# Topology Optimization of Anisotropic Materials with Smooth Fiber Orientation
## 2 Model Derivation
### Adaptive Filtering of the Fiber Angle  
In the method of concurrent optimization of topology and fiber angle, many scholars have not filtered the fiber angle, which leads to poor continuity of the fiber; in addition, the traditional filtering method uses a distance-weighted filter to adjust the fiber angle. This filtering scheme will lead to excessive smoothing of the fiber, thereby reducing the stiffness of the overall structure. Therefore, this section proposes an adaptive filtering method for the fiber angle. This method uses the characteristics of the Gaussian function to dynamically adjust the filtering strength according to the real-time optimization effect and then moderately smooth the fiber angle. Since the introduction of the constraint function in Section 2.2 improves the continuity of the fiber, but at the same time leads to a decrease in the overall structural stiffness, the role of adaptive filtering is to ensure the continuity of the fiber while improving the stiffness of the overall structure. Firstly, the traditional filtering methods are reviewed as follows:  
\[\hat{f}_{e}=\frac{\sum\limits_{m=1}^{N}\hat{H}_{m}t_{m}}{\sum\limits_{m=1}^{N} \hat{H}_{m}}, \tag{13}\]  
where \(\hat{f}_{e}\) is the fiber angle of the central unit after filtration; \(t_{m}\) is the fiber angle of the neighborhood of the central unit before filtering; \(N\) is the number of neighborhood units; and \(\hat{H}_{m}\) is the weight factor, which is defined as follows:  
Figure 2: Adjustment of the fiber angle difference \(\varphi_{e,q}\) between adjacent units. \[\hat{H}_{m}=r_{\rm min}-dist(e,m),\{m\in N_{r},|dist(e,m)\leq r_{\rm min}\}, \tag{14}\]  
where \(dist(e,m)\) is the distance between unit \(m\) and central unit \(e\) and \(r_{\rm min}\) is the filter radius. The adaptive filtering of fiber angle is based on the characteristics of the Gaussian function. According to the convergence degree of fiber angle, the standard deviation of the Gaussian function is corrected in real time, and then the filtering strength is adjusted. The convergence degree of the fiber angle is described by the maximum change value \(\Delta\theta_{\rm max}\) of the design variable. The specific calculation is as follows:  
\[\Delta\theta_{\rm max}=\max\{\theta_{n-1}-\theta_{n-2}\}, \tag{15}\]  
where \(n\) represents the number of iterations and \(\theta_{n-1},\ \theta_{n-2}\) are the fiber angle values of all elements in this iteration and the previous iteration results, respectively. Figure 3 is the calculation diagram of \(\theta_{n-1},\ \theta_{n-2}\). The difference between the fiber angles of all units in the two iteration results is calculated, and the maximum difference is selected as \(\Delta\theta_{\rm max}\), which indicates the convergence degree of fiber angle optimization. It is also used as a correction factor for the standard deviation \(\vartheta_{\vartheta}\) of the Gaussian function for the real-time correction of fiber angle filtering. The modified equation is as follows:  
\[\vartheta_{\vartheta}=\Delta\theta_{\rm max}.\sigma_{\vartheta}, \tag{16}\]  
Here, \(\vartheta_{\vartheta}\) is the standard deviation of the modified Gaussian function and \(\sigma_{\vartheta}\) is the initial value of the predefined standard deviation. After correcting the standard deviation of the Gaussian function, the weight of the Gaussian kernel function is calculated as follows:  
\[W_{\theta}(r_{z})=\frac{1}{2\pi\vartheta_{\vartheta}{}^{2}}e^{-\frac{r_{z}}{2 \vartheta_{\vartheta}{}^{2}}},\ r_{z}\leq r_{\rm min}, \tag{17}\]  
where \(W_{\vartheta}\) is the Gaussian kernel function used for fiber angle filtering and \(r_{z}\) is the distance between all units in the filtering area and the central unit. The size of the standard deviation \(\vartheta_{\vartheta}\) can determine the degree of influence of the surrounding units on the central unit in the filtering area. The larger the standard deviation, the greater the influence of the surrounding units on the central unit, and the more obvious the filtering effect.