# Design and Mechanical Properties of Negative Poisson's Ratio Structure-Based Topology Optimization
## 2 Design of a Negative Poisson's Ratio Structure Based on the Variable Density Method
### Optimization Examples and Analysis  
The example in Figure 1 uses N_x and N_y to represent the number of elements in the x and y directions of the initial structure, respectively. The volume fraction is defined as the percentage of the optimized structure's volume to the initial volume. For example, if volfrac = 0.5, the volume of the optimized structure is 50% of the initial volume. The variable density method requires a penalty factor, which penalizes intermediate density units to minimize the number of density units and make the density tend toward 0 or 1. An appropriate penalty factor is crucial since a small penalty factor creates many gaps that make material manufacturing difficult, while large penalty factors increase optimization iterations and time costs. The filtering radius affects the material density distribution and stability of optimization results. A larger filtering radius increases smoothness at the expense of the accuracy and details of the optimization results, creating refinement and instability. Choosing the right optimization parameters is, therefore, crucial. There are two types of filtering options: density filtering and sensitivity filtering. Density filtering involves modifying the sensitivity information of the objective function and volume constraints to obtain pseudo-densities for each element. On the other hand, sensitivity filtering directly filters the sensitivity information of the objective function using the design variables, which represent the pseudo-densities of the elements. During topological optimization, the interior of the structure is set as a \"blank area\" to maintain material connectivity and ensure physical feasibility. Material density in these blank areas is relatively low. N_x and N_y determine the initial size of the structure but have minimal influence on the optimized structure. The volume fraction and blank region radius directly impact the morphology of the optimized structure. Therefore, fixed values are chosen for the penalty factor and filtering radius due to their widespread applicability. Different filtering methods have a significant impact on both the structure and the objective function, so they are chosen as parameters to be varied. Taking into account the effects of these parameters, six sets of data in Table 1 are selected for MATLAB programming to implement the topology optimization algorithm. The final values of the objective function are compared to determine the most suitable parameters. Figure 2 presents the numerical variation charts for the objective function obtained by comparing the six sets of data. The charts show that the value of the objective function rapidly decreases as the number of iterations increases and then stabilizes at a converged value. However, groups E and F exhibit oscillations between two objective function values, which may be due to the optimization algorithm being trapped in a local minimum. Sensitivity filtering smooths the objective function during the optimization process, but changing the initial value or parameters is necessary to achieve convergence. To improve the optimization results, the value of \(\beta\) will be changed in the next step of this article to achieve optimal convergence.