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## Generation of Topological Interlocking Configurations from a Geometric Approach

thesis

posted on 28.04.2020 by Andres M Bejarano Posada#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

A Topological Interlocking Configuration (TIC) is an assembly where the shape and alignment of the blocks define the kinematic constraints. Conventional TICs are single-layered structures made of convex blocks. The interface between the blocks in an assembly is face-to-face contact. The traditional convention disregards the use of joinery, adhesive, or other mechanisms that keep two pieces next to each other. However, TICs require a support structure that prevents the lateral strain of the blocks.

The generation process of a TIC starts with a surface tessellation that describes a geometric domain. Each tile in the tessellation represents a traversal section of a block. For regular tessellations and uniform generation parameters, such sections lie in the middle of their respective blocks. Additionally, such conditions guarantee the blocks align adequately with each other. If one of such conditions does not hold, then the resultant blocks may not be aligned. Furthermore, there could be overlapping between the blocks, which makes a TIC invalid.

Traditionally, the generation parameters are angle values set at the edges of the tiles. The angles must match between tiles such that each block in the assembly has a geometry that imposes kinematic constraints to its neighboring blocks. Using the same angle values on regular and semi-regular tessellation produces feasible blocks. That is not the case for non-regular tessellations, curvilinear surfaces, and free-form 3D meshes. In such cases, the generation method must find specific angle values to design the blocks and reduce overlapping.

In this thesis, we propose a TIC generation framework focused on the generation of valid interlocking assemblies based on multiple types of surface tessellations. We start with the Height-Bisection method, a TIC generation approach that uses the distances from a tile to its respective evolution sections as the generation parameters. The method considers the bisector vectors between two tiles to define the parameters that generate aligned blocks to each other. We expand the generation model to a complete pipeline process that finds feasible generation parameters. The pipeline includes clipping parameters and methods in case that overlapping between blocks cannot be avoided.

Additionally, we describe a generalization of the mid-section evolution concept to include multiple evolution steps during the generation process. Our approach considers the angles and distances required to generate infinitely many TICs, including shapes that are not possible using the traditional generation method and the Height-Bisection method. Finally, we consider the interlocking assemblies that cannot maintain static equilibrium due to the shape of the surface tessellation. We consider the Structure Feasibility Analysis method to find the location and magnitude of the minimum tension forces that guarantee a TIC will reach a static equilibrium state. We describe how to update the generation parameters according to the results of the feasibility analysis. Our results show that the proposed pipeline generates valid TICs based on different surface tessellations, including closed and free-form shapes.

The generation process of a TIC starts with a surface tessellation that describes a geometric domain. Each tile in the tessellation represents a traversal section of a block. For regular tessellations and uniform generation parameters, such sections lie in the middle of their respective blocks. Additionally, such conditions guarantee the blocks align adequately with each other. If one of such conditions does not hold, then the resultant blocks may not be aligned. Furthermore, there could be overlapping between the blocks, which makes a TIC invalid.

Traditionally, the generation parameters are angle values set at the edges of the tiles. The angles must match between tiles such that each block in the assembly has a geometry that imposes kinematic constraints to its neighboring blocks. Using the same angle values on regular and semi-regular tessellation produces feasible blocks. That is not the case for non-regular tessellations, curvilinear surfaces, and free-form 3D meshes. In such cases, the generation method must find specific angle values to design the blocks and reduce overlapping.

In this thesis, we propose a TIC generation framework focused on the generation of valid interlocking assemblies based on multiple types of surface tessellations. We start with the Height-Bisection method, a TIC generation approach that uses the distances from a tile to its respective evolution sections as the generation parameters. The method considers the bisector vectors between two tiles to define the parameters that generate aligned blocks to each other. We expand the generation model to a complete pipeline process that finds feasible generation parameters. The pipeline includes clipping parameters and methods in case that overlapping between blocks cannot be avoided.

Additionally, we describe a generalization of the mid-section evolution concept to include multiple evolution steps during the generation process. Our approach considers the angles and distances required to generate infinitely many TICs, including shapes that are not possible using the traditional generation method and the Height-Bisection method. Finally, we consider the interlocking assemblies that cannot maintain static equilibrium due to the shape of the surface tessellation. We consider the Structure Feasibility Analysis method to find the location and magnitude of the minimum tension forces that guarantee a TIC will reach a static equilibrium state. We describe how to update the generation parameters according to the results of the feasibility analysis. Our results show that the proposed pipeline generates valid TICs based on different surface tessellations, including closed and free-form shapes.