Abstract
An assembly comprises parts joined together to achieve a specific form or functionality. Compared to monolithic objects, assemblies have many benefits in terms of fabrication, transportation, and adaptability. Parts of assemblies are always geometrically simple to fabricate with digital techniques, can be efficiently packed for transportation, and offer adaptability through flexible replacement or modification. Hence, assemblies are widely used in our daily lives that most of our consumer products, industry machines, and architectural structures are assemblies. This thesis focuses on the computational design of complex assemblies with discrete equivalence classes, where the assembly parts are inversely computed instead of being pre-defined, a process known as post-rationalization. Parts falling into the same discrete equivalence class should have exactly the same shape. A discrete equivalence class, known as a template element, can take various forms, such as shells, rods, nodes, or tiles, depending on the type of assemblies. An assembly with discrete equivalence classes is formed by assembling the instances of each template element together. These assemblies have clear advantages of reducing the fabrication cost and simplifying the physical construction due to the reuse of template elements. In this thesis, we propose computational approaches to model different types of complex assemblies with discrete equivalence classes, aiming to maximize the reusability of template elements. We first introduce two error-driven mesh optimization approaches to model masonry shell structures with a small set of template shell elements and wireframe meshes with a small set of template node and rod elements, respectively, where gaps and overlaps are minimized yet exist. To preserve the perfect contacts between instance elements, we adopt domain decomposition approaches to produce assemblies without gaps and overlaps, known as tilings. We propose a domain decomposition approach to create K-hedral tiling of 2D finite domains with the minimized number K of prototiles, called inverse tiling. We observe that directly generalizing our inverse tiling approach to 3D finite domains is not practical in terms of fabrication since either assemblability or structural stability cannot be ensured. To address these challenges, we present another domain decomposition approach to model high-level interlocking puzzles (i.e. a kind of 3D tilings) that require multiple moves to take out the first subassembly to avoid falling apart easily. We validate our computational approaches by fabricating a series of physical prototypes, demonstrating that our approaches have great potential to be applied for various assembly design problems ranging from small-scale such as puzzles and pixelized art to large-scale such as architectural rationalization.