# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 2 Materials and Methods
### NURBS Free-Form Deformation Approach for Conformal Wireframe  
The modeling of the conformal wireframe is based on the morphing of a lattice structure inside a design space and is performed in Rhinoceros 3D CAD software (release 7.8.2) and the relative visual programming environment Grasshopper.  
First, a regular lattice structure is created by repeating the unit cell with a linear array along with the X-, Y-, and Z-axes. Four different beam-based unit cells are used: simple cubic (SC), body-centered cubic (BCC), modified Wallach-Gibson (WG), and diamond (DA) unit cell [56] (Figure 2). These unit cells are selected to show different behavior of bending and stretching-dominated lattices, according to the Maxwell rule [8], and to consider symmetry and the number of elements. The SC and the DA cells have a bending-dominated behavior, whereas the BCC and the WG cells a stretching-dominated one. SC and BCC belong to the same cubic cell family, sharing the beams at the edges of the cube; the main difference is based on the additional beams of the BCC, from the vertices to the center, which reinforce the cell. As opposed to the SC and BCC cells, the WG and DA ones are asymmetric, so the cell orientation affects the results related to the load condition. Finally, BCC and WG have more elements in the unit cell.  
Figure 1: Front view (**a**), side view (**b**), and perspective view (**c**) of the piston rod.  
Then, each array is morphed by the NURBS-FFD method to fit inside the design space of the piston road, while the straight elements of the cells are transformed into curve elements. More in detail, the NURBS-FFD technique is available in several commercial CAD software such as Rhinoceros, Blender, Autodesk Maya, and Autodesk 3 ds Max and provides a user interface that simplifies the transformation of the regular lattice into a conformal one. Indeed, it is sufficient to move the vertices of a trivariate NURBS to adapt a box to the design space, simultaneously transforming the rods contained within it.  
The deformation aims to maintain the cell aspect ratio similar to the undeformed cells in the central zone of the design space and, at the same time, cells that conform to the borders at the boundaries. In detail, this is performed by creating a trivariate NURBS volume around the object and moving its control points. The detailed procedure is as follows:  
* A regular wireframe based on the repetition along the X-, Y-, and Z-axis is generated. The number of instances is 14 cells along the Z-direction, 10 cells along the X-direction, and 4 cells along Y-direction for the BC, BCC, and WG. However, for the DA, the instances along the X-direction are 6 to maintain the aspect ratio (Figure 3a);
* A 3D NURBS volume is built around the wireframe using the Cage-Edit command. In the case study, the cage is governed by 5 control points in X-direction, 2 control points in Y-direction, and 3 control points in Z-direction (Figure 3b).  
The control cage is adapted to the vertices of the design space and the curves of the boundaries by moving the control points (Figure 3c).  
Figure 3: NURBS-FFD approach applied to the diamond lattice structure (green lines) (**a**) via a control cage (red lines) (**b**) to conform the lattice to the design space (black lines) (**c**).  
Figure 2: Unit cell geometry: simple cubic (SC), body-centered cubic (BCC), modified Wallach-Gibson (WG), and diamond (DA).  
This approach can be automated by defining a specific transformation of the cage, and it adapts to different geometric configurations of the design space. Moreover, it allows warping transformations if densification of the cells is needed.