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

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 $$\mathbb {R}^{d}$$Rd 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 $$n = 50$$n=50 spheres in dimensions $$d = 2, 3, 4, 5$$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 $$\mathbb {R}^{2}$$R2, with new (conjectured) packings for previously unexplored generalizations of the same model-class in $$\mathbb {R}^{d}$$Rd with $$d= 3, 4, 5.$$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.