Crashworthiness analysis and multiobjective optimization for circular tubes with functionally graded thickness under multiple loading angles

Abstract

In this article, the nonlinear finite-element analysis software LS-DYNA is used to study the crashworthiness of functionally graded thickness circular tubes under multiple loading angles (0°, 10°, 20°, 30°, and 40°). The specific energy absorption (SEA) and maximum crashing force ( $F_{\max }$ ) are used as evaluation indices with which to analyze the effect of the length-to-diameter ratio and thickness gradient index P on the crashworthiness characteristics of functionally graded thickness circular tubes. Taking the length-to-diameter ratio and thickness gradient as design variables, the full factorial experiment was designed. A surrogate model of the integrated specific energy absorption ( $\mathrm{SE}{\mathrm{A}}_{\alpha }$ ) and maximum crashing force ( $F_{\max }^{0\circ }$ ) when the impact angle is 0° was constructed using the response surface method; the nondominated sorting genetic algorithm II was used to optimize the surrogate model, and the Pareto-front solution set was obtained. The results show that the thickness gradient index and the length-to-diameter ratio have a greater influence on the crashworthiness characteristics of functionally graded thickness circular tubes under oblique impact load, and the appropriate geometric parameters can improve the crashworthiness characteristics of functionally graded thickness circular tubes.

Introduction

Train, automobile, and large ship collisions have not only caused great economic losses but have also posed a serious threat to the safety of passengers. As an efficient energy-absorbing structure, the thin-walled structure has been widely used in automobiles, ships, aircraft, and other vehicles. This structure can absorb most of the impact energy from collisions through plastic deformation. In recent years, the crashworthiness characteristics of different forms of thin-walled structures, such as circular, square, foam-filled and windowed, ¹ and surface-grooved tubes, have been analyzed in detail. Due to its high strength, low weight, and good manufacturing process, aluminum tubes are widely used in energy-absorption (EA) devices.

In the collision process, thin-walled structures not only bear axial load but can also be subjected to oblique loads of different angles. Under oblique loading, the EA characteristics of thin-walled tubes decrease sharply, and the combined mode of axial progressive and global buckling is produced. Compared to axial progressive collapse, the global buckling mode is unstable. Therefore, the design of thin-walled structures that can absorb more energy and produce a stable deformation under oblique load has attracted more attention. Gao et al. ² investigated the crashworthiness of thin-walled structures with different shapes under oblique load. Tarlochan et al. ³ studied the crashworthiness of thin-walled structures with different cross-sectional shapes under axial and oblique loading. Ahmad et al. ⁴ found that conical tubes exhibit good crashworthiness under oblique load. These thin-walled structures have the same wall thickness, but their inherent weakness is that they are not able to make full use of the material to meet the requirements for lighter weight. ⁵ Therefore, it is widely considered that a new structure with varying thicknesses should be used to improve material usage. Tailor rolling blanks (TRBs), which lead to a continuous thickness variation in a workpiece, have been developed in recent years. ⁶ TRB technology can realize the continuous change of the thickness of the structure and obtain the ideal stiffness. Compared with the traditional uniform thickness of thin-walled tubes, functionally graded thickness (FGT) structures can provide more control of crashworthiness.

Sun et al. ⁷ used a finite-element (FE) model to study the crashworthiness of a functionally graded thin-walled square tube under axial impact and found that the impact behavior of the functionally graded tube is better than a uniform-thickness tube. Zhang and Zhang ⁸ carried out experimental and numerical studies on the impact characteristics of tapered tubes with variable thickness under axial load. Xu et al. ⁹ carried out experimental studies on the crashworthiness of thin-walled structures with FGT. Xu ¹⁰ found that FGT tubes had the advantage of reducing the initial peak force and improving EA compared to circular and conical tubes of uniform thickness. Li et al. ¹¹ investigated the crashworthiness of circular tubes with FGT under axial and oblique impact using the FE method and found that the FGT tube had better EA performance than a conical tube under oblique loading. Li et al. ¹² studied the impact characteristics of hollow and foam-filled thin-walled structures under multiple loading angles. Zhang et al. ¹³ investigated the axial impact characteristics of double functionally graded tubes and found that it possessed great EA characteristics. Baykasoglu and colleagues^14,15 studied the axial impact characteristics of functionally graded aluminum tubes and optimized their length-to-diameter (LD) ratio and thickness gradient. Xu ¹⁶ analyzed the axial impact characteristics of an FGT tube and optimized its length, diameter, and gradient index.

Based on the above-mentioned research, many scholars have analyzed the impact crashworthiness of FGT circular tubes. However, the effect of the LD ratio and thickness gradient index on the crashworthiness of FGT tubes under oblique impact load has not been studied thoroughly. This article mainly studies the crashworthiness of FGT tubes under axial and oblique impact load and analyzes the effect of the LD ratio and thickness gradient index P on crashworthiness and EA characteristics of FGT circular tubes. The results show that the two parameters of FGT tubes have a greater influence on crashworthiness characteristics. The response surface method is used to construct the surrogate model of the integrated SEA $\mathrm{SE}{\mathrm{A}}_{\alpha }$ and maximum crashing force $F_{max}^{0\circ }$ when the impact angle is 0°, and the genetic algorithm is used for the multiobjective structure optimization.

Numerical modeling

Geometric model

The thickness of the FGT tube in this study is changed along the axial direction, and the thickness of each position is described as equation (1), where x is the distance from the collision end of the tube, L the length of the circular tube, and P the thickness gradient index. The change in the value of P can control the thickness distribution in the tube. Under the action of the thickness gradient index P, the thickness of the tube changes continuously in the axial direction, and the convexity and concavity of the thickness function also change. As can be seen from Figure 1, when P < 1, the thickness gradient function is convex; when P > 1, the thickness gradient function is concave. The ideal deformation mode of a thin-walled tube is axial progressive crushing from the collision end to the other end. Therefore, the thickness of the tube at the collision end is $t_{\min }$ and the thickness at the other end is $t_{\max }$ ; the thickness distribution is expressed by

$$t\left(x\right)=t_{min}+(t_{max}-t_{min}){\left(\frac{x}{L}\right)}^P$$

(1)

Figure 1.

Schematic diagram of thickness distributions t(x) versus distance.

FE modeling

The FE model of the FGT tube was established using the FE preprocessing software HyperMesh; the FE model is shown in Figure 2. The tube is meshed with Belytschko–Tsay four-node shell elements. Convergence testing shows that the shell elements with dimension of 2.5 mm × 2.5 mm can predict the crushing behavior of an FGT tube. ¹³ The nonlinear explicit FE solver LS-DYNA is used to simulate the FGT tube under oblique impact. An automatic single contact is chosen to simulate the contact of the circular tube itself, and the static and dynamic coefficients of friction are set as 0.3 and 0.2, respectively. One end node of the tube is constrained by 6 degrees of freedom, and the rigid wall is defined by the keyword *rigidwall_planar_moving.

Figure 2.

Schematic diagram of oblique impact simulation.

The tube wall is made of aluminum alloy (AA6060-T4) with density $\rho =2700\mathrm{kg}/{\mathrm{m}}^3$ , Young’s modulus $E=68\mathrm{GPa}$ , Poisson’s ratio $\mu =0.3$ , yield strength ${\sigma }_{\mathrm{y}}=70\mathrm{MPa}$ , and ultimate strength ${\sigma }_{\mathrm{u}}=173\mathrm{MPa}$ . ² Since the aluminum is insensitive to the strain rate, the influence of strain rate on the material parameters is not considered in the modeling, and the keyword 24 material (*MAT_PIECEWISE_LINEAR_PLASTICITY) is used to simulate it.

In order to simulate oblique impact scenario of the circular tube, the tube is impacted by a 250-kg rigid wall with a velocity of 10 m/s, and the angle between the normal direction of rigid wall and the tube is $\alpha$ . In order to simulate the continuous change of thickness, the FE model is divided into 21 layers; each layer is 10 mm deep. In this article, $t_{\min }=0.6\mathrm{mm}$ and $t_{\max }=2\mathrm{mm}$ are selected, and the thickness expression of the discretized gradient thickness is given by expression (2), in which the length of the tube is L = 210 mm

$$t\left(x\right)=0.6+1.4{\left(\frac{n-1}{20}\right)}^P$$

(2)

Validation of the FE model

In order to verify the correctness of the FE model, the FE results of the average impact force ( $F_{\mathrm{avg}}$ ) under the axial impact load of the uniform-thickness tube are compared to the theoretical formula (3) proposed in the literature ¹²

$$F_{\mathrm{avg}}=\lambda 13.06{\sigma }_0{b_w}^{1/3}{t}^{5/3}$$

(3)

where ${\sigma }_0$ is the flow stress of the material, which can equal the average yield strength and ultimate strength of the material, $b_w$ the inner sectional width of the tube, t the thickness of the pipe, and $\lambda$ the dynamic increase factor. In total, four different wall thicknesses of 0.5, 1.0, 1.5, and 2.0 mm are used for verification. The comparison between the FE results and the empirical formula is shown in Figure 3, from which it can be seen that the fitting degree of the FE calculation results and the theoretical formula is good.

Figure 3.

Comparison between FE results and the theoretical formula.

In order to verify the FGT tube FE model, the FE calculation results and the test results from the literature ¹⁰ were compared. The material and dimension parameters of the FGT tube are from the literature. ¹⁰ The material of the FGT tube is AA6061-T5, while L = 210 mm, D = 45 mm, $t_{\min }=1.2\mathrm{mm}$ , $t_{\max }=1.5\mathrm{mm}$ , and P = 1 are used in the experiment. The comparison of the force–displacement curve and deformation mode between numerical simulation and experiment is shown in Figure 4. It can be seen from this figure that the deformation model of the numerical simulation is basically in agreement with experiment, the force–displacement curve is basically the same as the experimental result, and the relative error (RE) of the initial peak force is <5%. The correctness of the FE model is validated by comparing the deformation mode and the force–displacement curve, and the follow-up parameter influence study and optimization design can then be carried out.

Figure 4.

Comparison of numerical and experimental results ¹⁰ for the FGT circular tube.

Numerical analysis

Structural crashworthiness criteria

The typical force–displacement curve of the FGT circular tube under axial impact is shown in Figure 5. In order to evaluate the EA characteristics of FGT circular tubes under oblique impact loading, the EA, maximum impact force ( $F_{\max }$ ), and SEA are selected as evaluation indices. The EA is an important index for measuring the energy-absorbing structure’s ability to absorb impact energy, which can be defined as

$$\mathrm{EA}=\underset{0}{\overset{x}{\int }}F\left(x\right)\mathrm{dx}$$

(4)

where $x$ is the effective impact length, which is usually taken as 0.75L, with L the total length of the energy-absorbing structure; $F(x)$ is the instantaneous crushing force of the displacement $x$ . The SEA is the absorbed energy per unit mass of the energy-absorbing structure, which can be expressed as

$$\mathrm{SEA}=\frac{\mathrm{EA}}{M}$$

(5)

where M is the mass of the energy-absorbing structure. The SEA in this case is greater, and the EA characteristics of the thin-walled structure are better.

Figure 5.

Typical force versus displacement of FGT tubes.

In this study, in order to consider the load uncertainty and the crashworthiness of the FGT circular tube under axial and oblique loading comprehensively, the $\mathrm{SE}{\mathrm{A}}_{\alpha }$ is introduced as a new evaluation index, which can be described as follows

$$\mathrm{SE}{\mathrm{A}}_{\alpha }=\sum _{i=1}^n\mathrm{SE}{\mathrm{A}}^{{\alpha }_i}{w}^{{\alpha }_i}$$

(6)

where ${\alpha }_i$ is the angle of impact loading and $\mathrm{SE}{\mathrm{A}}^{{\alpha }_i}$ the SEA of the energy-absorbing structure when the impact angle is ${\alpha }_i$ .

Parameter influence analysis

Effect of LD ratio

In order to study the influence of different LD ratios on the EA of FGT tubes under different impact angles, the thickness gradient index P is set to 1 and the LD ratio is set to 2.0, 2.5, 3.0, 3.5, and 4.0. The length of the FGT tube is fixed at 210 mm. The effect of five different LD ratios on the crashworthiness of FGT tubes under different angles is shown in Figure 6. As can be seen from this figure, with increasing impact angle, the absorption energy and SEA of the FGT circular tube are decreased. Under large impact angle, the EA and SEA of the FGT tube are drastically reduced because the FGT tube undergoes a global buckling deformation under large angle. The EA decreases with increasing LD ratio, and the SEA increases with increasing LD ratio. It can be seen that the tube with a large LD ratio has a relatively small EA but a relatively high SEA because of its relatively small mass. It can be seen from Figure 6(c) that the maximum crashing force of the FGT tube exhibits a decreasing trend with the increasing impact angle and LD ratio. At a large impact angle ( $\alpha =40\circ$ ), the maximum impact force of the FGT tube is basically not affected by LD ratio.

Figure 6.

Comparison of (a) energy absorption, (b) specific energy absorption, and (c) maximum crashing force for different LD ratio values with increasing load angle.

Effect of thickness gradient index

In order to study the influence of different thickness gradient indices on the EA of FGT tubes under different impact angles, the LD ratio is set to 2.5 and the thickness gradient indices are set to 0.2, 0.4, 0.6, 0.8, 2.0, 4.0, 6.0, 8.0, and 10.0. The effect of 10 different thickness gradient indices on the crashworthiness of FGT tubes under different angles is shown in Figure 7. As can be seen from this figure, the absorption energy and SEA of the FGT circular tube dramatically decrease with increasing impact angle, and the maximum crashing force exhibits a decreasing trend. It can be seen from Figure 7(a) and (b) that with increasing thickness gradient index, the absorption energy decreases, and the SEA exhibits a decreasing trend. When the thickness gradient P < 1, the EA and SEA of the FGT tube are obviously larger than P > 1 because the FGT circular tube’s section thickness gradient function is convex when P < 1, and the corresponding mass of the circular tube is larger than the P > 1. However, a smaller thickness gradient P of FGT circular tubes does not result in a larger SEA. For example, there is almost no difference in the SEA of the FGT circular tube between thickness gradient P of 0.6 and 0.8, and the SEA with a thickness gradient P of 4.0 is larger than that of 2.0. It can be seen from Figure 7(c) that the maximum crashing force of the FGT tube shows a decreasing trend with increasing impact angle and thickness gradient index.

Figure 7.

Comparison of (a) energy absorption, (b) specific energy absorption, and (c) maximum crashing force for different P values with increasing load angle.

Multiobjective optimization

Experimental design

A full factorial experiment is used in determining the two design variables of the FGT circular tube. The thickness gradient index P is selected as 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 4.0, 6.0, 8.0, and 10.0, and the LD ratio as 2.0, 2.5, 3.0, 3.5, and 4.0, and 50 tests are designed. Since the SEA and maximum crashing force are significantly different when the gradient index P < 1 and P > 1, and the FGT tube is more efficient in increasing SEA values when $0<P⩽1$ and more efficient in reducing maximum crashing force when $1<P⩽10$ , two cases ( $0<P⩽1$ and $1<P⩽10$ ) are adopted for the FGT circular tube optimization. The $\mathrm{SE}{\mathrm{A}}_{\alpha }$ and maximum impact force $F_{max}^{0\circ }$ when the impact angle is 0° are selected as evaluation indicators. The experimental design and results are shown in Table 1.

Table 1.

Experimental design and numerical results.

No.	LD ratio	$0<P⩽1$			$1<P⩽10$
		P	$\mathrm{SE}{\mathrm{A}}_{\alpha }$ (kJ/kg)	$F_{max}^{0\circ }$ (kN)	P	$\mathrm{SE}{\mathrm{A}}_{\alpha }$ (kJ/kg)	$F_{max}^{0\circ }$ (kN)
1	2.0	0.2	7.777	52.455	2.0	5.657	41.877
2	2.0	0.4	7.383	48.722	4.0	5.158	34.072
3	2.0	0.6	7.202	49.644	6.0	4.758	31.698
4	2.0	0.8	7.004	55.489	8.0	4.650	25.021
5	2.0	1.0	7.098	50.535	10.0	4.604	24.878
6	2.5	0.2	9.323	46.125	2.0	6.523	44.244
7	2.5	0.4	8.950	50.317	4.0	6.239	30.618
8	2.5	0.6	8.791	47.368	6.0	5.836	32.128
9	2.5	0.8	8.531	47.420	8.0	5.592	27.093
10	2.5	1.0	7.977	46.931	10.0	5.496	21.284
11	3.0	0.2	9.646	45.068	2.0	7.526	35.756
12	3.0	0.4	9.350	31.554	4.0	6.404	28.626
13	3.0	0.6	9.285	33.959	6.0	5.956	23.265
14	3.0	0.8	8.909	37.392	8.0	5.677	21.174
15	3.0	1.0	8.952	40.802	10.0	5.303	20.343
16	3.5	0.2	9.800	41.972	2.0	8.471	31.088
17	3.5	0.4	10.116	32.972	4.0	7.220	29.879
18	3.5	0.6	9.694	31.977	6.0	6.435	21.367
19	3.5	0.8	9.722	33.123	8.0	5.990	19.836
20	3.5	1.0	9.571	33.815	10.0	5.964	20.227
21	4.0	0.2	10.295	41.653	2.0	8.685	27.960
22	4.0	0.4	10.602	34.622	4.0	7.278	20.572
23	4.0	0.6	10.201	30.006	6.0	6.696	18.648
24	4.0	0.8	10.064	31.709	8.0	6.342	21.280
25	4.0	1.0	10.381	29.276	10.0	5.999	16.076

LD: length-to-diameter.

In order to analyze the influence of the LD ratio and the thickness gradient index of the design variables on the response, the results of the main effect test are analyzed. The main effect analysis does not consider the influence of other design variables and analyzes the influence of a design variable on the response. The greater the absolute value of the main effect, the greater the influence of the design variable on the response. The main effect analysis results are shown in Table 2. It can be seen from this table, in the case of $0<P⩽1$ , the effect of the LD ratio on the integrated SEA and the maximum impact force is comparatively larger, with the influence degree being 4.49 and 5.43 times that of the thickness gradient index, respectively. In the case of $1<P⩽10$ , both LD ratio and thickness gradient index have a greater effect on the integrated SEA, and the thickness gradient index has greater influence on the maximum impact force.

Table 2.

Main effect analysis.

	$0<P⩽1$		$1<P⩽10$
	$\mathrm{SE}{\mathrm{A}}_{\alpha }$ (kJ/kg)	$F_{max}^{0\circ }$ (kN)	$\mathrm{SE}{\mathrm{A}}_{\alpha }$ (kJ/kg)	$F_{max}^{0\circ }$ (kN)
LD ratio	1.420	−9.738	0.989	−5.560
P	−0.316	−1.795	−0.922	−7.424

LD: length-to-diameter.

Surrogate model

Based on the results of the full factorial experiment, an approximate model of the integrated SEA and maximum crashing force is constructed based on the response surface method. Figure 8 shows the response surface of the integrated SEA and the maximum crashing force at various values of the LD ratio and thickness gradient index. It can be seen from this figure that the integrated SEA of the FGT tube decreases with increasing gradient index and increases with increasing LD ratio when 0 < P ⩽ 1. The maximum crashing force first decreases and then increases with increasing thickness gradient index P and decreases with increasing aspect ratio. The integrated SEA of the FGT tube increases with increasing LD ratio when 1 < P ⩽ 10 and decreases with increasing P. The maximum crashing force decreases with increasing LD ratio and decreases with increasing P. The integrated SEA and maximum crashing force are expressed in equations (7) and (8)

$$\mathrm{SE}{\mathrm{A}}_{\alpha }=\{\begin{array}{c}-60.837+87.566\mathrm{LD}+2.621P-41.309\mathrm{L}{\mathrm{D}}^2-12.264P^2+0.5\mathrm{LD}\times P\hfill \\ \multicolumn{1}{c}{+8.645\mathrm{L}{\mathrm{D}}^3+11.493P^3-0.671\mathrm{L}{\mathrm{D}}^4-3.504P^4(0<P⩽1)}\\ \multicolumn{1}{c}{-98.656+144.086\mathrm{LD}-0.580P-72.760\mathrm{L}{\mathrm{D}}^2+0.094P^2-0.109\mathrm{LD}\times P}\\ \multicolumn{1}{c}{+16.24\mathrm{L}{\mathrm{D}}^3-0.00243P^3-1.338\mathrm{L}{\mathrm{D}}^4+0.0000549P^4(1<P⩽10)}\end{array}$$

(7)

$$F_{max}^{0\circ }=\{\begin{array}{c}-897.809+1344.974\mathrm{LD}+2.232P-687.535\mathrm{L}{\mathrm{D}}^2-104.463P^2-7.607\mathrm{LD}\times P\hfill \\ \multicolumn{1}{c}{+151.322\mathrm{L}{\mathrm{D}}^3+253.591P^3-12.177\mathrm{L}{\mathrm{D}}^4-140.216P^4(0<P⩽1)}\\ \multicolumn{1}{c}{-669.151+1040.021\mathrm{LD}-14.404P-537.973\mathrm{L}{\mathrm{D}}^2+2.337P^2+0.577\mathrm{LD}\times P}\\ \multicolumn{1}{c}{+120.101\mathrm{L}{\mathrm{D}}^3-0.211P^3-9.865\mathrm{L}{\mathrm{D}}^4+0.00713P^4(1<P⩽10)}\end{array}$$

(8)

Figure 8.

Approximated response surfaces of $\mathrm{SE}{\mathrm{A}}_{\alpha }$ and $F_{\max }$ : (a) $0<P⩽1$ and (b) $1<P⩽10$ .

In order to evaluate the accuracy of the surrogate model, the RE, root-mean-square error (RMSE), maximum absolute error (MAX), and R² values ¹⁷ are selected to evaluate the error between the response value and the FE numerical calculation. The expression of each error analysis is given by equations (9)–(12), respectively

$$\mathrm{RE}=\frac{y\left(x\right)-\tilde{y}\left(x\right)}{y\left(x\right)}$$

(9)

$$\mathrm{RMSE}=\sqrt{\frac{\sum _{i=1}^n{(y_i-{\tilde{y}}_i)}^2}{n}}$$

(10)

$$\mathrm{MAX}=max|y_i-{\tilde{y}}_i|,i=1,2,\dots ,n$$

(11)

$$R^2=1-\frac{\sum _{i=1}^n{(y_i-{\tilde{y}}_i)}^2}{\sum _{i=1}^n{(y_i-{\overline{y}}_i)}^2}$$

(12)

where n is the number of design sample points, and $y(x)$ and $\tilde{y}(x)$ are the values of the FE element simulation and surrogate model, respectively. If the values of RMSE and MAX are smaller or the value of R² is larger, the surrogate model is more accurate.

The surrogate model error analysis is shown in Table 3. As can be seen from this table, the surrogate model has high precision, which can be carried forward in the subsequent optimization design.

Table 3.

Accuracies of the surrogate model.

	Function	RMSE	MAX	R 2	RE (%)
$0<P⩽1$	$\mathrm{SE}{\mathrm{A}}_{\alpha }$ (kJ/kg)	0.14785	0.3418	0.981	(−4.285, 4.150)
	$F_{max}^{0\circ }$ (kN)	2.3738	5.3036	0.913	(−16.808, 9.120)
$1<P⩽10$	$\mathrm{SE}{\mathrm{A}}_{\alpha }$ (kJ/kg)	0.1433	0.4089	0.981	(−6.269, 2.421)
	$F_{max}^{0\circ }$ (kN)	1.7752	3.8820	0.937	(−10.164, 14.356)

RMSE: root-mean-square error; MAX: maximum absolute error; RE: relative error.

Multiobjective optimization

Optimization techniques

After using the response surface method to construct the surrogate model of the integrated SEA and maximum crashing force of the FGT circular tube, an optimization algorithm is needed to optimize the agent model. In this article, a nondominated sorting genetic algorithm II (NSGA-II) with elitist strategy is used to solve the multiobjective optimization problem; the optimal mathematical model can be described as

$$\{\begin{array}{c}\mathrm{Min}\{-\mathrm{SE}{\mathrm{A}}_{\alpha },F_{max}^{0\circ }\}\hfill \\ \multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.2.0⩽\mathrm{LD}⩽4.0}\\ \multicolumn{1}{c}{\mathrm{Case}1:0<P⩽1;\mathrm{case}2:1<P⩽10}\end{array}$$

(13)

When the genetic algorithm is used to solve for the maximum and minimum values of the surrogate model, an initial population is randomly generated to encode the individual population, and the objective function is transformed into a fitness function; each individual is assessed for fitness, and the degree of merit of each individual is also assessed. The selection, crossover, and mutation algorithms are then selected to produce a better new-generation population. Finally, the termination conditions are judged. This genetic algorithm optimization process is illustrated in Figure 9, and the specific parameters of the surrogate model used in this article are shown in Table 4.

Figure 9.

Genetic algorithm optimization flowchart.

Table 4.

NSGA-II algorithm parameters.

Parameter	Value
Population size (multiples of 4)	20
Number of generations	50
Crossover probability	0.9
Crossover distribution index	10
Mutation distribution index	20
Maximum number of failed runs	5

Optimization results

In order to verify the correctness of the FE model, this article studies the minimum and maximum impact forces that the structure can achieve under specific constraints. For $1<P⩽10$ , an integrated SEA of not <10.4 kJ/kg is selected as a constraint, and 7.5 kJ/kg is selected for $1<P⩽10$ . Figure 10 shows the Pareto-front solutions for $-\mathrm{SE}{\mathrm{A}}_{\alpha }$ and $F_{\max }$ . Table 5 presents the multiobjective optimization result, the FE calculation result, and the comparative error analysis. The errors of the SEA and maximum impact force are both below 5%, indicating that the optimization results have a higher accuracy.

Figure 10.

Pareto-front solutions from the NSGA-II algorithm: (a) $0<P⩽1$ and (b) $1<P⩽10$ .

Table 5.

Optimization results of minimizing $F_{max}^{0\circ }$ with the $\mathrm{SE}{\mathrm{A}}_{\alpha }$ constraint.

	Design variables	NSGA-II		FEA result		Accuracies
		$\mathrm{SE}{\mathrm{A}}_{\alpha }$	$F_{max}^{0\circ }$	$\mathrm{SE}{\mathrm{A}}_{\alpha }$	$F_{max}^{0\circ }$	$\mathrm{R}{\mathrm{E}}_{\mathrm{SE}{\mathrm{A}}_{\alpha }}$ (%)	$\mathrm{R}{\mathrm{E}}_{F_{\max }}$ (%)
$0<P⩽1$	LD ratio = 3.9989, P = 0.438	10.4	32.7373	10.425	33.18	0.24	1.33
$1<P⩽10$	LD ratio = 3.9839, P = 4.0679	7.5	21.975	7.622	22.387	1.60	1.84

LD: length-to-diameter; FEA: finite-element analysis; NSGA-II: nondominated sorting genetic algorithm II.

Conclusion

The crashworthiness characteristics of an FGT circular tube under axial and oblique loading were studied in this article. Taking the EA, SEA, and maximum impact force as the main indices, the influence of the variable thickness gradient index P and LD ratio on the EA characteristics of an FGT circular tube under different loading angles was analyzed. The response surface method was used to construct a surrogate model of the integrated SEA and the maximum impact force, and the genetic algorithm was applied to optimize the agent model. The main conclusions of this study are summarized as follows:

1. The SEA decreases with increasing thickness gradient index and exhibits an increasing tendency with increasing LD ratio. The maximum impact force shows a decreasing trend with increasing thickness gradient index and LD ratio. A thickness gradient index in the range of $0<P⩽1$ has an advantage in maximizing the integrated SEA, and in the range of $1<P⩽10$ , it has an advantage in minimizing the maximum impact force.

2. In the case of $0<P⩽1$ , the effect of the LD ratio on the SEA and the maximum impact force is comparatively large. In the case of $1<P⩽10$ , both LD ratio and thickness gradient index have a greater effect on the integrated SEA, and the thickness gradient index has greater influence on the maximum impact force.

3. In order to maximize the integrated SEA and minimize the maximum crashing force, the NSGA-II was used to optimize the FGT tube, and the Pareto-front solution is obtained. The errors of the integrated SEA and the maximum impact force are within 5%, which verifies the accuracy and effectiveness of the optimization method.