# Crasworthiness Analysis and Multi-Objective Optimization for Concave I-Shaped Honeycomb Structure
## 1 Introduction
After nearly fifty years of development, materials with negative Poisson's ratio have penetrated micro and macro scales of different sizes and are used in a wide range of applications in military medicine, life, and vehicle safety. The molecular level is in the range of \(10^{-7}\) to \(10^{-5}\) mm with some crystalline structures and liquid polymorphs. The microscopic sizes from \(10^{-4}\) to \(10^{-1}\) mm include a wide range of composites and ceramics, as well as metallic foams, honeycomb materials, etc. The macroscopic level is mainly negative Poisson's ratio multi-cell structures with sizes between \(10^{-4}\) and \(10\) mm. The cell structure has obvious periodicity. Thus, the different sizes of the material structure have different ranges of applications, and their potential research technology value for industries and national defense and security has very far-reaching significance [15]. Today's negative Poisson's ratio material structures mainly include concave structures [16], chiral or anti-chiral structures [17], switchable architected structures [18], and origami and fold structures [19]. The traditional concave hexagonal honeycomb structure has been modified by many researchers into a new negative Poisson's ratio structure. These researchers theoretically calculated the relationship between the equivalent elastic modulus and Poisson's ratio of these new structures with each structural parameter. Then, they used simulation software to analyze and compare the analytical and finite element solutions to determine the effects of these structural parameters on the mechanical system. [PERSON] et al. [20] proposed a novel ring-shaped negative Poisson's ratio structure and investigated the impact of its structural dimensions on the equivalent elastic modulus and negative Poisson's ratio using the design approach of mechanical metamaterials. To determine the link between the equivalent elastic modulus, Poisson's ratio, and each structural parameter, [PERSON] et al. [21] developed a negative Poisson's ratio structure made up of circular curves. Global automobile ownership is continuously rising along with the industry's fast expansion. According to the World Health Organization's \"Global Report on Road Safety 2015\", 1.25 million persons died in road accidents globally in 2013 [22], and approximately half of these victims were pedestrians. Studies have shown that pedestrians are more likely to suffer head and lower limb injuries in a collision with a car; while lower limb injuries are generally not fatal, head injuries are up to 62% more likely to be fatal [23]. Therefore, the material and structure of energy-absorbing devices for cars must be strong and lightweight but also have a good energy-absorption effect. When compared to other materials of the same mass, the negative Poisson's ratio material has a greater energy absorption effect, making it ideal for the investigation of protective devices for human body protection. Furthermore, there is more and more research on the multi-objective optimization of a cellular structure with a negative Poisson's ratio. By employing the sampling method, which is a uniform experimental design, [PERSON] et al. [24] were able to obtain the fitting equations for the four evaluation indices of energy, specific energy absorption, peak crushing force, and equivalent Poisson's ratio for the geometrical parameters. They optimized the design based on the fitting equations. A metamodel-based optimization method, integrating finite element simulation, kriging metamodel, and multi-objective genetic algorithm, was employed to optimize the design of the hierarchical honeycomb by [PERSON] [25]. The optimization process aimed to achieve the maximum value of specific energy absorption and minimum value of peak crushing force. The majority of academics will use the surrogate model technique for highly nonlinear situations owing to the computational expense. Using nonlinear functions and collision simulation elements, [PERSON] et al.