The turbine parametric design starts with the definition of the torus shape. Various design parameters, including active diameter, aspect ratio and two arc radii are needed to determine the shell profile of an automotive torque converter torus. As the flow area in the circular path, in the proposed torque converter model, is assumed constant, only the design parameter area factor $f_{\text{a}}$ is used to determine the design path and core of torus. Consequently, by defining the turbine inlet radius $R_1$ and outlet radius $R_2$, along with the shell and core, the torus profile of turbine can be obtained as shown in Figure 1 .

Each turbine blade profile is formed by a three dimensional (3D) curve of the shell and a 3D curve of the core. The 3D curve can be calculated from the torus and a two dimensional (2D) design curve using the conformal transformation principle. Figure 2 shows the schematic representation of a 2D design curve. With the origin at the starting point of the curve, the coordinate of $P_0$ is (0, 0). The curve expression can be written with the following equation:

$A{x}^2+B{y}^2+Cx+Dy+Exy=0$(1)

where A, B, C, D and E are the coefficients of the variables. It should be noted that the value of D is 1.0 and the parametric equation of the 2D design curve can be obtained as

$A{x}_1^2+B{y}_1^2+C{x}_1+y_1+E{x}_1{y}_1=0$(2)

$A{x}_2^2+B{y}_2^2+C{x}_2+y_2+E{x}_2{y}_2=0$(3)

$\tan \left({\alpha }_1-90\right)+C=0$(4)

$\left(E{x}_2+2B{y}_2+1\right)\tan \left({\alpha }_2-90\right)+2A{x}_2+E{y}_2+C=0$(5)

where ${\alpha }_1$ is the inlet angle of turbine, and ${\alpha }_2$ is the exit angle of turbine. In the present study, four design parameters including inlet angle ${\alpha }_1$, exit angle ${\alpha }_2$, offset size $d$, and conic factor $f_{\text{c}}$ are provided to calculate the 2D design curve. The coordinates of $P_c$ and $P_1$ can be obtained as

$y_c=x_c\tan \left({\alpha }_1-90\right)$(6)

$y_c=y_2-\left(x_2-x_c\right)\tan \left({\alpha }_2-90\right)$(7)

$x_1=x_2/2+\left(x_c-x_2/2\right)f_{\text{c}}$(8)

$y_1=y_2/2+\left(y_c-y_2/2\right)f_{\text{c}}$(9)

where $P_c\left(x_c,y_c\right)$ is the intersection of the two tangent lines that across the curve’s starting and ending points, respectively ( Figure 2 ). According to the definition of 2D design curve, $x_2$ value and $y_2$ value equal to the length of torus $l$ and offset size $d$, respectively.

The 2D design curve can be easily constructed by having the torus and the four design parameters defined. After the definition of 3D curves of shell and core, another design parameter bias angle $\beta$ is used to obtain the blade profile. Given the turbine blades are constant-thickness stamped sheet metal, the blade thickness $t$ is defined. Once the torus shape and five blade design parameters including inlet angle ${\alpha }_1$, exit angle ${\alpha }_2$, offset size $d$, conic factor $f_{\text{c}}$, bias angle $\beta$ and blade thickness $t$ are determined, the basic blade geometry of turbine is generated by means of parametric equations and Creo software. The schematic representation of the construction of turbine blade is shown in Figure 3 .

After the definition of the torus and a blade, the construction of the whole turbine geometry is easy and is based on the rotation of the generic blade around the axis; for doing so the number of blades $z$ should be provided by the user. The completed turbine parametric geometry is shown in Figure 4 . The design parameters considered for the parametric study are shown in Table 1 , provided that the torus of turbine is determined.

An automotive torque converter is selected as a reference to generate the parametric model. The reference torque converter has an active diameter of 250 mm and the number of blades in the impeller, turbine, and stator are 31, 29, and 21, respectively. STAR-CCM+ software is used to generate the computational mesh and perform the internal flow calculations appropriately. The computational mesh is given in Figure 5 , where only one blade passage is modeled for each element to illustrate the mesh distribution in the computational field when approximately 103, 533 grid cells in total are used. The polyhedral cells with five prism layers is used on the model. The leakage between the elements and also between an element and the core flow is disregarded. A cyclic boundary condition is imposed on both peripheral boundaries outside a blade passage. A no-slip wall boundary condition is also imposed on all the walls bounding the domain,

with a spin applied as necessary. The interfaces between elements have been handled by using the mixing plane method. A second-order upwind differencing scheme is utilized and the SST k-w model is also used for the turbulence. Steady state simulations are performed for a range of speed ratios from 0.0 to 0.9 while maintaining an impeller speed of 2000 rpm.

Figure 6 compares the measured and calculated overall performance including efficiency $\eta$, torque ratio $Tr$, and impeller torque factor ${\lambda }_{\text{I}}$ for the parametric model of the torque converter. As indicated here, the tendencies of the experimental data correlated relatively well with the calculated results, confirming that the parametric model and computational method are valid in general. It is of note that the CFD underestimates the impeller torque factor ${\lambda }_{\text{I}}$ at low speed ratios. The discrepancy between measurements and calculations are reasonable because of the overestimated incidence losses at low speed ratios.

The ability of the SVR to make reasonable estimations is mostly dependent on input parameters selection. Adequate consideration of the factors of turbine effecting the torque converter performance studied is therefore crucial to developing a reliable network. As mentioned above, 11 design parameters are used to obtain an turbine parametric model. Blade scroll angle $\gamma$ is introduced as another design parameter to investigate the effect of the turbine geometry on the performance of a torque converter. The scroll angle is defined as the angle between the two planes containing the intersection of the design path and the entering and leaving edges of the blade when that blade does not lie in one axial plane. The scroll angle can be determined by four design parameters including inlet angle ${\alpha }_1$, exit angle ${\alpha }_2$, conic factor $f_{\text{c}}$, and offset size $d$, provided that the torus is defined. Matlab software is used to develop a simple-to-use GUI to calculate scroll angle as shown in Figure 7 .

The blade scroll angles at shell and core can be represented by the mean scroll angle and change simultaneously with the change of the mean value. Similarly, the inlet angles and exit angles at shell and core can also be represented by the mean value of inlet angle and mean exit angle. Finally, the 11 design parameters of turbine are translated into 6 parameters including inlet angle ${\alpha }_1$, exit angle ${\alpha }_2$, scroll angle $\gamma$, bias angle $\beta$, blade thickness $t$, and blade number $z$. The six design parameters are to be defined as input for the learning techniques in this research study. The range of design parameters and base values of the turbine are summarized in Table 2 .

Orthogonal design approach is a collection of mathematical and statistical techniques to reduce the number of experiments in order to find the effect of parameters affecting a response in a process, thereby aiming for a reduction in both costs and time. Orthogonal design sets out configurations (or arrangements) to be conducted using an appropriate orthogonal array; the terminology used in these arrays includes “factors” —an item that is to be varied during the simulations, “level” —the number of times a factor is to be varied during the simulations and “configuration number” —the number of simulations that are required to be run to complete the analysis [24] . For this paper six main geometrical parameters mentioned above are selected as design variables (factors) and five different values (levels) are assigned for each design parameter. So 25 (L₂₅[5⁶]) configurations with different combinations are generated for orthogonal design. Stall torque ratio $T{r}_0$ and peak efficiency ${\eta }^{\ast }$ are used as the dynamic characteristic and economic characteristic, respectively, to evaluate the performance of torque converters. The above mentioned CFD simulation method is applied and the corresponding calculation results for 25 cases are presented in Figure 8 . The 25 sets of calculation data are used for SVR modeling training.

The Latin Hypercube design is a popular choice of experimental design when computer simulation is used to study a physical process. These designs guarantee uniform samples for the marginal distribution of each single input [25] . Five cases are obtained using LHD method and the corresponding calculation results are shown in Figure 9 . The 5 sets of data as testing data to analyze the prediction performance of SVR.

SVR in its essence is closely related with SVM theory and its parameters such as kernel and operating process are not disparate. The $\epsilon$-insensitive loss function is the most widely used cost function. It can be assumed as a tube equal to the approximation accuracy that surrounds the training data [26] . The objective held by SVR is to search for the function with the most $\epsilon$ deviations from the real destination vector for all training information received and which must be as flat as possible [27] . For linear regression problems, the regression model can be expressed by Vapnik [28] and is defined by the following function:

$f\left(x\right)=\langle w,x\rangle +b$(10)

where $f\left(x\right)$ is an unknown target function, $\langle .,.\rangle$ denotes the dot product and $w$ is the weight vector. The $\epsilon$-insensitive loss function is defined by the following function:

$L_{\epsilon }\left(y\right)=\{\begin{array}{ll}0,\hfill & \text{for}\left|f\left(x\right)-y\right|\le \epsilon \hfill \\ \left|f\left(x\right)-y\right|-\epsilon ,\hfill & \text{otherwise}\hfill \end{array}$(11)

As a convex optimization problem, it can be written as:

Minimize $\frac{1}{2}{\Vert w\Vert }^2+C{\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\xi }_i+{\xi }_i^{*}\right)}$(12)

Subject to $\{\begin{array}{l}{y}_i-\langle w,x_i\rangle -b\le \epsilon +{\xi }_i\\ \langle w,x_i\rangle +b-y_i\le \epsilon +{\xi }_i^{*}\\ {\xi }_i,{\xi }_i^{*}\ge 0\end{array}$(13)

where ${\xi }_i$ and ${\xi }_i^{*}$ variables are to satisfy the function constraints. The corresponding dual optimization problem is defined as [29] :

$\underset{\alpha ,\alpha *}{\max }-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{l}{\sum }}{\displaystyle \underset{j=1}{\overset{l}{\sum }}\left({\alpha }_i^{*}-{\alpha }_i\right)\left({\alpha }_j^{*}-{\alpha }_j\right)}}\langle x_i,x_j\rangle -{\displaystyle \underset{i=1}{\overset{l}{\sum }}{y}_i}\left({\alpha }_i^{*}-{\alpha }_i\right)-\epsilon {\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i^{*}+{\alpha }_i\right)}$(14)

with constraints:

$0\le {\alpha }_i,{\alpha }_i^{*}\le C,i=1,\cdots ,l$

${\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i-{\alpha }_i^{*}\right)}=0$(15)

where ${\alpha }_i$ and ${\alpha }_i^{*}$ are Lagrange variables while $w$ and $b$ are denoted by the following equation and $x_r$ and $x_s$ are support vectors [28] .

$w={\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i^{*}-{\alpha }_i\right)}{x}_i$

$b=-\frac{1}{2}\langle w,\left(x_r+x_s\right)\rangle$(16)

For nonlinear regression problems, a nonlinear mapping $\varphi$ of the input space onto a higher dimension feature space can be used, and then linear regression can be performed in this space [30] , the nonlinear model is written as

$f\left(x\right)=\langle w,\varphi \left(x\right)\rangle +b$(17)

where

$w={\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i-{\alpha }_i^{*}\right)\varphi \left(x_i\right)}$(18)

$\langle w,\varphi \left(x\right)\rangle ={\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i-{\alpha }_i^{*}\right)}\langle \varphi \left(x_i\right),\varphi \left(x\right)\rangle ={\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i-{\alpha }_i^{*}\right)K\left(x_i,x\right)}$(19)

$b=-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{l}{\sum }}\left({\alpha }_i-{\alpha }_i^{*}\right)}\left(K\left(x_i,x_r\right)+K\left(x_i,x_s\right)\right)$(20)

where $x_r$ and $x_s$ are support vectors.

In this paper, the non-linear RBF kernel and polynomial kernel are investigated and are defined as [31] :

RBF: $K\left(x_i,x_j\right)=\exp \left(-\gamma {\Vert x_i-x_j\Vert }^2\right),\gamma >0$(21)

Polynomial: $K\left(x_i,x_j\right)={\left(\gamma x_i^{\text{T}}{x}_j+r\right)}^d,\gamma >0$(22)

where $\gamma$, $r$, and $d$ are kernel parameters.

The root mean square error $\text{RMSE}$ and correlation coefficient $R$ are used to estimate the accuracy of the prediction model:

$\text{RMSE}=\sqrt{\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(y_{\text{sim}}-y_{\text{pre}}\right)}^2}}{n}}$(23)

$R=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(y_{\text{sim}}-\overline{y_{\text{sim}}}\right)\left(y_{\text{pre}}-\overline{y_{\text{pre}}}\right)}}{\sqrt{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(y_{\text{sim}}-\overline{y_{\text{sim}}}\right)}^2{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(y_{\text{pre}}-\overline{y_{\text{pre}}}\right)}^2}}}}$(24)

in which $y_{\text{sim}}$ is the value of simulation and $y_{\text{pre}}$ is the value of prediction.

The SVR parameters must be set carefully in order to construct the SVR model efficiently [32] - [34] . Inappropriately chosen SVR parameters will result in over-fitting or under-fitting, and different parameter settings may also cause significant differences in performance [35] . Thus, selecting the optimal parameters is an important step in SVR design. Cross-validation together with grid-search method is applied to obtain the SVR optimal parameters. In $v$-fold cross-validation, we first divide the training set into $v$ subsets of equal size. Sequentially one subset is tested using the classifier trained on the remaining $v-1$ subsets. Thus, each instance of the whole training set is predicted once so the cross-validation accuracy is the percentage of data that are correctly classified [31] . In our scheme, 5-fold cross validation is selected and the values of epsilon and tolerance are 0.01 and 0.001 seem to perform well. To select optimal combinations of $C$ and $\gamma$ for radial basis function kernels, a coarse grid-search is used first. The contours of coarse grid search results for stall torque ratio and peak efficiency predictions are shown in Figure 10 .

After identifying a “better” region on the grid, a finer grid search on that region can be conducted. The three dimensional plot of finer grid search results for stall torque ratio and peak efficiency predictions are presented in Figure 11 .

It can be seen that the grid optimization process for stall torque ratio yields to a minimum $\text{RMSE}$ when $\left(C,\gamma \right)=\left(3.0314,0.37893\right)$ while the best $\left(C,\gamma \right)$ for peak efficiency is (4.5948, 0.26794). The similar grid-search method is also employed for SVR with polynomial kernel function. Table 3 provides the optimal values of parameters for this data set with polynomial and RBF kernel-based SVR.

The training data is used to establish the RBF and polynomial kernel-based SVR model for stall torque ratio and peak efficiency predictions. Results in Figure 12 and Figure 13 compare the RBF based SVR (SVR_rbf) with polynomial kernel based SVR (SVR_poly) models for both training data and testing data for stall torque ratio. The root mean squared error ( $\text{RMSE}$) and correlation coefficient ( $R$) served to evaluate the differences between the predicted and CFD simulation values for SVR_rbf and SVR_poly. A comparison between the SVR with RBF kernel results and the SVR with polynomial kernel results reveals that SVR with RBF kernel outperforms the SVR with polynomial kernel function in terms of estimation prediction for stall torque ratio.

Figure 14 represents the results of the peak efficiency of CFD and predicted data in the preparation stage, while Figure 15 demonstrates the results for the testing stage. Both the SVR with RBF kernel and SVR with polynomial kernel perform well in estimating the peak efficiency. For stall peak efficiency prediction, the SVR with RBF kernel function has very small $\text{RMSE}$ during training and this value is slightly larger in testing. The SVM with polynomial kernel have larger $\text{RMSE}$ in training data and smaller $\text{RMSE}$ in testing data than RBF. Table 4 compares the SVR_rbf and SVR_poly models. It can be concluded that the SVR with RBF kernel is better than the SVR with polynomial kernel at estimating the performance characteristics of a torque converter.