# Multistep Evolution Method to Generate Topological Interlocking Assemblies
## 2 Related Work
### Non-Planar and Curvilinear TI Assemblies  
The design of TI assemblies based on non-planar surfaces has been of interest to researchers on the topic.  
[PERSON] and [PERSON] considered how curvature in the geometric domain affects the resultant pieces of a non-planar Abeille-based TIC [27]. They observed the interdependency between curvature in the geometric domain and the resultant assembly. They found that both curvature and tilting angles are inversely proportional. Additionally, the distribution of curvature and piece vertices is directly proportional (resulting in smaller pieces that approximate the steep curvature).  
[PERSON] et al. [16] adapted the generation method formulated by [PERSON] et al. [3] to work on curvilinear tessellations. Their implementation includes an additional step that adapts the pieces to the curvature of the surface. In such a case, the assembly of the blocks must follow a specific order. [PERSON] and [PERSON] described approaches to designing interlocking pieces based on parametric design logic and discrete combinatorial processes [28; 29]. The former returns the geometry of each block according to its location in the tessellation. The latter focuses on exploiting the combinatorial capabilities of repeating block shapes in an assembly. They applied such approaches to generate TICs bounded by two NURBS surfaces. The boundary information helps to generate distorted, trimmed tetrahedra that align with the curvature of the surfaces.  
[PERSON] et al. [30] presented a geometric generation method for creating TICs specifically for cylindrical structures. The proposed non-planar interlocking element was designed with a symmetrical geometry featuring six concave-convex side surfaces, facilitating interlocking with adjacent elements. Creating an element starts with a regular hexagon and its circumcircle. Then, three non-adjacent arcs are reflected. These arcs are connected to form the top surface of the planar interlocking element, and the bottom surface is obtained by rotating the same shape by 60 degrees. The respective side surfaces are created by lofting the edges of the top and bottom surfaces. Finally, the planar interlocking element is morphed to fit the target cylindrical surface using a \"_flow along surface_\" technique. This transformation results in a non-planar interlocking element suitable for assembling tubular structures. The assembled structure, consisting of these elements, displays an identical pattern on the inner and outer surfaces. Each element is mechanically interlocked with its six neighbors, restricting movement in all directions purely through geometric constraints.  
[PERSON] and [PERSON] [31] proposed creating TI flat vaults using truncated octahedra with enhanced joint details to improve structural performance. The process begins with creating a hexagonal tessellation. Truncated octahedra are generated from this tessellation using the Starfish plug-in for Grasshopper. This plug-in facilitates the creation of parametric patterns, which are essential for generating the base geometric forms. X-joints are added to the truncated octahedra to enhance the contact surface area and improve structural performance. These X-joints are incorporated through volumetric modifications using Rhino software. Each X-joint increases the contact surface area by approximately ten cm\({}^{2}\), significantly improving the structural stability of the interlocking blocks. Two flat vault configurations are designed--one using truncated octahedra and the other using truncated octahedra with X-joints. The design process involves ensuring that the interlocking edges of the blocks fit together seamlessly, providing a robust and stable assembly. The structural performance of the designed flat vaults is analyzed using finite element analysis (FEA) in SimScale. The FEA assesses the displacement and von Mises stress distributions within the vaults under applied loads. This analysis helps quantify the improvements in structural performance due to the addition of X-joints.