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.     

Packing equal circles into a circle with circular prohibited areas

The paper considers the problem of packing a maximal number of identical circles of a given radius into a multiconnected domain. The domain is a circle with prohibited areas to be finite unions of circles of given radii. We construct a mathematical model of the problem and investigate its characteristics. The starting points are constructed in a random way or on the ground of the hexagonal lattice. To find the local maxima, a modification of the Zoutendijk method of feasible directions and a strategy of active inequalities are applied. We compare our results with the benchmark instances of packing circles into circular and annular containers. A number of numerical examples 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.     

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.     

求解不等圆布局问题的一类遗传算法物方法

在已有求解不等圆布局问题算法的基础上 ,根据问题特点提出了一类遗传算法 ,通过将拟物方法与标准遗传算法结合使用 ,较好地解决了对布局优化函数进行全局最优求解的问题 最后通过实例计算验证了本算法的有效性 .     

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.     

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.     

材料二维微结构仿真随机概率圆优化填充算法

几何建模是影响有限元法计算材料性能精确性的主要因素之一。利用圆等简单的 几何实体近似模拟离散化实体效果良好,不仅能大量简化几何计算,还能构造出误差在可接受 范围内的近似模型。材料特性不仅取决于平均颗粒尺寸,受颗粒尺寸分布的影响也十分显著。 本文设计的随机圆填充算法保证了填充圆半径服从某一概率分布,重点分析了常规圆产生方法 和依据干涉圆的干涉类型生成新圆。该算法不仅完全杜绝了圆的干涉还极大的减小了圆的相离, 相对于现有的填充算法,填充度有了显著提升。     

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。     

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.     

An improved energy landscape paving algorithm for the problem of packing circles into a larger containing circle

The problem of packing circles into a larger containing circle is a kind of NP-hard problem. It is of high theoretical and practical value. Lacking powerful optimization method is the key obstacle to solving this problem. The energy landscape paving (ELP) method is a class of heuristic global optimization algorithm and a generation of Monte Carlo method. By incorporating new configuration update mechanism into ELP method, an improved energy landscape paving (ELP+) algorithm is put forward. The computational results, on two sets of instances taken from the literature, show the effectiveness of the proposed algorithm.     

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.     

An effective hybrid algorithm for the problem of packing circles into a larger containing circle

Simulated annealing is a powerful stochastic search method, but it still has the disadvantage of blind search. Tabu search (TS) which can prevent cycling and enhance diversification, is an adaptive strategy based on tabu list. By reasonably combining simulated annealing with TS, an effective hybrid algorithm for the problem of packing circles into a larger containing circle is presented. Based on a special neighborhood and tabu strategy, some benchmark problem instances can be well solved by the presented hybrid algorithm, and the computational results can compete with the best literature results.     

颗粒复合材料代表体元并行建模算法研究

为了利用细观力学方法研究复合固体推进剂材料的力学性能,需要建立具有代表性的推进剂细观胞元模型,针对当前算法普遍存在的计算效率低下问题,依据分子动力学思想生成颗粒堆积模型的性能特性,通过分析负载均衡和消息通信,提出了并行模型的三个准则,设计了区域分解的并行策略,并利用共享存储并行和分布式存储并行两级并行手段实现了并行算法。最后在IBMBladeCenter集群平台上通过实例证明算法可以缓解负载均衡并缩减通信开销,上述试验数据验证了算法的高效性,达到了提高胞元生成效率的目的。     

面向布局优化问题的多量子态量子进化算法及其应

针对演化算法在求解带平衡约束的圆形布局问题上所出现的早熟现象,提出一种有利于保持种群多样性的多量子态量子进化算法,并结合高效的定位定序启发式方法进行求解。为了高效优化布局顺序,在量子进化算法的基础上：引入多量子态编码和基于平均收敛概率的收敛标准以提高求解速度;引入基于禁忌策略和启发信息的观测方法,使其所得到的n进制解为互不相同的整数串,同时保证优先布局质量大、半径大的小圆;引入动态量子进化策略,有效地引导种群向最优个体进化。在定位规则中引入定位概率函数提高解的精度,数值实验结果表明,该算法能够有效求解带平衡约束的圆形布局问题。     

Note on: An improved algorithm for the packing of unequal circles within a larger containing circle

This note proposed an improved version of the algorithm proposed by Wang et al. [Wang, H. Q., Huang, W. Q., Zhang, Q., & Xu, D. M. (2002). An improved algorithm for the packing of unequal circles within a larger containing circle. European Journal of Operational Research, 141, 339–347] for solving the disk packing problem with equilibrium constraints. An efficient strategy of accelerating the search process is introduced in the gradient method to shorten the execution time. A number of computational results are presented, showing the effectiveness of the proposed method.     

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.     

Iris center location using Hough transform with two-dimensional parameter space

An algorithm of approximate location of iris center in image is presented. It is based on Hough transformation for circles. Problem is set in a way to detect only iris center position without evaluating its size. Such formulation allows to reduce the dimension of parameter space compared to commonly used approaches that detect center position and radius simultaneously. Apart, omitting radius estimation gives an opportunity to use points of both pupil and iris circles in Hough transform, increasing the stability of method especially for images with poor pupil contour quality. The algorithm was tested for more than 95000 iris images from public domain databases.