). It can be seen that after approximately 64 generations of optimization progress of the improved DE, the values of the design variables, and the mass and braking torque become stable. The magnetic distribution of the MRB at the optimum in the case of the improved DE is shown in figure 4(d). It can be seen that the magnetic density is not uniformly distributed in the housing of the MRB. This is because the envelope of the shaft is rectangular, and then the magnetic density reaches maximum value at both sides and at the outer cylinder of the coil. As shown in the figure, the magnetic density at this position almost reaches magnetic saturation of the housing material.

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Second, the polygonal-shaped conventional MRB is taken into account. The geometry of the polygonal-shaped conventional MRB is shown in figure 5. In this case, besides the above-mentioned design variables (the coil height h_c, the coil width w_c, the disk thickness t_d, the inner radius of the disk R₁, the outer radius of the disk R_d, the outer radius of the MRB R and the housing thickness t_h), the geometric parameters of the control point _Pi (R_i, t_hi) are also set as design variables. Both the original DE and improved DE are applied to determine the optimal design of this MRB. Figure 6 shows the optimal solution of the polygonal-shaped conventional MRB, the result of which is summarized in table 4. From the results, it can be seen that the improved DE can provide a better solution than the original DE, in which the minimum mass is 1.027 and 1.023 kg, respectively. The number of wire layers along the width in the case of the original is 13, while in the case of the improved DE it is 12. This results in the number of coil turns being 91 and 84, respectively. Recognizably, the minimum mass of the polygonal-shaped conventional MRB gained by the DE-based methods is smaller than that obtained from the ANSYS optimization tool, which is 1.27 kg. However, similar to the rectangular envelope MRB, the number of wire layers along the height and the width of the coil obtained from the ANSYS optimization tool is not an integer, respectively 7.5 and 11.92. In addition, the inner radius of the disk (R₁) obtained from the ANASYS optimization tool is 15.4 mm, which is much smaller than that obtained from the DE-based methods (around 29.5 mm). The main reason is that the design variable R₁ is trapped in a local extremum near its initial value (10 mm). Therefore, a further optimal solution of the MRB is performed in which the number of wire layers along the height and the width of the coil are fixed at 7 and 12, respectively, based on the previous optimal solution and the initial value of R₁ is set at 25 mm. The results are shown in brackets in the 'first order' column in table 4. It can be seen that the results are now very close to those obtained from the DE-based methods, especially the optimal value of R₁, which is no longer trapped in the local extremum. On the other hand, it can be emphasized that the improved DE reaches the optimal result after 292 generations in 4.339 h, while to gain the optimal result, the original DE needs a greater computation cost with 795 generations in 12.035 h. The geometric parameters of the optimal result are presented in table 4 with the constraint being greater than 10 Nm. It can be seen that the two DEs provide different geometric configurations, but they have almost the same values of the objective function. This discrepancy usually occurs when the number of design variables is numerous. In this case, the problem of MRB optimization becomes very complex. Therefore, the objective function can contain some extrema that have the same value as the global optimal result. The convergence history of design variables, mass and braking torque of this type of MRB are performed in figures 6(a)–(c). It can be realized that after approximately 260 generations of the optimization process of the improved DE, the values of the design variables including mass and braking torque move to the steady state. The magnetic distribution of a 7-control point envelope MRB at the optimum in the case of the improved DE is shown in figure 6(d). It can be observed that the magnetic density is almost uniformly distributed in the housing of the MRB from control point P2. From control point P1 to P2, the magnetic density is much smaller, because the minimum housing thickness is constrained to be greater than 2 mm.

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Similar to the conventional MRBs, the side-coil MRBs considered in this study include the rectangular-shaped MRB and 7-segment polygonal MRB. First, the rectangular-shaped side-coil MRB is considered. In this case, significant geometric dimensions such as the coil height h_c, the coil width w_c, the disk thickness t_d, the inner radius of the disk R₁, the outer radius of the disk R_d, the outer radius of the MRB R, the housing thickness t_h and the inner radius of the coils R_c are set as design variables. Figure 7 shows the geometry of the MRB with the rectangular-shaped envelope. Two DE-based algorithms are applied to determine the optimal configuration of this MRB. The optimal results are shown in table 5. The optimal results obtained from the ANSYS optimization tool are also presented in the table. The results show that the minimum mass, the off-state torque and the number of coil turns obtained from the ANSYS optimization tool are respectively 1.24 kg, 0.278 Nm and 68 turns, while the original DE and improved DE reveal, respectively, that the minimum mass for both is 1.129 kg, the off-state torques are 0.291 and 0.297 Nm and the number of coil turns for each coil is 60. It can be seen that the two DEs offer a better solution than the first-order method in ANSYS. In addition, the number of wire layers along the height and the width of the coil obtained from the ANSYS optimization tool is not an integer, respectively 3.16 and 18.75 (the number of coil turns is 59.25). Therefore, a further optimal solution of the MRB is performed in which the number of wire layers along the height and the width of the coil is fixed at 3 and 20, respectively, based on the previous optimal solution (the number of coil turns is 60). The results are shown in brackets in the 'first order' column in table 5. It is also observed that the improved DE outperforms the original DE in computation cost (107 generations in 1.382 h for the improved DE, 309 generations in 3.991 h for the original DE). The convergence history of the design variables, mass and braking torque of this MRB are shown in figures 8(a)–(c). It can be observed that after approximately 68 generations of the optimization process of the improved DE, the values of the design variables, mass and braking torque have become stable. The magnetic distribution of the rectangular-shaped side-coil MRB at the optimum is displayed in figure 8(d). Similar to the rectangular-shaped conventional MRB, it can be seen that the magnetic density is not uniformly distributed in the housing of the MRB. At both sides of the coil, the magnetic density almost reaches magnetic saturation of the housing material.

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Second, the 7-control point polygonal envelope side-coil MRB is considered. Figure 9 shows the geometry of the MRB with the 7-control point polygonal envelope. In this figure, the control points of the MRB envelope are defined by their radius R_i and corresponding thickness t_hi. Besides the previously considered design variables such as the coil height h_c, the coil width w_c, the disk thickness t_d, the inner radius of the disk R₁, the outer radius of the disk R_d, the outer radius of the MRB R, the housing thickness t_h and the inner radius of the coils R_w, the design variables in this case also include the geometry of control points _Pi (R_i, t_hi) from P₁ to P₇. The results are shown in figure 10 and table 6. In this case, the results are attained with the MRB mass of 0.894 kg for both the original DE and improved DE, while the number of coil turns for each coil is 66 for the original DE and 54 for the improved DE. These results are compared with those obtained from [18], which have the minimum mass of 0.963 kg, and the number of coil turns of 61.7. Again, the number of wire layers along the height and the width of the coil obtained from the ANSYS optimization tool is not an integer and a further optimal solution of the MRB is performed in which the number of wire layers along the height and the width of the coil is fixed at 6 and 10, respectively, based on the previous optimal solution (the number of coil turns is 60). The disadvantage of DE regarding the computation cost is clearly shown in the figure where the number of generations of the original DE reached the maximum (1000 generations), while the improved DE just needed 294 generations to obtain the optimal solution. The computation cost for the optimal design of this MRB is ameliorated effectively after using the improved DE (4.884 h for the improved DE, 16.611 h for the original DE). It is noteworthy that the two DEs provide different geometric configurations, but they have the same values of the objective function. This discrepancy occurs when the number of design variables is numerous. In this case, the problem of MRB optimization becomes very complex. Therefore, the objective function can contain some extrema that have the same value as the global optimal result. At the optimum, the magnetic distribution of the 7-control point polygonal envelope side-coil MRB is presented in figure 10(d). The magnetic density is distributed more uniformly than the rectangular envelope side-coil MRB. From control point P1 to P2, the magnetic density is much smaller, because the minimum housing thickness is constrained to be greater than 2 mm.

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In order to evaluate the benefits of the proposed optimization method for MRBs over other gradient-free global optimization, the optimal design of the MRBs using the genetic algorithm (GA) and particle swarm optimization (PSO) method are conducted and the results are presented in table 7. In setting of the GA and PSO, all parameters that impact significantly on the optimal search process such as population size, convergence rate and boundary of design variables are similar to those of the improved DE. To avoid prolixity, in this work, only rectangular-shaped MRB is considered. From the results, it can be seen that the optimal mass of the MRB gained by the improved DE is the smallest, while the optimal one obtained by the GA is the highest (1.291 kg for the improved DE, 1.318 kg for PSO and 1.355 kg for GA). The effectiveness of the improved DE in obtaining the global optimal results is validated in comparison with the GA and PSO. Moreover, the improved DE is outstanding with the GA and PSO in the number of analyses (97 generations for improved DE, 500 and 180 generations for the GA and PSO, respectively) and computation cost (1.159 h for the improved DE, 11 h and 2.833 h for the GA and PSO, respectively). Considering the compactness based on the MRB size (brake length and radius), the MRB configuration obtained by the improved DE is also better than those of others. In addition, the off-state torque of the MRB gained by the improved DE is slightly higher than the one of the GA, but lower than the one of the PSO. Generally, with the optimization problem investigated in this study, the improved DE can find MRB configurations that can be better than those of the GA and PSO, as well as having less computational cost than other global optimization methods.

From the above, some advantages and conclusions of the proposed optimization procedure in the optimal design of MRBs can be summarized as follows:

• The values of the objective functions of the MRBs gained by the proposed optimization procedure are the global optimal solutions thanks to the robustness and reliability of the DE-based algorithm. It can be seen that the integrated methodology overcomes some disadvantages existing in ANSYS optimization tool related to the local optimal results and the infeasible solutions.
• Essential discrete variables can be taken into account in the optimal design of the MRBs in order to improve the accuracy and reliability of the results as well as to be able to apply the approach to a real-life design model.
• Based on the optimal results and the shortcomings in computation costs of the three optimization methods used in this research: first order and two DE-based algorithms, it can be seen that the improved DE is the most suitable one for solving the MRB optimal problems.