# Space Node Topology Optimization Design Considering Anisotropy of Additive Manufacturing
## 2 Materials and Methods
### Secondary Development of the BESO Method
In general, when using an optimization program based on finite element software, it cannot be calculated to define a void region material as zero; therefore, the void material is defined as a material with lower performance. In this way, two materials need to be defined in the optimization, and the formula is defined as  
\[\alpha_{i}= \begin{cases}\frac{1}{2}[1-\frac{\text{E}_{2}}{\text{E}_{1}}] \mathbf{U}_{i}^{\text{T}}\mathbf{K}_{i}^{\text{T}}\mathbf{U}_{i}&\text{for solidarea}\\ \frac{1}{2}[\frac{x_{min}^{p-1}(\text{E}_{1}-\text{E}_{2})}{x_{min}^{p}\text{ E}_{1}+(1-x_{min}^{p})\text{E}_{2}}]\mathbf{U}_{i}^{\text{T}}\mathbf{K}_{i}^{ \text{T}}\mathbf{U}_{i}&\text{for hole area}^{\prime}\end{cases} , \tag{5}\]  
which can be simplified as  
\[\alpha_{i}= \begin{cases}\frac{1}{2}[1-\frac{\text{E}_{2}}{\text{E}_{1}}] \mathbf{U}_{i}^{\text{T}}\mathbf{K}_{i}^{\text{T}}\mathbf{U}_{i}&\text{for solidarea}\\ 0&\text{for hole area}^{\prime}\end{cases}  \tag{6}\]  
where \(\alpha\) is the element sensitivity, \(\text{E}_{1}\) and \(\text{E}_{2}\) are the element elastic moduli of the solid area and the hole area, respectively, \(\mathbf{K}\) is the stiffness matrix calculated from the elastic moduli \(\text{E}_{1}\) and \(\text{E}_{2}\), and \(\text{P}\) is the penalty index, whose increase will decrease the sensitivity \(\alpha\), thus achieving the effect of increasing the penalty. According to the material interpolation method described above, the elements in the hole area are assigned a sensitivity value that is approximate and not equal to 0, while the element's sensitivity in the solid area receives the appropriate adjustments. A sensitivity filter needs to be added because the structure cannot be kept smooth. Although the use of the material interpolation method enables the elements to better participate in the iterative calculation, island elements are prone to appear adjacent to the solid element and the hole area element. Figure 1 is a schematic diagram of the sensitivity filter. The sensitivity filtering method is introduced to solve the checkerboard effect caused by the discontinuity of the grid sensitivity of the elements. A spherical filter is usually used in the three-dimensional optimization algorithm, which can ensure the smoothness of the structure after filtering. In the figure, the blue elements are the elements involved in filtering, and these elements are all within the scope of the sphere. By processing the elements within a certain radius Rmin near the target element according to the distance weighted average, the grid becomes stable and continuous. In this way, even element values with large differences can be made continuous. The weights are calculated as follows [33]:  
\[ \begin{array}{c}\omega(r_{j})=r_{min}-r_{j}\\ j=1,2\ldots m\end{array}  \tag{7}\]  
where \(r_{min}\) is the filtering range, \(r_{j}\) is the distance between unit \(j\) and the target unit, and \(m\) is the number of units within the range of \(r_{min}\). The unit sensitivity is calculated according to the following formula [33]:  
\[ \begin{array}{c}a_{i}=\frac{\sum\limits_{j=1}^{n}a_{i}^{7}\omega(r_{j})}{ \sum\limits_{i=1}^{n}\omega(r_{j})}\\ i=1,2,\ldots n_{i}j=1,2,\ldots m\end{array}  \tag{8}\]  
where \(i\) represents the serial number of the target unit, and \(j\) represents the serial number of the unit participating in the filtering within the filtering range. As the iterative process progresses, the boundaries of soft and solid elements become smooth. The sensitivity filtering method not only solves the checkerboard problem but also effectively reduces the dependence on the grid, which is one of the cores of the topology optimization method. Lastly, the solid elements and the pore elements are divided according to the volume fraction using the bisection method. This step requires finding a critical value that distinguishes solid elements from hole area elements. The maximum and minimum sensitivities of all units are summed before taking the average.