# Geometrical Degrees of Freedom for Cellular Structures Generation: A New Classification Paradigm
## 2 Design and Geometrical Consideration
### RVE Variation Factors
#### 2.4.2 RVE Variation Based on Load Conditions  
In the context of CSs, the designer typically deploys such architectures towards structural performance customization. One example is the use of CSs in the frame of structural topology optimization to balance lightweight and compliance requirements (see Figure 12). When it comes to \"load adaptive\" cellular structures, the shape of the cellular structure wireframe is based on a response to loads applied on the surface. [PERSON] [45] explains how space warping can be used to properly distribute the load. For example, in order to maximize strength, the CS will be deformed using a warping function. Another example is given by [PERSON] and [PERSON] [50]. The authors developed a method that uses stress fields in order to create CSs. [PERSON] et al. [100] described an example of a multi-step optimization procedure where geometry is topologically optimized and then discretized using a hexahedral mesh. The mesh is then populated with unit cells. Figure 11: Representative volume element variation based on boundary information—illustration of the difference between a cellular structure cell sizes are (**a**) not influenced by the boundary and (**b**) a cellular structure in which the cells’ size vary based on their distance to the boundary. Figure 10: Representative volume element variation based boundary information—illustration of the difference between uniform and conformal lattice structure [99]. A similar concept can be applied in other types of \"loads\". Another type of assumed \"load\" is permittivity for dielectric structures, as mentioned by [PERSON] et al. [47]. The idea is to create dielectric structures with spatially varying electromagnetic properties via AM. The unit RVE structures obtained by the researchers are made based on a curve wrapped within a rectangle. The curve is wrapped into multiple cells, such that when they are combined, they are able to change the permittivity of the cells by changing how dense the curve is wrapped at each of the cell's locations (see Figure 13). An example of stiffness maximization using lattice structure is given by [PERSON] et al. [11]. In their paper, they have redesigned a micro aerial vehicle, used for surveillance in hazardous areas. They have done so using a relative density mapping. Their technique consists of the following steps. First, they analyze the boundary conditions as well as the loads acting upon the vehicle. Then, they topologically optimize the part via the SIMP method. Afterward, they select the type of microstructure required. Finally, they fill each element on the part with a unit cell. The density of the element is what dictates the size of the unit cell struts. Furthermore, a minimum strut thickness can be imposed in order to properly fill every element. This addresses the problem of discontinuity in topologically optimized parts caused partially by elements with zero density. Figure 12: Representative volume element variation based on structural load conditions starting with (**a**) design space with boundary condition(s), followed by (**b**) topology optimization results, then finally (**c**) generated cellular structure. The CS parameters (or implicitly the local RVE properties) can be changed in order to meet a specific optimization objective. One of those objectives is thermal applications. In the article written by [PERSON] et al. [7], the authors show an example of a two-stage optimization of a CS. The first optimization is meant to reach certain customized structural properties and the second optimization to improve heat transfer without affecting the structural properties of the part. To do a multifunctional optimization, the typical force equation was replaced with the following: \(K\cdot D=F+G\), Where \(K\) is the global stiffness matrix for \(N\) elements stiffness matrices frame elements, \(D\) is the vector of global displacements, \(F\) is the vector of applied nodal loads and \(G\) is the vector of loads that account for thermal heating.