# Design and Mechanical Properties of Negative Poisson's Ratio Structure-Based Topology Optimization
## 1 Introduction  
The mechanical properties of materials have been a primary area of practical application and extensive study. Nevertheless, many material designs remain theoretical as they are not producible or experimentally verifiable due to manufacturing process limitations. However, the advent of additive manufacturing and machine learning design technologies has presented a practical manufacturing method for modern material design. Additive manufacturing technology offers greater freedom for material design with its superior mechanical properties and high precision compared to traditional machining processes like turning, milling, casting, forging, and welding. Consequently, the concept of metamaterials has emerged. The term \"metamaterials\" derives from the prefix \"meta\" meaning \"beyond\", implying that the properties or structure of these materials surpass those of traditional materials. The introduction of this terminology originated from a paper published by [PERSON] et al. [1], which investigated a material with negative permeability and dielectric constant at microwave frequencies. Progressively, researchers gained a better understanding of the metamaterial concept. In essence, metamaterials are composite materials and artificial composite structures designed and created with specific parameters, mimicking the attributes and properties of natural materials. The resulting material exceeds the abilities of the natural ones that make it up, and its macroscopic characteristics are largely determined by the microstructure of the material. [PERSON] et al. 's [2] work focused on the evolution and advancement of metamaterials, including their connections with emerging technology. Consequently, they showed that human thought plays a pivotal role in constructingnew theories, as stated by [PERSON]'s [3] deductive and falsificationist methods in the early 20 th century. In recent years, metamaterials research has rapidly progressed across various disciplines, including fundamental physics, optics, materials science, mechanics, and electrical engineering. Metamaterials research is roughly categorized into extraordinary electromagnetic materials [4], optical metamaterials [5; 6; 7], acoustic metamaterials [8; 9], thermal metamaterials [10], and mechanical metamaterials. [PERSON] et al. 's [11] taxonomy classified mechanical metamaterials into four types: (1) strong lightweight materials concerning Young's modulus, (2) tunable stiffness materials concerning shear modulus, (3) negative compressibility materials concerning bulk modulus, and (4) negative Poisson's ratio materials concerning Poisson's ratio. In recent years, the study of mechanical metamaterial structures has led to significant interest among researchers in negative Poisson's ratio materials. Regarding the structure of negative Poisson's ratio materials, researchers have conducted a lot of theoretical and applied research [12; 13]. The analysis of negative Poisson's ratio materials can be approached from the following aspects. [PERSON] [14] was the first to propose an approximate calculation method to determine the average elastic properties of periodic unit cell framework structures, and he discovered the construction of a framework that has negative Poisson's ratio properties. [PERSON] [15] named materials with a negative Poisson's ratio effect as \"auxetic materials\". Negative Poisson's ratio materials have higher stiffness, strength, energy absorption, and damage resistance than materials with a positive Poisson's ratio [16]. Some complex porous structures made up of ligaments with complex shapes and density gradients have higher structural efficiency than some traditional materials. For example, a comparison can be drawn between the pyramid complex in Giza, Egypt, and the Eiffel Tower in Paris. Even though the structure of the Eiffel Tower is similar to that of the pyramid complex, it has twice the height and is three orders of magnitude lighter in weight [17].