Packing ellipsoids into volume-minimizing rectangular boxes

Abstract

A set of tri-axial ellipsoids, with given semi-axes, is to be packed into a rectangular box; its widths, lengths and height are subject to lower and upper bounds. We want to minimize the volume of this box and seek an overlap-free placement of the ellipsoids which can take any orientation. We present closed non-convex NLP formulations for this ellipsoid packing problem based on purely algebraic approaches to represent rotated and shifted ellipsoids. We consider the elements of the rotation matrix as variables. Separating hyperplanes are constructed to ensure that the ellipsoids do not overlap with each other. For up to 100 ellipsoids we compute feasible points with the global solvers available in GAMS. Only for special cases of two ellipsoids we are able to reach gaps smaller than 10 −4 .

Clusters of polyhedra in spherical confinement

“…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.…”

Packing ellipsoids by nonlinear optimization

“…Moreover, while the models presented in [27], [38], and [49] deal with two-dimensional problems and rectangular containers and the model presented in [44] deals with spheroids and rectangular containers, the models introduced in the present work deal with n-dimensional problems with arbitrary ellipsoids and ellipsoidal and polyhedral containers. Models introduced in [36] (that was published online after the submission of the present manuscript) tackle problem (e) above and have some similarities with one of the models presented in this work. However, in [36], global optimization techniques are employed and only feasible solutions are delivered for medium-sized instances.…”

“…Models introduced in [36] (that was published online after the submission of the present manuscript) tackle problem (e) above and have some similarities with one of the models presented in this work. However, in [36], global optimization techniques are employed and only feasible solutions are delivered for medium-sized instances. On the other hand, using multi-start strategies combined with local minimization solvers, good-quality local solutions are reported in the present work.…”

Packing ovals in optimized regular polygons

“…Uhler and Wright (2013) relax these assumptions, and propose a model that minimizes a measure of overlap between ellipses (while overlaps still remain possible). Kallrath (2017) extends the work presented in Kallrath and Rebennack (2014) to pack non-overlapping ellipses of arbitrary size and orientation into optimized rectangular containers: the key modeling idea is to use separating lines to ensure that the ellipses do not overlap with each other. Birgin et al (2017) extend the work presented in Birgin et al (2016) for packing arbitrary ellipses in convex containers: they propose a multi-start strategy combined with starting guesses and a local optimization solver, in order to find good quality packings with up to 1000 ellipsoids.…”