In consideration of the significant role the brake plays in ensuring the fast and safe running of vehicles, and since the present parameter optimization design models of brake are far from the practical application, this paper proposes a multiobjective optimization model of drum brake, aiming at maximizing the braking efficiency and minimizing the volume and temperature rise of drum brake. As the commonly used optimization algorithms are of some deficiency, we present a differential evolution cellular multiobjective genetic algorithm (DECell) by introducing differential evolution strategy into the canonical cellular genetic algorithm for tackling this problem. For DECell, the gained Pareto front could be as close as possible to the exact Pareto front, and also the diversity of nondominated individuals could be better maintained. The experiments on the test functions reveal that DECell is of good performance in solving highdimension nonlinear multiobjective problems. And the results of optimizing the new brake model indicate that DECell obviously outperforms the compared popular algorithm NSGAII concerning the number of obtained brake design parameter sets, the speed, and stability for finding them.
The safe reliability of the vehicles is attracting more and more attentions with the sharp increase in the vehicles. Since the improvement of the brake design and manufacturing is of great significance in ensuring the brake quality thus the vehicle’s safe running, the brake parameter optimization design which is a key part in the whole brake design procedure is becoming increasingly important. Therefore, the research in this field is becoming a hot topic.
In general, in the brake parameters optimizing procedure, the first step is to set up parameter optimization model and the next step is to calculate the model with an optimization algorithm. A lot of researches have been done in this field [
Based on foregoing statement, this paper establishes a multiobjective optimization design model of drum brake, which aims at maximizing the braking efficiency and minimizing the volume and temperature rise of the drum brake. It is expected that this model could meet the requirements of practical engineering design better. Besides, in order to deal with the model more efficiently, a differential evolution cellular multiobjective genetic algorithm (DECell) is proposed by introducing a differential evolution strategy into the cellular genetic algorithm in which an individual may only interact with individuals belonging to its neighborhood. The proposed algorithm is then applied to solve the brake model.
The rest of the paper is organized as follows. The next section shows the details of the multiobjective optimization model of drum brake. Section
Drum brake is one of the most commonly used brakes in vehicle design; it can be categorized into leading and trailingshoe brake, twoleadingshoe brake, twotrailingshoe brake, and servo brake concerning the arrangement of the brake shoes. The optimization design object in this paper is the leading and trailingshoe brake, which is shown in Figure
Structural parameters and force diagram.
Figure
One has
The optimization target is the comprehensive performance of the brake. And the subobjectives are maximizing the braking efficiency factor and minimizing the volume and temperature rise of the drum brake. The subobjectives are presented as follows.
One has the following [
With the same braking force, the bigger the braking efficiency is, the bigger the braking torque is, and the better the brake performs, hence the more efficient the brake is. Therefore, the enhancement of braking efficiency is of great significance to the safe running of vehicle. On this basis, the braking efficiency factor is selected as the optimization objective, and the objective function is
With the same function and efficiency, the smaller the brake is, the fewer materials and the less space it will take up. Hence, the volume of the brake drum should be as small as possible, and it is taken as the optimization objective as follows:
It can be regarded that the heat energy generated through braking is absorbed by the front brake and rear brake due to the fast braking process. The heat energy is then distributed to the front brake and rear brake in accordance with their distribution coefficients. The brake drum is installed at the rear wheel, so the rear brake temperature can be calculated by the following formula:
The possibility of the selflocking for the brake shoe should be taken into consideration when designing the drum brake. To avoid selflocking, the following constraint should be satisfied:
The maximum pressure on the frictional liners should be less than the defined pressure value:
The nonuniform distribution of pressure would cause uneven wear of the frictional liners, which would change the contact area, location, and pressure characteristic of the friction interface. Then the braking efficiency and stability is greatly lowered. The constraint is as follows:
During emergency braking, the specific energy dissipating rate of the frictional liners should be less than the defined rate:
While braking, the frictional characteristic would be better if the positive pressure and energy load of the brake shoe lining area get lower. In cars, the general value range of the friction area of drum brake lining should obey the following constraints:
To calculate the heating load of the brake, the temperature rise of brake from the initial speed of
Since the diameter of car brake drum is usually
When choosing the liner width, the pressure in a unit area should not be oversized since too large width would cause nonuniform contact. In general, the ratio of liner width against brake drum radius should meet the following constrains:
In order to solve the abovementioned multiobjective optimization model much more efficiently, we introduce differential evolution (DE) strategy into the cellular genetic algorithm and thus form a new algorithm called differential cellular multiobjective genetic algorithm (DECell), which is able to minimize the distance from gained Pareto front set to exact optimal front while maintaining good uniformity and distribution range.
DE strategy is well suited for solving multiobjective problems of multidimension and nonlinear as the main feature characterizing it is performing global parallel direct search by using the information of the distance and direction among the individuals in the population, and what is more, it can be easily implemented [
As for each individual
Cellular genetic algorithms (cGAs) are a kind of decentralized genetic algorithms (GAs) in which the population is structured in such way that the interaction of the individuals composing it is limited to a certain subset of individuals in the immediate vicinity of their location. Usually, the population is arranged in a 2D toroidal mesh in cGAs, and individuals are allowed to interact only with their nearby neighbors (see Figure
Breeding loop in DECell.
Based on cGA, the detailed principle of DECell is presented in Figure
In the genetic operation process for each cellular individual, the nondominated individuals are stored in an external archive at the same time, and all the individuals in an external archive are ranked according to their crowding distance. If the number of the superior individuals exceeds the specified capacity of the archive, the individuals with smaller crowding distance will be removed. At the end of each generation, some of the individuals in the external archive are chosen randomly to replace the individuals of the same number in the original population so as to update the population constantly. By this method, the nondominated individuals in the external archive are able to obtain a reasonably good approximation to the Pareto optimal front when maintaining the diversity. The main pseudocode of DECell is in Pseudocode
population
archive
//Select the neighbors for the current individual (MOORE is adopted).
neighborhood
//Select two parent individuals from the neighbors.
parent1
parent2
//Make sure the two parent individuals are different.
//Crossover operator with DE strategy and Pseudocode
the detailed operation.
offspring
position(parent2));
offspring
// The superior offspring replaces the present individual.
// Put the nondominated individuals into the external archive.
population
To assess how competitive the DECell is, a set of unconstraint, multivariable, and multiobjective test functions (i.e., DTLZ family) [
This paper uses three assessment indicators: generational distance [
Generational distance is used to measure the distance between the obtained Pareto front and the Pareto optimal front. The computational formula is as follows:
The spread measurement proposed by Deb is an indicator that measures the distribution and spread of the obtained Pareto front. The calculation formula is as follows:
Hypervolume is adopted to calculate the volume covered by the individuals of the obtained Pareto front. It is defined as
The parameters settings of DECell, NSGAII, MOCell, PESA2, and PAES are as follows: all the algorithms adopt real coding and polynomial mutation. NSGAII, MOCell, PESA2, and PAES employ SBX [
Average (avg) and standard deviation (std) of the GD indicator of the Pareto front obtained by five algorithms for different problems.

Average (avg) and standard deviation (std) of the spread (

Average (avg) and standard deviation (std) of the HV indicator of the Pareto front obtained by five algorithms for different problems.

We consider first the generational distance (GD). As the test results of the computation on the benchmarks in Table
Regarding the spread value (Table
The values of hypervolume indicator (HV) in Table
To compare the performance of algorithms more clearly, we present Figures
DTLZ2 solved by DECell.
DTLZ2 solved by MOCell.
DTLZ2 solved by NSGAII.
DTLZ2 solved by PESA2.
DTLZ2 solved by PAES.
Another analysis method is needed to further analyze the influence of the evaluation number on the algorithm performance. The simple way is to observe the evolution of three given indicators during the whole execution of the algorithms from which we can tell what is happening in every generation. Figure
Evolution of the three indicators for different algorithms in ZDT1.
A more detailed analysis can be performed if we show the results using boxplots, which constitutes a useful way of depicting groups of numerical data. Figure
Boxplot representation of GD, spread, and HV values of the fronts computed by five algorithms in ZTLZ3.
It can be concluded that in general DECell performs better than the other algorithms when solving complicated multiobjective problems, while NSGAII takes the second place. Though DECell is outstanding in accuracy, it is worse than some other MOEAs in efficiency. That is to say, DECell costs more computational time than NSGAII, PESA2 for the problems, because DECell is a kind of decentralized cellular genetic algorithms, where individual interacts only with its nearby neighbors and diffusion of solutions to the population happens slowly in a smooth fashion.
The computational procedures of the brake model by using DECell are as follows.
Randomly generate initial population in the constraint range of optimization parameters by real coding. The optimization parameters are
The tournament method is used, which selects the parent individuals based on the individual rank and crowding distance. The detailed procedures are as follows: select the neighbors of the current individual; compare the ranks of the neighbors, and remain the individuals with lower ranks since the individuals with lower ranks performs better; if the neighbors are of the same rank, compare their crowding distances, and reserve the ones with bigger crowding distances. At last, two better individuals are selected from the neighbors.
Crossover operation is performed to generate more new individuals and improve the space searching ability. DE strategy is introduced to the crossover operation, and the detailed procedures are explained in formulas (
proc crossoverInDE(
//
index of the parameters
// perform differential crossover operation
for (
if (rand(0,1) < CR or
else
end if
end for
return
end_proc crossoverInDE
This operation aims to avoid local convergence.
If the maximum evaluation generations are reached, output the results or go back to step
For a vehicle, some of its parameters and its leading and trailingshoe brake are shown in Table
Model parameters.
Parameters  The weight 
Distribution coefficient 
Rim radius 
Initial velocity 
Friction coefficient 
Specific heat capacity 

Values  1560  0.77  177.8  30  0.35  482 
To search optimal solutions of the multiobjective problems, Table
Domain of decision variables.
Variable 







Upper limit  10  90  36  120  120  85 
Lower limit  45  120  75  180  160  120 
 
Variable 






 
Upper limit  60  90  5  15  6  
Lower limit  130  120  16  35  12 
To evaluate the competitiveness of this algorithm, the most commonly used multiobjective genetic algorithm NSGAII is also adopted to compute the brake model besides DECell. The parameters settings are as follows: population size
After each evolution, the superior solutions gained by NSGAII are put in the final population, while those obtained by DECell are stored in the external archive. Since the population size and the archive capacity are both 100, each algorithm obtains 100 superior solutions after computation. However, there are still some dominated solutions existing in the 100 individuals, which means that there would be 100 or less non dominated ones after each computation.
Let the maximum evaluation generations be
Even though we cannot easily and specifically tell which algorithm performs better in terms of the three objectives from Figure
Distribution of the Pareto front.
Pareto front set schematic diagram between volume and temperature rise.
In Figure
Figure
Pareto front set schematic diagram between volume and efficiency factor.
To further assess the algorithms performance in searching Pareto optimal solutions, different maximum evolution generations, which are got from 1000 to 10000 with the interval 1000, are adopted. For every maximum evolution generation, the aforementioned two algorithms are applied to compute the model of brake for 20 times, respectively, and the averages of the number of the nondominated individuals gained by each algorithm during the 20 times computations are plotted in Figure
The number of the nondominated individuals obtained by the two algorithms.
Figure
To further analyze the computational stability and accuracy of the algorithm, the maximum evolution generation is set as
Performance comparison.

Each data in the grey blanks presented in Table
As is shown in Table
Meanwhile, as for the deviation of three indicators, two of them obtained by DECell are lower than those obtained by NSGAII, which means that DECell is of low volatility and good stability. As for each objective, all the optimal values found by DECell are better than those got by NSGAII. It indicates that DECell can converge to the much better Pareto solutions and it is of higher accuracy.
Therefore, it can be concluded that differential cellular multiobjective genetic algorithm performs better than NSGAII in terms of solution space searching ability and speed for finding much more and better solutions when solving the brake optimization design model. Moreover, the solutions could be kept effectively. What is more, this algorithm is of good stability and high accuracy, in which much more good design parameters of brake could be easily found for designers to choose.
This paper established a multiobjective optimization design model of drum brake with the goals of maximizing the efficiency factor of braking, minimizing the volume of drum brake, and minimizing the temperature rise of brake, in order to better meet the requirements of engineering practice.
We present a new optimization algorithm by introducing the differential strategy into the canonical cellular genetic algorithm to solve the model effectively, which is called differential evolution cellular multiobjective genetic algorithm (DECell). In DECell, all the population individuals are structured in a bidimensional grid, and each individual may only interact with its neighborhood. Besides, three parent individuals are selected for each individual to take part in the crossover operation together. With this method, the obtained Pareto front would be as approximate as possible to the Pareto optimal front. Besides, their distribution uniformity and coverage are well maintained. The evaluation on test function indicates that the proposed algorithm outperforms the other four popular multiobjective algorithms (i.e., NSGAII, MOCell, PESA2, and PAES) greatly when solving highdimensional nonlinear multiobjective problems. The experimental computation of DECell and NSGAII on parameter optimization design of drum brake suggests that the DECell can find much more and better solutions stably. It is an effective algorithm that can be applied to solve the drum brake parameters optimization and other complicated engineering problems.
This work was supported by the National Science Foundation of China (51275274).