## 1 Introduction
In later studies, researchers focus on the design of the cellular structure with specific performances [36; 37; 38] or gradient properties [30; 39] by using the discipline of topology optimization. Particularly, for an FGM composed of cellular structures, the connectivity issue along the gradient direction has to be considered. Otherwise, the fabrication of the FGM structure is not able to be guaranteed [40; 41]. As a result, many researches have been done for tackling this specific issue in the topology optimization of FGMs. For instance, [PERSON] and [PERSON] [9] systematically proposed three different approaches to solve the connectivity issue between adjacent cellular structures, and made an in-depth discussion about the connection effect. [PERSON] et al. [10] devised a way to optimize all three cells simultaneously, and developed a filtering scheme to enhance the connectivity between different material microstructures. [PERSON] et al. [11] used a kinematically connective constraint approach to guarantee the connectivity of the neighboring cellular structures. [PERSON] et al. [42; 43] proposed a VCUT level set method to generate functional graded cellular structure, where a high-order continuity between the graded microstructures can be achieved. [PERSON] et al. [44] investigated the surface-based FGM, and illustrated the connection issue between completely different unit cells. Above-mentioned studies revealed that while considering the connectivity, the smoothness of the boundary makes a significant impact on the mechanical property and manufacturability of the cellular FGMs [40; 41]. However, the existing methods used to tackle the connectivity problem are usually coupled with the optimization process, which may reduce the efficiency of the optimization process and increase the complexity of the design formulation. An efficient method that is independent of the optimization process can alleviate this issue. Level set methods [45], owing to their perfect description of the smooth boundary, are effective ways to divide solid and void parts of the structure in topology optimization. This favorable feature of the level set methods makes it to be considered as a favorable candidate to handle the connectivity issue in the topology optimization design for FGMs. The main concept of LSM is to describe a low-dimensional structural geometry with the high-dimensional level set function, and the zero level set represents the structural boundary. In the classic level set method, the structural boundary propagation is governed by solving the Hamilton-Jacobi partial differential equation (H-J) PDE) [19; 46], which usually requires complex numerical schemes. Furthermore, it is not easy to incorporate the classic level set method with the well-established structural optimization algorithms, like the optimality criteria (OC) method [47], the method of moving asymptotes (MMA) [48], and so on. In order to use the existing optimization algorithms and reduce the computational complexity, several alternative LSMs [49; 50; 51] have been developed. Especially, the parametric level set methods (PLSM) use the radial basis functions (RBFs) [52] to interpolate the level set function, and thus the time and spatial variables in the level set function are decoupled. In this fashion, the complicated H-J PDE-driven topology optimization problem in the traditional LSM can be converted into a compact parameterized optimization problem, which can be easily solved by the gradient-based algorithms. This study adopts the PLSM due to its high efficiency. Based on the above considerations, this paper systematically investigates the topological optimization design of FGMs, from the generating of cellular structures to the connectivity of adjacent microstructures. In this study, the cellular FGMs endowed the property of extreme bulk or shear modulus are devised by using the incorporation of the TO method with numerical homogenization theory.