Methodology to Solve Multi-Dimentional Sphere Packing Problems

This paper discusses the problem of optimally packing spheres of various dimensions into containers of arbitrary geometrical shapes. According to the international classification, this problem belongs to Sphere Packing Problems (SPPs). The aim of this work is to create an integrated methodology for solving SPPs.В статті розглядається задача оптимального розміщення куль різної розмірності в контейнерах довільних геометричних форм. Згідно з міжнародною класифікацією ця задача належить до класу SPP (Sphere Packing Problems). Метою даної роботи є створення єдиної методології розв’язання задач SPP.В статье рассматривается задача оптимального размещения шаров различной размерности в контейнерах произвольных геометрических форм. Согласно международной классификации эта задача относится к классу SPP (Sphere Packing Problems). Целью данной работы является создание единой методологии решения задач SPP

Упаковка большого числа конгруэнтных шаров в цилиндре

Розглядається задача пакування максимальної кiлькостi конгруентних куль одиничного радiуса в прямому круговому цилiндрi заданих розмiрiв. Запропоновано математичну модель задачi. Вважається, що радiуси куль є змiнними. Глобальний максимум задачi забезпечує пакування всiх куль. Алгоритм побудови початкового пакунку грунтується на гратчастому пакуваннi куль. Результати порiвнюються з кращими вiдомими результатами. Наведено чисельнi приклади.The paper considers the problem of packing a maximal number of congruent spheres of the unit radius into a straight circular cylinder of given sizes. A mathematical model of the problem is suggested. Radii of all spheres are supposed to be variable A global maximum of the problem ensures the packing of all spheres. An algorithm of constructing a starting package is based on a lattice packing of spheres. Results are compared with the best known ones. A number of numerical examples are given

Structure and thermodynamics of platelet dispersions

Various properties of fluids consisting of platelike particles differ from the corresponding ones of fluids consisting of spherical particles because interactions between platelets depend on their mutual orientations. One of the main issues in this topic is to understand how structural properties of such fluids depend on factors such as the shape of the platelets, the size polydispersity, the orientational order, and the platelet number density. A statistical mechanics approach to the problem is natural and in the last few years there has been a lot of work on the study of properties of platelet fluids. In this contribution some recent theoretical developments in the field are discussed and experimental investigations are described.Comment: 23 pages, 18 figure

The 3D Object Packing Problem into a Parallelepiped Container Based on Discrete-Logical Representation

The problem of 3D geometric objects irregular tight packing into minimal height cuboid is considered. Main approaches to solving this problem are described. The no-fit-polyhedron based algorithm using discrete-logical representation is proposed. Some examples and computational results are also given for public input data. © 2016The work was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.000

Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples

Optimal Packing of Irregular 3D Objects For Transportation and Disposal

This research developed algorithms, platforms, and workflows that can optimize the packing of 3D irregular objects while guaranteeing an acceptable processing time for real-life problems, including but not limited to nuclear waste packing optimization. Many nuclear power plants (NPPs) are approaching their end of intended design life, and approximately half of existing NPPs will be shut down in the next two decades. Since decommissioning and demolition of these NPPs will lead to a significant increase in waste inventory, there is an escalating demand for technologies and processes that can efficiently manage the decommissioning and demolition (D&D) activities, especially optimal packing of NPP waste. To minimize the packing volume of NPP waste, the objective is to arrange irregular-shaped waste objects into one or a set of containers such that container volume utilization is maximized, or container size is minimized. Constraints also include weight and radiation limits per container imposed by transportation requirements and the waste acceptance requirements of storage facilities and repositories. This problem falls under the theoretical realm of cutting and packing problems, precisely, the 3D irregular packing problem. Despite its broad applications and substantial potential, research on 3D irregular cutting and packing problems is still nascent, and largely absent in construction and civil engineering. Finding good solutions for real-life problems, such as the one mentioned above, through current approaches is computationally expensive and time-consuming. New algorithms and technologies, and processes are required. This research adopted 3D scanning as a means of geometry acquisition of as-is 3D irregular objects (e.g., nuclear waste generated from decommissioning and demolition of nuclear power plants), and a metaheuristics-based packing algorithm is implemented to find good packing configurations. Given the inefficiency of fully autonomous packing algorithms, a virtual reality (VR) interactive platform allowing human intervention in the packing process was developed to decrease the time and computation power required, while potentially achieving better outcomes. The VR platform was created using the Unity® game engine and its physics engine to mimic real-world physics (e.g., gravity and collision). Validation in terms of feasibility, efficiency, and rationality of the presented algorithms and the VR platform is achieved through functional demonstration with case studies. Different optimal packing workflows were simulated and evaluated in the VR platform. Together, these algorithms, the VR platform, and workflows form a rational and systematic framework to tackle the optimal packing of 3D irregular objects in civil engineering and construction. The overall framework presented in this research has been demonstrated to effectively provide packing configurations with higher packing efficiency in an adequate amount of time compared to conventional methods. The findings from this research can be applied to numerous construction and manufacturing activities, such as optimal packing of prefabricated construction assemblies, facility waste management, and 3D printing

Deformation behavior of cylindrical block copolymer bicrystals : pathway to understanding block copolymer grain boundaries

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2006.Includes bibliographical references (leaves 166-169).Model bicrystals made by adhering pieces of near-single-crystal styrene-isoprene-styrene (SIS) cylindrical block copolymer (BCP), produced by a roll-casting process; yield various types of pure tilt grain boundaries. The study of the deformation of the bicrystals, each containing one grain boundary, enables a deeper understanding of the influences grain boundaries and the incompatibilities between them have on mechanical behavior. Mechanical properties and deformation of near-single-crystal systems provide a reference base for the expected bicrystal behavior. We consider various aspects of incompatibility that can arise from joining two highly anisotropic grains together (i.e. Young's modulus, Poisson's ratio, deformation mode(s)). Experimentally, the structure of grain boundaries was characterized using atomic force microscopy (AFM). In deformation experiments, optical microscopy was employed to examine the deformation gradient in the specimen and in situ small angle x-ray scattering (SAXS) was used to monitor the microdomain structural evolution. Finally, finite element simulations illustrated the state of strains of the bicrystal. The symmetric (45-45) bicrystal turns out to be the most complex system, despite of the simplest geometry.(cont.) Due to the opposite orientation of the grains, the deformation of the symmetric bicrystal results in a rigid translation of the grain boundary. A triangular shaped influence region indicates that the influence distance varies along the grain boundary length. A portion of the influence region has limited expansion and is slightly sheared along the grain boundary. Another portion of the influence region experiences high tension. For the asymmetric (90-45) bicrystal, the deformation is mostly influenced by the difference in deformation modes: dilation vs. shearing. The distortion due to deformation in the diagonal grain induces rotation and advances the deformation in the perpendicular grain near the grain boundary. For the asymmetric (90-0) bicrystal, the most influential factors are the Young's modulus and the Poisson's ratio. The much softer perpendicular grain assumes most of the deformation by extensive dilation and lateral contraction. Near the grain boundary, the perpendicular grain is constrained by the rigid parallel grain such that its deformation is impeded. The influence region in the perpendicular grain is narrow and invariant along the grain boundary length.by Panitarn Wanakamol.Ph.D