Packing ellipsoids by nonlinear optimization

Abstract

In this paper, continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space are introduced. Two different models for the non-overlapping and models for the inclusion of ellipsoids within half-spaces and ellipsoids are presented. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, illustrative numerical experiments are presented.

Clusters of polyhedra in spherical confinement

“…Countless studies, inspired by the ubiquity of biopolymer packing in confinement, have investigated polymer behavior inside spheres and rod-like containers (119,120), in boxes (121), in slit-like channels and tubes (122,123), in quasi-2D confinement (124), and even on curved 2D surfaces (125). Recently, ellipsoid packing and assembly in spherical and ellipsoidal confinement (56,58,126) has been used to mimic the effects of cell nucleus confinement on the behavior of ellipsoid-like nucleosomes and higher-order chromosome territories (3,127,128). The richness of our results for polyhedra in spherical confinement suggests that further investigation into the interplay between particle shape and container shape in these new packing in confinement problems will be interesting and informative.…”

“…However, to our knowledge, only a handful of studies have addressed 3D dense packings of anisotropic particles inside a container. Of these, almost all pertain to packings of ellipsoids inside rectangular, spherical, or ellipsoidal containers (56)(57)(58), and only one investigates packings of polyhedral particles inside a container (59). In that case, the authors used a numerical algorithm (generalizable to any number of dimensions) to generate densest packings of N = ð1 − 20Þ cubes inside a sphere.…”

A study of ellipse packing in the high-dimensionality problems

“…In the fundamental investigation [16], issues of packing both ellipses and ellipsoids in various convex regions were considered. When modeling conditions of nonintersection of objects, two approaches were studied: the first one was based on the idea of a dividing straight line (plane) [10], and the second was based on the use of affine space transformations R n , n=2,3.…”

“…Followed the aforementioned transformations, the idea of the method developed in [8] for modeling geometrical relations of a circle and an ellipse was used to formalize conditions of nonintersection of the obtained objects. Generation of "good" starting points and application of the Algencan solver [17] in solution of nonlinear programming problems allowed the authors of paper [16] to improve the majority of results of study [10].…”

The Ellipses Packing in a Rectangle of the Minimal Size

“…В фундаментальном исследовании [12] рассмотрены вопросы упаковки как эллипсов, так и эллипсоидов в различных выпуклых областях. При моделировании условий непересечения объектов исследуются два подхода: первый основан на идее разделяющей прямой (плоскости) из [6], а второй -на использовании аффинных преобразований пространства R n , n = 2, 3.…”