# Smooth Design of 3D Self-Supporting Topologies Using Additive Manufacturing Filter and SEMDOT
## 1 Introduction  
Design for additive manufacturing (DfAM) represents a range of design methods through which performance and/or other key considerations such as manufacturability, reliability and cost can be optimized subject to the capabilities of additive manufacturing (AM) technologies [1; 2; 3; 4]. In recent years, topology optimization for AM has become one of the most important branches in DfAM, as topology optimization has great potential to fully exploit the significant benefits provided by the increased design freedom offered by AM [5; 6]. As traditional element-based algorithms such as solid isotropic material with penalization (SIMP), rational material with penalization (RAMP) and bi-directional evolutionary structural optimization (BESO) will inevitably form non-smooth boundaries, post-processing or redesign methods have to be used to obtain accurate boundary information for the purpose of engineering applications [7; 8; 9], meaning that extra efforts have to be made after topology optimization. Given the significance of accurate boundary representation, some element-based algorithms that are capable of forming smooth boundaries such as multiresolution topology optimization (MTO) methods [10; 11; 12], elemental volume fraction-based methods [13; 14; 15; 16] and a method using floating projection [17; 18] have been developed in recent years. Optimized topologies are fabricated layer-by-layer and have to be sufficiently supported to prevent the component distortion caused by high bending stresses in some AM technologies such as fused deposition modeling (FDM) and selective laser melting (SLM). Even though introducing support structures to topological designs can resolve this issue, more materials need to be wasted, and extra efforts have to be made to remove supportstructures after manufacturing [19; 20; 21]. Therefore, optimizing self-supporting topologies directly is an effective way to mitigate overhang angle limitations, save materials and avoid additional effort. Even though there are a great number of investigations regarding 2D self-supporting topologies [22; 23; 24], these studies have more theoretical value than application value compared to the investigations on 3D self-supporting designs. Studies on 3D self-supporting design have attracted scholarly attention in recent years. The early work in this field was [PERSON]'s gradient-based AM filter [25], followed by an improved method capable of executing simultaneous topology optimization and support structures considering metal AM constraints and post-print machining requirements [26]. [PERSON] et al. [27] proposed hybrid additive-subtractive manufacturing constraints for 3D continuum structures based on BESO. [PERSON] and [PERSON] [28] proposed a second-order measure of boundary oscillations for overhang control to enhance the manufacturability of designs in AM, and a 3D cantilever beam case was tested. [PERSON] et al. [29] formulated the self-supporting requirements as an explicit quadratic continuous constraint, and 3D benchmark problems were tested. [PERSON] and [PERSON] [30] integrated the proposed constraint that was imposed on the overhang angle, directional-dependent overhang angle and horizontal minimum length scale into the MTOP method to conduct the smooth design of 3D self-supporting topologies. [PERSON] et al. [31] presented a 3D front propagation-based overhang filter through which printable topologies can be obtained. Most recently, [PERSON] et al. [32] developed a new layer-wise geometric self-supporting constraint for 3D continuum structures based on BESO, and an AM experiment was conducted to validate the effectiveness of the proposed constraint by printing a hinge frame used in the aerospace field. The authors of this study merely focused on the smooth design of 2D self-supporting topologies through the combination of the Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) algorithm and [PERSON]'s AM filter in previous works [33; 34].