## 2 Design of TPMS Structures
### Design of Functionally Graded TPMS Structures  
The volume fraction of the TPMS structure plays a crucial role in determining its performance. The lattice structure uniformity and regular arrangement make it essential to consider the volume fraction when analyzing its properties. By using a unit size of 6 mm, the volume fraction can be calculated through triple integration using MATLAB. This calculation helps establish the relationship between the volume fraction (\(\rho^{*}\)) and the constant \(t\). The fitting accuracy of the data is represented by the R2 value, indicating a high level of accuracy, as shown in Figure 2. This is supported by the fact that the fitting results closely resemble those of a previous study [29]. Figure 1: TPMS-based cellular structures, unit cells, and SLM-made AlSi10 Mg uniform TPMS-based cellular structures. The most commonly utilized functional gradient structures based on TPMS in the existing literature are the volume fraction and cell size gradients [30; 31]. For this study, only volume fraction gradients (Sh-D) were considered, as it has been found that cell size gradients do not have a significant impact on mechanical properties [9; 31]. The volume fractions at the ends of the structure were determined based on the average volume fraction of the design, and the corresponding \(t\) values were calculated using the fitting equation, and \(t\) varied linearly in the corresponding directions to realize the gradient design. In the spatial coordinate system, the parameters \(t\) vary with the coordinate system, and the parameters \(t\) as a function of \(x\), \(y\), and \(z\) can be expressed as follows:  
\[t(x,y,z)=Af_{x}(x)+Bf_{y}(y)+Cf_{z}(z)+t_{0} \tag{3}\]  
\(f_{x}(x)\), \(f_{y}(y)\), \(f_{z}(z)\) are the distribution functions of volume fractions in different directions, \(t_{0}\) is the control mean volume fraction; \(A\), \(B\), \(C\) are the variations of volume fractions in different directions. \[A=D_{x}\left(\frac{\mathrm{d}\rho^{*}}{\mathrm{d}t}\right)^{-1}=\frac{D_{x}}{ 0.3382} \tag{4}\]  
\[B=D_{y}\left(\frac{\mathrm{d}\rho^{*}}{\mathrm{d}t}\right)^{-1}=\frac{D_{y}}{ 0.3382} \tag{5}\]  
\[C=D_{x}\left(\frac{\mathrm{d}\rho^{*}}{\mathrm{d}t}\right)^{-1}=\frac{D_{x}}{ 0.3382} \tag{6}\]  
\(D_{x}\), \(D_{y}\), \(D_{x}\) are the changes in volume fraction in the X, Y, and Z directions. If the average volume fraction is \(\rho^{*}\), which can be obtained since it is only controlled by \(t_{0}\). \[t_{0}=\frac{\rho^{*}+0.001}{0.3382} \tag{7}\]  
Since the volume fraction \(\rho^{*}\) varies linearly with the parameter \(t\), the volume fraction distribution function in the X, Y, Z direction can be expressed by a univariate absolute value function and satisfies the following conditions:  
\[\left\{ \begin{array}{l}f_{x}(0)=\frac{1}{2}\\ f_{x}\left(-\frac{H}{2}\right)=f_{x}\left(\frac{H}{2}\right)=-\frac{1}{2} \end{array} \right. \tag{8}\]  
Figure 2: The diagram between constant \(t\) and volume fraction (\(\rho^{*}\)) for D unit cells. \[\left\{ \begin{array}{l}f_{y}(0)=\frac{1}{2}\\ f_{y}\Big{(}-\frac{H}{2}\Big{)}=f_{y}\Big{(}\frac{H}{2}\Big{)}=-\frac{1}{2}\\ \end{array} \right. \tag{9}\]  
\[\left\{ \begin{array}{l}f_{z}(0)=\frac{1}{2}\\ f_{z}\Big{(}-\frac{H}{2}\Big{)}=f_{z}\Big{(}\frac{H}{2}\Big{)}=-\frac{1}{2}\\ \end{array} \right.