## 2 Materials and Methods  
### TPMS Unit Design  
This paper realizes the construction of porous units based on the implicit function expression of surface units. The general implicit function expression of TPMS is as follows [36]:  
\[\Phi(r)=\sum_{k-1}^{k}A_{k}\cos(2\pi(h_{k}gr)/\lambda_{k}+p_{k})+t=0 \tag{1}\]  
where \(A_{k}\) is the amplitude factor, \(k\) is the periodic wavelength, \(r\) is the position vector in Euclidean space, \(h_{k}\) is the kth grid vector in the reciprocal space, \(p_{k}\) is the phase offset, and \(t\) is the distance constant of the surface.  
The utilization of implicit functions allows for the construction of TPMS structures with precise periodicity and phase. A visual representation of the typical TPMS unit morphology is depicted in Figure 1 for various selected parameters \(t\), with its corresponding implicit function expression (Table 1).  
Figure 1: The schematic diagram of triply periodic minimal surface (TPMS) unit morphology.  
The utilization of implicit functions to express implicit surfaces is not feasible in current additive manufacturing systems. The TPMS unit's physical representation is completed using the STL data exchange format, making it easily applicable to the additive manufacturing system. The isosurface drawing function of Matlab is used to complete the point set calculation and visualization of the implicit surface. The Delaunay triangulation principle is then utilized to triangulate the point set. The STL solid unit is created by using triangulation (Figure 2).  
The distance constant of implicit surface t indicates the distance between a point on the surface \(\Phi(X,Y,Z)\) defined by the implicit function and the corresponding point on the minimal surface derived from the function. Based on the condition that the integrity of the solid units built on the surface is maintained, the \(t\) values of these units exhibit varying  
\begin{table}
\begin{tabular}{c c} \hline
**TPMS** & **Implicit Function Expression** \\ \hline Gyroid & \(\Phi(X,Y,Z)=\sin X\cos Y+\sin Y\cos Z+\sin Z\cos X+t=0\) \\ Diamond & \(\Phi(X,Y,Z)=\cos X\cos Y\cos Z-\sin X\sin Y\sin Z+t=0\) \\ I-WP & \(\Phi(X,Y,Z)=2\cos X\cos Y+\cos Y\cos Z+\cos Z\cos X)\) \\  & \(-(\cos 2X+\cos 2Y+\cos 2Z)+t=0\) \\ Neovius & \(\Phi(X,Y,Z)=3\cos X+\cos Y+\cos Z)+4\cos X\cos Y\cos Z) \\  & \(+t=0\) \\ Primitive & \(\Phi(X,Y,Z)=\cos X+\cos Y+\cos Z+t=0\) \\ Fischer-Koch S & \(\Phi(X,Y,Z)=\cos 2X\sin Y\cos Z+\cos X\cos 2Y\sin Z\) \\  & \(+\sin X\cos Y\cos 2Z+t=0\) \\ F-RD & \(\Phi(X,Y,Z)=4\cos(X\cos Y\cos Z)-(\cos 2X\cos 2Y+\cos 2Y\cos 2Z\) \\  & \(+\cos 22\cos 2X)+t=0\) \\ PMY & \(\Phi(X,Y,Z)=2\cos X\cos Y\cos Z+\sin 2X\sin Y+\sin X\sin 2Z\) \\  & \(+\sin 2Y\sin Z+t=0\) \\ \end{tabular} where \(X=2\pi x/L\), \(Y=2\pi y/L\), \(Z=2\pi z/L\), \(L\) is the period of change of the implicit surface, and \(t\) is the distance constant of the implicit surface.  
\end{table}
Table 1: TPMS (triply periodic minimal surface) units and their implicit function expressions.  
Figure 2: The materialization of triply periodic minimal surface (TPMS) element based on the Delaunay triangulation principle and 3D-printing TPMS unit model of 30% porosity.  
ranges and impacts on porosity. Defects such as self-intersection and unit breakage can occur due to incorrect \(t\) values. Thus, it is crucial to initially determine the range of \(t\) values for each unit. Under the premise of ensuring the continuity of the solid elements constructed by the surface, we choose the same porosity of 30% (Figure 2), and the formability of SLM technology is verified next in Section 2.2.