## 3 Parameterization Schemes for Fiber Orientation
### Material Parameterization  
Further relaxation in the feasible design space can introduce through material heterogeneity. For example, heterogeneous composite [123] materials consist of two or more materials and are engineered to vary the spatial composition and structure continuously. Thus, such variation pushes the envelope of material property space beyond natural limits. In addition, recent studies [124; 125; 126] have shown that CF4 is ready to manufacture FRC structures with continuous yet spatially varying fiber paths and fiber volume fractions. Thus, if properly optimized, variable FRC material properties may perform better than a fixed FRC material volume fraction. Therefore, a composite structure comprising heterogeneous FRC material distribution provides considerably larger design freedom to CF4. Accordingly, [PERSON] et al. [127] considered a SIMP-based sequential TO approach to design FRC structures by considering fiber and material fractions in a given design space. A sequential process begins with designing an isotropic-material matrix with voids, selecting fiber fractions, and optimally orienting the fibers. However, the sequential approach sacrifices the exploration of new topologies that might be optimal for variable FRC structures. In that spirit, [PERSON] et al. 's [128] work investigated the simultaneous design of matrix topology, fiber material layout, and orientation using an anisotropic topological derivatives framework. In addition, the dense arrangement of fibers was evenly spaced for the part's manufacturability while retaining their specific patterns. However, the structural performance resulting from simplifying the dense fiber arrangement was not evaluated, thus questioning the reliability of the printed part. The work, as mentioned earlier, implemented mono-scale approaches to optimize the distribution and orientation of the FRC material. However, CF4 also provides an effective way to fabricate mono-scale structures and multi-scale structures. Multi-scale approaches can be classified into lattice-based and hierarchical-based topology optimization based on their micro-scale optimization methodology. The first approach uses offline homogenization to calculate effective elastic coefficients of a fixed or a set of multi-variable micro-structures. The computed sets of effective elastic coefficients are then interpolated to attain continuous variations in these coefficients for the corresponding variations in the micro-structure's parameters. In contrast, the hierarchical approach solves a spatially varying micro-structural and macro-scale optimization problem at each material point, leading to high computational costs and connectivity issues. Nevertheless, since FRCs have inherent multi-scale characteristics, spatially varying material distributions and geometric patterns spanning at least two or more scales hold a promising future for designing next-generation lightweight structures. In addition, interested readers can refer to [PERSON] et al. [129] review paper to understand the general framework for multi-scale topology optimization. On the other hand, the multi-scale strategy for anisotropic materials is challenging for the following reasons: length scale controls, connectivity across adjacent micro-structures, ability to produce models for damage criteria to capture actual anisotropic behavior, and unique treatments at the boundaries of the domain, for example. One must investigate these factors through experiments or appropriate numerical tools to estimate the actual performance of printed parts. Only a few works address the multi-scale approach for FRCs based on the authors' knowledge. [PERSON] et al. [130] adopted the homogenization method for simultaneously designing spatially varying fiber volume fractions and orientations as shown in Figure 6. Their work used SIMP to design the macro-structure's composite topology. Finally, the de-homogenization procedure [131] applied to fiber micro-structures obtained in the coarser mesh was visualized by projecting at a finer mesh.