## 3 Parameterization Schemes for Fiber Orientation
### Continuous Parameterization  
The continuous parameterization of fiber orientation (CFO) design uses the angle itself as the design variable [74; 75]. The design variable is the continuous and independent parameter that provides flexibility in changing the orientation across the design points, relaxing orientation design space, as shown in Figure 3. The rotation stiffness tensor, \(\overline{\mathbf{C}}(\theta)\), is derived from the base anisotropic stiffness tensor, \(\mathbf{C}\), using rotation tensor \(\mathbf{T}(\theta)\), where \(\theta\) coincides with the direction of the fiber, and \(c\) and \(s\) stands for \(\cos\theta\) and \(\sin\theta\):  
\[\overline{\mathbf{C}}(\theta)=\mathbf{T}^{-1}(\theta)\cdot\mathbf{C}\cdot \mathbf{T}^{\prime}(\theta)=\left[ \begin{array}{ccc}\bar{C}_{11}&\bar{C}_{12 }&\bar{C}_{16}\\ \bar{C}_{12}&\bar{C}_{22}&\bar{C}_{26}\\ \bar{C}_{16}&\bar{C}_{26}&\bar{C}_{66}\end{array} \right] \tag{11}\]  
\[\mathbf{T}(\theta)=\left[ \begin{array}{ccc}c^{2}&s^{2}&2 cs\\ s^{2}&c^{2}&-2 cs\\ -cs&cs&c^{2}-s^{2}\end{array} \right],\quad\mathbf{T}^{\prime}(\theta)=\left[ \begin{array}{ccc}c^{2}&s^{2}&cs\\ s^{2}&c^{2}&-cs\\ -2 cs&2 cs&c^{2}-s^{2}\end{array} \right] \tag{12}\]  
Figure 2: Illustration of DTO considering anisotropic material via fiber orientation parameterization. Paramaterized fiber orientation, \(\theta\), in the optimization process continuously spanned in the angle range \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Thus, handling a continuous fiber orientation design presents difficulties due to a fourth-order transform tensor that rotates to a given angle composed of multivalued sine and cosine functions, resulting in a non-convex optimization problem. Furthermore, optimizing the fiber orientation is susceptible to the initial fiber configuration, thus causing difficulties in obtaining the optimized solution. As illustrated in [76], suboptimal solutions are the persistent outcome of a continuous fiber orientation design problem. One brute-force way to avoid it is by further relaxing the design space. For instance, free material optimization (FMO) [77; 78] parameterizes each stiffness tensor element independently as the design variable. This secures the scheme from the complexity of the orientation design variable of the design space. However, as compensation, point-wise nonlinear constraints ensure the positive semi-defintiveness of the obtained stiffness tensor and link it to the feasible physical design, making this approach challenging. [PERSON] et al. [79] formulated an orientation design variable as a tensor field to simplify the first tensor invariant constraint and remove nonlinear constraints successfully introduced due to the second tensor invariant. Still, as commented, the violation of these constraints is observed at the joint point of the structural members where the orientation shows the discontinuous distribution. Early studies utilizing the analytically derived optimally criterion [80] for optimizing fiber orientation date back to the pioneering work of [PERSON] on the strain-based method [81; 82; 83]. In that work, strain energy density was transformed into principal strain, and it was concluded that material orientation axes that lie along principal strain axes always give stationary energy density. However, [PERSON] [84] argued that the discussion is limited to a unit cell case where the orientation variable is separated from the design domain to obtain extreme strain energy. After that, a similar deduction using iterative optimality criteria [85; 86] formulated the stress-based method [87] by exercising an invariant stress field for material orientation. Finally, [PERSON] and [PERSON] [88] extended the stress-based method to determine the optimal orientation optimization problem corresponding to multiple loads. Despite their similarity, the stress-based method produces a slightly stiffer structure than the strain method because strong couplings exist among the orientational variables when the strain field is used [84].