# Design and Mechanical Properties of Negative Poisson's Ratio Structure-Based Topology Optimization
## 3 Structural Geometric Construction and Equivalent Elastic Properties
### Comparison of Analytical and Finite Element Results
#### 3.2.2 The Influence of Various Parameters on the Equivalent Poisson's Ratio  
In order to analyze the influence of various parameters on the equivalent Poisson's ratio in the analytical solution, we conducted an analysis where we examined the effects of different parameters on the equivalent Poisson's ratios while keeping the other parameters constant.  
To begin with, Figure 12a shows the effect of the transverse edge a on the equivalent Poisson's ratio. With the gradual increase in a, the equivalent Poisson's ratio also increases steadily, and the rate of increase is swift, while the negative Poisson's ratio feature gradually disappears.  
The influence of the oblique edge b on the equivalent Poisson's ratio is the next aspect to consider, as shown in Figure 12b. As b increases, the effective Poisson's ratio experiences a gradual decrease, displaying negative Poisson's ratio characteristics, albeit with a moderate reduction rate. It is important to note that while the effective elastic modulus and effective Poisson's ratio are negatively correlated with other parameters, they exhibit a positive correlation with the oblique edge b.  
Parameter c plays a crucial role in the deformation behavior of the structure under substantial forces. In Figure 12c, it is evident that as c increases, the equivalent Poisson's ratio of the structure gradually decreases, resulting in a more pronounced negative Poisson's ratio characteristic, aligning with the expected behavior.  
Parameter h, which represents the vertical edge of the unit cell, exerts an influence on the equivalent Poisson's ratio and acts as a moderating factor. This is depicted in Figure 12d.  
Parameters \(\alpha\) and \(\gamma\), which represent the angles between edges, significantly impact the trend in the equivalent Poisson's ratio. Figure 12e demonstrates that the effect of \(\gamma\) is more pronounced. As the angles between the edges increase, the connecting area in the unit cells also increases, resulting in a decrease in the macro-level equivalent Poisson's ratio. The decrease highlights negative Poisson's ratio characteristics.  
Based on previous experience, the discussion above demonstrates that there is a certain difference between the analytical and simulation solutions that can be categorized into two explanations. Primarily, the energy method used in theoretical derivation neglects the impact of tensile and shear forces, leading to significant disparities between the analytical and simulation solutions for certain results. Additionally, the simulation model used in this paper assumes that all segments are either horizontal or vertical. This assumption may underestimate the impact of forces on some segments during theoretical analysis, leading to underestimated results during segmented integration. Nevertheless, during the simulation process, each segment is capable of undergoing bending deformation as a result of its elasticity, which may cause the simulation solution to exceed the analytical solution in some situations.