Abstract
This paper deals with a problem the packing polyhex clusters in a regular hexagonal container. It is a common problem in many applications with various cluster shapes used, but symmetric polyhex is the most useful in engineering due to its geometrical properties. Hence, we concentrate on mathematical modeling in such an application, where using the “bee” tetrahex is chosen for the new Compact Muon Solenoid (CMS) design upgrade, which is one of four detectors used in Large Hadron Collider (LHC) experiment at European Laboratory for Particle Physics (CERN). We start from the existing hexagonal containers with hexagonal cells packed inside, and uniform clustering applied. We compare the center-aligned (CA) and vertex-aligned (VA) models, analyzing cluster rotations providing the increased packing efficiency. We formally describe the geometrical properties of clustering approaches and show that cluster sharing is inevitable at the container border with uniform clustering. In addition, we propose a new vertex-aligned model decreasing the number of shared clusters in the uniform scenario, but with a smaller number of clusters contained inside the container. Also, we describe a non-uniform tetrahex cluster packing scheme in the proposed container model. With the proposed cluster packing solution, it is accomplished that all clusters are contained inside the container region. Since cluster-sharing is completely avoided at the container border, the maximal packing efficiency is obtained compared to the existing models.
Highlights
A problem of packing objects in the container of a given shape is common in many applications such as computer science, manufacturing, industrial engineering, and production [1]
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]
Model, The two default variants of multi-resolution grids with that are used and in the literature where the small-cell vertex overlaps with the container centroid are CA model, where small hexagonal cells share their midpoints with the container, and VA model, We apply a hexagonal tessellation to solve a packing problem of hexagonal clusters where the small-cell vertex overlaps withapproach the container centroid insideWe a regular hexagonal container
Summary
A problem of packing objects in the container of a given shape is common in many applications such as computer science, manufacturing, industrial engineering, and production [1]. We believe that these shared or partial objects should be discussed, as they cannot be considered the inner packed items They are unpacked, but they take place in the container and prevent us from putting the whole cluster inside the container border. 4-hexagonal cluster tetrahex allows periodic tessellation, and it is minimal in perimeter and rhomboid in shape. We find named tetrahex allows periodic tessellation, and it is to minimal in applicability perimeter and in it optimal for our application, and many authors use it due its broad inrhomboid engineering shape.on. We find it optimal for our application, and many authors use it due to its broad applicability in engineering based onclusters its geometry and symmetrical structure [21].container, completely avoiding shared thatToallthe hexagonal are contained the inner of the best of our knowledge, the inproblem ofpart packing polyhex clusters composed of the clusters atcells the inside container border. (c) from [29]) to solve the packing problem such that all hexagonal clusters are contained in the inner part of the Figure
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