## 4 Numerical Examples in 2D Case
### Comparison of FGMs with C\({}^{0}\) and C\({}^{1}\) Smoothness  
For comparison, two different gradient connection methods are considered to construct the FGM, where the interfaces between different CBSs are respectively characterized with C\({}^{0}\) and C\({}^{1}\) smoothness. First of all, FGM is devised by using the proposed transitional cell method, in which the FGM is consisting of CBSs and TCSs. Here, CBSs are optimized with the extreme performance of maximum bulk modulus, and the property gradient is governed by the gradually changing volume fraction along the predefined direction. In this example, each CBS is discretized into \(20\times 20\) four-node quadrilateral elements. The Young's modulus and Poisson's ratio of the solid material are set as E = 910 MPa and \(\
u\) = 0.3, while in case of computational singularity, for the void material, E\({}_{\text{void}}\) = 0.001 MPa and \(\
u_{\text{void}}\) = 0.3. The volume fraction of each CBS and TCS are displayed in Table 1. The optimized FGMwith the specified gradient in volume fraction is presented in Figure 4. With the construction of transition cells, a smooth connection is achieved between the neighboring cellular structures. At the same time, the required characteristic of the volume fraction gradient is maintained. Under the condition that each layer of the cellular structures in the FGM has the same volume fraction, the conventional FGM constituted by C\({}^{0}\) connection is designed. In particular, the FGM with C\({}^{0}\) connection can be implemented by respectively devising the CBSs with the corresponding gradient volume constraints. Then, the tailored CBSs are resembled along the gradient direction without constructing the TCS layers. In order to compare the effects of the two gradient connection methods on the structural mechanical properties, statics analysis in ANSYS Workbench is conducted. To ensure the validity of simulation, the gradient structure is tiled into 9 columns (Figure 5) and modelled with 1 mm thickness. The loading and boundary conditions are shown in Figure 6, where the bottom of the structure fixed and a set of uniformly distributed forces F = 20 N is applied at the top. The Young's modulus and Poisson's ratio of the solid material are set to be the same with the optimization configurations. The simulation results are plotted in Figure 7, where the stresses on two selected joints in both the FGMs obtained by considering different interface smoothness are displayed. For the design with the C\({}^{0}\) connection, the magnitude of the stresses is 5.9 MPa and 12.7 MPa. While for the design with C\({}^{1}\) smoothness, the magnitude of the stresses is 3.6 MPa and 5.9 MPa at the same sample points. From the results of the simulation analysis, it can be clearly seen that with the force applying on the top, the obvious stress concentration occurs at the interface between two adjacent cellular structures. When comparing to the result obtained by using the proposed transitional connection method, conventional FGM design with C\({}^{0}\) connection displayed more serious stress concentration. Because the C\({}^{0}\) connection only ensures that the similar CBSs can be connected, the connected interface lacks a smooth transition. In such a case, the connection surfaces in the FGM with C\({}^{0}\) connection will be more likely to collapse. This proves that the proposed method can guarantee the higher-order continuity at the interfaces between the optimized CBSs, and thus alleviate the stress concentration at those interfaces. In this study, the strain energy caused by the applied loads is selected as an indicator to illustrate the structural stiffness of the devised FGM. In other words, the smaller internal strain energy of a structure is, the better structural stiffness it will be [41].