###### Abstract  
Topology Optimization (TO) recently gained importance due to the development of Additive Manufacturing (AM) processes that produce components with good mechanical properties. Among all additive manufacturing technologies, continuous fiber fused filament fabrication (CF4) can fabricate high-performance composites compared to those manufactured with conventional technologies. In addition, AM provides the excellent advantage of a high degree of reconfigurability, which is in high demand to support the immediate short-term manufacturing chain in medical, transportation, and other industrial applications. CF4 enables the fabrication of continuous fiber-reinforced composite (FRC) materials structures. Moreover, it allows us to integrate topology optimization strategies to design realizable CFRC structures for a given performance. Various TO strategies for attaining lightweight and high-performance designs have been proposed in the literature, exploiting AM's design freedom. Therefore, this paper attempts to address works related to strategies employed to obtain optimal FRC structures. This paper intends to review and compare existing methods, analyze their similarities and dissimilarities, and discuss challenges and future trends in this field.  
topology optimization; continuous fiber-reinforced composites; additive manufacturing; structure optimization; fused filament fabrication +
Footnote †: journal: Applied  
_Applied_  
_sciences_  
_Review_  
_Applied_Applied  
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}^{*},\quad i=1,\dots,Q\\ &:H(\phi)=\begin{cases}0,\quad\phi(\mathbf{x},\tau)<0\\ 1,\quad\phi(\mathbf{x},\tau)\geq 0\end{cases}\end{split}\] (5)  
LSTO, as classified above, operates on the boundaries instead of local density, which is the zero-level set of a scalar function (LSF), and topological changes are based on the LSF's evolution. Moreover, several LSMs have emerged, classified based on LSF parameterization, mapping the level-set-based geometry onto the mechanical model, and the strategy for solving the optimization problem, as detailed in the review paper [35]. The identification of the material interfaces, the structural response's accuracy, and the optimization process's convergence are influenced by the structural boundaries in the discretized mechanical model. The structural topology is mapped using conforming mapping, immersed boundary techniques, or density-based mapping. The first two approaches generally provide a crisp representation of the mechanical model's boundaries and are considered to solve variational problems (2) to perform general topology optimization (5). On the other, density-based
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mapping replaces the Heaviside function with density distribution \(\rho(\phi)\) to approximate (5) as \(\int_{\Omega}fH(\phi)dV\approx\int_{\Omega}f\rho(\phi)dV\), where the density distribution can map the LSF point-wise or element-wise.  
Finally, we rest our synopsis on the LSTO on the following grounds. First, most of the works reviewed in the articles follow density-based TO (as discussed in Section 2.2) to optimize anisotropic materials' orientation and distribution; moreover, DTO has also been applied to several other applications in academia and industrial sectors. Secondly, LSTO often adopts Eulerian mesh with esratz material (point-wise density distribution) and a DTO formulation because of the DTO's simplicity and ease of implementation. Still, the crisp boundary description is maintained throughout the optimization process, allowing shape sensitivity analysis and design updates by solving the HJ equation, i.e., different from DTO.  
Notably, the above deduction is not biased toward following a particular TO approach as no comparative studies are performed on their methodologies, numerical efficiency, numerical verification of the attained optimized design, or their realizability and applicability. Hence, the readers are referred to citations marked earlier for different TO approaches.