# Geometrical Degrees of Freedom for Cellular Structures Generation: A New Classification Paradigm
## 2 Design and Geometrical Consideration  
### Cellular Structure Design Classification  
There are numerous classifications of CS that have already been established, such as the work by [PERSON] and [PERSON] [26]. The basis for the classification, presented in the present work, is inspired by these existing classifications and attempts to further harmonize the available literature in this space. [PERSON] and [PERSON] [26] have divided CSs into three structural design classes: foam structures, two-dimensional (2D) CSs referring to extruded 2D cells and three-dimensional (3D) CSs referring to structures that periodically repeat in 3D. They then described an emerging classification for 3D CSs, which is based on how ordered the CSs are, ranging from randomized to pseudo-periodic and periodic structures. The classification for this review work will consider both randomized and ordered classifications. Other classifications, such as the one by [PERSON] et al. [27], rely on the different types of design variables for CSs: patterns (referring to the characteristics of the minimum CS unit replicated), surface limits (referring to the size of the shape of the CS boundary), progressivity (referring to the variation of the CS unit cell's thickness) and conformity (referring to whether or not the CS unit cells are varied according to strict parameters). Many advances in the design of CSs have shown that there is a growing body of potential unit cells, with a push towards implicit modeling for the design of CSs. Therefore, in the present review, the CSs will not be classified based on their unit cells. In addition, some reviewers such as [PERSON] et al. [19] have divided CSs stochastic structures into open cells and closed cell foams. That concept also has its limitations, as there are examples of triply periodic minimal structures with a varying volume fraction, which can be seen as open on one end and completely closed on the other. Hence, it is more important to isolate the sources of morphology variation inside a CS rather than trying to classify them in different distinct, non-intersecting groups. The present review challenges the current paradigm, which aims at grouping CSs into multiple distinct and non-overlapping groups. Instead, the current approach identifies the main geometric degrees of freedom (GDoF) that come into play when designing a CS. It is important to first define some concepts that will recur throughout the article. The design space refers to the volume that contains the CS (the object being lightweight), and the Representative Volume Element (RVE) refers to the smallest element that is replicated throughout the design space. There are multiple types of RVEs and some of the most famous will be discussed later in Section 2.2. The RVE can vary throughout the design space in many ways: it can be transformed (scaled or rotated), its volume fraction can be changed and its topology/type can also be changed. The RVE can be modified to optimize the design space as a response to multiple factors: based on the boundary of the design space (or the surface of the design), based on the loads that are applied to the CS and based on random or external factors (such as trying to replicate the density of bones). All of those have been considered when creating Table 1.  
The RVEs inside the CS may also vary based on both the loads and the boundary conditions, whereas for other CSs, their RVEs may be transformed and changed based on imposed volume fractions. Thus, certain case studies can appear in multiple entries on the table. This is the case for the work of [PERSON] et al. [30], wherein the authors have generated a strut-based CS that considers the boundary of the design space where the strut-based RVE are then modified towards the surface of the part. However, the thickness of the struts is varied based on the load applied to the design space. That paper, therefore, appears in two locations in the CSs classification table. It is important to notice that several design paths can lead to the same design result.