## 1 Introduction  
Improving crashworthiness can reduce injuries [1, 2]; therefore crashworthiness design is a necessary means and method to improve crashworthiness. In order to reduce passenger injuries from collision impacts, energy-absorbing structures are often designed and installed in unoccupied areas at the front end of the train [3, 4, 5, 6], where collision energy is absorbed through the deformation of circular tubes. Thin-walled circular tubes can be compressed by models to produce a specific deformation pattern with stable reaction forces and high specific energy absorption [7, 8, 9], which makes them ideal energy-absorbing structures and has led to their wide use in automotive, railway transportation and aerospace industries [10, 11, 12, 13]. Some deformation modes of circular tubes have attracted a large number of scholars, such as expansion [14, 15, 16, 17, 18, 19], outward turning [20, 21, 22, 23, 24, 25] and splitting [26, 27, 28, 29, 30]. [PERSON] [31] found that the energy absorption efficiency of shrink circular tubes is significantly greater than that of expanded circular tubes for different dimensional parameters by means of numerical simulation combined with experimental methods. Compared with the long rod structure of the expanding tube, the shrink tube has a more compact structure, a more stable energy absorption process, a higher mean force and a greater energy absorption efficiency, making it suitable to be used as the energy-absorption structure for the train anti-climbing device. The deformation mechanism of the shrink circular tube is similar to that of the expanding circular tube, but there are fewer studies on shrink tubes. [PERSON] et al. [32] proposed a straight taper shrinking (STS) tube and used quasi-static experiments and finite element simulations to study the crashworthiness of the STS tube. They found that the STS tube could form two deformation modes, shrinking (S) and buckling (B), and that matching the structural parameters, such as taper angle and tube wall thickness, could lead to different deformation modes. Additionally, the cone angle had the most significant effect on the crashworthiness of STS tubes. [PERSON] et al. [33] proposed a theoretical model for the shrinkage of metal tubes by classifying the deformation into three modes based on the relationship between the actual mold radius, the critical mold radius and the conical vertebral angle. By comparing the results with numerical simulations over a wide range of geometrical parameters, it was found that the range of applicability of the theoretical model was conical angles less than or equal to 40\({}^{\circ}\) and diameter-thickness ratio greater than or equal to 10, respectively. [PERSON] et al. [34] investigated the shrinkage process of round tubes as a forming method to produce round tubes with specific diameters. [PERSON] et al. [35] addressed the forming limit of aluminum tubes with expansion and contraction and observed experimentally and through simulation the occurrence of toughness fractures, wrinkles and local buckling as phenomena. They found that the geometric parameters and lubrication conditions were the key factors for the success of the shrinkage process. [PERSON] et al. [36] proposed a novel energy absorber for high-speed trains using a combined shrinkage round tube and found that the rate of increase in the maximum crushing force was much greater than the rate of increase in specific energy absorption as the wall thickness increased. They chose to use multi-objective particle swarm optimization to obtain a Pareto bound for the dual optimization objective. [PERSON] et al. [37] found that by combining the simultaneous action of both the shrinkage and splitting modes, the combined mode energy absorption efficiency was about 60% higher than that of a single contraction absorber. From the above review, it can be seen that the shrink tube has high mean force and specific absorption energy, light mass and stable deformation.