# Topology Optimization of Anisotropic Materials with Smooth Fiber Orientation
## 1 Introduction
In the topology optimization of anisotropic materials, [PERSON] [21] combined a filtering technique applied to the material stiffness tensor, which is not limited to the actual parameterization of the design variables and can control the smoothness of the fiber path. [PERSON] [22] combined the characteristics of two design variables, regarded the density material variable in the element as the weight factor of fiber orientation, and completed the adjustment of fiber orientation to create a continuous fiber trajectory direction. Similarly, [PERSON] [23] greatly reduced the curvature of the fiber path by keeping the filter radius of the fiber angle consistent with the mesh resolution and applied it to the optimization of the three-dimensional structure. [PERSON] et al. [24] completed the material layout and fiber angle optimization of orthotropic materials through the density-based method (SOMP) and demonstrated the EQS method, streamline method, and offset method fiber filling schemes based on density-based topology optimization. The manufacturability is high, but this post-processing method often leads to lower structural stiffness. [PERSON] [25] discussed the optimal design methods of variable stiffness laminated continuous fiber-reinforced composites and transformed these designs into actual manufacturing processes. More prospectively, [PERSON] [26] introduces a new method called FRC-TOuNN, which uses neural networks to perform topology optimization of continuous fiber-reinforced composites. This method combines machine learning techniques and topology optimization algorithms to improve the efficiency and accuracy of the design process, especially when dealing with complex fiber paths and load conditions. However, for the concurrent optimization of topology and the fiber angle, although the above fiber angle optimization strategy and filtering technology significantly improve fiber continuity, there are two obvious defects: it is impossible to quantify the degree of fiber continuity and achieve accurate optimization goals and in the pursuit of fiber smoothness, too much of the stiffness of the overall structure is sacrificed. In view of the above problems, this paper proposes a topology optimization method for anisotropic materials with smooth fiber orientation, which introduces the fiber angle constraint strategy and adaptive filtering technology. The fiber angle constraint strategy can quantify the degree of fiber continuity and effectively improve fiber continuity. At the same time, the application of adaptive filtering technology can adjust a reasonable fiber angle distribution on the basis of smoothing fibers, thereby enhancing the stiffness of the overall structure. Therefore, the combination of the fiber angle constraint strategy and adaptive filtering technology can accurately improve fiber continuity while ensuring overall structural stiffness. Specifically, the proposed method aims to meet the following criteria:  
1. The fiber angle constraint function is created and integrated into the Method of Moving Asymptotes (MMA) [25] to accurately improve fiber continuity;
2. The Gaussian function is used to filter the fiber angle adaptively, and the strength of the filter is controlled by dynamically adjusting the standard deviation of the Gaussian function so as to realize the reasonable distribution of the fiber angle and further improve the structural stiffness;
3. The whole optimization process is realized by MATLAB R2021a, and the code structure is easy to understand and implement (see Appendix A). The organizational structure of this paper is as follows. Section 2 first introduces the concurrent optimization framework based on density. On this basis, it expounds the constraint strategy and adaptive filtering technology of the fiber angle and shows the complete optimization process and MATLAB code implementation. In Section 3, a large number of numerical examples are analyzed.