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Multiobjective Optimization of Carbon Fiber-Reinforced Plastic Composite Bumper Based on Adaptive Genetic Algorithm.

1. Introduction

People nowadays have the increasing awareness of energy conservation and emission reduction and focus more on passive safety performance of cars in collisions. Therefore, the crashworthiness and lightweight design of automobile body structure are regarded as one of the most important parts of vehicle design. Substantial researchers devoted themselves to conduct research studies on improvement of crashworthiness and reduction weight of vehicles. To sum up, there are mainly three ways to achieve these goals: adjustment of material usage, improvement of manufacturing processes, and optimization design of vehicle structure. It is proved that the combination of material adjustment and optimized design of body structure is the most effective way to improve the crashworthiness and realize lightweight [1, 2].

In the latest research studies, carbon fiber-reinforced plastic (CFRP) composites have been increasingly utilized instead of conventional metallic materials in the automotive industry due to their advantageous properties like high strength, high stiffness, lightweight, and excellent energy absorption [3-8]. Kiran et al. proposed the method of using composite instead of metallic materials in automobile chassis and analyzed the static structure of equivalent stress and displacement of composite structural parts [9]. Kim et al. designed carbon fiber-reinforced composites and hybrid fiberglass-reinforced composite hood by finite element analysis and proved that composite hood had better performance in crashworthiness and weight reduction than traditional steel and aluminum hood [4].

As the main load-bearing and energy-absorbing part in the low-speed collision of the car, the bumper system plays a vital role in protecting other parts of the car and the safety of the occupants [4, 10-12]. Therefore, it is of great significance to make the optimization design of the car bumper. Combining genetic algorithm, linear perturbation eigenvalue analysis method, and static RIKS analysis technique, Kim et al. designed a carbon fiber-reinforced composite automotive lower arm, which is characterized by twice the stiffness and buckling strength of the steel lower arm and 50% decline in weight [13]. To find out the suitable material of bumper beam, Kim et al. used hybrid glass/carbon fiber-reinforced composite that had 33% lighter weight and better crashworthiness to replace conventional glass mat-reinforced thermoplastic (GMT) [13].

Recently, the integration of finite element analysis and optimization algorithms has been an advanced way to improve the crashworthiness and lightweight performance of the automotive body structure [14-17]. Fang et al. formed a complete optimized design flow that can be utilized as references by introducing DOE (design of experiment), surrogate model, and multiobjective optimization analysis methods into the crashworthiness and lightweight design of body parts [18, 19]. The selection of optimization algorithm will dominate the quality and accuracy of the result, so a variety of intelligent algorithms are proposed by many researchers [11, 20]; i.e., Sun et al. presented a novel optimization algorithm named ICGA (integer-coded genetic algorithm) and applied it on the configurational optimization of multicell topologies [21]. It can be concluded that increasing intelligent algorithms are applied to multiobjective optimization design of the vehicle, such as particle swarm optimization (PSO), genetic algorithms, and simulated annealing algorithms. The genetic algorithm is one of the most common algorithms, but its limitations, i.e., low convergence speed and premature convergence to the local optimum, are often reported in research studies [22, 23]. Therefore, the improvement of GA is necessary and significant.

In this study, design optimization of the carbon fiber-reinforced composite bumper is carried out with the combination of finite element analysis, the Kriging surrogate model, and multiobjective optimization. The carbon fiber stacking angle sequence is determined to be the optimal design variables, with the purpose of improving the crashworthiness and lightweight performance of CFRP composite bumper. For the multiobjective optimization problem, the selection of the algorithm is the most critical, so an adaptive genetic algorithm (AGA) is presented to generate the best possible design of the carbon fiber stacking angle sequence.

2. Adaptive Genetic Algorithm

2.1. Limitations and Improvement of GA. Genetic algorithm (GA) is widely employed to search the global optimal solution in optimization problems; however, it has some defects in solving complicated optimization problem [24], such as the premature convergence to the local optimum and much time consuming. The crossover probability [P.sub.c] and mutation probability [P.sub.m] greatly determine the quality of solutions and the convergence speed of GA [25]. The moderately large values of [P.sub.c] promote the extensive recombination of schemata, while small values of [P.sub.m] are necessary to prevent the disruption of the solutions. Instead of fixed values of [P.sub.c] and [P.sub.m], AGA utilizes the population information in each iteration and adaptively adjusts the [P.sub.c] and [P.sub.m] to maintain the population diversity as well as to sustain the convergence capacity. And the roulette wheel selection method is employed as the selection operator.

It is essential to identify whether the GA is converging to an optimum. The difference in the average and maximum fitness values of the population [26], [F.sub.max] - [F.sub.avg], is used to be a yardstick for detecting the convergence of GA. And when GA converges to a local optimum, the value of [F.sub.max] - [F.sub.avg] decreases. Therefore, the values of [P.sub.c] and [P.sub.m] are supposed to be varied adaptively depending on the value of [F.sub.max] - [F.sub.avg] to prevent premature convergence of the GA to local optimum. In addition, to preserve the "good" solutions of every populations, lower values of [P.sub.c] and [P.sub.m] should be set for high fitness solutions and higher values of [P.sub.c] and [P.sub.m] should be set for low fitness solutions [27, 28]. While the high fitness solutions aid in the convergence of the GA, the low fitness solutions prevent the GA from getting stuck at a local optimum. So the values of [P.sub.m] should depend on the fitness values F of the solution too. Similarly, the values of [P.sub.m] should also depend on the fitness values F' of both the parent solutions. The closer the F(F') is to [F.sub.max], the smaller the [P.sub.c]([P.sub.m]) should be. From the above, the adaptive value of [P.sub.c] and [P.sub.m] can be expressed as follows:

[mathematical expression not reproducible], (1)

[mathematical expression not reproducible], (2)

where [P.sub.c1] and [P.sub.m1] are less than 1.0 to constrain [P.sub.c] and [P.sub.m] to the range (0.0 - 1.0), F is the individual fitness, F' is larger of the fitness values of the solutions to be crossed, and [F.sub.avg] and [F.sub.max] is the average and maximum fitness values of the population, respectively.

However, the adapting crossover probability [P.sub.c] and mutation probability [P.sub.m] defined by equations (1) and (2) have a drawback: if F(F') = [F.sub.max], the values of [P.sub.c] and [P.sub.m] are zero that will lead to premature convergence to the local optimum. As an effective method to overcome this drawback, improved equations to calculate [P.sub.c] and [P.sub.m] are proposed in this paper; they can be corrected as

[mathematical expression not reproducible], (3)

where [P.sub.c2] and [P.sub.m2] are the added parameters that can avoid the condition of [P.sub.c] = [P.sub.m] = 0, which will avoid premature convergence to the local optimum. The process of adaptive algorithm is depicted in Figure 1.

2.2. Performance Test of Adaptive Genetic Algorithm. In this study, two typical test functions are employed to test and evaluate the adaptive genetic algorithm and genetic algorithm and thus verify their optimization performance.

2.2.1. Schaffer Function.

Z = 0.5 - [[(sin [square root of ([x.sup.2] + [y.sup.2])]).sup.2] - 0.5]/ [(1 + 0.001([x.sup.2] + [y.sup.2])).sup.2]. (4)

The function is a multipeak function with multiple local maxima, which is mainly utilized to test the optimal accuracy of the algorithm [29]. The function can be expressed as equation (4), and then, the optimal problem for the test is constructed as follows:

[mathematical expression not reproducible]. (5)

The global maximum of Schaffer function is f(0, 0) = 1, and there are countless local maxima at the value of 0.99028 around the global maximum. The graphical representation of function and contour line is shown in Figure 2.

2.2.2. Rosenbrock Function.

f(x) = [n-1.summation over (i=1)] (100 [([x.sup.2.sub.i] - [x.sub.i- 1]).sup.2] + [([x.sub.i] - 1).sup.2]). (6)

This function is a nonconvex function, which is widely used as a performance test function for optimization algorithms [30]. The function can be expressed as equation (4), and then, the optimal problem for the test is constructed as follows:

[mathematical expression not reproducible]. (7)

The global minimum is inside a long, narrow, parabolic-shaped flat valley. To find the valley while trivial and to converge to the global minimum are also difficult. The graphical representation of function is shown in Figure 3.

The genetic algorithm and adaptive genetic algorithm are tested by the Schaffer function, respectively, for comparison. To provide a fair comparison and test the robustness of the AGA and GA, the population size is set at 10, 15, and 20, respectively, and the maximum number of iterations for two algorithms is fixed at 100. Referring to the previous research [26], the values of [P.sub.c] and [P.sub.m] can be set as 1 and 0.5 for GA, while the values of [P.sub.c1] and [P.sub.m1] are also set as 1 and 0.5 for AGA. And the values of [P.sub.c2] and [P.sub.m2] are 0.5 and 0.01 to constrain the [P.sub.c] and [P.sub.m] to the best range [31]. Each test function is tested 30 times in MATLAB, and then, the optimal, worst, and average values of fitness are obtained in AGA and GA. The performance comparison between AGA and GA under different test functions is shown in Table 1, and the fitness curves are plotted in Figures 4 and 5.

From Table 1 and Figures 4 and 5, the performance of AGA and GA can be concluded as follows:

(1) In terms of Schaffer function, both AGA and GA can jump to the local optimal solution and reach the global optimal solution at value of 1, but the convergence speed of AGA is faster than GA

(2) For Rosenbrock function, the ability of searching global optimal solutions of AGA is better than GA; however, the convergence speed of AGA and GA is close

All in all, compared with the GA, the novel AGA proposed in this paper improves in both convergence speed and the ability to search global optimal solution.

3. Finite Element Model

In the whole process of a car frontal collision, the deformation of the car mainly concentrated in the front cabin while the deformation of the components behind a-pillar is small or almost no deformation occurs. Therefore, the finite element model of the body-in-white car is built as a reduced model, which uses a particle to replace the inessential components behind a-pillar. The size of the mesh is vital to the accuracy of CAE [32]; therefore, the element size of 5 x 5 mm is determined to be sufficient for the front cabin, because this part has a great influence on crashworthiness, and 10 x 10 mm is determined to the other parts. After preprocessing in Hypermesh software, the FE model is modeled by quadrilateral shell elements and the total number of elements is 226503, in which the triangular elements only account for 3.8%, and it is shown in Figure 6.

Nonlinear finite element Ls-Dyna is utilized to simulate the crash behavior of automobile in this study. In the initial model, aluminum is used as the material of the bumper beam. Based on the latest C-NCAP (China-New Car Assessment Program) [33], the collision velocity of the car is determined to be 50 km/h, the acceleration is 9.8 m/[s.sup.2], and the total time is 0.12 s.

Then, the material of bumper beam is replaced by the GFRP composite. The bumper beam is the main frontal part of automotive structure during a frontal crash and is generally subjected to bending load. Therefore, stiffness of the initial bumper beam is supposed to be the same as the GFRP bumper beam [34], and the equal stiffness equation can be expressed as equation (9):

[E.sub.a][I.sub.a] = [E.sub.l][E.sub.l], (8)

where [I.sub.l] and [E.sub.l] are the inertia moment and elastic modulus of CFRP composite bumper, respectively, while the [I.sub.a] and [E.sub.a] are the inertia moment and elastic modulus of the aluminum bumper. And I can be calculated by

I = b[h.sup.3]/12. (9)

Then, the thickness of CFRP composite bumper can be calculated by

[h.sub.l] = [h.sub.a] [cube root of ([E.sub.a]/[E.sub.l])], (10)

where [h.sub.l] is the thickness of the CFRP composite bumper and [h.sub.a] is the thickness of the aluminum bumper.

The thickness of the aluminum bumper beam is 1.80 mm; therefore, the thickness of the CFRP bumper beam is 1.92 mm calculated by using equation (10). In Hypermesh software, the material model 54 (CFRP) is selected instead of aluminum in the bumper beam. The failure criteria of CFRP were introduced in the research [35]. After replacing the aluminum bumper beam by the CFRP bumper beam, the weight of the bumper beam is reduced from 3.790 kg to 2.153 kg that achieves 43.19% weight reduction. And the material properties of CFRP are listed in Table 2 [36].

4. Multiobjective Optimization of CFRP Composite Bumper

4.1. Definition of the Optimization Problem. The structures of composite have great flexibility in design, because they can be customized in different ways to meet specific design requirements. By selecting the best parameters such as fiber and matrix material, fiber orientation, or layer thickness, the fatigue strength of the laminated composite can be significantly improved, thereby improving the structural properties [37, 38]. The CFRP composite studied in this research can be regarded as a multidirectional laminate plate composed by a single-layer plate that is laminated in different directions. The microstructure of CFRP can be shown in Figure 7.

The stiffness of CFRP is greatly affected by the different stacking angle sequences [39, 40]; therefore, further optimization is carried out to find the optimal stacking angle sequence, with the purpose of improving the crashworthiness and reducing the quality of bumper.

4.1.1. Design Variables. The thickness of the bumper beam of the above model is 1.92 mm, and CFRP laminates are designed as symmetric based on the center of the laminate and composed of ten layers in total, so the thickness of each ply is 0.192 mm. Then, the stacking angle of each layer is determined to be the design variables; therefore, five independent design variables can be considered as [[x.sub.1]/[x.sub.2]/[x.sub.3]/ [x.sub.4]/[x.sub.5]]s, which represent the centrosymmetric stacking angle sequence of CFRP laminates. And the value of design variable x can be taken as [0/[+ or -] 45/90] based on the global coordinates. The schematic diagram of 10-plies CFRP based on the global coordinates is vividly depicted in Figure 8.

4.1.2. Constraints. The peak impact force F, the intrusion L, and the energy absorption Q of bumper beam are considered to be the constraints of optimization design, and the values of them are supposed to be lower than those of initial aluminum model.

4.1.3. Objectives. According to C-NCAP, upward displacement automotive steering column [S.sub.1] and rearward displacement automotive steering column [S.sub.2] can be used to evaluate the crashworthiness and they are shown in Figure 9. And the purpose of optimization design is to minimize them.

The optimization problem can be defined mathematically as follows:

[mathematical expression not reproducible]. (11)

4.2. Kriging Model. Design optimization of crashworthiness and lightweight is a nonlinear problem that usually costs much time. In the optimization problem above, the objective function and constraint function are complicated nonlinear function. As an effective approach, the Kriging surrogate model is employed to represent multimodal and nonlinear functions in this paper [41-43]. It can be expressed as

y(x) = f(x) + Z(x), (12)

where f(x) is a constant approximate model, and it is similar to the polynomial response surface that provides design space for a global model and Z(x) is a stochastic process with mean zero and variance [[sigma].sup.2] and it creates localized deviations to assist the Kriging model generating interpolation among sample points. The covariance matrix of Z(x) is expressed as

Cov[Z([x.sup.i]), Z([x.sup.j])] = [[sigma].sup.2] [R ([x.sup.i], [x.sup.j])], (13)

where R([x.sup.i], [x.sup.j]) is a correlation matrix of Gauss-Markov theorem.

In this study, the Kriging surrogate model is used to approximate F, L, Q, [S.sub.1], and [S.sub.2] functions for each case. Design of experiment (DOE), as a widely used technique to address how to select the minimum number of training points for mapping the entire design space properly, is the first step of surrogate modeling technique. The optimal Latin hypercube sampling (OLHS) [44] can efficiently generate uniformly distributed sampling points, so it is applied to generate 120 baseline designs. Then, the F, L, Q, [S.sub.1], and [S.sub.2] at these training points are calculated by Ls-Dyna software. All the calculation results are based on the global coordinate. Based on the simulation results, the Kriging models of the above optimal problem are constructed. To calculate easily, 1, 2, 3, and 4 are substituted for 0[degrees], -45[degrees], 45[degrees], and 90[degrees]. The 120 baseline designs are shown in Table 3.

The fitting accuracy of the Kriging surrogate model is crucial to the result of an optimal problem, and the [R.sup.2], RMAE (relative maximum absolute error), and RMSE (Root-mean-square error) are employed as the criterion to evaluate the accuracy of it [45]. They can be calculated by

[mathematical expression not reproducible], (14)

where [bar.y] is the mean response value of test points, [[??].sub.i] is the predicted response value and [y.sub.i] is the real test response value and N is the number of test samples. The ideal surrogate model is supposed to meet the requirements that [R.sup.2] > 0.9, RMAE < 0.3, and RMSE < 0.1. After evaluating the Kriging surrogate model in I-sight software, it was obtained that all the values of [R.sup.2], RMAE, and RMSE meet the requirements, which indicates the Kriging model in this paper has high accuracy and reliability.

4.3. Implement of Adaptive Genetic Algorithm. The adaptive genetic algorithm is employed in the optimization problem mentioned in Section 4.2. The stacking angles 0[degrees], -45[degrees], 45[degrees], and 90[degrees] of each layer in CFRP are the genes, and they are encoded as 1, 2, 3, and 4. The stacking angle sequence [[x.sub.1]/[x.sub.2]/ [x.sub.3]/[x.sub.4]/[x.sub.5]] is the chromosomes. The Kriging surrogate models of [S.sub.1] and [S.sub.2] are the fitness functions while the Kriging surrogate models of F, L, and Q are the constraint functions. The values of [P.sub.c] and [P.sub.m] are 1 and 0.5 for GA, and the values of [P.sub.c1], [P.sub.c2], [P.sub.m1], and [P.sub.m2] are 1, 0.5, 0.5, and 0.01, respectively, for AGA. To compare the optimal result between AGA and GA, the population size is set as 20 and maximum iteration is 2000 in both GA and AGA.

In terms of AGA, 24 Pareto solutions are obtained and 14 Pareto solutions are obtained using GA. The Pareto-optimal fronts of AGA and GA are shown as Figure 10. In practical engineering requirement of crashworthiness, the upward displacement automotive steering column [S.sub.1] is as important as the rearward displacement automotive steering column [S.sub.2]. Therefore, in this paper, the Pareto solution that has the minimum sum of [S.sub.1] and [S.sub.2] is selected as the optimal solution. Then, the optimal solutions of AGA and GA and the comparison to the initial aluminum model are shown in Table 4.

It can be concluded from Figure 10 that a more widespread and even distribution nondominated solution set is obtained by AGA. Both the number and the distributivity of Pareto solutions of AGA are better than those of GA. And the results also indicate that the crashworthiness indicators of the AGA-optimized bumper have slightly improvement than those of GA.

4.4. Verification of the Results to FE Model. To verify the optimal result of AGA in practical application, the best stacking angle consequence is applied in the FE model and then calculated in Ls-Dyna. The whole process of collision can be considered as the conversion of kinetic energy to internal energy, and the C curve of total energy basically maintains a horizontal shape, which represents that the total energy satisfies energy conservation. The maximum of the hourglass energy and slip energy should be less than 5% of the maximum kinetic energy, which means that they are within the acceptable range, which is depicted in Figure 11. It is a necessary precondition of collision simulation, and the optimized FE model meets the requirements. And the optimal results of the Kriging surrogate model are compared to the simulation results by FEA. The accuracy of surrogate model is shown in Table 5.

It can be concluded from the above results that the Kriging surrogate model is of high accuracy and the optimization algorithm is effective. And the comparison of crashworthiness performance in Ls-Dyna is shown in Figure 12.

5. Conclusion

In this study, a novel multiobjective optimization method considering the adaptive genetic algorithm and Kriging surrogate model is proposed for improving crashworthiness and reducing weight of the automotive bumper. The CFRP material is employed in the bumper beam instead of aluminum, and the optimal stacking angle sequence of each layer in CFRP is obtained by AGA. The following conclusion can be summarized:

(1) Compared with the genetic algorithm, the adaptive genetic algorithm proposed in this paper has higher convergence speed and stronger ability to search global optimal solutions. And applying them to the optimization problem in crashworthiness, the comparison results indicate the crashworthiness of the CFRP bumper optimized by AGA is better than that of GA.

(2) After the use of CFRP material in the bumper beam and optimization by AGA, the new bumper achieves 43.19% weight reduction and exhibits better crashworthiness, compared to the initial aluminum bumper.

Furthermore, better optimization algorithm and more design variables of material will be considered into the research of the optimization of automotive bumper in the future.

Data Availability

No data were used to support this study.


The views expressed in this paper are those of the authors and do not necessarily reflect the official views of affiliations.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This work was supported by the research fund of the Shanghai Automotive Industry Technology Development Foundation (1744), National Natural Science Foundation of China (Grant no. 51809168), and Young Teacher Initiation Program of Shanghai Jiao Tong University (17X100040060).


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Dawei Gao [ID], (1) Haotian Liang, (1) Guijie Shi [ID], (2) and Liqin Cao (3)

(1) School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

(2) State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, China Strategy Institute of Ocean Equipment Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

(3) College of Environment and Chemical Engineering, Yanshan University, Qinhuangdao 066004, China

Correspondence should be addressed to Guijie Shi;

Received 19 April 2019; Revised 3 October 2019; Accepted 30 October 2019; Published 15 November 2019

Academic Editor: Francesco Aymerich

Caption: Figure 1: Flowchart of adaptive genetic algorithm.

Caption: Figure 2: Graphics and contour line of Schaffer function: (a) graphics; (b) contour line.

Caption: Figure 3: Graphics and contour line of Rosenbrock function.

Caption: Figure 4: Fitness curves of GA and AGA under Schaffer function. (a) Fitness curve of Schaffer function at 10 population size. (b) Fitness curve of Schaffer function at 20 population size. (c) Fitness curve of Schaffer function at 30 population size.

Caption: Figure 5: Fitness curves of GA and AGA under Rosenbrock function. (a) Fitness curve of Rosenbrock function at 10 population size. (b) Fitness curve of Rosenbrock function at 20 population size. (c) Fitness curve of Rosenbrock function at 30 population size.

Caption: Figure 6: Finite element model.

Caption: Figure 7: Microstructure of CFRP.

Caption: Figure 8: Schematic diagram of 10-plies CFRP based on the global coordinates.

Caption: Figure 9: Diagram of [S.sub.1] and [S.sub.2] in the FE model.

Caption: Figure 10: Pareto-optimal fronts of AGA and GA.

Caption: Figure 11: Energy change curve.

Caption: Figure 12: Comparison of crashworthiness performance in Ls-Dyna. (a) Initial aluminum model. (b) Model optimized by GA. (c) Model optimized by AGA.
Table 1: Comparison of algorithm performance.

Test                     Population
function     Algorithm      size        Optimal solution

Schaffer        AGA          10                1
function                     20                1
                             30                1
                GA           10                1
                             20                1
                             30                1
Rosenbrock      AGA          10       6.4090 x [10.sup.-4]
function                     20       6.4038 x [10.sup.-7]
                             30       5.3694 x [10.sup.-9]
                GA           10       2.8100 x [10.sup.-3]
                             20       4.7154 x [10.sup.-5]
                             30       6.5108 x [10.sup.-6]

Test                     Population
function     Algorithm      size         Worst solution

Schaffer        AGA          10              0.9982
function                     20                1
                             30                1
                GA           10              0.9801
                             20              0.9852
                             30              0.9993
Rosenbrock      AGA          10       6.4128 x [10.sup.-4]
function                     20       8.4196 x [10.sup.-7]
                             30       6.3283 x [10.sup.-9]
                GA           10       2.6120 x [10.sup.-3]
                             20       6.2234 x [10.sup.-5]
                             30       6.8200 x [10.sup.-6]

Test                     Population
function     Algorithm      size        Average solution

Schaffer        AGA          10              0.9999
function                     20                1
                             30                1
                GA           10              0.9984
                             20              0.9991
                             30              0.9999
Rosenbrock      AGA          10       6.4108 x [10.sup.-4]
function                     20       7.2016 x [10.sup.-7]
                             30       5.7953 x [10.sup.-9]
                GA           10       2.7800 x [10.sup.-3]
                             20       5.2868 x [10.sup.-5]
                             30       7.5864 x [10.sup.-6]

Table 2: Material properties of CFRP used in the FE model.

              Property               Description              Value

Mass            [rho]                  Density                 1455
             []     Longitudinal tensile strength     1409.9
             []    Transverse compressive strength    764.1
Strengths    []     Longitudinal tensile strength      42.5
(MPa)        []    Transverse compressive strength    121.6
             [X.sub.lts]       In-plane shear strength         68.9
             [E.sub.1]      Longitudinal elastic modulus     129.3
Elastic      [E.sub.2] =      Transverse elastic modulus       9.11
properties    [E.sub.3]
             [G.sub.12]             Shear modulus              5.44
             [Nu.sub.12]           Poisson's ratio             0.3

Table 3: 120 baseline designs.

SN     [x.sub.1]   [x.sub.2]   [x.sub.3]   [x.sub.4]   [x.sub.5]

1          1           1           3           2           3
2          2           4           3           2           2
3          1           2           1           1           2
4          4           1           2           2           2
5          3           1           4           3           1
6          4           4           3           4           1
7          4           3           4           3           3
8          2           4           1           4           4
9          4           3           4           2           3
10         4           3           4           4           3
...       ...         ...         ...         ...         ...
118        4           1           2           4           2
119        3           3           1           4           1
120        4           2           3           3           1

                   L        Q      [S.sub.1]   [S.sub.2]
SN      F (N)     (mm)     (kJ)      (mm)        (mm)

1      1430.73   153.6    7.8585     75.04       81.26
2      1375.26   136.96   7.8545     83.11       89.24
3       1224     201.25   7.8031     71.39       95.76
4      901.32    196.71   7.811      82.86       86.3
5      1208.88   177.8    7.9812     77.56       87.56
6      1405.52   174.02   7.8783     74.92       93.87
7      1365.18   176.29   7.5458     77.06       97.02
8      1153.42   143.01   7.9337     78.45       88.4
9      997.11    202.76   7.7793     71.13       89.87
10     1062.66   158.89   7.8189     84.87       87.35
...      ...      ...      ...        ...         ...
118    1355.1    147.55   7.6764     78.32       82.73
119    1178.63   165.7    7.5893     71.51       97.65
120    891.23    143.77   7.8308     75.67       98.07

Table 4: Comparison of optimal solutions results.

             aluminum      GA-optimized        AGA-optimized
              model            model               model

F           1006.920 N       984.457 N           906.591 N
L           240.583 mm      231.896 mm          225.178 mm
Q            7.239 KJ        7.741 kJ            7.802 KJ
[S.sub.1]   87.625 mm       80.5465 mm           75.490 mm
[S.sub.2]   98.566 mm        88.363 mm           87.911 mm
m            3.790 kg        2.153 kg            2.153 Kg
Design          --       [90/45/90/45/-45]   [90/90/45/-45/45]

            Compared     Compared to
            to GA (%)   initial model

F             -7.49         -9.96
L             -2.98         -6.4
Q             0.78          7.78
[S.sub.1]     -6.7         -13.85
[S.sub.2]     -0.51        -10.81
m               0          -43.19
Design         --            --

Table 5: Accuracy assessment of the Kriging surrogate model.

Objective    FE model    Kriging model   Accuracy error (%)

F           908.403 N      906.591 N           -0.12
L           240.583 mm    238.799 mm           -0.74
Q            7.396 kJ      7.802 kJ             4.66
[S.sub.1]   74.647 mm      75.490 mm            1.13
[S.sub.2]   85.997 mm      87.911 mm            2.23
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Title Annotation:Research Article
Author:Gao, Dawei; Liang, Haotian; Shi, Guijie; Cao, Liqin
Publication:Mathematical Problems in Engineering
Date:Nov 1, 2019
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