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Open AccessArticle

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

1
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture (FESB), University of Split, 21000 Split, Croatia
2
University Department of Professional Studies, University of Split, 21000 Split, Croatia
3
Fermi National Accelerator Laboratory (Fermilab), Batavia, IL 60510-5011, USA
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(5), 700; https://doi.org/10.3390/sym12050700
Received: 30 January 2020 / Revised: 20 February 2020 / Accepted: 28 February 2020 / Published: 1 May 2020
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. View Full-Text
Keywords: cluster packing; polyhex; regular hexagon; tetrahex cluster packing; polyhex; regular hexagon; tetrahex
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MDPI and ACS Style

Prvan, M.; Burazin Mišura, A.; Gecse, Z.; Ožegović, J. A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container. Symmetry 2020, 12, 700. https://doi.org/10.3390/sym12050700

AMA Style

Prvan M, Burazin Mišura A, Gecse Z, Ožegović J. A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container. Symmetry. 2020; 12(5):700. https://doi.org/10.3390/sym12050700

Chicago/Turabian Style

Prvan, Marina; Burazin Mišura, Arijana; Gecse, Zoltan; Ožegović, Julije. 2020. "A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container" Symmetry 12, no. 5: 700. https://doi.org/10.3390/sym12050700

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