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Deployment, Patrolling and Foraging
Published in Yasmina Bestaoui Sebbane, Multi-UAV Planning and Task Allocation, 2020
In 2D space scenarios, the maximal coverage problem can be mapped to a circle packing formulation. The problem turns into the sphere packing problem in 3D, and the strategies designed for 2D become NP-hard in 3D. The problem of coverage in 3D space is often a critical part of the scenario for the observation of an environment. The number of nodes and their locations are restricted by the investigated environment and the reception range of node. Moreover, the dynamic UAV network topology and flight must be handled efficiently considering the communication constraints of the UAVs. In [10], a node positioning strategy for UAV networks is proposed with a wireless sensor and actor network structure according to different capabilities of the nodes in the network. The positioning algorithm utilizes the valence shell electron pair repulsion (VSEPR) theory of chemistry, based on the correlation between molecular geometry and the number of atoms in a molecule. By using the rules of VSEPR theory, the actor nodes in the proposed approach use a lightweight and distributed algorithm to form a self-organizing network around a central UAV, which has the role of the sink.
The last chapter
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
The Newton number or kissing number problem is a local variant of the sphere packing problem. Fix a sphere of radius a in Euclidean n-space. The Newton number τn is the maximal number of spheres of radius a which touch the given sphere and do not penetrate each other. Because of scaling, this does not depend on the radius a. Consider an n-dimensional integer lattice. Each point has the same number of closest neighbors. This number is then a lower bound on τn. The E8 lattice and the Leech lattice are exceptional even in this respect. In fact, we saw that each point in the E8 lattice has 240 neighbors. It can be proved that higher kissing numbers are impossible: τ8 = 240.
Comprehensive experimental study on the sand retention media of pre-filled sand control screens
Published in Particulate Science and Technology, 2021
Chengyun Ma, Jingen Deng, Xingliang Dong, Dongzheng Sun, Zhe Feng, Xinjiang Yan, Cheng Hui, Deliang Tian
Geometrically, sphere packing refers to the way in which the least overlapping spheres are placed within a certain range. The size of these balls is usually considered the same. At this time, the ratio of the total volume of the sphere to the size of the space is called the density. In theory, the densest packing density formed by the same sphere in three dimensions can reach 74%. By contrast, the randomness (e.g. throwing a few balls into the box at random) has an average density of only 64%. Two of the most common densest packing methods, namely, face-centered cube and hexagonal closest packing, must be triangular at the bottom to pile up the smallest pyramid possible. When the smallest pyramid should be piled up, the bottom must be hexagonal. The three most often used packing structures are shown in Figure 4. Table 2 provides an overview of packing parameters for the three lattice structures.
Clustering methods for large scale geometrical global optimization
Published in Optimization Methods and Software, 2019
Francesco Bagattini, Fabio Schoen, Luca Tigli
For the sphere packing problem, indeed, the literature reports that the fcc (face-centred cubic) and hcp (hexagonal close-packed) are the most dense known packings of equal spheres. However, computational results show that non-rigid pieces (rattlers) lead to irregular solutions even for small instances. In fact, provably optimal configurations are known for packings of up to 10 spheres only – see [26] and [27]. For this reason, we considered as enough representative for parameter calibration a set of instances starting from 20 up to 70 spheres. For sphere packing we tested three different batch sizes : 10, 20 and 50. For atomic clusters was set to 50 as some early experiments showed that in this case changing this parameter was not very significant. The set of values for σ was chosen in the same way as in the LJ case; the maximum number of concentration steps was raised to 4. For this problem, we also included the decision whether to consider only absolute values for BOP features or not; as the outer shell for sphere packing is forced by the packing constraint, we choose not to evaluate the effect of computing the BOP only for the internal spheres. Moreover the number of spheres in the experiments was relatively small, so that distinguishing between core and surface spheres is no more significant.