# Space Node Topology Optimization Design Considering Anisotropy of Additive Manufacturing
## 2 Materials and Methods  
### Secondary Development of the BESO Method  
The BESO method is one of the most popular topology optimization methods of continuum structures. BESO is optimized by continuously reducing the volume, which can avoid falling into the local optimal solution, with broad prospects for secondary development. Compliance is usually taken as the objective function when optimizing the space nodes that require stronger stiffness, and optimizing the structural shape with the smallest deformation while taking into account the lightweight design. This paper uses the BESO method based on the finite element method, which can ensure the efficiency and effect of the optimization, using the equal-volume regular hexahedron square as the variable element to get the energy density of the element as the design variable. Its mathematical description is as follows:  
\[\left\{ \begin{array}{ll}\text{Minimize:}&\text{C}\left(X_{i}\right)=\mathbf{ F}^{\text{T}}\mathbf{U}\\ \text{s.t.}:&V^{*}-\sum\limits_{i}^{n}V_{i}=0\\ &X_{i}=1\quad\text{when }i\in\ \Omega_{1}\\ &X_{i}=x_{min}\text{ when }i\in\ \Omega_{0}\end{array} \right., \tag{1}\]  
where C is the compliance, \(\mathbf{F}\) is the force matrix, \(\mathbf{U}\) is the displacement (deformation) matrix, \(V^{*}\) is the target volume fraction, \(V_{i}\) is the volume of the \(i\)-th element, \(X_{i}\) is the design variable of the \(i\)-th element, \(n\) is the total number of solid elements, \(x_{min}\) is a nonzero minimum value, \(\Omega_{1}\) is the solid region, and \(\Omega_{0}\) is the hole region. The formula takes compliance C as the objective function and strives to minimize the compliance, which means that the deformation \(\mathbf{U}\) is the smallest when the load \(\mathbf{F}\) is unchanged, and the structural configuration at \(V^{*}\) volume is obtained, so as to achieve lightweight and maximize stiffness. When \(i\) belongs to the solid area, the design variable \(X_{i}\) is itself; when \(i\) belongs to the hole area, it is necessary to carry out the \(X_{i}\) penalty. This is to make it approximate to air, but not exactly equal to 0, to avoid matrix singularity [31]. To make the results continuous in the finite element calculation, the most commonly used BESO method introduces the material interpolation formula of the SIMP method. The material interpolation method can make the hole area unit get the sensitivity penalty in the iterative calculation, and then effectively distinguish the solid unit and the hole area unit. \[\text{E}(x_{i})=x_{i}^{p}\text{E}_{1}, \tag{2}\]  
where \(\text{E}_{1}\) denotes the Young's modulus of the solid material, \(x_{i}\) is the design variable of the \(i\)-th element, and p is the penalty exponent. After the model is calculated by finite element software, the energy density of each element is obtained. Generally, the large value is not corrected, but the material method is used to penalize the value of the element in the hole area to reduce the weight of sensitivity filtering. This paper adopts a solid-void design; hence, according to the relevant literature, it can be expressed by the following formula [32]:  
\[\frac{\partial C}{\partial x_{i}}=-\frac{px_{i}^{p-1}}{2}\mathbf{U}_{i}^{ \text{T}}\mathbf{K}_{i}\mathbf{U}_{i}, \tag{3}\]  
where \(\mathbf{K}_{i}\) is the elemental stiffness matrix. The design variable is discrete because materials in solid and void are different, and every design variable of the individual element is represented by a sensitivity, thereby the formula can be defined by:  
\[\alpha_{i}=-\frac{1}{p}\frac{\partial C}{\partial x_{i}}= \begin{cases}\frac{1 }{2}\mathbf{U}_{i}^{\text{T}}\mathbf{K}_{i}\mathbf{U}_{i}&when\ x_{i}=1\\ \frac{x_{min}^{p-1}}{2}\mathbf{U}_{i}^{\text{T}}\mathbf{K}_{i}\mathbf{U}_{i}& when\ x_{i}=x_{\text{min}}\end{cases}  \tag{4}\]  
Then the sensitivity number will decline to nearly 0 with the penalty exponent increasing to infinity.