A note on circle packing

The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parallelized in two different ways and its performance is reported for a triangular, planar domain test case. The implementation uses the highly parallel graphics processing unit (GPU) on commodity hardware. The speedups so achieved are discussed based on a number of experiments.     

Optimal Packing Configuration Design with Finite-Circle Method

The Finite-circle Method (FCM) is further developed to solve 2D and 3D packing optimization problems with system compactness and moment of inertia constraints here. Instead of using the real geometrical shape as in existing solutions, we approximate the components and the design domain with circles of variant radii. Such approximation makes it possible to transform the original problem into a basic packing problem of FCM approximated components. Meanwhile, the overlapping between different components can be easily avoided by limiting the distance between corresponding circles in terms of their radii. With this formulation, the FCM provides a general and systematic approach and makes gradient-based optimization algorithms applicable. Furthermore, FCM has been extended to 3D packing problems by simply replacing circles with spheres in this paper. Several examples designing the compactness and moment of inertia of the component systems are presented to show the effect of FCM.     

Constrained circles packing test problems: all the optimal solutions known

Taking a satellite module layout design as engineering background, this paper gives constrained test problems for an unequal circle packing whose optimal solutions are all given. Given a circular container D with radius R, the test problem can be constructed in the following steps. First, M=217 circles are packed into D without overlaps by ‘packing with a tangent circle’ to get the values of radii and centroid coordinates of the circles, which are expressed by R. Then the 217 circles are arranged in descending sequence of radius and are divided into 23 groups according to the radius. Finally, seven test problems are constructed according to the circles of q=1, 2, …, 7 groups. The optimal solution to the test problems as well as its optimality and uniqueness proof are also presented. The experimental results show that the test problems can effectively evaluate performances of different evolutionary algorithms.     

Evolutionary computation solutions to the circle packing problem

In this work, we present an evolutionary omputation-based solution to the circle packing problem (ECPP). The circle packing problem consists of placing a set of circles into a larger containing circle without overlaps: a problem known to be NP-hard. Given the impossibility to solve this problem efficiently, traditional and heuristic methods have been proposed to solve it. A naïve representation for chromosomes in a population-based heuristic search leads to high probabilities of violation of the problem constraints, i.e., overlapping. To convert solutions that violate constraints into ones that do not (i.e., feasible solutions), in this paper we propose two repair mechanisms. The first one considers every circle as an elastic ring and overlaps create repulsion forces that lead the circles to positions where the overlaps are resolved. The second one forms a Delaunay triangulation with the circle centers and repairs the circles in each triangle at a time, making sure repaired triangles are not modified later on. Based on the proposed repair heuristics, we present the results of the solution to the CPP problem to a set of unit circle problems (whose exact optimal solutions are known). These benchmark problems are solved using genetic algorithms, evolutionary strategies, particle swarm optimization, and differential evolution. The performance of the solutions is compared to those known solutions based on the packing density. We then perform a series of experiments to determine the performance of ECPP with non-unitary circles. First, we compare ECPP’s results to those of a public competition, which stand as the world record for that particular instance of the non-unitary CPP. On a second set of experiments, we control the variance of the size of the circles. In all experiments, ECPP yields satisfactory near-optimal solutions.     

An action-space-based global optimization algorithm for packing circles into a square container

This paper proposes an action-space-based global optimization (ASGO) approach for the problem of packing unequal circles into a square container such that the size of the square is minimized. Starting from several random configurations, ASGO runs the following potential descent method and basin-hopping strategy iteratively. It finds configurations with the local minimum potential energy by the limited-memory BFGS (LBFGS) algorithm, then selects the circular items having the most deformations and moves them to some large vacant space or randomly chosen vacant space. By adapting the action space defined for the rectangular packing problem, we approximate each circular item as a rectangular item, thus making it much easier to find comparatively larger vacant spaces for any given configuration. The tabu strategy is used to prevent cycling and enhance the diversification during the search procedure. Several other strategies, such as swapping two similar circles or swapping two circles in different quadrants in the container, are combined to increase the diversity of the configurations. We compare the performance of ASGO on 68 benchmark instances at the Packomania website with the state-of-the-art results. ASGO obtains configurations with smaller square containers on 63 instances; at the same time it matches or approaches the current best results on the other five instances.     

A New Upper Bound for the Cylinder Packing Problem

This paper introduces a new upper bound to the problem of fitting identical circles into a rectangle. This problem is usually referred to as the 'cylinder packing problem' or 'cylinder palletization'. In practice, it arises when it is desired to maximize the number of cylindrical items packed in an upright position onto a rectangle/pallet. The upper bound developed consists in determining the reduced pallet area by deducting a lower bound for the unused pallet area from the total area of the pallet. The upper bound for the number of identical circles to pack into the pallet is computed by the ratio reduced pallet area/circle area . The results obtained for five distinct sets of problems are analyzed and compared with previous bounds found in the published literature.     

Generation of triangular mesh with specified size by circle packing

This paper describes an algorithm for the generation of a finite element mesh with a specified element size over an unbound 2D domain using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within the open domain. As soon as a circle is added to the generation front, triangular elements are directly generated by properly connecting frontal segments with the centre of the new circle. Circles are packed closely and in contact with the existing circles by an iterative procedure according to the specified size control function. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection between frontal segments. If the mesh generation of a physical object is required, the object boundary can be introduced. The boundary recovery procedure is fast and robust by tracing neighbours of triangular elements. The finite element mesh generated by circle packing can also be used through a mapping process to produce parametric surface meshes of the required characteristics. The sizes of circles in the pack are controlled by the principal surface curvatures. Five examples are given to show the effectiveness and robustness of mesh generation and the application of circle packing to mesh generation over curved surfaces.     

Comments on the hierarchically structured bin packing problem

We study the hierarchically structured bin packing problem. In this problem, the items to be packed into bins are at the leaves of a tree. The objective of the packing is to minimize the total number of bins into which the descendants of an internal node are packed, summed over all internal nodes. We investigate an existing algorithm and make a correction to the analysis of its approximation ratio. Further results regarding the structure of an optimal solution and a strengthened inapproximability result are given.     

求解不等圆Packing问题的一个启发式算法

求解具有NP难度的圆形packing问题具有很高的理论与实用价值.现提出一个启发式方法,求解了货运中常遇到的矩形区域内的不等圆packing问题.此算法首先将待布局圆按半径大小降序排列,然后用占角动作来逐个放置.通过试探性地放入一个或多个待布局圆,给出了占角动作的度以及更全局的有限枚举策略来评价占角动作的优度.在放置每一个圆时,以贪心的方式选取当前具有最大优度的占角动作来放置.最后用测试算例验证了算法的高效性.     

求解圆形packing问题的拟人退火算法

Circles packing problem is an NP-hard problem and is difficult to solve. In this paper, a hybrid search strategy for circles packing problem is discussed. A way of generating new configuration is presented by simulating the moving of elastic objects, which can avoid the blindness of simulated annealing search and make iteration process converge fast. Inspired by the life experiences of people, an effective personified strategy to jump out of local minima is given. Based on the simulated annealing idea and personification strategy, an effective personified annealing algorithm for circles packing problem is developed. Numerical experiments on benchmark problem instances show that the proposed algorithm outperforms the best algorithm in the literature.     

Packing unequal circles using formulation space search

In this paper we present a heuristic algorithm for the problem of packing unequal circles in a fixed size container such as the unit circle, the unit square or a rectangle. We view the problem as being one of scaling the radii of the unequal circles so that they can all be packed into the container. Our algorithm is composed of an optimisation phase and an improvement phase. The optimisation phase is based on the formulation space search method whilst the improvement phase creates a perturbation of the current solution by swapping two circles. The instances considered in this work can be categorised into two: instances with large variations in radii and instances with small variations in radii. We consider six different containers: circle, square, rectangle, right-angled isosceles triangle, semicircle and circular quadrant. Computational results show improvements over previous work in the literature.     

求解矩形和圆形装填问题的最大穴度算法

在超大规模集成电路设计，裁缝裁剪布料，玻璃切割等工作中提出了矩形和圆形装填问题，即把不同大小的矩形块和圆饼装入一个矩形容器中，以最大化容器的面积利用率为优化目标。对这一问题，可采用模拟退火，遗传算法等国际流行算法进行求解，但这些方法计算时间较长，计算结果的优度也不甚理想。利用人类的智慧和经验，提出了一种求解此问题的最大穴度算法。并对3个随机生成的测试实例进行了实算测试。所得结果的平均面积利用率为90．80％，平均计算时阊为8．38s。测试结果表明，算法对求解矩形和圆形装填问题是行之有效的。     

Reconstruction of a digital circle

Since digitization always causes some loss of information, reconstruction of the original figure from a given digitization is a challenging task. Reconstruction of digital circles has already been addressed in the literature. However, an in-depth analysis of an OBQ image of a continuous circle as well as a solution to its domain construction problem is still lacking. In this paper a detailed analysis of digital circles has been carried out. A modified I_R method is formulated to numerically compute the domain of each digital quarter circle for a given radius. Several properties of the OBQ image of a circle reveal that in many cases it is possible to split a digital circle into four digital quarter circles, such that the domains of the individual quarter circles can be combined to obtain the domain of the full circle. Moreover, the domain of a quarter circle is geometrically characterized.     

正三角形容器内等圆Packing问题的启发式算法

等圆Packing问题研究如何将n个单位半径的圆形物体互不嵌入地置入一个边长尽量小的正三角形容器内,作为一类经典的NP难度问题,其有着重要的理论价值和广泛的应用背景.模拟退火算法是一种随机的全局寻优算法,通过将启发式格局更新策略与基于梯度法的局部搜索策略融入模拟退火算法,并与二分搜索相结合,提出一种求解正三角形容器内等圆Packing问题的启发式算法.该算法将启发式格局更新策略用来产生新格局和跳坑,用梯度法搜索新产生格局附近能量更低的格局,并用二分搜索得到正三角形容器的最小边长.对41个算例进行测试的实验结果表明,文中算法改进了其中38个实例的目前最优结果,是求解正三角形容器内等圆Packing问题的一种有效算法.     

A two-dimensional vector packing model for the efficient use of coil cassettes

We consider the problem of efficiently packing steel products, known as coils, into special containers, called cassettes for shipping. The objective is to minimize the number of cassettes used for packing all the given coils where each cassette has capacity limits on both total payload weight and size. We model this problem as a two-dimensional vector packing problem and propose a heuristic. We also analyze the worst-case performance of the proposed algorithm under a special condition which, in fact, holds for the particular real-world case that we handled. Our computational experiment with real production data shows that the proposed algorithm performs quite satisfactorily in practice.     

Packing circles within ellipses

The problem of packing circles within ellipses is considered in the present paper. A new ellipse‐based system of coordinates is introduced by means of which a closed formula to compute the distance of an arbitrary point to the boundary of an ellipse exists. Nonlinear programming models for some variants of 2D and 3D packing problems involving circular items and elliptical objects are given. The resulting models are medium‐sized highly challenging nonlinear programming problems for which a global solution is sought. For this purpose, multistart strategies are carefully and thoroughly explored. Numerical experiments are exhibited.     

New heuristics for packing unequal circles into a circular container

We propose two new heuristics to pack unequal circles into a two-dimensional circular container. The first one, denoted by A1.0, is a basic heuristic which selects the next circle to place according to the maximal hole degree rule. The second one, denoted by A1.5, uses a self look-ahead strategy to improve A1.0. We evaluate A1.0 and A1.5 on a series of instances up to 100 circles from the literature and compare them with existing approaches. We also study the behaviour of our approach for packing equal circles comparing with a specified approach in the literature. Experimental results show that our approach has a good performance in terms of solution quality and computational time for packing unequal circles.     

Generation of finite element mesh with variable size over an unbounded 2D domain

With the advance of the finite element, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of finite element mesh of variable element size over an unbounded 2D domain by using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing circles is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as a circle is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new circle. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection with frontal segments, and a linear time complexity for mesh generation can be achieved. In case the boundary of the domain is needed, simply generate an unbounded mesh to cover the entire object. As the element adjacency relationship of the mesh has already been established in the circle packing process, insertion of boundary segments by neighbour tracing is fast and robust. Details of such a boundary recovery procedure are described, and practical meshing problems are given to demonstrate how physical objects are meshed by the unbounded meshing scheme followed by the insertion of domain boundaries.     

Packing congruent spheres into a multi‐connected polyhedral domain

In this paper, we have considered a problem of packing the maximal number of congruent spheres into a multi‐connected polyhedral domain. A mathematical model of the problem has been formulated on the basis of Φ functions. This paper proposes a special way of constructing starting points. To find local maxima, a modification of the Zoutendijk method of feasible directions and a strategy of active inequalities are used. We developed a special approach to search for an approximation to a global maximum. A number of numerical examples are also provided.