## 1 Introduction  
Organisms like bamboo, to adapt the environment [1], often consist of a gradient structure. Inspired by the materials that already exist in nature [2], functionally graded materials (FGM) is usually composed of continuously varying cellular structures in one direction and self-repeated in the other direction. Due to their excellent performances, FGM has been extensively studied and reported [3, 4]. In particular, this study focuses on the man-made FGMs constructed by a series of porous microstructures due to their design flexibility and high performances [5]. With the development of additive manufacturing, the fabrication of complex cellular structures has become possible and artificial FGMs are also widely utilized in engineering applications [6; 7; 8]. However, cellular FGM often contains complex and spatially varying microstructures. It is difficult to simultaneously devise all the different microstructures and ensure their connection at the boundaries. Because different microstructures usually have different topologies, and there often lacks a proper mechanism on the macro scale to connect all the microstructures, especially when tailoring the FGMs with the topology optimization [9; 10; 11]. Over the past few years, for the design of the cellular structures, researchers have widely adopted topology optimization (TO) technology and the numerical homogenization method [12]. Topology optimization is regarded as an effective structure optimization method by using the finite element method (FEM) and advanced optimization algorithms. Under the given boundary and loading conditions, TO takes the structural performances as the design objective and gradually achieves the optimal material distribution within the design domain [13]. Up to now, several TO methods have been developed, e.g., the homogenization method [14], solid isotropic material with penalization (SIMP) method [15; 16], evolutionary structural optimization (ESO) [17; 18], level set methods (LSM) [19; 20; 21; 22], and so on. There are many other ways to describe the configuration of the structure, such as the SIMP approach based on non-uniform rational basis spline (NURBS) framework [23; 24; 25]. Moreover, the numerical homogenization method has been developed as an efficient tool to evaluate effective properties of the periodic porous materials [26]. Considering the scale effect, the above cellular solid performs more like a material whose elasticity tensor can be approximately calculated by imposing periodic boundary assumptions. For the topological optimization design of periodic cellular materials, the incorporation of the topology optimization scheme with the homogenization theory can automatically devise the geometrical configuration of cellular structures with extreme performances, such as maximum bulk or shear modulus [27], negative Poisson's ratio [28; 29; 30], thermoelastic property [31], and so on. On the other hand, cellular structures with desired properties can also be tailored by utilizing an inverse homogenization procedure [32; 33]. The above studies mainly assume that the whole design domain consists of a number of uniform cellular materials. However, FGMs with spatially varying cellular microstructures always have more design freedom and can facilitate the exploration of combined functions for real-world applications [6; 34]. FGM can be regarded as a kind of artificial composite that can continuously change their volume fraction, composition or microstructures, so as to adapt to the service environments [35]. Therefore, mechanical performances (e.g., stiffness, strength, and toughness) of the FGM can be greatly improved when compared to the traditional materials with uniform cellular structures [3]. However, how to achieve FGM properties via the change of material constituents or porous microstructures over layer or volume is always a difficult task.