Coverage and Connectivity in Three-Dimensional Networks

Most wireless terrestrial networks are designed based on the assumption that the nodes are deployed on a two-dimensional (2D) plane. However, this 2D assumption is not valid in underwater, atmospheric, or space communications. In fact, recent interest in underwater acoustic ad hoc and sensor networks hints at the need to understand how to design networks in 3D. Unfortunately, the design of 3D networks is surprisingly more difficult than the design of 2D networks. For example, proofs of Kelvin's conjecture and Kepler's conjecture required centuries of research to achieve breakthroughs, whereas their 2D counterparts are trivial to solve. In this paper, we consider the coverage and connectivity issues of 3D networks, where the goal is to find a node placement strategy with 100% sensing coverage of a 3D space, while minimizing the number of nodes required for surveillance. Our results indicate that the use of the Voronoi tessellation of 3D space to create truncated octahedral cells results in the best strategy. In this truncated octahedron placement strategy, the transmission range must be at least 1.7889 times the sensing range in order to maintain connectivity among nodes. If the transmission range is between 1.4142 and 1.7889 times the sensing range, then a hexagonal prism placement strategy or a rhombic dodecahedron placement strategy should be used. Although the required number of nodes in the hexagonal prism and the rhombic dodecahedron placement strategies is the same, this number is 43.25% higher than the number of nodes required by the truncated octahedron placement strategy. We verify by simulation that our placement strategies indeed guarantee ubiquitous coverage. We believe that our approach and our results presented in this paper could be used for extending the processes of 2D network design to 3D networks.Comment: To appear in ACM Mobicom 200

Lowest order Virtual Element approximation of magnetostatic problems

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=\mu\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions

Local non-planarity of three dimensional surfaces for an invertible reconstruction: k-cuspal cells

International audienceThis paper addresses the problem of the maximal recognition of hyperplanes for an invertible reconstruction of 3D discrete objects. k- cuspal cells are introduced as a three dimensional extension of discrete cusps defined by R.Breton. With k-cuspal cells local non planarity on discrete surfaces can be identified in a very straightforward way

Determination Of Spatial Grain Size Distribution Of A Sintered Metal

The determination of the spatial grain size distribution of a sintered metal from the size distribution estimated from a sample obtained in the section plane is a stereological problem. This problem is discussed with reference tohomothetic particles (cubes of two different sizes) and to a system of three types of grains (fine and coarse cubes and coarse triangular prism). Two models are developed to solve the problem, one taking into account the size of the grains and section profiles only and one that includes shape considerations. The models are tested with simple and artificial examples, as well as with simulated data

A systematic procedure for the virtual reconstruction of open-cell foams

Open-cell foams are considered a potential candidate as an innovative catalyst support in many processes of the chemical industry. In this respect, a deeper understanding of the transport phenomena in such structures can promote their extensive application. In this contribution, we propose a general procedure to recover a representative open-cell structure starting from some easily obtained information. In particular, we adopt a realistic description of the foam geometry by considering clusters of solid material at nodes and different strut-cross sectional shapes depending on the void fraction. The methodology avoids time-consuming and expensive measuring techniques, such as micro-computed tomography (μCT) or magnetic resonance imaging (MRI). Computational Fluid Dynamics (CFD) could be a powerful instrument to enable accurate analyses of the complex flow field and of the gas-to-solid heat and mass transport. The reconstructed geometry can be easily exploited to generate a suitable computational domain allowing for the detailed investigation of the transport properties on a realistic foam structure by means of CFD simulations. Moreover, the proposed methodology easily allows for parametric sensitivity analysis of the foam performances, thus being an instrument for the advanced design of these structures. The geometrical properties of the reconstructed foams are in good agreement with experimental measurements. The flow field established in complex tridimensional geometries reproduces the real foam behavior as proved by the comparison between numerical simulations and experiments