On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

Abstract

In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. We revisit this problem by using some well-known formulations concerning regular hexagons. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. Archimedean {3,122} semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Efficiency remains satisfactory, as we show that, by producing the proposed irregular hexagon sensors from the same wafer as a regular hexagon, we can obtain almost the same SE.

Modeling and Analysis of HAPS-based Cellular IoT Systems for Protected Areas

“…(2 + 1), where r represents the radius of a cell. Finally, the HAPS coverage, , is obtained by ( 2) [19].…”

A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms

“…In most publications on sphere packing, spheres are defined by the Euclidean norm. However, many applied and theoretical packing problems, e.g., producing square, hexagonal or dodecagonal CMS sensors [34] or tiling non-overlapping distinct squares in a square container [35], can be considered as sphere packing for spheres defined in a suitable norm. To the best of our knowledge, using non-Euclidean norms to define distances in sphere packing problems was first proposed in [36][37][38].…”

“…Irregular packing problems typically require special sophisticated modeling approaches and techniques to represent placement (non-overlapping, containment) conditions (see, e.g., [24] and the references therein). However, in many applications, the shapes involved are not spherical and possess similar properties, e.g., have certain levels of central symmetry [34,35]. Our objective in this paper is to describe and investigate a class of irregular packing problems where placement conditions can be stated as simple, as in sphere packing.…”

A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container

“…Even though most of the research supports the approximation of different container shapes with the minor change of the cost function, special attention is devoted to packing polygons into the circular or polygonal region of interest (ROI) [2,10]. It is crucial in applications such as sensor manufacturing when polygonal sensors are cut out from the circular wafer [11]. In some cases, inner objects are not independent, in sense, they have to be grouped in a number of clusters before being embedded inside the container.…”