Performance Study of Origami Crash Tubes Based on Energy Dissipation History

Abstract

Thin-walled tubes are widely used as energy-absorbing components in traffic vehicles, which can absorb part of the energy in time by using the plastic deformation of the components during collision so as to reduce the damage of the vehicle body and improve the overall safety and reliability of traffic vehicles. The prefolded design of thin-walled tube components can guide it to achieve the ideal energy dissipation performance according to the preset damage path, so the related research based on origami tubes has attracted a lot of attention. Since the geometry of the origami tubes is controlled by many parameters and stress and deformation is a complex nonlinear damage process, most of the previous studies adopted the method of case analysis to carry out numerical simulation and experimental verification of the relevant influence parameters. This paper makes a new exploration of this kind of problem and focuses on solving the related technical problems in three aspects: 1. The automatic model modeling and 3D display based on parameters are proposed; 2. System integration using Python programming to automatically generate the data files of ABAQUS for finite element simulation was realized, and we sorted the finite element analysis results into an artificial intelligence analysis data set; 3. Clustering analysis of the energy consumption history of the data set is carried out using a machine learning algorithm, and the key design parameters that affect the energy consumption history are studied in depth. The sensitivity of the energy absorption performance of the origami tubes with multi-morphology patterns to the crease spacing is studied, and it is shown that the concave–convex crease spacing distribution with a distance larger than 18 mm could be used to activate specific crushing modes. In the optimal case, its initial peak force is reduced by $66.6\%$ compared to uniformly spaced creases, while the average crushing force is essentially the same. Furthermore, this paper finds a new path to optimizing the design of parameters for origami tubes including a multi-morphology origami pattern from the perspective of energy dissipation.

1. Introduction

As a kind of energy dissipation element, a thin-walled tube is widely used in transportation machinery such as vehicles [1,2], trains [3,4,5], helicopters [6] and so on. It absorbs the interference energy of equipment in a low-speed collision through plastic deformation and improves safety performance. Previous studies [7,8,9,10] have shown that the introduction of specific forms of defects on the surface of thin-walled tubes, such as slots, ripples and small windows, can effectively control the collapse mode and reduce the initial peak value but cannot obtain higher specific energy absorption at the same time.

Therefore, researchers [11] introduced the rigid origami pattern [12,13] to the thin-walled tube as another geometric defect to obtain a lower initial bending force and more uniform crushing force than the traditional thin-walled tube. Since then, research on the energy dissipation performance of this kind of prefolded thin-walled tube has been ongoing [14,15,16]. For example, by introducing a prefolded diamond pattern (a rigid origami pattern that could be spread out completely to a flat sheet) in the corners of the square original tube, Ma et al. [17] in their work improved Zhang’s [18] design, significantly reduced the sensitivity of tubes to imperfections and doubled the number of traveling plastic hinge lines, allowing it to absorb more energy during a collision. Zhou et al. [19] studied the energy absorption performance of origami tubes with different geometries, loading rates and tup masses based on low-velocity impact testing, and the three collapse modes were compared analytically. The results indicated that the complete diamond model was highly sensitive to geometric imperfections, and it was the most efficient one in terms of energy absorption. In addition, the mean crushing force decreases with the increase in the number of buckling points. Yang et al. [20] used a 3D printing technique to manufacture the proposed tubes and investigated the influence of the origami circular tube with two different isometric origami patterns (called diamond and full-diamond patterns) on energy dissipation performance through experiments and numerical analysis. They found that the initial peak force of origami tubes would be significantly reduced, whereas its energy absorption capacity could be improved or maintained. Wang et al. [21] introduced appropriate geometric imperfection into the finite element model and focused on the imperfection-sensitivity of origami tubes. Parametric analysis indicates that an appropriate geometry contributed to improving the compliance of the origami tubes, which led to stable collapse behavior with higher energy absorption performance. Moreover, they proposed a bulkhead-reinforced origami crash box as a low imperfection-sensitive energy absorption device. One class of square thin-walled tubes with equal-spaced trapezoidal crease patterns was carefully designed by Zhou et al. [22]. The quasi-static numerical simulation revealed that the complete diamond pattern was successfully triggered when three key parameters (modulus M, dihedral angle $\theta$ and area ratio $\omega$) were selected appropriately. Compared with the diamond origami crash boxes, the trapezoid origami tubes were the most desirable in terms of energy absorption. Yuan et al. [23] designed and analyzed the effect of origami triggers on various (triangular, square, hexagonal and decagonal tube) profiles, including tapered ones. Through a few quasi-static impact experiments and numerical simulations, it was demonstrated that these origami tubes collapsed into a diamond-shaped mode with a doubled number of traveling plastic hinge lines. Meanwhile, a significant increase in overall energy absorption and an effective reduction in peak reaction forces. Wang et al. [24] introduced the origami structure into the train collision safety design of high-speed trains and proposed a combined multi-cell thin-walled energy-absorbing structure made from aluminum, which was optimized using the NSGA-II algorithm. It was shown that the optimal structure is of great advantage in improving the initial peak crushing force (IPCF) for the energy absorption structure. In addition, Acar and Altin [25,26] further explored the optimization of the crash performances for thin-walled tubes. Meanwhile, a variety of indexes [27,28] were used to evaluate the energy absorption properties of prefolded thin-walled tubes. It included the initial peak force ($P_{\max }$), total energy dissipation (${\mathit{E}}_{\mathrm{total}}$), the mean crushing force (${\mathit{P}}_{\mathrm{m}}$), the efficiency of crushing force (ECR), the specific energy absorption SEA and initial elastic stiffness (IES). Overall, these works have collectively contributed to the current understanding of the advantages of employing origami patterns. However, the effectiveness of different origami patterns and various tube shapes varies, even though the initial peak force was seen to be reduced to some extent. Moreover, the study [29,30] of origami tubes with multi-morphology patterns and corresponding theoretical efforts and analyses are less developed. Therefore, exploring a new approach that combines the versatility of multi-morphology origami patterns and the multiple possibilities of combinations to find the response relationship between energy dissipation performance and multiple parameters is the focus of this study.

One possible solution is to automatically mine the abstract mapping relationship within the data from the complex data results with the help of artificial intelligence and reasonable machine learning methods. In particular, since LeCun [31] published the revolutionary research in 2015, the research and application of machine learning have entered a new era. For example, Takiyama [32] used a deep convolution neural network to classify medical images. Goldberg [33] studied the natural language processing technology based on neural networks. Nabian et al. [34] used deep learning technology to study the reliability of transportation networks. Lin et al. [35] and Zhang [36] proposed a convolution network that can be used to detect the damage to vibrating structures. Lei et al. [37] developed a real-time topology optimization method driven by machine learning. Based on machine learning methods, Rafiei and Adeli [38,39,40] successively proposed a new model for estimating the sales price of real estate units and a new algorithm for damage detection of high-rise buildings and developed an unsupervised learning method for a global and local health assessment of structures. In addition, Yu et al. [41] recently applied machine learning technology to the deterioration analysis of cementitious composites and then achieved high-fidelity and efficient stochastic chemophysical modeling. YÄślmaz et al. [42] used machine learning tools to predict the Rayleigh damping coefficients of the free-metal sheet damping coating layer based on thicknesses. Kunwar et al. [43] combined finite element numerical simulation with neural network analysis to unravel the partial of the growth mechanism of the $C{u}_{6}S{n}_5$ intermetallic compound (IMC) under thermomigration. The above methods have achieved good results in their respective applications; therefore, more researchers in different fields should introduce machine learning to help exploration and discovery.

In this paper, we use a prefolded multi-morphology pattern to design origami tubes. By combining parametric numerical simulations with machine learning methods, an analytical method is explored for the energy absorption of origami tubes with multi-morphology patterns. The structure of the paper is as follows: firstly, the multi-morphology patterns shape and parametric numerical simulation method of the origami tubes and the specific analysis paths are introduced in the second section. Then, in the third section, the machine learning methods and processes used in the research are described. Subsequently, in Section 4, the relationship between the energy dissipation performance and parameters of the origami tube with multi-morphology patterns is analyzed based on the results after the machine learning analysis, and the effect of multi-morphology patterns on the collapse pattern and the applicability of the method are discussed. Finally, the conclusion of this paper is summarized in the fifth section.

2. Numerical Simulation

The proposed analysis method is shown in Figure 1. For efficient identification of the spatial attitude of the origami tubes with multi-morphology patterns and building an accurate finite element model, a visualized 3D geometric model is first built based on the uniform parametric mode, and the numerical model is constructed according to the selected finite element software and geometric model parameters, as well as the accurate finite element simulation. Next up, the calculated values are normalized and cleaned down to obtain a data set. Finally, during the data analysis step, a machine learning algorithm is used to calculate and analyze the data set. Note that the reason for choosing the crushing force–displacement history curves as a sample of the data set is that compared to the usually combined metrics such as $P_{max}$ and $P_m$, the crushing force–displacement history curve is the most direct characterization between the geometric parameters of creases and the material properties and energy absorption capacity.

2.1. 3D Geometry Model

In Figure 2, the multi-morphology pattern used in this article is shown. According to the research of H.Wang [29,44], it has the characteristics of non-uniform interval distribution and multiple geometric patterns, and the origami tube based on a crease graph can be constructed by a uniform parameterization method. The crease pattern illustrated in Figure 2a is introduced into the tube’s surface. When each layer of the crease pattern is finally folded and closed in the circumferential direction, various origami tubes, as shown in Figure 2c, are constructed in the space. According to the unified parameter method reported in [44], the spatial position of each vertex ${\mathit{V}}_{\mathit{ij}}$ can be represented by five parameters, namely, the total length ($\mathit{s}$) of the unfolding origami, the height (${\mathit{h}}_{\mathit{j}}$) between adjacent layers, the number of layers ($\mathit{m}$), the number of folds of each layer ($\mathit{n}$) and a side length at the corresponding position of each layer (${\mathit{a}}_{\mathit{j}}$ or ${\mathit{b}}_{\mathit{j}}$), as shown in Figure 2a,c. In the cylindrical coordinate system, the vertex ${\mathit{V}}_{\mathit{ij}}({\mathit{R}}_{\mathit{ij}},{\theta }_{\mathit{ij}},{\mathit{Z}}_{\mathit{ij}})$ can be expressed as the following relationship:

$$\begin{array}{ccc}\hfill & R_{ij}=\frac{\sqrt{{\left(\frac{s}{n}\right)}^2-a_j\left(\frac{s}{n}-a_j\right)\left(1-cos\frac{\pi }{n}\right)}}{2sin\frac{\pi }{n}}\hfill & \hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill & {\theta }_{ij}=\left\{\begin{array}{cc}\hfill \frac{i-1}{n}\pi +\frac{a_j}{2},& i=\mathrm{even}\hfill \\ \hfill \frac{i}{n}\pi -\frac{a_j}{2},& i=\mathrm{odd}\hfill \end{array}\right.\hfill & \hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill & Z_{ij}=\left\{\begin{array}{cc}0,\hfill & j=1\hfill \\ \sum _2^j\sqrt{{\left(h_j^{j+1}\right)}^2-\frac{{(a_j-a_{j+1})}^2}{4}{tan}^2\frac{\pi }{2n}},\hfill & 1<j<m\hfill \end{array}\right.\hfill & \hfill \end{array}$$

(3)

after the linear transformation, ${\mathit{V}}_{\mathit{ij}}^{\mathit{Z}}$ can be obtained in a Cartesian coordinate system:

$$\begin{array}{ccc}\hfill V_{ij}^Z=& (X_{ij}^Z,Y_{ij}^Z,Z_{ij}^Z)\hfill & \hfill \\ \hfill =& \left[\begin{array}{ccc}{R}_{ij}& R_{ij}& 1\end{array}\right]\left[\begin{array}{ccc}cos\theta & 0& 0\\ 0& sin\theta & 0\\ 0& 0& Z_{ij}\end{array}\right]\hfill & \hfill \end{array}$$

(4)

the key fold shown in Figure 2d can be expressed as follows:

$$\begin{array}{ccc}\hfill & \Delta T_1=\Delta T_2=\frac{|a_j-a_{j+1}|}{2}tan\frac{\pi }{2n}\hfill & \hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill & {\lambda }_{ij}=arccos\left(\frac{|a_j-a_{j+1}|}{2h_j^{j+1}}tan\frac{\pi }{2n}\right)\hfill & \hfill \end{array}$$

(6)

Here, it should be noted that the corresponding zone between the two adjacent layers of the closed-loop tube can be folded flat along the axial direction only if the deployed height between the two adjacent layers meets the relationship $h_j^{j+1}=\Delta T_1=\Delta T_2$.

To highlight the analytical methods, the crease pattern spacing distribution (spacing height ($h_j$)) is chosen as the main control factor of the multi-morphology patterns’ shape to study the energy dissipation ability of the multi-morphology patterns origami tube. Based on this, a total of 33 origami tubes were designed, numbered D2-1∼33, where the crease with uniformly spaced distribution D2-1 is a benchmark. Specifically,

Table 1. Distribution of crease parameters and numerical results.

Model	${\mathit{h}}_1$	${\mathit{h}}_2$	${\mathit{h}}_3$	${\mathit{h}}_4$	${\mathit{P}}_{\mathbf{max}}$	${\mathit{P}}_{\mathbf{max}}$	${\mathit{P}}_{\mathbf{m}}$	${\mathit{P}}_{\mathbf{m}}$	Cluster	Collapse
	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{kN}\right)$	Reduction	$\left(\mathbf{kN}\right)$	Increase		Mode
D2-1	30	30	30	30	19.3	-	20.0	-	3	D ${}^{\mathrm{a}}$
D2-2	40	20	40	20	12.6	34.8%	15.7	−21.5%	4	M ${}^{\mathrm{b}}$
D2-3	45	30	15	30	7.9	59.0%	19.3	−3.6%	5	M
D2-4	30	45	30	15	11.5	40.3%	16.8	−16.0%	6	M
D2-5	30	45	15	30	8.0	58.3%	16.7	−16.6%	8	M
D2-6	45	15	40	20	8.2	57.6%	18.8	−6.2%	4	M
D2-7	40	20	45	15	12.5	35.3%	15.0	−25.3%	4	M
D2-8	30	40	30	20	19.3	−0.3%	18.1	−9.5%	6	M
D2-9	40	30	20	30	12.0	37.9%	17.6	−12.3%	2	M
D2-10	20	45	40	15	12.5	34.9%	21.0	4.7%	1	M
D2-11	45	20	15	40	7.7	60.0%	19.8	−1.0%	1	M
D2-12	45	45	15	15	8.0	58.6%	19.9	−0.6%	5	M
D2-13	40	20	30	30	12.2	36.8%	16.9	−15.5%	4	M
D2-14	30	30	40	20	20.2	−4.8%	18.2	−9.3%	3	M
D2-15	45	40	15	20	7.9	59.2%	18.9	−5.8%	5	M
D2-16	45	40	20	15	10.5	45.5%	17.9	−10.5%	8	M
D2-17	40	45	20	15	10.5	45.4%	18.6	−7.4%	8	M
D2-18	40	45	15	20	7.9	59.1%	17.4	−13.3%	8	M
D2-19	40	15	45	20	8.2	57.6%	18.4	−7.9%	2	M
D2-20	40	30	30	20	17.9	7.4%	19.4	−3.1%	6	M
D2-21	40	40	20	20	12.0	37.9%	17.0	−15.1%	2	M
D2-22	40	20	15	45	7.8	59.4%	19.9	−0.6%	7	M
D2-23	20	40	45	15	12.8	33.6%	19.7	−1.8%	3	M
D2-24	40	20	20	40	10.9	43.3%	19.1	−4.7%	2	M
D2-25	20	40	40	20	20.1	−4.3%	18.5	−7.6%	8	M
D2-26	45	15	15	45	7.5	60.9%	18.9	−5.4%	5	M
D2-27	15	45	45	15	12.6	34.8%	20.0	−0.3%	5	M
D2-28	45	15	30	30	8.0	58.4%	19.9	−0.7%	7	M
D2-29	30	30	45	15	12.7	33.9%	18.9	−5.8%	6	M
D2-30	45	30	30	15	11.4	40.7%	20.7	3.5%	6	M
D2-31	45	15	45	15	8.2	57.5%	17.8	−11.4%	4	M
D2-32	30	20	40	30	12.3	36.1%	16.3	−18.4%	2	M
D2-33	45	20	40	15	12.1	37.3%	15.0	−25.2%	4	M

^a D: diamond mode. ^b M: diamond mode and concertina mode.

lists the parameter settings for the crease pattern spacing of all 33 tubes, where each tube pattern has the same number of folds (n = 4), number of layers (m = 5), side length ($a_{1-5}$ = 0, 60 mm, 0, 60 mm, 0), total width (s = 240 mm) and total height (H = 120 mm) so that four crease patterns are formed in the area between the two adjacent layers, and the crease patterns of these 33 tubes have the same surface area, as shown in Figure 3 for five of the tubes. Note that the model for this size parameter was chosen to allow easy comparison with previous studies.

2.2. Finite Element Simulation

Machine learning technology needs to rely on a certain amount of data for support, and the modeling and post-processing of a single-folded tube takes a long time. Therefore, in this paper, a script program in Python language was compiled and substituted into Abaqus [30,45,46] for batch modeling and post-processing. The specific process in the geometric model part and the finite element part is shown in Figure 1. In the geometric model part, a program is compiled according to Algorithm 1, which can automatically calculate the $V_{ij}$ coordinates of the spatial vertices of the origami tube. Then, by loading the Mayavi [47] library, it can easily and succinctly visualize the 3D data of the geometric model, as shown in Figure 2c and Figure 3. In addition, the above operations can be used as the preprocessing of the next three-dimensional finite element generation method.

Algorithm 1: Compute 3D spatial coordinates of the vertex $V_{ij}$

1: Input: total length $\left(s\right)$; list of the heights for each floor $\left\{h\right\}$; number of layers $\left(m\right)$; 2:           number of prefolded points in each layer $\left(n\right)$;list of edge lengths for each layer $\left\{a\right\}$. 3: Output: $V_{ij}^Z$ 4: Initialize an empty array with size $\left(m,2\times n,3\right)$ 5: for each layer $i=1,\cdots ,m$ do 6:      Calculate the coordinate values of $X_{ij}$, $Y_{ij}$ and $Z_{ij}$, 7:      for each vertex in each layer according to Equations (1)–(4), 7:      $V_{ij}^Z=(X_{ij}^Z,Y_{ij}^Z,Z_{ij}^Z)$ 8: end for

In the finite element simulation, the commercial finite element analysis software Abaqus/Explicit [45,48,49,50] was used to simulate the axial crushing process of an origami tube under a quasi-static state. As shown in Figure 1, in order to apply the python script in Abaqus to generate the finite element model, it is necessary to add the custom data conversion algorithm shown in Algorithm 2 to the program with $V_{ij}$ coordinates compiled above to meet the data requirements when Abaqus generates the model. Figure 4 is a typical simulation of the axial crushing process. Specifically, the origami tube of the whole model was placed in the middle of two rigid plates, in which the lower steel plate was fixed as a support, and the compression process was realized by the axial downward displacement of the upper rigid plate. A smooth amplitude curve was defined in Abaqus to control the displacement loading rate of the upper steel plate. Here, note that the loading rate is controlled by the length of the tube, the remaining height after the collapse and the time of loading, so it is not constant. After the completion of axial crushing, all models had the remaining height of 35 mm [17] for study and comparison. The completely fixed boundary conditions were applied to the upper and lower rigid plate, and when the upper rigid plate moved down, only the degree of freedom in the moving direction was released. The three translational degrees of freedom of the upper end and the lower end of the origami tube were coupled with the upper and lower rigid plates. The stationary rigid panel was completely fixed, whereas all of the degrees of freedom of the moving rigid panel were constrained except for the translational one in the axial direction of the tube. Four-node shell elements with reduced integration $\mathrm{S}4\mathrm{R}$ were used to mesh the tube, supplemented by a few triangular elements ($\mathrm{S}3\mathrm{R}$) to avoid excessively small or distorted elements.

Algorithm 2 Data reconfiguration ( ${}^{\mathrm{a}}$${\left\{V_{ij}^Z\right\}}_m^T$ represents the transpose of row m for the ${\left\{V_{ij}^Z\right\}}_m$ array, the same as below. ${}^{\mathrm{b}}$ Permutation () is a method to exchange the order of alignment, for example: array $e{g}_1=\left[1,2,3,4\right] ,Permutation\left(e{g}_1\right)=\left[2,3,4,1\right]$.)

1: Input: the spatial coordinates array $\left\{V_{ij}^Z\right\}$ of $V_{ij}$; number of layers $\left(m\right)$; 2:           number of prefolded points in each layer $\left(n\right)$. 2: Output: the array $\left\{W_{ij}\right\}$ 3: Initialize an empty array with size $\left(m,2\times n,3\right)$ 4: for $i=1,\cdots ,m$-1 do 5:      Initialize an empty array $\left\{N\right\}$ with size $\left(2\times n,4\right)$ 6:      $\left\{N\right\}=\left({}^{\mathrm{a}}{\left\{V_{ij}^Z\right\}}_m^T,{}^{\mathrm{b}}Permutation\left({\left\{V_{ij}^Z\right\}}_m^T\right)\right.,$ $\left.{\left\{V_{ij}^Z\right\}}_{m+1}^T,Permutation\left({\left\{V_{ij}^Z\right\}}_{m+1}^T\right)\right)$ 7:      ${\left\{W_{ij}\right\}}_m=\left\{N\right\}$ 8: end for

In addition, the interface point along the thickness direction of the element was set to 5 to improve the accuracy. When defining the contact, the upper and lower rigid plate and the origami tube wall adopt the surface contact pair, the self-contact pair was used between various parts of the origami tube and the friction coefficient was $0.25$ [51]. In terms of materials, the soft steel with the same parameters and properties as that in the study [17,51] was used. The specific mechanical properties of the material are as follows: $\rho =7800$ kg; Young’s modulus Elastic $E=210$ GPa; Poisson’s ratio $\upsilon =0.3$; ${\sigma }_y=200$ MPa; ${\sigma }_u=400$ MPa; ultimate strain ${\epsilon }_u=0.2$; strain strengthening index $n=0.34$. The stress–strain relationship curve obtained by the tensile test is shown in Figure 5.

Different from the previous models, the origami tube in this paper uses the full model, and the height between the layers was not equal; therefore, before the numerical model analysis, the grid density convergence test and analysis were carried out to ensure the accuracy of the results. As in the study of [17,20], it was verified using the following principle recommended by Abaqus [45]: in most processes, the kinetic energy of the deformed material does not exceed $5\%$ of its internal energy to ensure a correct quasi-static response. The ratio of pseudo-strain energy to total energy or plastic dissipation is less than $5\%$; otherwise, it is necessary to increase the grid density to reduce the effect of hourglass stiffness in the results.

Tube D2-1 was selected as the representative of the component, and the mesh size decreases from 3 mm to 1 mm. The force and displacement curve in the compression process is shown in Figure 6. It can be seen that the curve profile changes significantly when the mesh size was larger than 1.5 mm, but when the grid size was 1.0–1.5 mm, they were similar. In addition, when the mesh size changed in 1.2–1.5 mm and 1.0–1.5 mm, the average attenuation amplitude of crushing force was less than $5\%$. Combined with the accuracy and calculation time, the mesh size of 1 mm and 0.02 s analysis time was selected to establish the finite element model.

2.3. Numerical Simulation Verification and Energy Absorption Evaluation

In terms of numerical simulation validation, the A1-1 model in [17] was used to verify that the specific geometric size of the model was 240 mm and the thickness was 1 mm. As shown in Figure 7a, the material physical properties were the same as above. After modifying the python compiler, the finite element model can be quickly established, and the numerical model analysis can be completed at the same time. The results are compared as shown in Figure 7b. As can be seen in the figure, the two matched well on the whole, and there was only a slight difference in the second half because a symmetrical $1/2$ model was used for A1-1, and its sensitivity to decimal error is lower than that of the full three-dimensional model [11]. However, it has little effect on the overall results, which is similar to that found in [17,22]. In addition, in terms of average crushing force, the A1-1 model and comparison model were 19.03 kN and 19.15 kN, respectively, which also show good consistency.

In practice, there were two typical methods [19,20,23] to practically manufacture the designs in this paper. One used a 3D printing technique to manufacture it. The other one used a male and female mould to stamp the sheet, the stamped sheet was folded along the creases to form a half-tube with the guide of a half-tube mould, then two half-tubes are connected by riveting or welding, and the completed tube is heat treated to release the residual stresses created during the folding.

In order to better understand the influence of crease pattern spacing distribution on tube energy consumption, a variety of evaluation criteria were adopted in this paper. In addition to the initial peak force ($P_{\max }$) and average crushing force ($P_{\mathrm{m}}$) used in study [17,20,22,23,52], the criteria reported in [29] were also selected: the efficiency of crushing force (ECR) and initial elastic stiffness (IES) sum to evaluate the stability and initial stiffness of the load-bearing capacity of the origami-based tube. The formula for all indicators is defined as follows:

$$\begin{array}{ccc}\hfill & P_m=\frac{{\int }_0^{\delta }F\left(s\right)ds}{\delta }\hfill & \hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill & ECR=\frac{P_m}{P_{max}}\hfill & \hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill & IEC=\frac{P_{max}}{{\delta }_i}\hfill & \hfill \end{array}$$

(9)

where $\delta$ is the final breaking distance, and ${\delta }_i$ is the crushing distance corresponding to the initial peak force $P_{\max }$.

3. Crushing Force–Displacement History Curve Clustering

As shown in Figure 8a, the introduction of an origami pattern significantly changes the mechanical properties of thin-walled tubes, and the change in crease geometric parameters would also affect their energy absorption capacities [11,17,22,53], as shown in the different crushing force–displacement curves. As shown in Figure 8b, with the change in crease parameters, the load–displacement curve and collapse mode change, and some of the curve changes show consistency. This study is to distinguish different types of energy consumption according to the crushing force–displacement history curve and then observe the response of the change in crease parameters. At the same time, because of the nature of the data, that is, the crushing force–displacement data obtained from the quasi-static axial compression simulation, it is actually a kind of time series. A targeted and appropriate processing method should have the following characteristics: firstly, it can learn effective discriminative information from existing data, and secondly, it can play a role in the absence of label training. Because the number of categories was not known beforehand, one of our goals was to reduce manual work in classification. Finally, it is more convenient to interpret the processing results. The resulting method should be able to automatically identify possible categories and should be easy to verify. Next, an unsupervised time series clustering method was used and shows that it could meet the requirements.

3.1. Clustering Method

The goal of data clustering (or clustering analysis) is to find its natural grouping in a set of objects (patterns, data points) [54]. When there is no feedback, unsupervised clustering can be used to realize the analysis. This paper needs to deal with the crushing force–displacement data with time tags, so it will involve two aspects: similarity measurement and clustering method.

3.1.1. Similarity Measurement between Different Curves

As shown in Figure 9a, the simplest distance measurement between two sequences is usually the Euclidean distance, but this distance does not accord with the general understanding of the true meaning of the sequence and is clumsy in capturing the similarity of different sequences. Therefore, study [55] proposes a time warping method (Time Warping) to correspond the points of a sequence at one time to multiple consecutive points at another time. Using the dynamic time warp distance (DTW) can flexibly capture the similarity of the two sequences. In this paper, a dynamic time warp distance with smoothing (soft-DTW) to measure the similarity between different data of time series was chosen, which is obtained by smoothing the recursion of Bellman in DTW:

$$\begin{array}{cc}\hfill L_{min}\left(i,j\right)=& min\left\{\begin{array}{c}{L}_{min}\left(i,j-1\right)\hfill \\ L_{min}\left(i-1,j\right)\hfill \\ L_{min}\left(i-1,j-1\right)\hfill \end{array}\right\}+\delta \left(i,j\right)\hfill \end{array}$$

However, in terms of classification, it has better performance than the dynamic time warp distance DTW [56], and it allows measuring time series data of different lengths. In addition, as shown in Figure 9b, the alignment loss is the length of the path (obtained from the sum of the elements on the path), where DTW considers the optimal alignment (orange circle path directed by the red arrow ), Euclidean alignment is diagonal path (gray arrow path), and soft-DTW considers all possible alignment matrices of delannoy $\left(n-1,m-1\right)$. Compared with DTW, soft-DTW considers all the soft minimum values (soft-minimum) that may cross the path cost in the alignment of two time series and is proven as a differentiable (differentiable) function [57] so that it can be better handled in the clustering iteration. The soft-DTW distance function is defined as:

$$\begin{array}{ccc}\hfill & {\mathbf{dtw}}_{\gamma }\left(x,y\right):={\min }^{\gamma }\left\{\langle A,\Delta \left(x,y\right)\rangle ,A\in A_{n,m}\right\}\hfill & \hfill \end{array}$$

(10)

where, $\gamma \ge 0$ is the smoothing parameter, and the time series $x=\left(x_1,\cdots ,x_n\right)\in R^{p\times n}$, $y=\left(y_1,\cdots ,y_n\right)\in R^{p\times m}$. In addition, $\Delta \left(x,y\right):={\left[\delta \left(x_i,y_j\right)\right]}_{ij}$ is a matrix composed of losses at the corresponding positions of time series x and y. The loss can be the Euclidean distance (ED) or Manhattan distance. ED is used in this paper. $A\in A_{n,m}\subset {\left\{0,1\right\}}^{n\times m}$ is a set of (binary) alignment matrices using the inner product $\langle A,\Delta \left(x,y\right)\rangle$ of A and $\Delta \left(x,y\right)$ as the value of the alignment path cost. In addition, the generalized minimum operator ${\min }^{\gamma }$ is defined as follows:

$$\begin{array}{ccc}\hfill & {\min }^{\gamma }\left\{a_1,\cdots ,a_n\right\}:=\left\{\begin{array}{cc}{\min }_{i\le n}{a}_i,\hfill & \gamma =0\hfill \\ -\gamma log\sum _{i=1}^n{e}^{-a_i/\gamma },\hfill & \gamma >0\hfill \end{array}\right.\hfill & \hfill \end{array}$$

(11)

here, $\gamma =0$ is the dynamic time warp distance (DTW).

3.1.2. Unsupervised Learning-Clustering Method

Different from online correlation analysis, the data clustering in this paper is actually a kind of offline analysis based on shape. Specifically, the overall shape of the load–displacement curve is very important to the analysis, but the details of its generation time are not important, so the simple, efficient and popular K-means [58,59,60] method was chosen, and it has strong adaptability. Usually, the K-means algorithm can be expressed as follows: first, set a cluster number k, according to the number of categories, select k initial center points from the sample data set; then, according to the similarity measure between the data, assign each data object to the class to which the nearest center point belongs, thus forming k clusters. After all the objects were assigned, the average value was calculated as the new central point value of the cluster according to the data sample objects contained in each cluster, and the two steps of assignment and update were repeated to iteratively update the distribution of the cluster until the distribution of samples in the cluster did not change. This paper divides the time series data objects and uses soft-DTW as the distance metric function between different samples, so the goal of the K-means clustering algorithm can be expressed as follows:

$$\begin{array}{ccc}\hfill & arg\underset{x}{min}J\left(x,y\right)=\underset{x_1,\cdots ,x_k\in R^{p\times n}}{\mathrm{argmin}}\sum _{i=1}^N\frac{1}{m_i}\underset{j\in \left[\left[k\right]\right]}{min}{\mathbf{dtw}}_{\gamma }\left(x_j,y_i\right)\hfill & \hfill \end{array}$$

(12)

where $\left[\left[k\right]\right]=\left\{0,\cdots ,k\right\}$ is the cluster label, $x_1,\cdots ,x_k\in R^{p\times n}$ is the cluster centroid time series array, $y_i$ is the i data sample, and $m_i$ is the length of $y_i$. It can be seen from Equation (12) that the K-means clustering algorithm selects k clustering centroids through continuous iteration, divides the whole data set into k disjoint subsets according to these k clustering centroids and requires the k subsets to be as close to the clustering centroids as possible. In addition, considering the non-convexity of Equation (10), we use the L-BFGS-B [61,62] algorithm to iteratively calculate the centroid in the minimum cluster that was used. In this case, the gradient of Equation (10) is:

$$\begin{array}{ccc}\hfill & {\nabla }_x{\mathbf{dtw}}_x\left(x,y\right)={\left(\frac{\partial \Delta \left(x,y\right)}{\partial x}\right)}^{T}E\hfill & \hfill \end{array}$$

(13)

where, according to the backward recursive algorithm [61], E is:

$$\begin{array}{ccc}\hfill E=& \left[e_{i,j}\right]=\left[\begin{array}{ccc}{e}_{i+1,j}& e_{i,j+1}& e_{i+1,j+1}\end{array}\right]\left[\begin{array}{c}{e}^{\frac{1}{\gamma }\left(r_{i+1,j}-r_{i,j}-{\delta }_{i+1,j}\right)}\\ e^{\frac{1}{\gamma }\left(r_{i,j+1}-r_{i,j}-{\delta }_{i,j+1}\right)}\\ e^{\frac{1}{\gamma }\left(r_{i+1,j+1}-r_{i,j}-{\delta }_{i+1,j+1}\right)}\end{array}\right]\hfill & \hfill \end{array}$$

(14)

where $r_{i,j}=\delta \left(x_i,y_j\right)+{\min }^{\gamma }\left\{\begin{array}{c}{r}_{i-1,j-1}\\ r_{i,j-1}\\ r_{i-1,j}\end{array}\right\}$ is the alignment loss between time series $x=\left(x_1,\cdots ,x_n\right)\in R^{p\times n}$ and $y=\left(y_1,\cdots ,y_n\right)\in R^{p\times m}$ different data points, which can be calculated recursively by Bellman. Here, the pseudocode of K-means time series clustering is shown in Algorithm 3.

Algorithm 3 Based on K-Means time series clustering ( ${}^{\mathrm{a}}$${\lambda }_i$ is the weight of each datum within the same cluster.)

1: Input: All time series data $\left\{T_N\right\}$ and number of clusters k 2: Output: The k clusters of data after clustering 3: Randomly initialize clustering centers $x_1,\cdots ,x_k\in R^{p\times n}$, by using k-means++ [63] algorithm 4: repeat 5:      for $i=1,2,\cdots ,N$ do 6:          Calculate the distance between each data and each clustering center according to Equation 12 7:          and assign the data to the cluster whose distance is closest to the clustering center. 8:     for  $j=1,2,\cdots ,k$ do 9:         Using L-BFGS-B algorithm, according to Equations (13) and (14) and ${}^{\mathrm{a}}$$\underset{X\in R^{p\times n}}{\min }{\sum }_{i=1}^n\frac{{\lambda }_i}{m_i}{\mathbf{dtw}}_{\gamma }\left(x,y_i\right)$ 10:        recalculate the clustering center $x_j$ of data $\left\{T_N\right\}$. Then update clustering center. 11: untill The clustering center does not change

3.1.3. Number of Clusters Setting

The K-means clustering algorithm needs to set the value of k first, which requires prior knowledge to set the value, which will have a great impact on the result. If the value of k is too small, the clustering effect is not significant, but if the value of k is too large, it will seriously deviate from the real number of clusters, thus losing the meaning of clustering. Therefore, the value of k is determined comprehensively by the elbow method [64] and contour coefficient [65]. The measurement standard of the elbow method is the sum of the square errors SSE. Observe the k-SSE line chart drawn, when the initial k is very small, the increase in k will make the value of SSE decrease rapidly; when k increases to a certain value, the decreasing speed of SSE will slow down, and they tend to be stable. At this time, select the value near the inflection point (elbow) in the image as the initial value of k, so a better clustering effect can be obtained. SSE is defined as:

$$\begin{array}{ccc}\hfill & SSE=\sum _{i=1}^k\sum _{p\in C_i}{\left|p-m_i\right|}^2\hfill & \hfill \end{array}$$

(15)

where $C_i$ represents the i cluster, p is the sample in $C_i$ and $m_i$ is the centroid of $C_i$.

However, not all cases are easy to find the inflection point of the mid-break line of the image. At this time, the selection of k needs the help of other methods because this paper uses an unsupervised clustering method. Therefore, the contour coefficient is selected as the auxiliary judgment standard. The silhouette coefficient is defined as:

$$\begin{array}{ccc}\hfill & S\left(i\right)=\frac{b_i-a_i}{max\left(b_i-a_i\right)}\hfill & \hfill \end{array}$$

(16)

here, the cohesion $a_i$ represents the average distance from the sample i to other samples in the same cluster, and the resolution $b_i$ represents the average distance from the sample i to all the samples in the nearest cluster $C_j$. $C_j$ can be expressed as:

$$\begin{array}{ccc}\hfill & arg\underset{C_k}{min}\frac{1}{n}\sum _{p\in C_k}{\left|p-X_i\right|}^2\hfill & \hfill \end{array}$$

(17)

the contour coefficients of all samples are averaged, i.e., $S=\frac{1}{n}{\sum }_{i=1}^{n}S\left(i\right)$. In general, when S takes the maximum value, the corresponding k is taken as the best clustering number.

3.2. Data and Operating Environment

In the process of numerical simulation, in order to obtain reasonable accuracy and a moderate amount of data, the sampling frequency of load and displacement data was set to $0.01\mathrm{MHz}$. As a result, the size of the two-dimensional time series (load–displacement curve) of a single model was $402\left(201\times 2\right)$, which met the requirements of cluster analysis and did not need downsampling to reduce the dimension. Finally, the size of the sample data set of the whole cluster analysis was 13,266 $\left(33\times 201\times 2\right)$.

In the computing platform environment, all the pre- and post-processing of the cluster analysis was carried out on the same platform. In terms of software, the main language of the program was Python, version Python3.7, and the packages Tensorflow, tslearn [66] and SciPy [67] were used to handle cluster analysis and weight initialization. Finally, the operating system is Windows 10. In addition, model computation and data processing in this paper were completed on a computer with an AMD Ryzen Threadripper 2950X 16-Core Processor 3.5 GHz and 128 GB DDR4 RAM. Indeed, such a hardware configuration is not always necessary, but better hardware will make the computations faster.

4. Results and Analysis

4.1. Number of Clusters Analysis

As shown in Figure 10, the average profile coefficient is largest when $k=2$, which indicates that the best cluster number is 2. However, it should be noted that the sum of the square of the clustering error SSE value is still very large, so this is an unreasonable cluster number. As can be seen from Equation (16), the reason is that the profile coefficient $S\left(i\right)$ takes into account the separation degree $b_i$; when the $S\left(i\right)$ is large, it is not necessarily the reason that the cohesion $a_i$ is small, or it may be that when both $b_i$ and $a_i$ are large, $b_i$ is much larger than $a_i$, then $a_i$ can take a large value, that is, the average distance between the sample and other samples in the same cluster will become larger, and the compactness inside the cluster will become weaker, so the sample in the cluster is far away from the cluster centroid. As a result, the square sum of the clustering error is larger. Therefore, $k=8$ was chosen as the cluster number, and the average profile coefficient was lower than that of $k=2$, but the SSE was already at a relatively low level, so the best cluster number of the load–displacement curve data set in this paper was 8.

4.2. Energy Absorption Performance Analysis of Each Cluster after Clustering

The results of the clustering for the model energy dissipation history curves are shown in Figure 11. With the constraints of imbalanced differences in the true distribution within the sample data and the volume of data, there are fewer models within Cluster 1, Cluster 3 and Cluster 7 but a higher similarity of curves within the clusters and fewer differences in the distance $\left(\delta \right)$ measured by the soft-DTW metric. The remaining clusters have a moderate number of internal curves and show good clustering of the curves, especially Cluster 2, Cluster 5 and Cluster 8, which exhibit good compactness not only in the first half of the force versus displacement curves, where they tend to be consistent, but also in the second half where the classification is relatively difficult. In addition, Cluster 4 is analyzed combined with Table 1 and Figure 11d, which reveals that the distribution of the crease spacing within the cluster exhibits a type of concave–convex spacing arrangement with one large and one small. Meanwhile, here, it should be noted that in this paper, the parameters of the model’s crease spacing are selected without corresponding previous knowledge, and the variation in each parameter is also random; however, the clustering can still produce good results.

From the initial peak force $\left(P_{\max }\right)$ in Table 1 and

Table 2. Numerical results in each cluster.

Model	${\mathit{P}}_{\mathbf{max}}$	${\mathit{P}}_{\mathbf{m}}$	IES	ECR	$\mathit{\delta }{}^{\mathbf{a}}$
	$\left(\mathrm{kN}\right)$	$\left(\mathrm{kN}\right)$	$\left(\mathrm{N}/\mathrm{mm}\right)$
(a) Cluster 1
D2-10	12.5	21.0	2699.57	1.67	9.39e8
D2-11	7.7	19.8	3177.53	2.57	9.39e8
(b) Cluster 2
D2-9	12.0	17.6	1955.91	1.47	7.83e8
D2-32	12.3	16.3	2259.96	1.33	1.64e9
D2-21	12.0	17.0	1600.84	1.42	1.72e9
D2-19	8.2	18.4	3108.94	2.26	1.73e9
D2-24	10.9	19.1	1323.27	1.75	2.31e9
(c) Cluster 3
D2-1	19.3	20.0	1140.67	1.04	8.79e8
D2-14	20.2	18.2	1301.61	0.90	1.05e9
D2-23	12.8	19.7	3365.79	1.54	1.29e9
(d) Cluster 4
D2-2	12.6	15.7	2200.32	1.25	6.63e8
D2-7	12.5	15.0	1478.99	1.20	1.15e9
D2-33	12.1	15.0	1741.56	1.24	1.60e9
D2-31	8.2	17.8	3480.01	2.17	1.66e9
D2-13	12.2	16.9	2108.07	1.39	1.77e9
D2-6	8.2	18.8	3368.71	2.30	1.85e9
(e) Cluster 5
D2-12	8.0	19.9	1324.74	2.49	8.69e8
D2-27	12.6	20.0	2089.04	1.59	1.56e9
D2-15	7.9	18.9	2382.30	2.40	2.00e9
D2-3	7.9	19.3	2962.28	2.44	2.38e9
D2-26	7.5	18.9	1325.05	2.51	2.63e9
(f) Cluster 6
D2-29	12.7	18.9	3532.30	1.48	2.63e8
D2-20	17.9	19.4	1281.86	1.09	2.04e9
D2-8	19.3	18.1	1124.70	0.94	3.16e9
D2-4	11.5	16.8	3188.97	1.46	3.67e9
D2-30	11.4	20.7	2956.80	1.81	5.13e9
(g) Cluster 7
D2-28	8.0	19.9	3011.79	2.48	2.04e9
D2-22	7.8	19.9	2975.39	2.54	2.04e9
(h) Cluster 8
D2-16	10.5	17.9	1513.66	1.71	7.00e8
D2-17	10.5	18.6	1441.57	1.76	8.27e8
D2-5	8.0	16.7	3545.03	2.08	8.83e8
D2-18	7.9	17.4	2556.22	2.21	2.08e9
D2-25	20.1	18.5	1517.19	0.92	2.57e9

^aδ is the soft-DTW distance between each curve and the center of mass.

, it can be seen that all tubes except D2-8, D2-14, D2-20 and D2-25 are significantly lower than the D2-1 model that has an equally spaced distribution. The reason is that geometrical defects in the design can usually cause early buckling of the tubes [68], and in this case, the non-uniform crease spacing is also a design imperfection. In particular, when a folded layer height of 15 mm is present in the origami tube, an early buckling of the tube generally takes place from that layer, as shown in Figure 12. While for the D2-8, D2-14, D2-20 and D2-25 models, the minimum folded layer height is 20 mm, and although early buckling also begins at this layer, the differences from D2-1 in $P_{\max }$ for these models are only ranging from −4.3% to 7.4%. These indicate that the $P_{\max }$ of such origami tubes can be effectively reduced by controlling the minimum value of the folded layer height in the uneven division of the crease spacing. Moreover, in the mean crushing force $P_{\mathrm{m}}$, also compared with the D2-1 model, there are various degrees of decreasing in $P_{\mathrm{m}}$ for each cluster internal model. The analysis of the reasons shows that although the initial stresses will be distributed according to the predetermined creases [69,70], during the process of origami tubes being crushed and folded continuously, if the difference between the neighboring folded layers is too large, buckling always takes place first in the layer with small spacing, and a folded layer that has collapsed prematurely will inhibit the travel of the plastic hinge in the other layers of the tube. Moreover, the change in the folding sequence of the folded layers also makes new and shorter inclined plastic hinge lines appear in the same layer of folding leaves, which creates a secondary folding situation. The result is that a non-uniformly spaced origami tube cannot follow the crease folding designed for uniformly spaced origami tubes, so it is necessary to have a new crease spacing distribution for non-uniformly spaced origami tubes to lead the tube folding.

In addition, the initial elastic stiffness $\left(\mathit{IES}\right)$ of each tube has significant variation, where model D2-1 with uniformly distributed crease spacing is lower than the majority of tubes, and the $\mathit{IES}$ of the model will be relatively smaller and closer to D2-1 when the crease spacing h in the model is all greater than 15 mm, such as D2-8, D2-14, D2-20, D2-24, D2-25, etc. It means that the distribution of the crease spacing significantly affects the $\mathit{IES}$ of the origami tube, besides a relatively small crease height 15 mm, which has a negative effect on the $\mathit{IES}$ of the model. The uniform or nearly uniformly distributed crease spacing causes the initial crushing of the origami tube with uniform distribution of stresses along the origami pattern [66], thus positively influencing the reduction in the origami tube $\mathit{IES}$. Here, it should be noted that the $\mathit{IES}$ of the model is varied for each cluster class internally, and no distinct regularity is found. For this reason, the soft-$\mathit{DTW}$ distance used for clustering gives more attention to the similarity of the global shape of the curves. As described in Section 3.1, it is measured with one moment matching multiple moments and thus includes the similarity of the curves after a certain degree of compression and stretching, which leads to different $\mathit{IES}$ values for models within the same cluster.

An effective classification allows cluster models with similar energy absorption performance, but also the energy dissipation performance of the models should be significantly different between different cluster classes. Here, the models in Cluster 5 (Table 2(e)) have a lower initial peak force and similar mean force (with a drop between 0 and 6%) as compared to D2-1 with equal spacing distribution, which is a class of crease spacing distribution with better energy absorption performance. Moreover, the crease spacing distribution of the models in Cluster 4 is concave–convex (one large and one small), as shown in Figure 13a, and the models that are closer to the cluster center of mass have relatively stable variations in $P_{\max }$, $P_{\mathrm{m}}$, and ECR, and the crushing process is also relatively stable, as discussed in the next section. On the other hand, as shown in Figure 13b, when comparing the mean crushing force of the models near the center of mass for different clusters, Cluster 5 > Cluster 6 > Cluster 8 > Cluster 4, and the differentiation is more apparent, which indicates that the clustering captures the Pm of the model better when the origami tube design is oriented to the mean crushing force as the demand, a variety of different models can be clustered according to this method to optimize the design. Regarding the efficiency of crushing force $\left(\mathit{ECR}\right)$, since the existence of a relatively small crease height 15 mm, which as analyzed in the previous section makes the $P_{\max }$ of the origami tube decrease significantly, the ECR of all others tubes except D2-14, D2-8 and D2-25 are larger than the D2-1 tube whose creases are distributed equally spaced. Meanwhile, as shown in Figure 14, the ECR of models that are near the center of mass in each cluster fluctuate differently, and the crease spacing distribution has a significant impact on the ECR of the model, but the clustering also revealed some better cluster classes, such as Cluster 4 and Cluster 2, which have an overall lower and less fluctuating ECR, and these types of origami tubes have a better performance in the case where the peak load may lead to fatal consequences.

4.3. Verification in Terms of Clustering Quality

In order to further illustrate the effectiveness of the clustering method, a class of models with similar concave–convex crease spacing distribution to Cluster 4 was additionally selected for numerical simulation and analysis, and the detailed parameters and results are shown in

Table 3. The design parameters and simulation results of the secondary clustering model.

Model	${\mathit{h}}_1$	${\mathit{h}}_2$	${\mathit{h}}_3$	${\mathit{h}}_4$	D ${}^{\mathbf{a}}$	${\mathit{P}}_{\mathbf{max}}$	${\mathit{P}}_{\mathbf{max}}$	${\mathit{P}}_{\mathbf{m}}$	${\mathit{P}}_{\mathbf{m}}$	IES	ECR	Cluster
	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{mm}\right)$	$\left(\mathbf{kN}\right)$	Reduction	$\left(\mathbf{kN}\right)$	Increase
D2-1	30	30	30	30	0	19.3	-	20		1140.67	1.04	-
D2-34 ${}^{\mathrm{b}}$	31.5	28.5	31.5	28.5	3	18.87	2.2%	18.24	−8.8%	1199.12	0.97	1
D2-35	33	27	33	27	6	18.07	6.4%	17.33	−13.4%	1331.44	0.96	1
D2-36	34.5	25.5	34.5	25.5	9	16.98	12.0%	16.63	−16.9%	1532.17	0.98	1
D2-37	36	24	36	24	12	15.88	17.7%	16.00	−20.0%	1640.95	1.01	1
D2-38	37.5	22.5	37.5	22.5	15	14.60	24.3%	15.32	−23.4%	1828.22	1.05	2
D2-39	39	21	39	21	18	13.45	30.3%	15.29	−23.6%	2095.99	1.14	2
D2-40	40.5	19.5	40.5	19.5	21	12.22	36.7%	16.22	−18.9%	2277.38	1.33	2
D2-41	42	18	42	18	24	10.93	43.3%	16.16	−19.2%	2644.38	1.48	2
D2-42	43.5	16.5	43.5	16.5	27	9.62	50.1%	17.04	−14.8%	3390.00	1.77	2
D2-43	45	15	45	15	30	8.21	57.5%	17.89	−10.5%	3485.15	2.18	3
D2-44	46.5	13.5	46.5	13.5	33	6.46	66.6%	19.15	−4.2%	3723.75	2.97	3

^a D: Crease spacing distance. ^b Models D2-34 to D2-44 have the same side lengths: [a₁₋₅ = 0, 60 mm, 0, 60 mm, 0].

. As shown in Figure 15, it can be found that the energy dissipation curve of the model gradually separates into three cluster classes with obvious discrepancies as the spacing distance increases, and the $P_{\max }$, $P_{\mathrm{m}}$, ECR and IEC of the model show a certain regularity after analysis using the clustering method that was proposed above. Specifically (shown in Figure 16), $P_{\max }$ gradually decreases with increasing spacing distance, while $P_{\mathrm{m}}$ decreases and then increases, and the turning point happens at the mean folded layer height of $0.6$ accompaniment 18 mm. The reason for this is that a large spacing distance inevitably leads to a small interlaminar crease height, and as was discussed above, the smaller the interlaminar crease height, the more likely it is to buckle. However, benefiting from the concave–convex crease distribution, when these short layers are buckled, the model crease distribution will be reshaped into a uniform distribution, making $P_{\mathrm{m}}$ increase. On the other hand, ECR, which is affected by the same factors, has a similar trend to $P_{\mathrm{m}}$, while IEC changes in the same way as $P_{\max }$. Accordingly, it can be seen that the distribution of concave and convex crease spacing, which was found in the original classification (Cluster 4), can be obviously found in the response regularity of the crease spacing distribution in the cluster to the energy dissipation index of the folded tube after the analysis. As a result, if the same secondary clustering analysis is conducted for the other clusters, it is expected that the correspondent change regularity can be found.

In light of this, although polymorphic origami patterns make the energy dissipation of origami tubes varied and cannot be quantified by a specific crease parameter, it is possible to cluster origami tubes with similar energy dissipation performance after analysis through the application of the method in Figure 1 and then choosing the corresponding cluster class according to one’s needs before targeting secondary analysis, and there is a possibility to find a certain pattern from it.

4.4. Crushing Process Analysis of Each Cluster after Clustering

The energy dissipation performance of the origami tube is directly influenced by the compression crushing process and the final damage mode. Although most of the models show a mixed damage mode, the crushing damage process is different between each cluster. A typical model closest to the center of mass is selected in each cluster, as shown in Figure 12, and stress maps of the model at a specific crush distance are captured. D2-1 of Cluster 3 (Figure 12b) is a typical model with uniform crease spacing distribution, which has a more homogeneous stress distribution than the model with non-uniform crease spacing distribution during the crushing process, plastic deformation compression of each folded layer at the same time, the buckling pattern follows the plastic hinge line originally designed [17] and the final damage mode is the complete diamond mode. As for the origami tube with non-uniformly distributed crease spacing, such as D2-16 of Cluster 8 (Figure 12h), buckling first happens in the layer with the smallest height of the folded layer, and the damage displays a concertina mode, but the plastic hinge line between the layers cannot be fully developed. In addition to the geometric design conditions, the vertex of the layer is subsequently twisted and then folded after further compression, finally showing a mixed damage mode, and the other models are similar. However, it should be noted that different clusters may also vary. For example, Cluster 4 (D2-2 Figure 12d) with a concave–convex crease spacing distribution, which also always buckles first in the small height folded layer during the crushing damage, making the $P_{\max }$ of the model smaller, and then its upper and lower folded layers come close together. However, it should be noted here that the folded layers that are adjacent to the top and bottom of the small height folded layer do not buckle immediately, and then the model is reconstructed into a typical model with a class of uniform crease spacing distribution, followed by a shift in the damage mode, which makes the $P_{\mathrm{m}}$ vary, with the results as analyzed in the previous section. Moreover, as shown in Figure 17, Cluster 5 and Cluster 4 also show some intra-cluster similarity in the final mode pattern after crushing.

5. Conclusions and Future Work

In this study, we have compiled a geometric and numerical simulation procedure for a class of origami tubes with multi-morphology patterns. A total of 44 axial compression crushing processes of origami tubes are simulated and used as sample data to analyze the energy dissipation history curves through machine learning methods in order to investigate the energy dissipation performance of origami tubes with multi-morphology patterns. The results show that, first of all, the unsupervised time series clustering method using soft-DTW as a distance metric can effectively classify the model energy dissipation history curve. Furthermore, if the secondary analysis is adopted, the response regularity of the corresponding distribution can be found more effectively. Secondly, the minimum fold layer height has a key effect on the $P_{\max }$, $P_{\mathrm{m}}$, IEC and ECR metrics of the model with multi-morphology patterns in this study. Furthermore, unlike previous studies, in the case of origami tubes with multi-morphology patterns, the clustering based on the energy dissipation history makes it easier to find multiple origami tubes (e.g., Cluster 4, Cluster 5 and Cluster 6) with similar energy dissipation performance and crushing modes. Finally, the concave–convex crease spacing distribution can be reconstructed into a class of uniform crease distribution during the crushing process of the origami tube, especially when the spacing distance is larger than 18 mm ($0.6$ accompanying the mean folded layer height), which makes the folding guided. The final damage modes do not have much difference, so this crease spacing distribution can also be used to activate specific crushing modes.

As discussed, the following conclusions can be concluded. The analysis method that is used in the paper would be applicable to study the energy dissipation performance for origami tubes with multi-morphology patterns. By using the energy dissipation history curve to combine the model parameter design, the parameter optimization can be targeted in the engineering design process to achieve a customizable energy consumption performance for the origami tube structures, e.g., to design energy-absorbing boxes for different types of vehicles and their important components, etc. Moreover, when designing and producing origami tubes with multi-morphology patterns, a specific spacing distribution can effectively improve energy absorption and activate specific crushing modes. However, in real applications, origami tubes are primarily subjected to dynamic impacts, and the loading environment of impact is also more complex than quasi-static. It is necessary to consider such complexity of an application environment for origami tubes, including the variation in impact loading velocity. Therefore, in our next studies, there is a focus on the energy absorption efficiency of origami tubes with multi-morphology patterns with more complex conditions and closer to the real application conditions (dynamic conditions). On the other hand, the design parameters of origami tubes with multi-morphology patterns are very diverse, and in the future, an end-to-end deep learning approach will be used to further analyze the energy dissipation performance of origami tubes with multi-morphology patterns under simultaneous variation in multiple design parameters.