## 3 Parameterization Schemes for Fiber Orientation
### Discrete-Continuous Parameterization  
Utilizing continuous and discrete methodology benefits is another alternative to fiber orientation optimization. The key idea in the following approaches is to fill the gaps by acknowledging the beneficial characteristics of both strategies to improve computational efficiency, reduce local optima, and/or resolve fiber continuity and/or manufacturability issues. Therefore, an approach to reduce the risk of falling into local optimal without sacrificing the fiber's continuity can use both discrete and continuous parameterization, as suggested by [PERSON] et al. [115]. Their work proposed a coarse-to-fine strategy, where the orientation design space is divided into discrete sub-intervals. After that, the CFO searches for an optimized solution in a sub-interval, where the sub-interval selection problem is solved using the DMO approach. However, no criterion is defined to determine the number of sub-intervals required in advance. Nevertheless, the proposed strategy provides flexibility to integrate alternatives suggested for DMO and CFO approaches. [PERSON] et al. [116] studied the Cartesian system for orientation design variables to improve initial design dependency and local optima issues encountered in the continuous parameterization approach. The parameterization scheme was further extended to yield an optimized discrete orientation design for a given discrete orientation set in their work. Moreover, the characteristics representing the orientation design variables in the vectorial form consider the \(2\pi\) ambiguity, which occurs due to the periodic nature of the orientation design variable. Introducing vectorial design variable as a point-wise quadratic inequality constraint yields more interpolated elasticity tensors than the single variable polar representation. However, the optimization algorithm treats continuous and discrete problems as two different problems; thus, investigating the coupled optimization framework might be an outlook for consideration. [PERSON] et al. [112] proposed a parameterizing scheme that continues the computational approach suggested by [PERSON] and [PERSON] [105]. Introducing a normal distribution function as a weighting function in their parameterizing strategy guarantees fiber convergence, low sensitivity to the initial fiber configuration, and continuity of the fiber orientation. [PERSON] and [PERSON] [117] develop a continuous global function by applying the shepherd interpolation method at scattered design points to represent the fiber orientation throughout the design domain. The interpolation function's benefit is that it ensures fiber continuity while considering a reduced orientation design space in contrast with CFO. Unfortunately, it suffers from the initial configuration and ends at the sub-optimal solution. Another work of [PERSON] [118] applied multilevel optimization for fiber orientation optimization and verified its efficiency against single-level optimization. Still, the optimization results in different fiber arrangements for different initial fiber orientations. As a result, the efficiency of the multilevel approach relies on the attained fiber orientation field at a coarse level since the optimization at the successive refined level starts from an initial design computed at its neighboring coarser level. A more recent effort by [PERSON] et al. [119] introduced a discrete-continuous framework taking inspiration from the available literature [112; 115]. However, no comparative studies are considered, and solutions fall to the local minima when the discrete-continuous interval is greater than two. Finally, [PERSON] et al. [91] present an approach to optimizing material orientations using multiple print planes (Figure 4) and demonstrate their optimization framework extends the material orientation design space by attaining lower compliance value. Apart from the numerical verification, the results are manufactured using nylon filaments with chopped carbon fibers.