# Topology Optimization of Anisotropic Materials with Smooth Fiber Orientation
## 1 Introduction  
Topology optimization [1] has attracted much attention in structural design, especially in the industrial field to deal with lightweight requirements. Traditional methods are not enough to meet the requirements, so fiber-reinforced composites [2] have become the focus of optimization, which means that in addition to topology optimization of the overall structure, the best orientation of the material must also be considered. The fiber orientation is generally defined by the fiber angle of the discrete element. The fiber angle optimization is divided into discrete material optimization (DMO) [3, 4, 5, 6] and continuous fiber angle optimization (CFAO) [7, 8, 9, 10, 11]. Unlike DMO, which is not easy to converge and has limited candidate angles, CFAO is more widely used. It is no longer limited to a set of candidate angles in a specified direction but takes the continuous rotation angle of the anisotropic substrate as a design variable. Therefore, CFAO is favored by many scholars in the concurrent optimization of topology and the fiber angle. Many scholars have proposed different methods in the study of CFAO topology optimization, aiming at improving fiber continuity. Based on the improved SIMP method [12], [PERSON] [13] used the volume control and stress control algorithm to optimize the topology in two stages to obtain the optimal structural topology of isotropic materials. After establishing the average load transfer trajectory in the optimal topology, the fiber reinforcement placement path is defined for the main continuous fiber filling area. The two-stagetopology design makes the structural mechanical performance better, and the streamline method makes the fiber laying smoother. [PERSON] et al. [14] conducted in-depth research on the optimal fiber orientation and fiber volume ratio under the constraint of the fiber cost function and proposed a classical compliance minimization solution, but the problem expression they used is not a traditional topology optimization equation. In the framework of cellular automata, [PERSON] [15] obtained a smooth distribution of fiber angles by averaging adjacent fiber angles. This method is similar to using a mean filter to adjust the fiber angle, which often leads to excessive smoothing of the fiber angle. Similarly, [PERSON] and [PERSON] [16] proposed a cooperative design strategy by imposing angle constraints between adjacent elements to ensure that bidirectional constraints are satisfied. This method effectively coordinates the design of fiber orientation so that it can meet the specific mechanical performance requirements while maintaining structural integrity. [PERSON] [17] described a three-phase method to optimize the orientation of continuous fibers in isotropic materials. This method allows the design of composite materials with variable stiffness, where the fiber volume fraction and the orientation of each node are considered stiffness design variables [18; 19]. [PERSON] et al. [20] skillfully applied the curvature constraint of the fiber angle in the optimization process and carefully smoothed the optimization results to enhance manufacturability. In the topology optimization of anisotropic materials, filtering or projection techniques are widely used, which play an important role in the whole optimization process. The application of these technologies can not only effectively avoid numerical instability but also improve the manufacturability of the optimization results. Therefore, the appropriate filtering method injects deep robustness and feasibility into the concurrent optimization method of structural topology and the smoothing fiber angle. It is of great significance to use the filtering technology to control the fiber orientation in the CFAO method, which can avoid the stress concentration caused by the excessive curvature of the fiber and solve the manufacturing constraints related to the curvature of the fiber in the subsequent printing.