scholarly journals Algorithms for Working with Orthogonal Polyhedrons in Solving Cutting and Packing Problems

2021 ◽  
Author(s):  
Vladislav Chekanin ◽  
Alexander Chekanin
Keyword(s):  
Complex Shape ◽  
Three Dimensional ◽  
Geometric Transformation ◽  
Cutting And Packing ◽  
Packing Problems ◽  
Basic Model ◽  
Composite Objects ◽  
Hard Problems ◽  
Orthogonal Polyhedron ◽  
Polygonal Model

In this paper problems of cutting and packing objects of complex geometric shapes are considered. To solve these NP-hard problems, it is proposed to use an approach based on geometric transformation of polygonal objects to composite objects (orthogonal polyhedrons) made up of rectangles or parallelepipeds of a given dimension. To describe the free space inside a voxelized container, a model of potential containers is used as the basic model that provides the ability of packing orthogonal polyhedrons. A number of specialized algorithms are developed to work with orthogonal polyhedrons including: algorithms for placing and removing composite objects, an algorithm for forming a packing with a given distance between objects to be placed. Two algorithms for the placement of orthogonal polyhedrons are developed and their efficiency is investigated. An algorithm for obtaining a container of complex shape presented as an orthogonal polyhedron based on a polygonal model is given. The article contains examples of placement schemes obtained by the developed algorithms for solving problems of packing two-dimensional and three-dimensional non-rectangular composite objects.

scholarly journals A survey on the cutting and packing problems

2020 ◽  
Vol 13 (4) ◽  
pp. 567-572
Author(s):  
Loris Faina
Keyword(s):  
Three Dimensional ◽  
Cutting And Packing ◽  
Unified Approach ◽  
Geometrical Method ◽  
Packing Problems ◽  
Enumeration Scheme

Abstract This paper presents a unified approach, based on a geometrical method (see Faina in Eur J Oper Res 114:542–556, 1999; Eur J Oper Res 126:340–354, 2000), which reduces the general two and three dimensional cutting and packing type problems to a finite enumeration scheme.

Packing Optimization by Enhanced Rubber Band Analogy

2005 ◽  
Author(s):  
Hong Dong ◽  
Georges M. Fadel ◽  
Vincent Y. Blouin
Keyword(s):  
Three Dimensional ◽  
Optimization Method ◽  
Rubber Band ◽  
Volume Relaxation ◽  
New Developments ◽  
Packing Problems ◽  
Rubber Balloon ◽  
Elastic Forces ◽  
Reaction Forces ◽  
Packing Optimization

In this paper, some new developments to the packing optimization method based on the rubber band analogy are presented. This method solves packing problems by simulating the physical movements of a set of objects wrapped by a rubber band in the case of two-dimensional problems or by a rubber balloon in the case of three-dimensional problems. The objects are subjected to elastic forces applied by the rubber band to their vertices as well as reaction forces when contacts between objects occur. Based on these forces, objects translate or rotate until maximum compactness is reached. To improve the compactness further, the method is enhanced by adding two new operators: volume relaxation and temporary retraction. These two operators allow temporary volume (elastic energy) increase to get potentially better packing results. The method is implemented and applied for three-dimensional arbitrary shape objects.

scholarly journals A three-dimensional model study of long-term mid-high latitude lower stratosphere ozone changes

2003 ◽  
Vol 3 (1) ◽  
pp. 1081-1107 ◽  
Author(s):  
M. P. Chipperfield
Keyword(s):  
High Latitude ◽  
Lower Stratosphere ◽  
Transport Model ◽  
Three Dimensional ◽  
Horizontal Resolution ◽  
Chemical Transport Model ◽  
Three Dimensional Model ◽  
Detailed Chemistry ◽  
Basic Model ◽  
Column Ozone

Abstract. We have used a 3D off-line chemical transport model (CTM) to study the causes of the observed changes in ozone in the mid-high latitude lower stratosphere from 1979–1998. The model was forced by European Centre for Medium Range Weather Forecasts (ECMWF) analyses and contains a detailed chemistry scheme. A series of model runs were performed at a horizontal resolution of 7.5°×7.5° and covered the domain from about 12 km to 30 km. The basic model performs well in reproducing the decadal evolution of the springtime depletion in the northern hemisphere (NH) and southern hemisphere (SH) high latitudes in the 1980s and early 1990s. After about 1994 the modelled interannual variability does not match the observations as well, which is probably due in part to changes in the operational ECMWF analyses – which places limits on using this dataset to diagnose dynamical trends. For mid-latitudes (35°–60°) the basic model reproduces the observed column ozone decreases from 1980 until the early 1990s. Model experiments show that the halogen trends appear to dominate this modelled decrease and of this around 30–50% is due to high-latitude processing on polar stratospheric clouds (PSCs). Dynamically induced ozone variations in the model correlate with observations over the timescale of a few years. Large discrepancies between the modelled and observed variations in the mid 1980s and mid 1990s can be largely resolved by assuming that the 11-year solar cycle (not explicitly included in the 3D model) causes a 2% (min-max) change in mid-latitude column ozone.

Raster-based mathematical morphology for cutting and packing problems

2021 ◽  
Author(s):  
Fernanda Miyuki Yamada ◽  
Hiroki Takahashi ◽  
Harlen Costa Batagelo ◽  
João Paulo Gois
Keyword(s):  
Mathematical Morphology ◽  
Cutting And Packing ◽  
Packing Problems

scholarly journals A Unified Method for Computing Position and Orientation Workspaces of General Stewart Platforms

2011 ◽  
Author(s):  
Oriol Bohigas ◽  
Llui´s Ros ◽  
Montserrat Manubens
Keyword(s):  
Complex Shape ◽  
Three Dimensional ◽  
Stewart Platform ◽  
Large Dimension ◽  
Connected Components ◽  
Cartesian Space ◽  
Position And Orientation ◽  
Stewart Platforms ◽  
Orientation Workspace ◽  
Unified Method

The workspace of a Stewart platform is a complex six-dimensional volume embedded in the Cartesian space defined by six pose parameters. Because of its large dimension and complex shape, such workspace is difficult to compute and represent, so that comprehension on its structure is being gained by studying its three-dimensional slices. While successful methods have been given to determine the constant-orientation slice, the computation and appropriate visualization of the constant-position slice (also known as the orientation workspace) has proved to be a challenging task. This paper presents a unified method for computing both of such slices, and any other ones defined by fixing three pose parameters, on general Stewart platforms involving mechanical limits on the active and passive joints. Additional advantages over previous methods include the ability to determine all connected components of the workspace, and any motion barriers present in its interior.

Ultrafast shape recognition for similarity search in molecular databases

2007 ◽  
Vol 463 (2081) ◽  
pp. 1307-1321 ◽  
Author(s):  
Pedro J Ballester ◽  
W. Graham Richards
Keyword(s):  
Biological Activity ◽  
Three Dimensional ◽  
Molecular Shape ◽  
Scientific Disciplines ◽  
Distance Distributions ◽  
Molecular Databases ◽  
Hard Problems ◽  
Data Explosion ◽  
Novel Drug ◽  
Drug Leads

Molecular databases are routinely screened for compounds that most closely resemble a molecule of known biological activity to provide novel drug leads. It is widely believed that three-dimensional molecular shape is the most discriminating pattern for biological activity as it is directly related to the steep repulsive part of the interaction potential between the drug-like molecule and its macromolecular target. However, efficient comparison of molecular shape is currently a challenge. Here, we show that a new approach based on moments of distance distributions is able to recognize molecular shape at least three orders of magnitude faster than current methodologies. Such an ultrafast method permits the identification of similarly shaped compounds within the largest molecular databases. In addition, the problematic requirement of aligning molecules for comparison is circumvented, as the proposed distributions are independent of molecular orientation. Our methodology could be also adapted to tackle similar hard problems in other fields, such as designing content-based Internet search engines for three-dimensional geometrical objects or performing fast similarity comparisons between proteins. From a broader perspective, we anticipate that ultrafast pattern recognition will soon become not only useful, but also essential to address the data explosion currently experienced in most scientific disciplines.

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