# Geometrical Degrees of Freedom for Cellular Structures Generation: A New Classification Paradigm
## 2 Design and Geometrical Consideration
### RVE Variation Factors
#### 2.4.3 RVE Variation Based on Random and External Sources  
Random and external sources are grouped together because they both have little to do with the geometry of the part, nor with what the part is going through. Many definitions have been given to randomized CSs; however, in this article, the following definition is given: a CS is called randomized if a designer goes through the process of designing a specific CS twice with the same inputs without any guarantee that the resulting structure will be exactly the same. When generating a randomized CS, one might not be able to predict the exact layout of the material; however, it can be possible to approximately predict the stiffness of a randomized CS. This can be done through the global or localized control of porosity or conversely, density. An example can be shown in Figure 14. There are many applications  
Figure 13: Representative volume element variation based on permittivity conditions. **(a)** Illustration showing examples of the space-filling curve geometry used to create spatially varying permittivity distributions. By varying the order N, the user can tune the local volume fraction of printed material and thus the local effective permittivity. **(b)** Illustration showing how the unit cell geometries can be connected end-to-end resulting in a single continuous curve with spatially varying effective EM properties [47]. of randomized CSs within the medical field, one of them is described by [PERSON] et al. [101]. The authors have developed a method to design biomimetic scaffolds using a Voronoi diagram. In the article, they describe an algorithm that takes as inputs the information from the scan of a patient's bone, the desired porosity, and a target pore size. Then, based on the scattered points within a volume, the proposed algorithm creates Voronoi cells. The edges of the Voronoi cells are then thickened. The resulting CS will mimic the structure of the human bones. Their process is randomized because when generating the points or the seeds of the Voronoi cells, the user does not precisely control the location of those points but they can control the distribution or density of those points inside the design space. Similar work is done in [PERSON] et al. [51], where they use an error distribution method to map a Voronoi diagram to a grayscale image. [PERSON] et al. [54] have also proposed an approach for a completely customized structure using the Voronoi tessellation of a specific region to realize cellular structure typical of metal foams. In their case, the random change affects the topology of the cells. In the work by [PERSON] et al. [46], the authors have classified cellular structures for AM in the biomedical field; they have classified the conformal lattice in the pseudo-random category and Voronoi in the random category in a few instances. A Voronoi diagram is not necessarily a randomized CS, it all depends on how the seeds are generated, if the seeds have a fixed position then the Voronoi diagram will always be the same; hence, it cannot always be classified as random. The main difference with Voronoi diagrams is that the design variables are the seeds' position and it is easier to generate random points than to directly generate randomly placed unit cells. In the latter case, one has to ensure connectivity, which is an issue that does not occur when working directly with a Voronoi diagram. According to [PERSON] et al. [102], the absence of a regular structure (for example, the stacking of more complex strut-based unit cells) affords for a simple approach to grade the foam geometry. To create randomized shapes, the vertices of existing unit cells can be varied in a random fashion, as is the case for [PERSON] et al. [61]'s work. A similar method is captured in the work of [PERSON] et al. [62]. They create a network of struts that it subtracted from an original design space. This shows that sometimes, the unit cells are not the solid, but the voids.