Crashworthiness Design of Thin-Walled Tubes Reinforced by Triply Periodic Minimal Surfaces

Abstract

This article aims to propose a kind of internally reinforced square tube based on triply periodic minimal surfaces (TPMSs), which can be obtained by computer-assisted techniques and additive manufacturing. To achieve this goal, a finite element simulation model is exploited to simulate the collision situation of reinforced thin-walled tubes. The design of reinforced tubes can be formulated into a multiparameter and multiobjective optimization problem, which can be solved using the well-known non-linear programming by quadratic Lagrangian (NLPQL) optimization method. Three types of TPMS-based tubes and two multicell tubes are compared under the same conditions to show the effectiveness of our method. Through the methods mentioned above, TPMS-reinforced tubes are found to be superior in crashworthiness. This means that the safety performance of the automobiles with lighter weight can be effectively improved. The optimal parameters of three types of TPMS-reinforced tubes under different conditions were obtained, providing the foundations and references for subsequent related studies. In addition, TPMS is first explored in the design of crashworthiness for automobiles in this article. Due to the controllability and implicit functional expression of TPMSs, TPMS-reinforced tubes can be easily controlled and optimized. Meanwhile, it is easy to manufacture them by three-dimensional printing technologies.

Introduction

A vehicle's crashworthiness is designed to protect the passengers and important components in the event of a crash. Safety equipment, strong structures, and other vehicle construction issues contribute to the crashworthiness of vehicles. Moreover, reducing the weight of automobiles can effectively reduce fuel and other forms of energy consumption and the lightweight design is of benefit to crashworthiness as well. Therefore, structural design and optimization for crashworthiness are important topics in the field of automobile manufacture.

For the crash-proof design of thin-walled tubes under axial load, the direct method is to change the wall thickness. Tubes with a simple structure are widely used due to the easy manufacture and certain anticollision effect. Baroutaji et al.¹ compared the anticollision performance of thin-walled tubes with different velocities and shapes.^2–4 Characteristics of simple structures cannot meet the safety requirements under high-speed collisions. Chen and Wierzbicki⁵ proposed to add additional plates inside the thin-walled tubes to form multiple cells. The multicell structures were well studied in several articles.^6–8 Multicell tubes have better energy absorption (EA) efficiency and crashworthiness than unit-cell tubes under various conditions, while the initial peak crashing force (PCF) of the multicell tubes decreases. Mahmoodi et al.⁹ studied the impact resistance of multicell tubes and found that increasing the number of cells would improve the impact resistance, but the rate was gradually decreasing. Qi et al.¹⁰ found that the loading angle, thin-walled tube thickness, and cone angle had a large impact on crashworthiness performance.

Porous metal foams were also applied to fill thin-walled tubes as reinforced structures to further enhance the EA capacity under the constraint of PCF, as suggested in several articles.^11–15 The high lightweight design and EA efficiency made the porous internal structures suitable for engineering applications. With the development of additive manufacturing, more complicated structures have been studied.^16–18 Triply periodic minimal surfaces (TPMSs) are minimal surfaces expressed by implicit functions (shown in Fig. 1a) and they have many good properties, such as smoothness, full interconnectivity without enclosed hollows, high surface-to-volume ratio, and high specific strength and stiffness.^19,20

FIG. 1.

Illustration of TPMSs. (a) Different types of TPMSs, including Schwarz's Primitive-Triply Periodic Minimal Surface (P-TPMS), Schoen's Gyroid-Triply Periodic Minimal Surface (G-TPMS), and Schoen's iWP-Triply Periodic Minimal Surface (IWP-TPMS), respectively. (b) P-TPMS with different C (from left to right: −0.5, 0, and 0.5, respectively).

In this article, three types of TPMSs, including Schwarz's Primitive-Triply Periodic Minimal Surface (P-TPMS), Schoen's Gyroid-Triply Periodic Minimal Surface (G-TPMS), and Schoen's iWP-Triply Periodic Minimal Surface (IWP-TPMS), are exploited to design the reinforcement tubes. TPMS structures can effectively reduce the PCF and collapse force fluctuation while meeting the requirements of a lightweight design, strong EA, and limited PCF.¹⁹ Due to parameter controllability (shape and thickness) and implicit functional expression of TPMSs, the TPMS structures are creatively used as the internal filling for thin-walled tubes. The proposed TPMS-reinforced tubes have better crashworthiness performance than the multicell reinforcement tubes. A feasible optimization scheme of the TPMS-reinforced tubes has also been introduced by controlling the shape and thickness parameters.

Materials and Methods

Construction of TPMS-reinforced tubes

TPMSs are minimal surfaces with the mean curvature vanishing at every point and they can be expressed by implicit functions. Some commonly used TPMSs can also be approximated by trigonometric functions,²⁰ including P-TPMS, G-TPMS, and IWP-TPMS:

where $r=\left(x,y,z\right)\in {ℛ}^3$ and $C\in \left[-1,1\right]$ is a constant that controls the shapes of TPMSs. In Figure 1, we illustrate different types of TPMSs and the impact of parameter C on a specific TPMS.

Taking P-TPMS as an instance (Fig. 2), a TPMS-reinforced square tube model was constructed by the combination of TPMS cells and a thin-walled square tube. The steps are listed as follows:

FIG. 2.

Construction of TPMS-reinforced thin-walled tube. (a) P-TPMS, (b) thin-walled tube, and (c) composite model.

• Importing a dual-cell geometry model of a TPMS established by Equation (1).

• Assembling the dual-cell surface with the thin-walled tube by mapping the TPMS boundary vertices onto the nearest tube walls.

• Merging two models by remeshing around the intersecting regions.

Compression experiments and simulations

Compression experiments

The TPMS models (STL format) were first obtained by our early work method, Hu et al.²⁰ Then, the TPMS-reinforced tubes can be constructed using the bidirectional bias of the TPMSs, as shown in Figure 3. The external rectangular frames are metal plates, which are used to fix the specimens during the axial crushing. Finally, the composite models were cut into slices by Fastlayer software to generate G-code files. The detailed fabrication parameters are listed in Table 1.

FIG. 3.

Overhead view of different TPMS-reinforced tubes. (a) P-TPMS, (b) G-TPMS, and (c) IWP-TPMS.

Table 1.

Fabrication Parameters of Selective Laser Melting (SLM) Printed Specimens Made with SS-316L

Manufacturing parameters	Parameter information
Powder size (μm)	45 ± 10
Laser point size (μm)	40
Scanning speed (m/s)	8
Ambient temperature (°C)	15–30
Layer thickness(μm)	30
The gas supply	Ar/N2 protection

All the models in our experiments have good printability and can be printed well without external supports. The specimens made of stainless steel 316L were fabricated by an Selective Laser Melting (SLM) three-dimensional (3D) printer (Fast Form FF-M140) whose printing size is 2500 mm × 1000 mm × 2100 mm. The 3D printer adopts the scanning system of a high-precision scanning galvanometer, 200 W/500 W fiber laser, Windows7 operating system, and Eplus 3D printing software. The supply voltage and power consumption are 380 V and 6 kW, respectively. The specimens were cooled slowly to room temperature in the furnace after 2 h of heat treatment at 650°C. In our experiments, the density of forming components was higher than 90%, which can be considered as a solid fill to obtain good product quality. According to the relevant design of the EA box for the anticollision beam^6–8 and the cost of experiments, the size of each TPMS model was set to 20 mm × 20 mm × 40 mm. The thickness of square tubes and TPMS structures was set to 1 mm, and the thickness of the metal fixed base at the bottom was set to1.2 mm.

After the specimens were 3D printed, we exploited an MTS testing machine (type WD-P6105) to execute the compression experiments. The speed of the compression process was set at 0.5 mm/min to satisfy the quasi-static compression tests. The effective compression displacement was set to 24 mm, which is 60% of the specimen length, considering the time in quasi-static solid experiments and the visibility in compression experiments.²¹ The printed specimens and the experiment setting are shown in Figure 4. Moreover, five identical metal models were printed for each reinforced tube for compression tests.

FIG. 4.

Illustration of experiment and modeling. (a) Solid compression experiment. (b) Three-dimensional finite element model.

Simulations

LS-DYNA was used to simulate the axial collision compression behavior of TPMS-reinforced tubes. The material of the tubular structure in the finite element (FE) model was MATL98 in axially crushing analyses.²² The material characterization of MATL98 was done as per the Johnson–Cook constitutive isotropic hardening model,²² which is often used to simulate adjustments based on material properties and experimental controls. In this model, mass density ρ = 7830 kg/m³, Young's modulus E = 200 GPa, Poisson's ratio μ = 0.3, A = 440 MPa, B = 1200 MPa, N = 0.8, and c = 0. To simulate the cracking effect, the effective plastic strain at failure was assigned to 0.9, which is set according to the breaking time and state of the metal material in the compression test. MATL20 was selected for the striker as a material model to describe the rigid body, with mass density ρ = 785,000 kg/m³, Young's modulus E = 200 GPa, and Poisson's ratio μ = 0.3. The fixed base was set as a rigid wall.

The reinforced tubes and tabular striker were modeled by using four-node shell continuum (S4R) elements with five integration points along the thickness direction of the element. The mesh-independent study was based on the element sizes of 0.6, 0.5, 0.44, and 0.4 mm (corresponding to the number of elements, 20,228, 26,521, 34,329, and 41,989, respectively). The ratio of the difference between two adjacent element sizes is <8% for PCF, while it is <2% for specific energy absorption (SEA). According to our experiments, it is appropriate to set the element size to 0.5 mm. The thickness of the striker was 0.04 mm and the thickness of the reinforced tubes was 1 mm. The general numbers of mesh elements used for the current analysis of P-TPMS, G-TPMS, and IWP-TPMS are 26,521, 32,933, and 29,265, respectively. There are two contacts, including the single side contact of the reinforced tube itself and the contact between the points on the striker and the reinforced tube surface.

The top of each reinforced tube was axially compressed with an 800-kg striker and constraints of six degrees of freedom in x, y, and z directions were set between its bottom and fixed base. The speed of the striker was nonuniformly increased from 0 to 4 m/s, which avoids unnecessary dynamics in the numerical solution.

To minimize hourglass and dynamic effects, the ratio of external work to kinetic energy was limited to 6%.²³ The FE model of simulation is shown in Figure 4b.

The collision simulations were verified to be consistent with the compression experiments, as shown in Figure 5 (left). We can see that the number of folds (1, 3, and 2 corresponding to P, G, and IWP, respectively) and the positions are almost the same for each TPMS-reinforced tube between simulations and compression experiments. The deformation of G-TPMS is slightly different, which may be caused by manufacturing defects and uneven heat treatment temperature distribution. Furthermore, we compared the load–displacement curves of three TPMS structures in Figure 5 (right). We can see that the collapse force trend of each structure is almost the same as the simulation. For the IWP-TPMS-reinforced tube, the load–displacement curve of simulation deviates from the result of the solid experiment, which is caused by the sudden warping on the right side of the solid tube due to the manufacturing defect. For the P-TPMS, G-TPMS, and IWP-TPMS-reinforced tubes, the increasing ratios of PCF are 0%, 2.6%, and 5.9%, respectively. The corresponding increasing ratios of SEA are 7.4%, 2.4%, and −7.7%, respectively. Table 2 provides detailed data on the average values of five test models for each reinforcement tube. Overall, the error of PCF is <6% and the error of SEA is <8%, so the collision simulations are accurate and feasible.

FIG. 5.

Illustration of compression experiments and collision simulations, including the model deformation and load–displacement curves. (a) P-TPMS, (b) G-TPMS, and (c) IWP-TPMS.

Table 2.

Experimental Results of Collision Simulations by LS-DYNA and Compression Experiments by the WD-P6105 Testing Machine

TPMS	Simulation		Experiment		Increasing ratio (%)
	PCF (kN)	SEA (J/g)	PCF (kN)	SEA (J/g)	PCF (kN)	SEA (J/g)
P-TPMS	46.6	22.6	46.6	21.0	0	7.4
G-TPMS	58.5	25.8	57.0	25.8	2.6	2.4
IWP-TPMS	59.0	22.0	55.7	23.9	5.9	−7.7

P-TPMS, Schwarz's Primitive-Triply Periodic Minimal Surface; G-TPMS, Schoen's Gyroid-Triply Periodic Minimal Surface; IWP-TPMS, Schoen's iWP-Triply Periodic Minimal Surface; PCF, peak crashing force; SEA, specific energy absorption.

Response surface methodology-based optimization

To obtain desired TPMS-reinforced tubes, the response surface methodology (RSM) was exploited to optimize the variables (thickness t and shape C) according to the simulation data. Two important indicators, the initial PCF and SEA, were usually used to evaluate the crashworthiness of automobile components. PCF is the peak value of the first wave of the load–displacement curve and it is used to estimate the dynamic characteristics of structures. The calculation of SEA can be expressed as follows:

$${\int }_0^{\delta }F\left(x\right)dx∕M$$ (2)

where $\delta$ is the total compression displacement, x is the displacement variable, F(x) is the load–displacement function, and M is the component mass. SEA is used to measure the EA efficiency and cost impact of mass.⁷

In this study, a fourth-order RSM of SEA and PCF with variables, thickness t and shape C, was established using the Isight platform. The mathematical expression referred in the study by Kurtaran et al.²⁴ is shown as follows:

$$\begin{array}{cc}F\left(y\right)& =a_0+{\sum }_{i=1}^m{b}_i{x}_i+{\sum }_{i=1}^m{\sum }_{j\ge i}^m{c}_{ij}{x}_i{x}_j+{\sum }_{i=1}^m{d}_i{x}_i^3\\ & +{\sum }_{i=1}^m{e}_i{x}_i^4,\end{array}$$ (3)

where x_i is the i-th component of the m-dimensional independent variable and a₀, b_i, $c_{ij}$, d_i, and e_i are unknown coefficients of the RSM function. The main reason we chose RSM is that it uses a few number of crashworthiness simulations to construct smooth approximations of objectives and constraint functions, and this is a proper balance between efficiency and accuracy. The accuracy and precision of the RSM model were judged by the RSM error between the simulation data and the corresponding RSM model data, which is calculated as root mean square error (RMSE).

Then, the optimization problem of the objective function with constraints can be formulated as follows:

where the weight factor ω was given differently in each optimization, and the ranges of thickness t and shape C were given empirically. $P\left(t,C\right)$ is the RSM function of PCF on variables t and C, and $S\left(t,C\right)$ is the RSM function of SEA on variables t and C. The one with minimal objective value is considered to have the best crashworthiness under the weight. To avoid falling into the local optimal solution, t = 0.6, 1.0, and 1.3 and C = −0.3, 0, and 0.3 were selected as the initial values for P-TPMS, G-TPMS, and IWP-TPMS-reinforced tubes, respectively.

Finally, a well-known, multiobjective optimization method named non-linear programming by quadratic Lagrangian (NLPQL) was used to solve the problem. NLPQL adopts the optimization method of sequential quadratic programming, which is suitable for optimization of multiobjective problems with a continuous and differentiable objective. This algorithm is mainly used for high-order nonlinear problems to obtain stable and fast convergence. The optimization parameters are listed in Table 3, and the flow chart of optimization can be seen in Algorithm 1.

Table 3.

Parameters of Response Surface Methodology-Based Optimization with Non-Linear Programming by Quadratic Lagrangian

Parameters	Data
Max iterations	40
Termination accuracy	1.0E6
Real step size	0.001
Min abs step size	1.0E4
Max failed runs	5
Failed run penalty value	1.0E30
Failed run objective value	1.0E30

Algorithm 1: RSM-based optimization
Input: simulation data t, C, PCF, and SEA, initial values t0 and C0, parameters.
Output: optimal results t, C, PCF, and SEA. Objective value.
Initialize: t = t0 and C = C0;
For each type of TPMS-enforced tube, do
construct response surfaces $P\left(t,C\right)$, $S\left(t,C\right)$ according to all the simulation data;
for weight factor = 0.1:0.1:0.9, do
define the optimization formula of the optimization problem;
solve the optimization problem based on NLPQL;
verify the optimization result by simulation;
end
End

Results

Based on the simulation data, the response surface models can be established. The errors of RSM models were calculated by the RMSE between the simulation data and the corresponding RSM model data. The errors of PCF response surfaces of P-TPMS, G-TPMS, and IWP-TPMS were 0.03, 0.027, and 0.065, respectively, and the errors of SEA response surfaces were 0.172, 0.089, and 0.092, respectively. The error precision was high enough so that the response surface models can be used for analysis and optimization.

The response surfaces can be visualized in Figure 6. It can be seen that the shape variable C has a large effect on SEA of three TPMS structures, but has little effect on PCF. However, the thickness t affects both PCF and SEA of all the structures, and the value of PCF increases linearly when t increases.

FIG. 6.

Response surfaces of TPMS-reinforced tubes. (a) Response surfaces of PCF for P-TPMS, G-TPMS, and IWP-TPMS, respectively. (b) Response surfaces of SEA for P-TPMS, G-TPMS, and IWP-TPMS, respectively. PCF, peak crashing force; SEA, specific energy absorption. Color images are available online.

According to the response surfaces, the NLPQL method²⁵ can be exploited to obtain optimized solutions. In Figure 7, we can see that the optimized solutions can be connected into curves. The nonoptimized points were above the curves of the optimized solutions, respectively. Therefore, we can conclude from the points in Figure 7 that the optimized solutions were usually located at the boundaries of the response surfaces. Therefore, the optimized solutions of the thickness t for three types of TPMS-reinforced tubes were at the lower boundary (t = 0.65). The optimized solutions of shape C with different weight factors are mostly taken around C = 0 for P-TPMS, C = 0.35 for G-TPMS, and C = −0.35 for IWP-TPMS, respectively. More detailed data are listed in Appendix Tables A1–A3. The TPMS-reinforced tubes with the above optimized C and t values are visible in Figure 8.

FIG. 7.

Distribution of the simulation and optimized results upon PCF–SEA. Color images are available online.

FIG. 8.

Illustration of three types of optimized TPMS-reinforced tubes (from left to right: P, G, and IWP).

Discussion

In our experiment, three types of TPMS-reinforced tubes with C = 0 were compared with two traditional multicell tubes⁶ under the same mass and external conditions. As a commonly used anticollision beam design structure, the multicell structures are well studied and have good controllability that can be easily used to compare with TPMS-reinforced tubes. The FE models and cross-sections of five reinforced tubes can be seen in Figure 9. Moreover, the related data in simulation for the five reinforced tubes are listed in Table 4.

FIG. 9.

Load–displacement curves of the simulation and the finite element models of reinforced tubes with cross-sections.

Table 4.

Detailed Data of Simulation for Five Reinforced Tubes

Tube	t (mm)	M (g)	SEA (J/g)	PCF (kN)	STDEV (kN)
P-TPMS	0.852	33.91	22.46	36.6	0.7727
G-TPMS	0.763	33.90	20.53	40.4	0.6047
IWP-TPMS	0.716	33.89	18.40	38.2	0.6241
Diag	1.0	33.91	23.48	57.3	0.8991
Polygon	0.657	33.92	32.0	58.3	0.8433

t, thickness parameter; M, mass of tubes; PCF, initial peak crashing force; STDEV, standard deviation used to quantize the smoothness.

To compare the crashworthiness of various structures, the Complex Proportional Assessment (COPRAS) method²⁶ was used for multiobjective decision-making and measurement of crashworthiness. The COPRAS can be indicated using the following procedure:

Step 1: Developing the initial matrix (X) and finding the relative coefficient matrix (R).

Step 2: Determining the weighted normalized decision matrix (D).

Step 3: Summing of beneficial and nonbeneficial attributes.

Step 4: Calculating the relative significance or priority (Q).

Step 5: Determining the quantitative utility (U).

According to the PCF and SEA data in Table 4, the numerical calculation of COPRAS can be expressed as follows:

The initial decision matrix X can be obtained:

where $x_{ij}$ is the value of the i-th alternative and the j-th decision indicator, m is the number of options, and n is the number of decision indicators.

The X matrix is normalized to the relative coefficient matrix R, which can be expressed as (Step 1):

$$R={\left[r_{ij}\right]}_{m\times n}={\left[\frac{x_{ij}}{{\sum }_{i=1}^m{x}_{ij}}\right]}_{m\times n}=\left[\begin{array}{c}\begin{array}{cc}0.15858& 0.19220\\ \end{array}\\ \begin{array}{cc}0.17504& 0.17562\\ \end{array}\\ \begin{array}{cc}0.16551& 0.15740\\ \end{array}\\ \begin{array}{c}\begin{array}{cc}0.24827& 0.20090\\ \end{array}\\ \begin{array}{cc}0.25260& 0.27389\\ \end{array}\\ \end{array}\\ \end{array}\right].$$ (6)

where $r_{ij}$ is the value of the relative coefficient matrix on the i-th alternative and the j-th decision indicator.

When we set ω_PCF = ω_SEA = 0.5, the weighted normalized decision matrix can be expressed as (Step 2):

$$D={\left[y_{ij}\right]}_{m\times n}={\left[r_{ij}\cdot {\omega }_j\right]}_{m\times n}=\left[\begin{array}{c}\begin{array}{cc}0.07929& 0.09610\\ \end{array}\\ \begin{array}{cc}0.08752& 0.08781\\ \end{array}\\ \begin{array}{cc}0.08275& 0.07870\\ \end{array}\\ \begin{array}{c}\begin{array}{cc}0.12413& 0.10045\\ \end{array}\\ \begin{array}{cc}0.12630& 0.13694\\ \end{array}\\ \end{array}\\ \end{array}\right],$$ (7)

where $y_{ij}$ is the normalized value of the i-th optional and the j-th decision indicator and ${\omega }_j$ is the weight of the j-th indicator.

SEA is a beneficial attribute, while PCF is a nonbeneficial attribute (Step 3). The relative priorities Q_i can be expressed as follows (Step 4):

$$\left[Q_i\right]=\left[{\sum }_{j=1}^n{y}_{+ij}+\frac{{min}_{i=1,2,\dots ,m}\left({\sum }_{j=1}^n{y}_{-ij}\right)\cdot {\sum }_{i=1}^m{\sum }_{j=1}^n{y}_{-ij}}{{\sum }_{j=1}^n{y}_{-ij}\cdot {\sum }_{i=1}^m\left(\frac{{min}_{i=1,2,\dots ,m}\left({\sum }_{j=1}^n{y}_{-ij}\right)}{{\sum }_{j=1}^n{y}_{-ij}}\right)}\right]={\left[0.21715,0.19747,0.19468,0.17777,0.21294\right]}^T,$$ (8)

Where $y_{+ij}$ and $y_{-ij}$ are weighted values of beneficial and nonbeneficial attributes, respectively.

The quantitative utility value U_i is used to rank schemes, which can be expressed as follows (Step 5):

$$\begin{array}{cc}\left[U_i\right]& =\left[\frac{Q_i}{max{\left(Q_i\right)}_{i=1,2,\dots ,m}}\right]\\ & ={\left[1,0.90937,0.89652,0.81865,0.98061\right]}^T.\end{array}$$ (9)

The structure with the highest U_i value has the highest degree of utility.

Figure 10 shows the U_i value with the given weight factor ω_PCF from 0.1 to 0.9. The crashworthiness of the P-TPMS-reinforced tube is better than that of G-TPMS, IWP-TPMS, and diagonal tubes. In addition, when the weight of PCF is <0.475, the crashworthiness of the polygonal reinforced tube is optimal. However, when the weight is larger than 0.475, the crashworthiness of P-TPMS is optimal. Thus, the P-TPMS-reinforced tube had better crashworthiness performance than the polygonal tube with the demand for lower PCF increases. Moreover, all three TPMS-reinforced tubes show an upward trend with the increase of ω_PCF, while the multicell tubes show a downward trend. This indicates that the TPMS-reinforced structures have a better effect on reducing PCF than the traditional multicell tubes due to the unique structures of TPMSs.

FIG. 10.

Relationship between the quantitative utility U_i and weight $\omega$ of PCF.

The simulation load–displacement curves of five tubes are shown in Figure 9. It can be found that the force fluctuations of TPMS tubes are smooth, while the ones of traditional multicell tubes are severe, which can be quantized on the standard deviation in Table 4. It means that TPMS-reinforced tubes are stronger than the traditional multicell tubes.

Conclusions

The article constructed three novel types of TPMS (P-TPMS, G-TPMS, and IWP-TPMS)-reinforced tubes, which can greatly improve the crashworthiness of traditional multicell tubes. The lower PCF of TPMS-reinforced tubes, especially the optimized P-TPMS tube, shows the potential advantages of 3D printed tubes with complex curved surfaces.

Moreover, due to the controllability of TPMS-reinforced tubes, the shape of TPMS structures and the thickness of TPMS-reinforced tubes can be optimized. The optimized parameters of three TPMS-reinforced tubes under different conditions were obtained through the NLPQL method.

To sum up, the crashworthiness and lightweight design study of reinforced tubes has certain significance for protecting personal safety, saving automobile costs, and reducing energy consumption.

Nomenclature

A

import material constant a in simulation

a ₀

unknown constant term coefficients of RSM function

B

import material constant b in simulation

b_i

unknown first-order term coefficients of RSM function

C

shape parameter

c

import material constant c in simulation

$c_{ij}$

unknown quadratic mixed term coefficients of RSM function

D

weighted normalized decision matrix of COPRAS

d_i

unknown cubic term coefficients of RSM function

E

Young's modulus

e_i

unknown quadratic term coefficients of RSM function

m

number of independent variables used for RSM function and the number of options in COPRAS

N

import material constant n in simulation

n

number of decision indicators in COPRAS

Q

relative significance or priority matrix of COPRAS

R

relative coefficient matrix of COPRAS

r

coordinate vector in the rectangular coordinate system of space

$r_{ij}$

value of relative coefficient matrix on the i-th alternative and the j-th decision indicator

t

thickness parameter

U

quantitative utility

X

initial matrix for the procedure of COPRAS

x

displacement variable

x_i

the i-th component of the independent variable in RSM function

$x_{ij}$

value of initial decision matrix on the i-th alternative and the j-th decision indicator

$y_{ij}$

normalized value of the i-th optional and the j-th decision indicator

$y_{+ij}$

weighted value of nonbeneficial attribute (positive value in $y_{ij}$)

$y_{-ij}$

weighted value of nonbeneficial attribute (negative value in $y_{ij}$)

Greek Symbols

$\delta$

total compression displacement

$\mu$

Poisson's ratio

ϕ _P

trigonometric function of P-TPMS

ϕ _G

trigonometric function of G-TPMS

ϕ _IWP

trigonometric function of IWP-TPMS

$\omega$

given weight factor in optimization

${\omega }_j$

weight of the j-th indicator

ω _PCF

weight of PCF

ω _SEA

weight of SEA

Appendices

Appendix Table A1.

 Optimized Results of Schwarz's Primitive-Triply Periodic Minimal Surface-Reinforced Tubes

$\omega$	C	t (mm)	PCF (kN)	SEA (J/g)	Objective
0.1	0.031	0.961	44.2	23.0	−16.28
0.2	0.052	0.823	34.1	21.1	−10.07
0.3	0.060	0.650	25.8	18.8	−5.44
0.4	0.049	0.650	25.8	18.8	−0.98
0.5	0.040	0.650	25.8	18.8	3.48
0.6	0.041	0.650	25.7	18.8	7.93
0.7	0.029	0.650	25.7	18.7	12.37
0.8	0.011	0.650	25.7	18.6	16.82
0.9	−0.010	0.650	25.7	18.5	21.24

ω, given weight factor; C, shape parameter, t, thickness parameter; PCF, initial peak crashing force; SEA, specific energy absorption; Objective, objective function value of optimization.

Appendix Table A2.

 Optimized Results of Schoen's Gyroid-Triply Periodic Minimal Surface-Reinforced Tubes

$\omega$	C	t (mm)	PCF (kN)	SEA (J/g)	Objective
0.1	−0.063	1.301	84.7	30.1	−18.61
0.2	−0.071	0.828	46.1	23.4	−9.49
0.3	−0.071	0.650	34.1	20.0	−3.77
0.4	0.347	0.650	22.8	13.1	1.24
0.5	0.334	0.650	22.5	12.9	4.81
0.6	0.321	0.650	22.4	12.8	8.33
0.7	0.321	0.650	22.4	12.7	11.85
0.8	0.320	0.650	22.4	12.7	15.36
0.9	0.321	0.650	22.4	12.7	18.87

Appendix Table A3.

 Optimized Results of Schoen's iWP-Triply Periodic Minimal Surface-Reinforced Tubes

$\omega$	C	t (mm)	PCF (kN)	SEA (J/g)	Objective
0.1	−0.264	1.260	76.5	27.0	−16.67
0.2	−0.281	0.796	42.6	21.2	−8.45
0.3	−0.294	0.650	33.0	18.5	−3.05
0.4	−0.312	0.650	32.5	18.2	2.06
0.5	−0.340	0.650	31.8	17.6	7.07
0.6	−0.350	0.650	31.5	17.3	11.96
0.7	−0.350	0.650	31.5	17.3	16.84
0.8	−0.350	0.650	31.5	17.3	21.73
0.9	−0.350	0.650	31.5	17.3	26.61

Author Contributions

Y.J. contributed to the conception of the study and wrote the manuscript. S.W. contributed significantly to analysis and manuscript preparation. H.W. performed experiments and equipment operations. B.L. performed the data analyses and put forward constructive comments. W.H. helped perform the analysis with constructive discussions.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This research was supported by the National Natural Science Foundation of China (Grant Nos. 61772104, 61720106005, 61772105, 61936002, and 61907005) and the Fundamental Research Funds for Central Universities (DUT20TD107, DUT20JC32, and 2019RQ106).