## 3 Transitional Cell for Connectivity  
In this article, the FGM is formulated by a family of cellular base structure (CBS) with different layouts or shapes. In order to enable the gradient change in the FGM, geometric differences between two connected cells along the gradient direction are bound to exist, so the connectivity issue must be considered to make sure the feasibility and manufacturability. Shape interpolation technology [12] can be used to connect the CBS, in which the C\({}^{0}\) connectivity is maintained due to the similar topologies. However, the smoothness of boundary isn't taken into consideration, as shown in Figure 1. Obviously, the boundary mismatch will cause a stress concentration or manufacturing issue for the cellular structural FGM.  
In order to ensure the smooth boundary, PLSM is adopted. Considering that the cellular structural boundary is embedded in the level set function \(\Phi(x)\), a specific level set determines the exact configuration of a cellular structure. Usually, for a periodic cellular structure, \(\Phi(x)\) is a square matrix, and the dimension is depended on the number of level set knots. To connect two adjacent cellular structures, a transitional cellular structure (TCS) can be generated via the interpolation of two level sets that respectively represent those two cellular structures. The constructed TCS can have partial features of both the adjacent cellular structures, so as to better fit the continuous change between different boundaries. A C\({}^{1}\) connection between gradient microstructures by using the proposed method is shown in Figure 2. Through the proposed method, the mismatch phenomenon can be completely avoided. To design the layer-wise FGM, this study creates a TCS between every two CBS over layers to maintain the C\({}^{1}\) connection. The macro configuration of the concerned FGM is shown in Figure 3, where every two CBS layers contain a transitional connecting TCS layer.  
Figure 1: C\({}^{0}\) connectivity in FGM.  
Figure 3: Configuration of the considered FGM.  
Figure 2: Transitional cell (middle) to connect two CBS (left and right) with C\({}^{1}\) continuity.  
To construct a TCS, the corresponding level set functions of two optimized CBS are required. Different from the existing work, the \(\Phi(x)\) of TCS is not optimized but numerically interpolated. Thus, the level set function \(\Phi(x)\) of TCS can be obtained by using a very small computational effort. The interpolation process for TCS can be mathematically expressed as:  
\[\Phi_{trans}(i)=(1-\beta(i))\cdot\Phi_{1}(i)+\beta(i)\cdot\Phi_{2}(i)\qquad(i=1,2...n) \tag{19}\]  
where \(\Phi_{trans}\) is the level set function of the TCS. \(\Phi_{1}\) and \(\Phi_{2}\) respectively represent the level set function of the left and right CBS in Figure 2. \(n\) is the number of columns in the matrix of the level set function. \(\beta\) is a weighting factor reflecting the degree of transition from left to right. In the interpolation process, the parameter \(\beta\), which is a continuously changing value from 0 to 1, governs the geometry varying process from one CBS to the other CBS. Considering all the CBS in the FGM design, a transition cell can be easily constructed by using the level set functions of every two adjacent CBS. Therefore, the geometric continuity between two layers of the FGM can reach up to C\({}^{1}\). More importantly, the construction of every TCS in a cellular FGM has nothing to do with the optimization for the CBS. This means the CBS can be topologically designed without considering the connectivity constraints, and the performances of all the CBS will not be affected.