In this work, the Solid Isotropic Material with Penalization (SIMP) topology optimization (TO) method is revisited and reformulated within the mathematical framework of NURBS functions. The SIMP method aims at computing a fictitious density value (defined between 0 and 1) for each element of a predefined mesh of the structure to be optimized. Such a densitiy affects the elements stiffness matrix: the smaller is the density, the stronger is the penalization of the material properties. In this work, the fictitiuous density is defined thorugh a suitable NURBS surface. This fact implies several advantages.

Firstly, being the NURBS surface defined through the locally supported blending functions, it implicetely provides a local filter zone, whose size depends on the NURBS degree and on the components of the knot vector. Therefore, numerical artifacts typical of the SIMP method (e.g. the “checkerboard effect”), are avoided. Secondly, NURBS have been recently employed for TO in the framework of iso-geometric analysis: it is remarked the possibility to perform the TO directly on the NURBS-based geometry, without defining a mesh for the FEM. However, the number of knot vector components, i.e. the number of optimization variables, must be enhanced to achieve satisfactory results. Conversely, in this work the number of design variables (i.e. the parameters defining the NURBS surface) is kept low, by smartly optimizing the values of the knot vector components. Furthermore, some generic constraints have been implemented in the classic SIMP method, such as the minimum length scale and the maximum length scale. These constraints are usually limited to standard problems. However, due to NURBS formalism the proposed approach allows for handling both non-conventional objective functions and AM constraints (which are non-standard requirements for TO analyses). Particularly, in this background it is possible to extract the analytical expression of the local normal/tangent vector and the local radius of curvature. These quantities can be related respectively to the presence of overhangs (and consequently the presence of supporting structures) and to the radius-to-thickness ratio. Finally, the NURBS formalism does not affect the generality of the problem and the TO can be carried out by including non-linearity (either geometric or material).

The mathematical formulation of the TO problem considered in this work consists in minimizing the compliance of the structure by imposing its volume. Moreover, other constraints are added in order to minimize (even avoid) supporting structures and to impose a minimum curvature radius. In this work the TO is applied to standard benchmarks, such as the cantilever beam and the MBB beam.

In conclusion, this study aims at proving the possibility of improving the classic SIMP approach in the context of the NURBS formalism. On the one hand, the NURBS allow precisely describing the geometrical features, so the integration of AM constraints is a trivial task; on the other hand, the NURBS overcome some limitations, which are typical of the SIMP method, such as the necessity of establishing a suitable filtering zone.