The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes.
In this paper, we study an instance of the G-closure problem for two-dimensional periodic metamaterials. Specifically, we consider composites with isotropic homogenized elasticity tensor, obtained as a mixture of two isotropic materials. We focus on the case when one material is has zero stiffness i.e., single-material structures with voids. This problem is important, in particular, in the context of designing small-scale structures for metamaterials that can be manufactured using additive fabrication. A range of effective metamaterial properties can be obtained this way using a single base material.We demonstrate that two closely related simple parametric families based on the structure proposed by O. Sigmund in [26] attain good coverage of the space of isotropic properties satisfying Hashin-Shtrikman bounds. In particular, for positive Poisson's ratio, we demonstrate that Hashin-Shtrikman bound can be approximated arbitrarily well, within limits imposed by numerical approximation: a strong evidence that these bounds are achievable in this case. For negative Poisson's ratios, we numerically obtain a bound which we hypothesize to be close to optimal, at least for metamaterials with rotational symmetries of a regular triangle tiling.
New fabrication technologies have significantly decreased the cost of fabrication of shapes with highly complex geometric structure. One important application of complex fine-scale geometric structures is to create variable effective elastic material properties in shapes manufactured from a single material. Modification of material properties has a variety of uses, from aerospace applications to soft robotics and prosthetic devices. Due to its scalability and effectiveness, an increasingly common approach to creating spatially varying materials is to partition a shape into cells and use a parametric family of small-scale geometric structures with known effective properties to fill the cells. We propose a new approach to solving this problem for extruded, planar microstructures. Differently from existing methods for two-scale optimization based on regular grids with square periodic cells, which cannot conform to an arbitrary boundary, we introduce cell decompositions consisting of (nearly) rhombic cells. These meshes have far greater flexibility than those with square cells in terms of approximating arbitrary shapes, and, at the same time, have a number of properties simplifying small-scale structure construction. Our main contributions include a new family of 2D cell geometry structures, explicitly parameterized by their effective Young's moduli E , Poisson's ratios v , and rhombic angle α with the geometry parameters expressed directly as smooth spline functions of E, v , and α. This family leads to smooth transitions between the tiles and can handle a broad range of rhombic cell shapes. We introduce a complete material design pipeline based on this microstructure family, composed of an algorithm to generate rhombic tessellation from quadrilateral meshes and an algorithm to synthesize the microstructure geometry. We fabricated a number of models and experimentally demonstrated how our method, in combination with material optimization, can be used to achieve the desired deformation behavior.
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