## 2 Design Methods Based on TO and GD Tools
### Topology Optimization  
TO is a computational design methodology [18; 19; 20] that by means of using finite element analysis allows to generate an optimal material layout in a given design domain subject by respecting given constraints, loading, and boundary conditions. In particular, the design domain is discretized into finite elements, and one of a number of optimization techniques is used to determine the distribution of the material. This is usually achieved by minimizing the mass while preventing yielding or maximizing the stiffness constraining the mass to a target value. TO experienced rapid growth at the beginning of the 1990s as it became easily distinguishable from Shape or Size Optimization concepts [21; 22]. In the following years, TO tools evolved rapidly from the academic to the industrial field, in fact, they are used effectively in various application areas [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34] and, in the everyday industrial practice, it is frequently used for the redesign of already existing products [35; 36; 37; 38; 39]. Recently, thanks to the evolution of CAD systems and the appeal of additive technologies, different  
Figure 1: Engineering design methods with the introduction of TO and GD tools. software houses have started to integrate TO modules within the modelling environment. This need arises from the limit of this optimization process which, as depicted in Figure 1, requires a redesign phase to achieve a manufacturable design or to merely make small changes to the geometry. This redesign phase results not only in an increase in the final mass of the piece, but also in an increase in the design processing time. This issue is due to the fact that TO requires a finite element model, and its results are represented using a tessellation model that cannot be converted directly into a parametric CAD part and, furthermore, it cannot be manipulated directly within the CAD systems especially when the mesh is very dense. Despite the various attempts and efforts made in this direction, all of them lack an efficient integration between TO tools and CAD systems [40]. For this reason, in the last decade, various manual and semi-automatic methods were proposed in order to support designers to interpret and convert topology optimization results into a parametric CAD model [41; 42; 43; 44; 45; 46; 47]. Recently, some CAD systems have provided functionalities to operate directly on the results of topological optimization analysis. These are mainly sculpting and free-form modeling tools to which mechanical engineers are less accustomed to than a traditional feature-based modeling approach. As a consequence, in daily industrial practice, engineers and designers perform a remodeling of TO results necessary to achieve a manufacturable feature-based geometry. Most of the time the procedure is operated manually, and the part is modeled in a CAD environment using the mesh as a guideline during the modeling [48]. As depicted in Figure 1, the design process starts with a simple geometry that defines the design space and functional surfaces. Then, the 3D model is imported into the TO environment in which, in addition to loads and constraints, the objective functions that are usually expressed in terms of minimization or maximization for the given design parameters must also be defined. After that phase, the simulation can be launched and the solver generates the optimized geometry. The topological optimization solver is an optimization algorithm. In this context, there are several kinds of algorithms currently used that can be classified in deterministic and stochastic algorithms [22; 23]. The most commonly available commercial software performs a TO analysis by using the gradient-based density approach that is characterized by a design variable, ranging from 0 (void elements) to 1 (solid material), assigned to discretized elements representing an artificial density of material [49; 50]. The solution is found through an iterative process [51] called nested formulation [52].