Multi-objective optimization of disc brake structure based on heat transfer performance

2.1. Establishment of finite element model

In this paper, ANSYS / workbench is used for the parametric modeling of ventilation disc with finite element analysis [8]. For the disc brake, the moving effect of friction heat source is not obvious under the condition of high speed. Therefore, in order to facilitate the parameter structure optimization of the ventilation disc, the brake pad model is omitted, and the friction surface within one week of the brake pad is separated as the bearing surface through freezing and plane cutting method, and the thermal load and boundary condition are considered to have axial symmetry characteristics. Without affecting the optimization results, such treatment effectively improves the calculation efficiency.

In the design modeler, the three-dimensional model of the ventilation disc is established according to the initial design size of the disc body, as shown in Fig. 1. According to the working requirements of the brake disc, the peak temperature $T_{max}$ and mass $m$ are taken as the optimization objectives, and the total thickness $l_s$, ventilation groove height $D_s$, fin side angle ${\alpha }_s$ and fin number $N$ are taken as the design variables.

Fig. 1Establishment of parameterized model

Fig. 2Convective coefficient distribution

Because the key content of the model optimization is the heat transfer performance of the ventilation disc, the stress problem of the model can be ignored, and the transient heat transfer problem of the model can be solved in the transit thermal module. The total time of heat transfer analysis is 120 s and the time step is 1 s. The temperature field distribution of the model is calculated under the condition of constant speed downhill braking for 10 s. The preprocessing of the model mainly includes mesh generation, load application and the definition of boundary conditions. In order to ensure the accuracy of calculation, the tetrahedral structure grid type with a large degree of correlation is used to mesh the model. Finally, the number of mesh elements is 94338, and the number of nodes is 152023 Through the calculation of heat flow density (1 MPa, 60 km/h), the heat flow load of friction surface in 10 s is defined as 8.2×10⁵ w/m² by tabular data. The thermal radiation direction is defined as 0.28, and ambient temperature is 20 ℃. Establish the cylindrical coordinate system, and define the convection heat transfer coefficient of the end face of the ventilation disc and the ventilation groove surface respectively according to the vehicle speed. The convection heat transfer coefficient is shown in Fig. 2.

2.2. Result analysis

Through the finite element calculation, the transient temperature field of the ventilation disc can be obtained as shown in Fig. 3. The temperature field of the ventilation disc shows obvious difference at different time, and the conduction rate of the temperature in the axial direction is greater than that in the radial direction. When the input of heat flow load is stopped, the convective heat transfer becomes more obvious with time. At the 80 s time, the phenomenon of “hot spot” appeared obviously. The temperature rise rate of $T_{max}$ is very large in the initial stage (0-1 s), and it is stable in the range of 1-10 s. $T_{min}$ showed a trend of increasing first and then decreasing in the whole process.

Fig. 3Temperature field of ventilated disk at different time

a) The 5th second

b) The 10th second

3.1. Response function fitting and accuracy checking

The response surface model is also called the fitting function, which can obtain objective function with discrete data, including design variables and target variables (optimization objectives). The CCD (Central Composite Design) method is applied to realize discrete data sampling during the construction of response function, which is the basis for the optimized mathematical model [9]. This method can save the sampling time and shorten the single sampling period on the premise of sufficient sample data. In addition, it can intuitively show the influence of parameter change on performance. Given the cost of manufacturing process, the range of design variables is defined as shown in Table 1. By using the values of the four optimization variables, 145 sample design points combination (including initial values) in the decision space can be constructed.

Given the interaction between structural parameters, the second-order polynomial is used to obtain the response surface function. Define $l_s$ as $x_1$, $D_s$ as $x_2$, ${\alpha }_s$ as $x_3$ and $N$ as $x_4$, the fitting objective function $\widehat{y}\left(x\right)$ can be expressed as follows:

where $b_0$, $b_i$, $b_{ij}$ are undetermined regression coefficient.

According to the least square method, the regression analysis of error $\epsilon$ can be obtained:

where $y_i$ is the actual discrete response value that obtained from FEA, $n$ is the number of samples. Then the regression coefficient of fitting function can be solved by Matlab. According to the fitting principle, it can be known that the error of response surface model is inevitable. In order to ensure the reliability of the optimized mathematical model, it is necessary to verify the fitting accuracy. If the error is within the allowable range, it can prove that the response surface function designed in this paper has good feasibility. The fitting accuracy of response surface function is evaluated by four parameters, including fitting decision coefficient $R^2$, correcting decision coefficient $R_{adj}^2$ and root mean square error RMS. The error checking results of response surface functions are calculated. It can be seen that the fitting function is equipped with good fitting accuracy and can effectively ensure the reliability of the optimization results.

3.2. Construction of optimized mathematical model

In order to improve the efficiency of optimization calculation, the multi-objective optimization problem is transformed into the single-objective optimization problem [10, 11] when the optimization mathematical model is constructed. In other words, the mass objective function should be transformed into constraints within a specific range. According to the mass and temperature peak value under the initial structure of the brake disc, the searching range of the mass is set as less than 9.6 kg and the temperature peak is set as less than 298 ℃. The objective function and constraint function can be represented respectively as follows:

where $X$ is the design variable matrix; $y_1\left(X\right)$ is the mass response function; $y_2\left(X\right)$ is the temperature peak response function; $Y_t$ is the initial value of the optimization target calculated for the parameter test sample; ${\underset{\_}{x}}_i$ is lower limit of design variables; ${\stackrel{-}{x}}_i$ is upper limit of design variables (search results of $x_4$ need to be integers).

Through continuous iteration operation, the Pareto solution sets [12] of temperature peaks in different mass ranges can be obtained as in Fig. 4, and all the solutions satisfy the optimization mathematical model. In order to verify the optimization results of heat transfer performance, a set of solutions is randomly selected for finite element calculation ($l_s=$20.04 mm, ${\alpha }_s=$2.29°, $D_s=$8.26 mm, $N=$30). Keeping the load and boundary conditions of the finite element model unchanged, the maximum and minimum temperature curves of the brake disc before and after optimization can be shown more clearly by setting the total analysis time of 120 s (braking pressure is canceled during 10th-120th second, no heat source input). It can be seen that the optimized structure is more conducive to heat transfer than the initial parameters, especially when the heat flow load is relieved, the temperature drops significantly, thus promoting the heat dissipation efficiency of the brake disc. It can shorten the time of continuous high temperature stage and effectively enhance the reliability of continuous braking.

The optimization results must meet the cost performance requirements of disc brakes, but the mass and temperature peaks in the optimization objectives are different dimension parameters [13], so it is impossible to evaluate the performance directly from the results. Therefore, a weighted normalization evaluation method is proposed to standardize the data of the optimization objectives [14].

Fig. 4Pareto solution of objective function in different mass ranges

a) 9 $<m\le$9.6

b)$m\le$9

3.3. Weighted normalization

In weighted normalization, the objective decision value of any Pareto solution of the optimization objective can be expressed as:

According to the manufacturer's agreement, the weighting coefficient of ${\lambda }_c$ is 0.5. The Pareto solutions in different mass ranges are weighted and normalized by expanding the mass interval, and finally the scatter plots of the target decision-making values in four mass ranges are obtained as shown in Fig. 5. The minimum of objective decision value $B_{min}$ and the corresponding solution results are shown in Table 3, including the rate change of target variables and structural parameters.

Fig. 5Scatter plot of target decision value under different mass

a) 11.2 $<m\le$12 kg

b)$m\le \mathrm{}$9 kg

As can be seen from Table 2, the $B_{min}$ value in Number 2 is the smallest. Compared with the initial value, the mass increases by 11.3 %, and the peak temperature decreases by 15.2 %. According to the corresponding structural parameters in serial Number 2, the brake disc is manufactured, and the noise level is tested with the same brake pad and load conditions.