# Design and Optimization of Lattice Structures: A Review
## 3 Design and Optimization of Uniform Lattice Structures
### Unit Cell Design Based on Mathematical Algorithm  
It is an accurate method to describe lattice structures by using mathematical algorithm to design unit cell. Triply periodic minimal surface (TPMS) is one of the mathematical methods that can successfully transform the theoretical mathematical model into the actual lattice structures. TPMS can be generated in multiple ways including Weistreass formula evaluations, nodal approximations of the Weistreass formula, and numerical generation [74]. TPMS is a smooth infinite minimal surface, the minimal surface is defined as a surface in hyperbolic space which possesses a mean curvature or zero. TMPS divides the space into two mazes without self-intersection and has periodicity in three independent directions [166; 167]. It can also be considered that an important subclass of TPMS is to divide the space into two disjoint but continuous interlaced regions. TPMS has two key features: (1) The average curvature on the surface is zero everywhere; (2) they are perimeter in all three coordinate directions (i.e., they extend infinitely in all directions and have the symmetry of one of the crystal-space groups). [PERSON] et al. [167] designed Schoen Gyroid unit cell to form lattice structure based on TPMS. The experimental results showed that Schoen Gyroid lattice structure had the characteristics of self-supporting and high manufacturability. Schoen Gyroid lattice structure made of titanium alloy showed good strength. [PERSON] et al. [168] pointed out that lattice structure designed by TPMS has a good combination of specific stiffness and axisymmetric stiffness, high surface volume ratio, and pore connectivity. TPMS method reduced the need for surface skin. Therefore, scholars use  
Figure 12: 3D unit cell structure library [157]: tetrahedron, octahedron, cube, vector (**top row**, from left to right); icosahedron, dodecahedron, tetrakaidecahedron, triacontahedron (**bottom row**, from left to right). Figure 13: 3D texture mapping method [157]. TPMS method to design the uniform lattice structures [98; 169; 170; 171; 172; 173; 174; 175; 176; 177; 178]. [PERSON] et al. [178] reviewed multifunctional mechanical-metamaterials lattice structures based on TPMS. They summarized the TPMS mathematical formulas for generating different unit cell shapes. Figure 14 shows the different unit cell based on TMPS. Schwarz Primitive \(\Phi_{\rm p}\):  
\[cosx+cosy+cosz=c \tag{1}\]  
Schoen-Gyroid \(\Phi_{\rm G}\):  
\[sinx\times cosy+siny\times cosz+sinz\times cosx=c \tag{2}\]  
Schwarz-Diamond \(\Phi_{\rm D}\):  
\[cosx\times cosy\times cosz-sinx\times siny\times sinz=c \tag{3}\]  
Schoen-I-WP \(\Phi_{\rm IWP}\):  
\[2(cosx\times cosy+cosy\times cosz+cosz\times cosx)-(cos2x+cos2y+cos2z)=c \tag{4}\]  
Fischer-Koch S \(\Phi_{\rm S}\):  
\[cos2x\times siny\times cosz+cosx\times cos2y\times sinz+sinx\times cosy\ cos2z=c \tag{5}\]  
Schoen-FRD \(\Phi_{\rm FRD}\):  
\[4(cosx\times cosy\times cosz)-(cos2x\times cos2y+cos2y\times cos2z+cos2z\times cos 2x)=c \tag{6}\]  
where \(x=2\pi X/\!L_{x}\), \(y=2\pi Y/\!L_{y}\), \(z=2\pi Z/\!L_{z}\), and \(L_{x}\), \(L_{y}\) and \(L_{z}\) are the unit cell sizes in the \(X\), \(Y\), and \(Z\) directions. Based on TPMS, [PERSON] and [PERSON] [175] used implicit surfaces with unit cells of Diamond and Gyroid. Implicit surfaces provide a compact representation for potentially complex surfaces, and also  
Figure 14: Unit cell based on TPMS: (**a**) [PERSON] [176]; (**b**) [PERSON] [176]; (**c**) [PERSON] [176]; (**d**) Schoen IWP [176]; (**e**) [PERSON] [172]; (**f**) Schoen FRD [172]. offer a number of advantages such as flexibility and well-defined Boolean operation.