## 1 Introduction  
Additive manufacturing (AM) has become one of the most important methods for functional components due to its ability to produce complex geometries with high precision by combining materials layer by layer. Especially in metal-based AM, developments over the last decade have elevated the quality and reliability of the technology to a level suitable for creating fully functional final products rather than just prototypes. Topology optimization, used to take full advantage of AM's design freedom, is considered a switch that expands the manufacturability limits [1, 2, 3]. Topology optimization is an advanced structural design method that can obtain the optimal structural configuration through the best material distribution under certain boundary conditions [4, 5, 6]. Using additive manufacturing technology enables unique and innovative designs in various industries that cannot be achieved with conventional manufacturing [7; 8; 9]. To take full advantage of both topology optimization and AM, it is first necessary to integrate design constraints into topology optimization [10]. Support structures are required to prevent the overhangs from collapsing in the AM process, and supports usually have to be removed after manufacturing. Support structures used during manufacturing increase material consumption, post-processing time, and difficulties [11; 12]. Therefore, it has been reported that support structures are the most prominent problem limiting manufacturability in AM [13]. Therefore, one of the most prominent typical AM constraints for topology optimization is the overhang constraint [14]. An extensive body of research has been dedicated to minimizing additional supports in the additive manufacturing process. This has been achieved by incorporating the overhang constraint into the topology optimization process, leading to the proposal of optimized designs that eliminate the need for any support structures [15]. Consequently, manufactured structures have evolved to be integral components of the design rather than just supports. The critical overhang angle of the Ti-6 Al-4V part, fabricated using Selective Laser Melting (SLM), was empirically determined, with successful production realized at angles of 20\({}^{\circ}\) or greater [16]. Through numerous SLM-based experiments, Thomas further validated 45\({}^{\circ}\) as the critical overhang angle [17]. Additionally, the research conducted by Zou et al. delved into topology optimization for additive manufacturing, considering the self-supporting constraint within the Solid Isotropic Material with Penalization (SIMP) framework [18]. In an effort to minimize the volume of additional supports, Morgan et al. employed the optimal build direction within their optimization approach [19]. To further reduce supports, Hu et al. proposed a direction-based shape optimization approach [20]. A pivotal area of current research is the integration of overhang constraints into topology optimization to maintain self-supporting structures [21]. Johnson and Gaynor used this method to construct self-supporting structures [22]. In a similar study, Langelaar proposed a self-supporting AM filtering procedure that could be integrated into topology optimization, focusing on generation [15; 23]. This layer-based filtering concept introduced by Langelaar has seen extensive application and examination in the academic literature [24; 25; 26; 27]. An approach combining geometric and mechanical constraints has been developed to effectively utilize self-supporting structures [28]. Complementing this, Guo et al. introduced two effective strategies based on the Moving Morphable Components (MMC) and Moving Morphable Voids (MMV) frameworks, respectively [29]. In light of these investigations, topology optimization designs for AM have gained considerable significance in engineering applications and academic research in recent years.