# Application of Improved Particle Swarm Optimization in Vehicle Crashworthiness

Authors: Dawei Gao; Xiangyang Li; Haifeng Chen
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the optimization design process, particle swarm optimization (PSO) is limited by its slow convergence, low precision, and tendency to easily fall into the local extremum. These limitations make degradation inevitable in the evolution process and cause failure of finding the global optimum results. In this paper, based on chaos idea, the PSO algorithm is improved by adaptively adjusting parametersr1 and r2. The improved PSO is verified by four standard mathematical test functions. The results prove that the improved algorithm exhibits excellent convergence speed, global search ability, and stability in the optimization process, which jumps out of the local optimum and achieves global optimality due to the randomness, regularity, and ergodicity of chaotic thought. At last, the improved PSO algorithm is applied to vehicle crash research and is used to carry out the multiobjective optimization based on an approximate model. Compared with the results before the improvement, the improved PSO algorithm is remarkable in the collision index, which includes vehicle acceleration, critical position intrusion, and vehicle mass. In summary, the improved PSO algorithm has excellent optimization effects on vehicle collision.


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## Body

## 1. Introduction

With the increase of car ownership, traffic accidents have occurred from time to time, and the passive safety performance of cars in collisions has attracted more and more attention. Improve the crashworthiness of vehicles and reduce the occupant injury risk in the traffic accident and property loss becomes an important goal of automotive design. Therefore, many researchers and manufacturers have conducted research on improving the collision safety of vehicles. At present, the crashworthiness of the vehicle is mainly improved in the following three ways. Improve manufacturing processes, optimize structural design, and adjust material usage. Among them, the use of higher quality materials and improved processing technology can significantly improve the collision performance, but it will also lead to an inevitable increase in manufacturing costs. While in practical applications, optimizing the body design, especially the structural design of the front of the car, is the most widely used improvement. In the front part of the car, the structure of the bumper, energy absorbing box, front longitudinal beam, fender, hood, and other components will impact the safety of the collision, which requires the use of algorithms in the design process, to carry out multiobjective optimization on these structures [1, 2].To this end, many researchers have contributed to this aspect, and a variety of intelligent algorithms have emerged. Fonseca and Fleming [3] first proposed a multiobjective genetic algorithm based on Pareto optimal concept in 1993. Later, Sun et al. [4] used the nondominated sorting genetic algorithm (NSGA-II) to optimize the shape of the thin-walled tube, which improved the absorbed energy and reduced the peak impact force. Kirkpatrick [5] proposed a simulated annealing algorithm for large-scale combination optimization problems in 1983. Suppapitnarm et al. [6] improved the optimization performance of the algorithm in engineering practice by improving the acceptance criteria and cooling accuracy table of the simulated annealing algorithm. The Italian scholar Dorigo [7] was inspired by the ant colony searching for food behavior in nature and proposed the ant colony algorithm (ACA). Karl [8] et al. further introduced the concept of pheromone vector into the ant colony algorithm to coordinate the relationship between the various targets and deal with multiobjective asset selection. The particle swarm algorithm appeared relatively late, which was first used by Jacqueline [9] in multiobjective optimization in 1999. Fang [10] et al. used the particle swarm optimization algorithm together with the response surface approximation model to optimize the multiobjective reliability of the vehicle door and achieved good results.In recent years, several intelligent algorithms, such as particle swarm optimization (PSO), genetic algorithms, and simulated annealing algorithms, have received widespread attention in the area of scheduling [11–13]. Research shows that these algorithms suffer from defects when applied to complex optimization problems. Hence, the optimization results for a single technique tend to be unsatisfactory. Therefore, algorithm improvements based on fusion ideas have been proposed by domestic and foreign scholars [14, 15].PSO has several advantages, including fast convergence, few setting parameters, and simple and easy implementation; hence, it can be used to solve nonlinear, nondifferentiable, and multipeak optimization problems, particularly in science and engineering fields [16, 17]. However, the algorithm suffers from major disadvantages, including “premature” phenomena and “unripe” phenomena [18, 19]. The insufficient convergence speed and insufficient accuracy of PSO hinder it from meeting requirements. In addition, its global search ability is poor, and it easily falls into the local extremum [20]. These drawbacks make degradation inevitable in the evolution process and cause the failure of finding the global optimum results. The features exhibited by chaotic thoughts show that chaotic sequences appear to be particularly random, the tracks of chaotic variables are regular, and the entire search space can be searched without repetition; therefore, chaotic sequences are superior to blind random search and avoid the disadvantage of evolutionary algorithms of falling into the local optimum [21, 22].It is because of the aforementioned defects of the PSO algorithm that the present study introduces the chaos idea and proposes an improved PSO algorithm that uses chaos theory to dynamically adjust its parametersr1 and r2. In order to verify the effect of the method in actual optimization, eight sheets which have great influence on the frontal collision safety performance of a minibus were selected, and the thickness of the eight panels was used as the design variable. The peak acceleration amax at the lower end of the B-pillar, the mass m of the vehicle, the backward intrusion amount D1 at the dashboard tube beam, the backward intrusion amount D2 at the steering column hole of the dash panel, and the backward intrusion amount D3 at the clutch plate of the lower dash panel were used as a study response. The improved particle swarm optimization algorithm is used for multiobjective optimization design. Compared with the traditional particle swarm optimization algorithm, the results show that the improved one is superior to the traditional particle swarm optimization algorithm in terms of optimization effect and solution speed.


## 2. Improved PSO Algorithm

In 1995, a group of researchers led by Kennedy and Eberhart developed a new calculation method for the study of bird predation behavior. This calculation method uses the iterative changes of particles to simulate bird predation behavior and searches a certain number of particles for optimal particles in the spatial solution set [23, 24].The PSO algorithm is similar to bird predation. Each solution to the optimization problem is similar to a bird that is preying on a defined space; in this scenario, the bird is the “particle.” Each particle has its own position, and the ith particle position is expressed asxi=(xi1,xi2,…,xiD). The speed of flight is expressed as =(vi1,vi2,…,viD). Moreover, each particle has an adaptive value determined by the objective function. In each iteration of the particle swarm, the particle needs to determine two extreme values: the individual extremum pbest (the optimal solution found by the particle itself) and the global extremum gbest (the optimal solution found by the group). After determining these two extreme values, the particle’s d-dimensional (1 ≤ d ≤ D) velocity vid and location xid are updated according to the following equation:(1)νidt+1=ω×νidt+c1×r1×Ρidt-χidt+c2×r2×Ρgdt-χidt(2)xidt+1=xidt+vidt+1,where ω is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are random values that vary between [0,1].


### 2.1. Previous Algorithm Defect

The PSO algorithm can easily fall into the local optimal solution in the search process. According to (1), when xid=pid=pgd, the flight speed of the particles depends only on ω and vid. If the values of vid and ω are unequal to zero, then xit+1≠xi, and the particles fly away from the original trajectory. If vid is equal to zero, then vid(t+1) is also equal to zero. Once all the particles achieve pgd, the particle stops flying and converges to the local optimal solution.At each timet, the particles search within a certain range of space under the influence of group information and their own historical experience. When a superior solution is found, the guidance information is updated, a new search is launched, and the optimization process is continued. However, the convergence speed in the late search slows down given that the flight direction of all particles is determined according to their entire experience. After reaching a certain level, the algorithm falls into stagnation, thereby hindering the achievement of an accurate solution.Although some achievements have been made in the convergence analysis of the PSO algorithm, the premise is based on the assumption that the individual and global extremum values are constant. For the relevant conclusions of parameter selection, the proposed parameter values are highly unsatisfactory in practical applications despite the existence of a certain reference role. According to the basic principle of the PSO algorithm, correlations exist among the method parameters, especially the inertia weight and learning factor. Their mutual influence is also highly important. However, this mutual influence is often ignored, which leads to debatable results [25]. The experiment study [26] found that the bigger the value of ω, c1, and c2, the greater the probability of no convergence, and the value of ω has the largest effect. Choosing the appropriate ω, c1, and c2 can thus guarantee the convergence of particles. Therefore, it is necessary to improve the global search ability of the PSO algorithm. The optimization of vehicle collision performance requires an algorithm with a fast running speed to meet the scheduling aging requirements and rapid convergence to the global optimal solution to ensure scheduling accuracy. Therefore, the PSO algorithm needs to be improved such that it eliminates inferior solutions as soon as possible, avoids overiteration near these inferior solutions, and solves the collision performance optimization model [27] with many decision variables and complex scenes.


### 2.2. Algorithm Improved by Chaos Thought

Chaos is a universal nonlinear phenomenon whose behavior is complex and similar to random phenomena but has delicate internal regularity. Optimization search using chaotic variables is superior to blind random search as it can avoid the disadvantage of evolutionary algorithms of falling into the local optimum due to the ergodicity of chaos [28]. A chaotic idea is used to adaptively adjust the parameters related to particle velocity update. When generating chaotic sequences, the following logistic model is used:(3)λit+1=μ×λit×1-λit;i=1,2,…,n,where λit is the value of λi after the chaotic evolution in step t, λi∈0,1, and 1≤μ≤4. When μ=4 and λi is not 0.25, 0.5, and 0.75, the system exhibits complete chaotic characteristics, the generated chaotic sequence shows excellent randomness, and the trajectory of the chaotic variable can traverse the entire search space without repetition. The chaotic optimization of parameters r1 and r2 is as follows:(4)rit+1=4.0×rit×1-ritrit∈0,1.0,i=1,2.The inertia weight ω can balance the global and local searches of the PSO algorithm by controlling the effect of historical speed on the current velocity of the particle. Simultaneously, a suitable ω can diminish the time needed to find the optimal solution. The inertia weight setting for each particle is as follows:(5)ωt=ωmax-ωmax-ωmin×ttmax,where ωmax and ωmin are the maximum and minimum values of the inertia weight, respectively, and t and tmax are the current algebra and maximum number of iterations, respectively [29]. The execution process of chaotic particle swarm optimization algorithm is as follows: Firstly, the chaotic idea is used to dynamically adjust the parameters r1 and r2 of the particle swarm optimization algorithm to produce a good group. Then, in the evolution, formula (1) and formula (2) are used to lead the particle swarm to the optimal solution search.


### 2.3. Performance Test of Improved Algorithm

This study uses four typical mathematical standard test functions to test and evaluate the improved PSO algorithm and thus verify its optimization performance.


#### 2.3.1. Sphere Function

(6) f x = ∑ i = 1 n x i 2 .The function is a unimodal one that is mainly used to test the optimization precision of the algorithm. The search space and global optimal solution are-100≤xi≤100 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 1.Figure 1
3D graphics of Sphere function.


#### 2.3.2. Rosenbrock Function

(7) f x = ∑ i = 1 n - 1 100 x i 2 - x i - 1 2 + x i - 1 2 .This function is a multipeak test one. A particularly narrow trough exists between the local optimum and the global optimal solution. Moreover, the function is commonly used to test the execution performance of the algorithm. The search space and global optimal solution are -10≤xi≤10 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 2.Figure 2
3D graphics of Rosenbrock function.


#### 2.3.3. Rastrigin Function

(8) f x = 100 n + ∑ i = 1 n x i 2 - 10 cos ⁡ 2 π x i .The function is a multipeak test function, and the distribution of minimum values is regular. The search space and global optimal solution are -10≤xi≤10 and fx=0; xi=0. The 3D graphical representation of the function is shown in Figure 3.Figure 3
3D graphics of Rastrigin function.


#### 2.3.4. Griewank Function

(9) f x = 1 4000 ∑ i = 1 n x i 2 + ∏ i = 1 n cos ⁡ x i i + 1 .Similar to the Rastrigin function, the Griewank function has multiple minimum values that exhibit a regular distribution. The search space and global optimal solution are-600≤xi≤600 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 4.Figure 4
3D graphics of Griewank function.The particle swarm and improved PSO algorithms are applied to run the four test functions. The initial parameters of the two algorithms are set as follows: the inertia weight of the particle swarm algorithm is 0.9, and the acceleration constant isc1=c2=1.4. The maximum and minimum inertia weight values of the improved PSO algorithm are ωmax=0.9 and ωmin=0.4, respectively; the acceleration constant is c1=c2=1.4; and μ=4. The two optimization algorithms have 0.5 step size, 2000 iterations, and 10 particles. Each test function is tested 20 times, and the optimal and worst solutions and the average value of the solution are obtained. The performance comparison between the two algorithms under different test functions is shown in Table 1. At the same time, the average value of the fitness function curve obtained in 20 tests was taken to generate a fitness function curve as shown in Figures 5–8.Table 1
Comparison of algorithm performance.
Test function Optimal solution Worstsolution Average solution Improved particle swarm optimization Sphere 3.6×10−5 4.5×10−5 3.9×10−5  Particle swarm optimization Sphere 6.7×10−2 8.2×10−2 7.1×10−2  Improved particle swarm optimization Rosenbrock 1.2×10−1 1.6×10−1 1.4×10−1  Particle swarm optimization Rosenbrock 1.9 2.4 2.0  Improved particle swarm optimization Rastrigin 9.8×10−4 1.7×10−3 1.3×10−3  Particle swarm optimization Rastrigin 9.7×10−2 2.1×10−1 1.8×10−1  Improved particle swarm optimization Griewank 4.2×10−2 9.3×10−2 5.8×10−2  Particle swarm optimization Griewank 4.0×10−1 5.1×10−1 4.8×10−1Figure 5
Test results of Sphere function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 6
Test results of Rosenbrock function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 7
Test results of Rastrigin function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 8
Test results of Griewank function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOThe test results are shown in Figures5–8. For the Sphere function, the improved PSO algorithm presents a remarkable reduction in the fitness value over the PSO algorithm. By the improved algorithm, the fitness value is close to zero, and the convergence speed is slightly increased. For the Rosenbrock function, the performance of the improved PSO algorithm is slightly improved, and the fitness value is diminished. However, the convergence speed of the improved PSO algorithm under this function is not as good as that of the PSO algorithm. For the Rastrigin function, the improved PSO algorithm is excellent in terms of fitness value and convergence speed. In addition, a large degree of improvement is observed. For the Griewank function, the improved PSO algorithm is slightly better than the PSO algorithm in terms of fitness value and convergence speed, but the difference is average. Evidently, for a single optimization, the improved PSO algorithm is better than the PSO algorithm under the four test functions.As shown in Table1, for the four test functions, the optimal and average solutions of the improved PSO algorithm are better than those of the PSO algorithm. Moreover, the difference between the optimal and worst solutions obtained by the improved PSO algorithm is smaller than that obtained by the PSO algorithm. These results indicate that the improved PSO algorithm is more stable than the PSO algorithm is in the optimization process.Based on the four common test functions, the improved PSO algorithm proposed in this study exhibits an improvement in global search ability and search speed more than the PSO algorithm.


## 2.1. Previous Algorithm Defect

The PSO algorithm can easily fall into the local optimal solution in the search process. According to (1), when xid=pid=pgd, the flight speed of the particles depends only on ω and vid. If the values of vid and ω are unequal to zero, then xit+1≠xi, and the particles fly away from the original trajectory. If vid is equal to zero, then vid(t+1) is also equal to zero. Once all the particles achieve pgd, the particle stops flying and converges to the local optimal solution.At each timet, the particles search within a certain range of space under the influence of group information and their own historical experience. When a superior solution is found, the guidance information is updated, a new search is launched, and the optimization process is continued. However, the convergence speed in the late search slows down given that the flight direction of all particles is determined according to their entire experience. After reaching a certain level, the algorithm falls into stagnation, thereby hindering the achievement of an accurate solution.Although some achievements have been made in the convergence analysis of the PSO algorithm, the premise is based on the assumption that the individual and global extremum values are constant. For the relevant conclusions of parameter selection, the proposed parameter values are highly unsatisfactory in practical applications despite the existence of a certain reference role. According to the basic principle of the PSO algorithm, correlations exist among the method parameters, especially the inertia weight and learning factor. Their mutual influence is also highly important. However, this mutual influence is often ignored, which leads to debatable results [25]. The experiment study [26] found that the bigger the value of ω, c1, and c2, the greater the probability of no convergence, and the value of ω has the largest effect. Choosing the appropriate ω, c1, and c2 can thus guarantee the convergence of particles. Therefore, it is necessary to improve the global search ability of the PSO algorithm. The optimization of vehicle collision performance requires an algorithm with a fast running speed to meet the scheduling aging requirements and rapid convergence to the global optimal solution to ensure scheduling accuracy. Therefore, the PSO algorithm needs to be improved such that it eliminates inferior solutions as soon as possible, avoids overiteration near these inferior solutions, and solves the collision performance optimization model [27] with many decision variables and complex scenes.


## 2.2. Algorithm Improved by Chaos Thought

Chaos is a universal nonlinear phenomenon whose behavior is complex and similar to random phenomena but has delicate internal regularity. Optimization search using chaotic variables is superior to blind random search as it can avoid the disadvantage of evolutionary algorithms of falling into the local optimum due to the ergodicity of chaos [28]. A chaotic idea is used to adaptively adjust the parameters related to particle velocity update. When generating chaotic sequences, the following logistic model is used:(3)λit+1=μ×λit×1-λit;i=1,2,…,n,where λit is the value of λi after the chaotic evolution in step t, λi∈0,1, and 1≤μ≤4. When μ=4 and λi is not 0.25, 0.5, and 0.75, the system exhibits complete chaotic characteristics, the generated chaotic sequence shows excellent randomness, and the trajectory of the chaotic variable can traverse the entire search space without repetition. The chaotic optimization of parameters r1 and r2 is as follows:(4)rit+1=4.0×rit×1-ritrit∈0,1.0,i=1,2.The inertia weight ω can balance the global and local searches of the PSO algorithm by controlling the effect of historical speed on the current velocity of the particle. Simultaneously, a suitable ω can diminish the time needed to find the optimal solution. The inertia weight setting for each particle is as follows:(5)ωt=ωmax-ωmax-ωmin×ttmax,where ωmax and ωmin are the maximum and minimum values of the inertia weight, respectively, and t and tmax are the current algebra and maximum number of iterations, respectively [29]. The execution process of chaotic particle swarm optimization algorithm is as follows: Firstly, the chaotic idea is used to dynamically adjust the parameters r1 and r2 of the particle swarm optimization algorithm to produce a good group. Then, in the evolution, formula (1) and formula (2) are used to lead the particle swarm to the optimal solution search.


## 2.3. Performance Test of Improved Algorithm

This study uses four typical mathematical standard test functions to test and evaluate the improved PSO algorithm and thus verify its optimization performance.


### 2.3.1. Sphere Function

(6) f x = ∑ i = 1 n x i 2 .The function is a unimodal one that is mainly used to test the optimization precision of the algorithm. The search space and global optimal solution are-100≤xi≤100 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 1.Figure 1
3D graphics of Sphere function.


### 2.3.2. Rosenbrock Function

(7) f x = ∑ i = 1 n - 1 100 x i 2 - x i - 1 2 + x i - 1 2 .This function is a multipeak test one. A particularly narrow trough exists between the local optimum and the global optimal solution. Moreover, the function is commonly used to test the execution performance of the algorithm. The search space and global optimal solution are -10≤xi≤10 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 2.Figure 2
3D graphics of Rosenbrock function.


### 2.3.3. Rastrigin Function

(8) f x = 100 n + ∑ i = 1 n x i 2 - 10 cos ⁡ 2 π x i .The function is a multipeak test function, and the distribution of minimum values is regular. The search space and global optimal solution are -10≤xi≤10 and fx=0; xi=0. The 3D graphical representation of the function is shown in Figure 3.Figure 3
3D graphics of Rastrigin function.


### 2.3.4. Griewank Function

(9) f x = 1 4000 ∑ i = 1 n x i 2 + ∏ i = 1 n cos ⁡ x i i + 1 .Similar to the Rastrigin function, the Griewank function has multiple minimum values that exhibit a regular distribution. The search space and global optimal solution are-600≤xi≤600 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 4.Figure 4
3D graphics of Griewank function.The particle swarm and improved PSO algorithms are applied to run the four test functions. The initial parameters of the two algorithms are set as follows: the inertia weight of the particle swarm algorithm is 0.9, and the acceleration constant isc1=c2=1.4. The maximum and minimum inertia weight values of the improved PSO algorithm are ωmax=0.9 and ωmin=0.4, respectively; the acceleration constant is c1=c2=1.4; and μ=4. The two optimization algorithms have 0.5 step size, 2000 iterations, and 10 particles. Each test function is tested 20 times, and the optimal and worst solutions and the average value of the solution are obtained. The performance comparison between the two algorithms under different test functions is shown in Table 1. At the same time, the average value of the fitness function curve obtained in 20 tests was taken to generate a fitness function curve as shown in Figures 5–8.Table 1
Comparison of algorithm performance.
Test function Optimal solution Worstsolution Average solution Improved particle swarm optimization Sphere 3.6×10−5 4.5×10−5 3.9×10−5  Particle swarm optimization Sphere 6.7×10−2 8.2×10−2 7.1×10−2  Improved particle swarm optimization Rosenbrock 1.2×10−1 1.6×10−1 1.4×10−1  Particle swarm optimization Rosenbrock 1.9 2.4 2.0  Improved particle swarm optimization Rastrigin 9.8×10−4 1.7×10−3 1.3×10−3  Particle swarm optimization Rastrigin 9.7×10−2 2.1×10−1 1.8×10−1  Improved particle swarm optimization Griewank 4.2×10−2 9.3×10−2 5.8×10−2  Particle swarm optimization Griewank 4.0×10−1 5.1×10−1 4.8×10−1Figure 5
Test results of Sphere function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 6
Test results of Rosenbrock function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 7
Test results of Rastrigin function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 8
Test results of Griewank function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOThe test results are shown in Figures5–8. For the Sphere function, the improved PSO algorithm presents a remarkable reduction in the fitness value over the PSO algorithm. By the improved algorithm, the fitness value is close to zero, and the convergence speed is slightly increased. For the Rosenbrock function, the performance of the improved PSO algorithm is slightly improved, and the fitness value is diminished. However, the convergence speed of the improved PSO algorithm under this function is not as good as that of the PSO algorithm. For the Rastrigin function, the improved PSO algorithm is excellent in terms of fitness value and convergence speed. In addition, a large degree of improvement is observed. For the Griewank function, the improved PSO algorithm is slightly better than the PSO algorithm in terms of fitness value and convergence speed, but the difference is average. Evidently, for a single optimization, the improved PSO algorithm is better than the PSO algorithm under the four test functions.As shown in Table1, for the four test functions, the optimal and average solutions of the improved PSO algorithm are better than those of the PSO algorithm. Moreover, the difference between the optimal and worst solutions obtained by the improved PSO algorithm is smaller than that obtained by the PSO algorithm. These results indicate that the improved PSO algorithm is more stable than the PSO algorithm is in the optimization process.Based on the four common test functions, the improved PSO algorithm proposed in this study exhibits an improvement in global search ability and search speed more than the PSO algorithm.


## 2.3.1. Sphere Function

(6) f x = ∑ i = 1 n x i 2 .The function is a unimodal one that is mainly used to test the optimization precision of the algorithm. The search space and global optimal solution are-100≤xi≤100 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 1.Figure 1
3D graphics of Sphere function.


## 2.3.2. Rosenbrock Function

(7) f x = ∑ i = 1 n - 1 100 x i 2 - x i - 1 2 + x i - 1 2 .This function is a multipeak test one. A particularly narrow trough exists between the local optimum and the global optimal solution. Moreover, the function is commonly used to test the execution performance of the algorithm. The search space and global optimal solution are -10≤xi≤10 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 2.Figure 2
3D graphics of Rosenbrock function.


## 2.3.3. Rastrigin Function

(8) f x = 100 n + ∑ i = 1 n x i 2 - 10 cos ⁡ 2 π x i .The function is a multipeak test function, and the distribution of minimum values is regular. The search space and global optimal solution are -10≤xi≤10 and fx=0; xi=0. The 3D graphical representation of the function is shown in Figure 3.Figure 3
3D graphics of Rastrigin function.


## 2.3.4. Griewank Function

(9) f x = 1 4000 ∑ i = 1 n x i 2 + ∏ i = 1 n cos ⁡ x i i + 1 .Similar to the Rastrigin function, the Griewank function has multiple minimum values that exhibit a regular distribution. The search space and global optimal solution are-600≤xi≤600 and fx=0; xi=0, respectively. The 3D graphical representation of the function is shown in Figure 4.Figure 4
3D graphics of Griewank function.The particle swarm and improved PSO algorithms are applied to run the four test functions. The initial parameters of the two algorithms are set as follows: the inertia weight of the particle swarm algorithm is 0.9, and the acceleration constant isc1=c2=1.4. The maximum and minimum inertia weight values of the improved PSO algorithm are ωmax=0.9 and ωmin=0.4, respectively; the acceleration constant is c1=c2=1.4; and μ=4. The two optimization algorithms have 0.5 step size, 2000 iterations, and 10 particles. Each test function is tested 20 times, and the optimal and worst solutions and the average value of the solution are obtained. The performance comparison between the two algorithms under different test functions is shown in Table 1. At the same time, the average value of the fitness function curve obtained in 20 tests was taken to generate a fitness function curve as shown in Figures 5–8.Table 1
Comparison of algorithm performance.
Test function Optimal solution Worstsolution Average solution Improved particle swarm optimization Sphere 3.6×10−5 4.5×10−5 3.9×10−5  Particle swarm optimization Sphere 6.7×10−2 8.2×10−2 7.1×10−2  Improved particle swarm optimization Rosenbrock 1.2×10−1 1.6×10−1 1.4×10−1  Particle swarm optimization Rosenbrock 1.9 2.4 2.0  Improved particle swarm optimization Rastrigin 9.8×10−4 1.7×10−3 1.3×10−3  Particle swarm optimization Rastrigin 9.7×10−2 2.1×10−1 1.8×10−1  Improved particle swarm optimization Griewank 4.2×10−2 9.3×10−2 5.8×10−2  Particle swarm optimization Griewank 4.0×10−1 5.1×10−1 4.8×10−1Figure 5
Test results of Sphere function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 6
Test results of Rosenbrock function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 7
Test results of Rastrigin function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOFigure 8
Test results of Griewank function.
(a)
Optimal individual fitness for improved PSO algorithm (b)
Optimal individual fitness for PSOThe test results are shown in Figures5–8. For the Sphere function, the improved PSO algorithm presents a remarkable reduction in the fitness value over the PSO algorithm. By the improved algorithm, the fitness value is close to zero, and the convergence speed is slightly increased. For the Rosenbrock function, the performance of the improved PSO algorithm is slightly improved, and the fitness value is diminished. However, the convergence speed of the improved PSO algorithm under this function is not as good as that of the PSO algorithm. For the Rastrigin function, the improved PSO algorithm is excellent in terms of fitness value and convergence speed. In addition, a large degree of improvement is observed. For the Griewank function, the improved PSO algorithm is slightly better than the PSO algorithm in terms of fitness value and convergence speed, but the difference is average. Evidently, for a single optimization, the improved PSO algorithm is better than the PSO algorithm under the four test functions.As shown in Table1, for the four test functions, the optimal and average solutions of the improved PSO algorithm are better than those of the PSO algorithm. Moreover, the difference between the optimal and worst solutions obtained by the improved PSO algorithm is smaller than that obtained by the PSO algorithm. These results indicate that the improved PSO algorithm is more stable than the PSO algorithm is in the optimization process.Based on the four common test functions, the improved PSO algorithm proposed in this study exhibits an improvement in global search ability and search speed more than the PSO algorithm.


## 3. Multiobjective Optimization of Minibus Collision

### 3.1. Design Variables and Responses Functions

A finite element model of a 100% frontal overlap deformable barrier collision of a minibus is shown in Figure9. According to C-NCAP’s new vehicle evaluation index, under the premise of meeting the crash performance of the whole vehicle, the front part of the vehicle body should be crushed and fully deformed during the collision process so that substantial collision kinetic energy is absorbed and the acceleration peak at the lower end of the B-pillar is effectively decreased [30, 31]. Therefore, the thickness of the eight sheets in the front part of the vehicle body is selected as the design variable [32, 33], as shown in Figure 10. The initial values of the design variable are provided in Table 2.Table 2
Initial values of design variable.
Design variable t1 t2 t3 t4 t5 t6 t7 t8  Initial value (mm) 1.10 1.25 1.10 1.10 1.65 1.63 0.89 0.70Figure 9
Finite element model of 100% frontal collision of minibuses.Figure 10
Design variables.Two minimum objective functions are set to reduce collision acceleration and body mass, including acceleration peakamax at the lower end of B-pillar and mass m of the vehicle. In order to control the damage of vehicle body deformation to the occupant’s chest, head, foot, and other parts, the response function at relevant positions is restrained within acceptable level, such as the intrusion D1, D2, and D3 separately at the instrument panel tube beam, the steering column hole, and the lower dash panel clutch pedal [34, 35].


### 3.2. Approximate Model

This study uses the Latin hypercube test method [36] to obtain 70 sets of training data combined with the design variables and crash performance indicators of the vehicle finite element model (Table 3). On the basis of these data, this study establishes the Kriging approximation model [37]; in order to verify the prediction accuracy of the approximate model, 20 sample points were randomly selected in Table 3, and the accuracy of the approximate model of the established 100% frontal collision model was evaluated based on the results obtained from these sample points. The evaluation results show that the established approximate model satisfies the evaluation index of each response value of the 100% positive overlap collision Kriging approximation model (such as the average relative error RAAE ≥ 95% and coefficient of determination R2 ≤ 15%). It is enough to replace the original finite element model and carry out the next multiobjective particle swarm optimization design [38–40].Table 3
Training data.
(a)
Design variable test
Number of groups t1/mm t2/mm t3/mm t4/mm t5/mm t6/mm t7/mm t8/mm 1 0.66 1.39 1.84 0.69 1.13 1.05 1.51 1.46  2 0.68 1.41 0.99 1.14 0.92 0.81 1.44 1.10  3 0.72 1.20 1.47 1.17 1.73 0.77 1.41 1.25  … … ... ... ... ... ... ... ...  70 2.00 1.17 0.68 2.00 1.52 1.81 0.75 1.89  (b)
Response test design
Number of groups a m a x/(m/s2) m/kg D1/mm D2/mm D3/mm 1 54.8 717.5 86.5 240.0 230.0  2 52.9 716.9 84.5 209.5 210.0  3 55.3 717.7 92.1 246.2 237.7  … ... ... ... ... ...  70 46.5 717.4 96.5 261.8 262.0


### 3.3. Multiobjective Optimization Design

In the multiobjective optimization design, the acceleration peak at the lower end of the B-pillar and intrusion amount of the key part should be kept as small as possible, and the vehicle weight [41] should be reduced. The mathematical model expression of the multiobjective optimization design is as follows:(10)min⁡amaxt1,t2,t3…t8min⁡mt1,t2,t3…t8D1t1,t2,t3…t8≤97.6mmD2t1,t2,t3…t8≤260mmD3t1,t2,t3…t8≤268.4mmt1,t2,t3…t8∈1.1,1.25,1.1,1.1,1.65,1.63,0.89,0.7.The PSO algorithm has an inertia weight of 0.9 and an acceleration constant ofc1=c2=1.4. The optimization algorithm has 0.5 step size, 2000 iterations, and 10 particles. The maximum and minimum inertia weight values of the improved PSO algorithm are ωmax=0.9 and ωmin=0.4, respectively; the acceleration constant is c1=c2=1.4; and μ=4. The improved PSO algorithm also has 0.5 step size, 2000 iterations, and 10 particles. The approximate model is optimized, and the results are compared, as shown in Table 4.Table 4
Comparison of optimization results.
(a)
Design variable
t1/mm t2/mm t3/mm t4/mm t5/mm t6/mm t7/mm t8/mm Original finite element model 1.10 1.25 1.10 1.10 1.65 1.63 0.89 0.70  Particle swarm optimization 1.20 1.00 1.30 0.90 1.10 1.30 0.92 0.51  Improved particle swarm optimization 1.15 1.02 1.23 0.92 0.96 1.30 0.90 0.49  (b)
Collision target response
a m a x /(m/s2) D1/mm D2/mm D3/mm m/kg Original finite element model 57.60 97.60 260.60 268.40 716.85  Particle swarm optimization 53.20 90.10 252.40 254.20 710.61  Improved particle swarm optimization 47.90 84.70 241.60 242.10 710.40  Particle swarm optimization percentage -7.64% -7.68% -3.15% -5.29% -0.87%  Improved particle swarm optimization percentage -16.84% -13.22% -7.29% -9.80% -0.90%As shown in Table4, the approximate model is optimized by the PSO algorithm. The results of the collision show that although the collision evaluation parameters have different degrees of improvement, they are still not ideal. The optimized vehicle acceleration and key parts of intrusion will still cause considerable damage to the passengers inside the vehicle. After the optimization of the improved PSO algorithm for the approximate model, the peak acceleration at the lower end of the B-pillar is decreased to 47.9 m/s2. The intrusion amount D1 of the instrument panel tube beam is decreased from 97.6mm to 84.7mm. Moreover, the backward intrusion amount D2 of the steering column hole is decreased from 260.6mm to 241.6mm, and the lower front panel clutch pedal is inwardly invaded. The amount of backward intrusion D3 of the lower front panel clutch pedal is decreased from 268.4mm to 242.1mm, and the entire vehicle quality is diminished from 716.85kg to 710.4kg. Relative to the PSO algorithm, the improved PSO has a remarkable improvement in terms of the effect of the performance evaluation index. This feature is in accordance with the expected results. After comprehensive consideration, this study finds that the improved PSO algorithm is better than the PSO algorithm for the Kriging approximation model. Moreover, the improved PSO algorithm is used as the optimization scheme for the 100% front collision of the car. The fitting effect of each response value after optimization is shown in Table 5. It can be seen that the fitting errors of each evaluation index are small. The approximate model prediction accuracy meets the requirements, and the obtained optimization result can represent the optimization result of the original finite element model.Table 5
Optimized response fitting effect table.
Evaluation index Approximate model prediction Optimized finite element results D1 84.7mm 86.2mm D2 241.6mm 243.3mm D3 242.1mm 243.4mm a m a x 47.9m/s2 48.5m/s2 m 710.4kg 710.41kg


## 3.1. Design Variables and Responses Functions

A finite element model of a 100% frontal overlap deformable barrier collision of a minibus is shown in Figure9. According to C-NCAP’s new vehicle evaluation index, under the premise of meeting the crash performance of the whole vehicle, the front part of the vehicle body should be crushed and fully deformed during the collision process so that substantial collision kinetic energy is absorbed and the acceleration peak at the lower end of the B-pillar is effectively decreased [30, 31]. Therefore, the thickness of the eight sheets in the front part of the vehicle body is selected as the design variable [32, 33], as shown in Figure 10. The initial values of the design variable are provided in Table 2.Table 2
Initial values of design variable.
Design variable t1 t2 t3 t4 t5 t6 t7 t8  Initial value (mm) 1.10 1.25 1.10 1.10 1.65 1.63 0.89 0.70Figure 9
Finite element model of 100% frontal collision of minibuses.Figure 10
Design variables.Two minimum objective functions are set to reduce collision acceleration and body mass, including acceleration peakamax at the lower end of B-pillar and mass m of the vehicle. In order to control the damage of vehicle body deformation to the occupant’s chest, head, foot, and other parts, the response function at relevant positions is restrained within acceptable level, such as the intrusion D1, D2, and D3 separately at the instrument panel tube beam, the steering column hole, and the lower dash panel clutch pedal [34, 35].


## 3.2. Approximate Model

This study uses the Latin hypercube test method [36] to obtain 70 sets of training data combined with the design variables and crash performance indicators of the vehicle finite element model (Table 3). On the basis of these data, this study establishes the Kriging approximation model [37]; in order to verify the prediction accuracy of the approximate model, 20 sample points were randomly selected in Table 3, and the accuracy of the approximate model of the established 100% frontal collision model was evaluated based on the results obtained from these sample points. The evaluation results show that the established approximate model satisfies the evaluation index of each response value of the 100% positive overlap collision Kriging approximation model (such as the average relative error RAAE ≥ 95% and coefficient of determination R2 ≤ 15%). It is enough to replace the original finite element model and carry out the next multiobjective particle swarm optimization design [38–40].Table 3
Training data.
(a)
Design variable test
Number of groups t1/mm t2/mm t3/mm t4/mm t5/mm t6/mm t7/mm t8/mm 1 0.66 1.39 1.84 0.69 1.13 1.05 1.51 1.46  2 0.68 1.41 0.99 1.14 0.92 0.81 1.44 1.10  3 0.72 1.20 1.47 1.17 1.73 0.77 1.41 1.25  … … ... ... ... ... ... ... ...  70 2.00 1.17 0.68 2.00 1.52 1.81 0.75 1.89  (b)
Response test design
Number of groups a m a x/(m/s2) m/kg D1/mm D2/mm D3/mm 1 54.8 717.5 86.5 240.0 230.0  2 52.9 716.9 84.5 209.5 210.0  3 55.3 717.7 92.1 246.2 237.7  … ... ... ... ... ...  70 46.5 717.4 96.5 261.8 262.0


## 3.3. Multiobjective Optimization Design

In the multiobjective optimization design, the acceleration peak at the lower end of the B-pillar and intrusion amount of the key part should be kept as small as possible, and the vehicle weight [41] should be reduced. The mathematical model expression of the multiobjective optimization design is as follows:(10)min⁡amaxt1,t2,t3…t8min⁡mt1,t2,t3…t8D1t1,t2,t3…t8≤97.6mmD2t1,t2,t3…t8≤260mmD3t1,t2,t3…t8≤268.4mmt1,t2,t3…t8∈1.1,1.25,1.1,1.1,1.65,1.63,0.89,0.7.The PSO algorithm has an inertia weight of 0.9 and an acceleration constant ofc1=c2=1.4. The optimization algorithm has 0.5 step size, 2000 iterations, and 10 particles. The maximum and minimum inertia weight values of the improved PSO algorithm are ωmax=0.9 and ωmin=0.4, respectively; the acceleration constant is c1=c2=1.4; and μ=4. The improved PSO algorithm also has 0.5 step size, 2000 iterations, and 10 particles. The approximate model is optimized, and the results are compared, as shown in Table 4.Table 4
Comparison of optimization results.
(a)
Design variable
t1/mm t2/mm t3/mm t4/mm t5/mm t6/mm t7/mm t8/mm Original finite element model 1.10 1.25 1.10 1.10 1.65 1.63 0.89 0.70  Particle swarm optimization 1.20 1.00 1.30 0.90 1.10 1.30 0.92 0.51  Improved particle swarm optimization 1.15 1.02 1.23 0.92 0.96 1.30 0.90 0.49  (b)
Collision target response
a m a x /(m/s2) D1/mm D2/mm D3/mm m/kg Original finite element model 57.60 97.60 260.60 268.40 716.85  Particle swarm optimization 53.20 90.10 252.40 254.20 710.61  Improved particle swarm optimization 47.90 84.70 241.60 242.10 710.40  Particle swarm optimization percentage -7.64% -7.68% -3.15% -5.29% -0.87%  Improved particle swarm optimization percentage -16.84% -13.22% -7.29% -9.80% -0.90%As shown in Table4, the approximate model is optimized by the PSO algorithm. The results of the collision show that although the collision evaluation parameters have different degrees of improvement, they are still not ideal. The optimized vehicle acceleration and key parts of intrusion will still cause considerable damage to the passengers inside the vehicle. After the optimization of the improved PSO algorithm for the approximate model, the peak acceleration at the lower end of the B-pillar is decreased to 47.9 m/s2. The intrusion amount D1 of the instrument panel tube beam is decreased from 97.6mm to 84.7mm. Moreover, the backward intrusion amount D2 of the steering column hole is decreased from 260.6mm to 241.6mm, and the lower front panel clutch pedal is inwardly invaded. The amount of backward intrusion D3 of the lower front panel clutch pedal is decreased from 268.4mm to 242.1mm, and the entire vehicle quality is diminished from 716.85kg to 710.4kg. Relative to the PSO algorithm, the improved PSO has a remarkable improvement in terms of the effect of the performance evaluation index. This feature is in accordance with the expected results. After comprehensive consideration, this study finds that the improved PSO algorithm is better than the PSO algorithm for the Kriging approximation model. Moreover, the improved PSO algorithm is used as the optimization scheme for the 100% front collision of the car. The fitting effect of each response value after optimization is shown in Table 5. It can be seen that the fitting errors of each evaluation index are small. The approximate model prediction accuracy meets the requirements, and the obtained optimization result can represent the optimization result of the original finite element model.Table 5
Optimized response fitting effect table.
Evaluation index Approximate model prediction Optimized finite element results D1 84.7mm 86.2mm D2 241.6mm 243.3mm D3 242.1mm 243.4mm a m a x 47.9m/s2 48.5m/s2 m 710.4kg 710.41kg


## 4. Conclusion

( 1 ) In this study, the PSO algorithm is improved by dealing with the method's various limitations, such as slow convergence speed, low precision, and tendency to easily fall into the local extremum. Then, the chaos idea is introduced, and the parameters related to particle velocity update are adaptively adjusted by using chaos thought to generate chaotic sequences. Four commonly used mathematical test functions with different characteristics are used to verify the performance of the improved PSO algorithm. The results show that the improved PSO algorithm has stronger optimization ability than the PSO algorithm.( 2 ) The PSO and improved PSO algorithms are applied to vehicle collision optimization. According to the comparison results, the evaluation index of the collision performance after optimization by the PSO algorithm improves, but the effect is unremarkable. This result indicates the need for further optimization. After using the improved PSO algorithm, the improvement effect of the vehicle crash performance evaluation index is remarkable. Therefore, the optimized design of the improved PSO algorithm is used as the final optimization method, thereby providing a new design for car crash optimization.



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*Source: 8164609-2019-03-17.xml*