# An Optimization Workflow in Design for Additive Manufacturing
## 2 Design Workflow
To avoid intermediate densities in the optimized solution, discrete methods, also called \"hard-kill\" methods, can be used; in the Evolutionary Structural Optimization (ESO) method proposed by [PERSON] and [PERSON] [43], a criterion parameter is calculated for each element and at each iteration the elements with the lowest criterion parameter value are eliminated. Furthermore, Bi-directional Evolutionary Structural Optimization (BESO) method [44] is an extension of ESO, in which new elements can be added next to those elements with a high criterion parameter at each iteration. Level-set Methods (LSMs) are another category of topology optimization methods where the iso-contours of a level-set function implicitly define the interface between the material phases [45,46]; this approach allows to obtain a sharp transition between void and solid regions. In the proposed workflow, a SIMP density method is adopted because this method requires relatively few iterations, is suitable for a combination of a wide range of design constraints, multiple load conditions, multi-physics problems, and extremely large (often 3D) systems, is extensively used in industrial software [38], and can be easily implemented with a simple code [47,48]. As a result of the SIMP method, a density map is obtained, which is contoured to a specific level of density (threshold), obtaining a mesh surface. Often, the resulting mesh cannot be directly used for the production phase due to the lack of connection zones or the presence of coarse regions due to the process discretization. The remodeling of the topology optimization mesh is a research topic of interest, indeed. [PERSON] and [PERSON] [49] presented a tool that generates suitable outputs for AM by using filters and the continuation approach on the penalization parameter; [PERSON] et al. [50] proposed a CAD-oriented topology optimization method able to perform the optimization directly on the CAD model instead of on the mesh. Most of the time the procedure is operated manually, and the part is modeled in a CAD environment using the mesh as guideline during the modeling. Alternatively, software tools for automatic remodeling are available; these mainly adopt quad-remesh and subdivision surface approaches. If the size optimization approach is followed, given a cell type and the unit cell dimension, a wireframe model is obtained filling the design space with a conformal lattice structure. In a conformal structure, the unit cell can deform to adapt to the boundary of the part or the lines of the stress field; this feature eliminates weakness at boundaries and provides stiffness and resistance to the entire model [51]. The size optimization is then performed on the wireframe [52,53]. The diameter of the beams iteratively varies until all the beams reach the target utilization, defined as the ratio between the maximum Von Mises stress measured on the beam and the admissible stress; as previously highlighted, the size of the beam is controlled to ensure that the beam diameter is thick enough to be manufactured and that it does not exceed the upper bound to avoid interferences with the surrounding beams. The optimized wireframe is then modeled with a boundary representation mesh-based approach as proposed by [PERSON] et al. [54]. The obtained coarse mesh is then smoothed adopting the Catmull-Clark subdivision surface algorithm [55]: each quad face is subdivided into four smaller quad faces at every iteration. The algorithm produces a surface with continuity in curvature (C2 surface), except at extraordinary vertices where they are C1. This allows to reduce stress concentration, especially at nodal points, enhancing the mechanical properties and the fatigue life of the lattice [56, 57, 58]. Alternatively, it is possible to combine topology and size optimization. The literature shows how the density map of the topology optimization can be used to assign an optimized dimension to the diameter of a beam-like lattice structure [59; 60] and to the thickness of a shell-like lattice structure [61].