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Abstract We present a novel methodology to generate mechanical structures based on fractal geometry using the chaos game, which generates self-similar point-sets within a polygon. Using the Voronoi decomposition of these points, we are able to generate groups of self-similar structures that can be related back to their chaos game parameters, namely, the polygonal domain, fractional distance, and number of samples. Our approach explores the use of forward design of generative structures, which in some cases can be easier to use for designing than inverse generative design techniques. To this end, the central hypothesis of our work is that structures generated using the chaos game can generate families of self-similar structures that, while not identical, exhibit similar mechanical behavior in a statistical sense. We present a systematic study of these self-similar structures through modal analysis and tensile loading and demonstrate a preliminary confirmation of our hypothesis. 

Abstract We present a novel methodology to generate mechanical structures based on the idea of fractal geometry as described by the chaos game. Chaos game is an iterative method that generates self-similar point-sets in the limiting case within a polygonal domain. By computing Voronoi tessellations on these point-sets, our method generates mechanical structures that adopts the self-similarity of the point-sets resulting in fractal distribution of local stiffness. The motivation behind our approach comes from the observation that a typical generative structural design workflow requires the ability to generate families of structures that possess shared behavioral (e.g. thermal, mechanical, etc.) characteristics making each structure distinct but feasible. However, the generation of the alternatives, almost always, requires solving an inverse structural problem which is both conceptually and computationally challenging. The objective of our work is to develop and investigate a forward-design methodology for generating families of structures that, while not identical, exhibit similar mechanical behavior in a statistical sense. To this end, the central hypothesis of our work is that structures generated using the chaos game can generate families of self-similar structures that, while not identical, exhibit similar mechanical behavior in a statistical sense. Furthermore, each family is uniquely identifiable from the parameters of the chaos game, namely, the polygonal domain, fractional distance, and number of samples. We present a systematic study of these self-similar structures through modal analysis and demonstrate a preliminary confirmation of our hypothesis. 

A novel methodology is introduced for designing auxetic (negative Poisson's ratio) structures based on topological principles and is demonstrated by investigating a new class of auxetics based on two‐dimensional (2D) textile weave patterns. Conventional methodology for designing auxetic materials typically involves determining a single deformable block (a unit cell) of material whose shape results in auxetic behavior. Consequently, patterning such a unit cell in a 2D (or 3D) domain results in a larger structure that exhibits overall auxetic behavior. Such an approach naturally relies on some prior intuition and experience regarding which unit cells may be auxetic. Second, tuning the properties of the resulting structures is typically limited to parametric variations of the geometry of a specific type of unit cell. Thus, most of the currently known auxetic structures belong to a selected few classes of unit cell geometries that are explicitly defined in accordance with a specified topological (i.e., grid structure). Herein, a new class of auxetic structures is demonstrated that, while periodic, can be generated implicitly, i.e., without reference to a specific unit cell design. The approach leverages weave‐based parameters (A–B–C), resulting in a rich design space for auxetics that is previously unexplored. 

An approach for modeling topologically interlocked building blocks that can be assembled in a water‐tight manner (space filling) to design a variety of spatial structures is introduced. This approach takes inspiration from recent methods utilizing Voronoi tessellation of spatial domains using symmetrically arranged Voronoi sites. Attention is focused on building blocks that result from helical stacking of planar 2‐honeycombs (i.e., tessellations of the plane with a single prototile) generated through a combination of wallpaper symmetries and Voronoi tessellation. This unique combination gives rise to structures that are both space‐filling (due to Voronoi tessellation) and interlocking (due to helical trajectories). Algorithms are developed to generate two different varieties of helical building blocks, namely, corrugated and smooth. These varieties result naturally from the method of discretization and shape generation and lead to distinct interlocking behavior. In order to study these varieties, finite‐element analyses (FEA) are conducted on different tiles parametrized by 1) the polygonal unit cell determined by the wallpaper symmetry and 2) the parameters of the helical line generating the Voronoi tessellation. Analyses reveal that the new design of the geometry of the building blocks enables strong variation of the engagement force between the blocks. 

In this work, we introduce an approach to model topologically interlocked corrugated bricks that can be assembled in a water-tight manner (space-filling) to design a variety of spatial structures. Our approach takes inspiration from recently developed methods that utilize Voronoi tessellation of spatial domains by using symmetrically arranged Voronoi sites. However, in contrast to these existing methods, we focus our attention on Voronoi sites modeled using helical trajectories, which can provide corrugation and better interlocking. For symmetries, we only use affine transformations based on the Bravais lattice to avoid self-intersections. This methodology naturally results in structures that are both space-filling (owing to Voronoi tessellation) as well as interlocking by corrugation (owing to helical trajectories). The resulting shapes of the bricks appear to be similar to a variety of pasta noodles, thereby inspiring the names, Voronoi Spaghetti and VoroNoodles.