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Globally optimized packings of non-uniform size spheres in \(\mathbb {R}^{d}\): a computational study

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Abstract

In this work we discuss the following general packing problem: given a finite collection of d-dimensional spheres with (in principle) arbitrarily chosen radii, find the smallest sphere in \(\mathbb {R}^{d}\) that contains the given d-spheres in a non-overlapping arrangement. Analytical (closed-form) solutions cannot be expected for this very general problem-type: therefore we propose a suitable combination of constrained nonlinear optimization methodology with specifically designed heuristic search strategies, in order to find high-quality numerical solutions in an efficient manner. We present optimized sphere configurations with up to \(n = 50\) spheres in dimensions \(d = 2, 3, 4, 5\). Our numerical results are on average within 1% of the entire set of best known results for a well-studied model-instance in \(\mathbb {R}^{2}\), with new (conjectured) packings for previously unexplored generalizations of the same model-class in \(\mathbb {R}^{d}\) with \(d= 3, 4, 5.\) Our results also enable the estimation of the optimized container sphere radii and of the packing fraction as functions of the model instance parameters n and 1 / n, respectively. These findings provide a general framework to define challenging packing problem-classes with conjectured numerical solution estimates.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA

    János D. Pintér

  2. Pintér Consulting Services, Inc., Halifax, NS, Canada

    János D. Pintér

  3. Physicist at Large Consulting LLC, Bryn Mawr, PA, USA

    Frank J. Kampas

  4. Lazaridis School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, Canada

    Ignacio Castillo

Authors
  1. János D. Pintér
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  2. Frank J. Kampas
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  3. Ignacio Castillo
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Correspondence to Ignacio Castillo.

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Pintér, J.D., Kampas, F.J. & Castillo, I. Globally optimized packings of non-uniform size spheres in \(\mathbb {R}^{d}\): a computational study. Optim Lett 12, 585–613 (2018). https://doi.org/10.1007/s11590-017-1194-x

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