TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one

Critical packing in granular shear bands

In a realistic three-dimensional setup, we simulate the slow deformation of idealized granular media composed of spheres undergoing an axisymmetric triaxial shear test. We follow the self-organization of the spontaneous strain localization process leading to a shear band and demonstrate the existence of a critical packing density inside this failure zone. The asymptotic criticality arising from the dynamic equilibrium of dilation and compaction is found to be restricted to the shear band, while the density outside of it keeps the memory of the initial packing. The critical density of the shear band depends on friction (and grain geometry) and in the limit of infinite friction it defines a specific packing state, namely the \emph{dynamic random loose packing}

Non-Universality of Density and Disorder in Jammed Sphere Packings

We show for the first time that collectively jammed disordered packings of three-dimensional monodisperse frictionless hard spheres can be produced and tuned using a novel numerical protocol with packing density $\phi$ as low as 0.6. This is well below the value of 0.64 associated with the maximally random jammed state and entirely unrelated to the ill-defined ``random loose packing'' state density. Specifically, collectively jammed packings are generated with a very narrow distribution centered at any density $\phi$ over a wide density range $\phi \in [0.6,~0.74048\ldots]$ with variable disorder. Our results support the view that there is no universal jamming point that is distinguishable based on the packing density and frequency of occurence. Our jammed packings are mapped onto a density-order-metric plane, which provides a broader characterization of packings than density alone. Other packing characteristics, such as the pair correlation function, average contact number and fraction of rattlers are quantified and discussed.Comment: 19 pages, 4 figure

On Multi-dimensional Packing Problems

We study the approximability of multi-dimensional generalizations of three classical packing problems: multiprocessor scheduling, bin packing, and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule n d-dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension accross all bins is bounded by a fixed quantity, say 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. Finally, packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a miximum number of vectors in a single bin of unit height. We obtain a variety of new algorithmic as well as inapproximability results for these three problems

Bulk dynamics of Brownian hard disks: Dynamical density functional theory versus experiments on two-dimensional colloidal hard spheres

Using dynamical density functional theory (DDFT), we theoretically study Brownian self-diffusion and structural relaxation of hard disks and compare to experimental results on quasi two-dimensional colloidal hard spheres. To this end, we calculate the self and distinct van Hove correlation functions by extending a recently proposed DDFT-approach for three-dimensional systems to two dimensions. We find that the theoretical results for both self- and distinct part of the van Hove function are in very good quantitative agreement with the experiments up to relatively high fluid packing fractions of roughly 0.60. However, at even higher densities, deviations between experiment and the theoretical approach become clearly visible. Upon increasing packing fraction, in experiments the short-time self diffusive behavior is strongly affected by hydrodynamic effects and leads to a significant decrease in the respective mean-squared displacement. In contrast, and in accordance with previous simulation studies, the present DDFT which neglects hydrodynamic effects, shows no dependence on the particle density for this quantity

Theoretical prediction and design for vortex generators in turbulent boundary layers

A theoretical study is presented of three-dimensional turbulent flow provoked in a boundary layer by an array of low-profile vortex generators (VGs) on the surface. The typical VG sits in the logarithmic region of the incident boundary layer, and the turbulence model used seems representative in this region. The governing equations yield a forward-marching three-dimensional vortex-type system, which is solved computationally and analytically for spanwise periodic VG arrays. Streamwise vortex patterns of various strengths are produced downstream, owing to three-dimensional distortion of the original logarithmic profile and to the turbulent stresses present. Predictions are given for certain basic VG shapes, e.g. triangular, with various spanwise spacings, and the predictions are found to agree favourably overall with recent experiments. In addition, the analytical formulae obtained prove useful in suggesting designs for favourable VG distributions, based on three factors: close spanwise packing, increased VG length, and suitably non-smooth spanwise shaping