Wrapping the cube and other polyhedra

An infinite series of twofold, two-way weavings of the cube, corresponding to 'wrappings', or double covers of the cube, is described with the aid of the two-parameter Goldberg- Coxeter construction. The strands of all such wrappings correspond to the central circuits (CCs) of octahedrites (four-regular polyhedral graphs with square and triangular faces), which for the cube necessarily have octahedral symmetry. Removing the symmetry constraint leads to wrappings of other eight-vertex convex polyhedra. Moreover, wrappings of convex polyhedra with fewer vertices can be generated by generalizing from octahedrites to i-hedrites, which additionally include digonal faces. When the strands of a wrapping correspond to the CCs of a four-regular graph that includes faces of size greater than 4, non-convex 'crinkled' wrappings are generated. The various generalizations have implications for activities as diverse as the construction of woven-closed baskets and the manufacture of advanced composite components of complex geometry. © 2012 The Royal Society.

High order schemes on three-dimensional general polyhedral meshes - Application to the level set method

In this article, we detail the methodology developed to construct arbitrarily high order schemes - linear and WENO - on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set. © 2012 Global-Science Press.

Structural and thermodynamic properties of different phases of supercooled liquid water.

Computer simulation results are reported for a realistic polarizable potential model of water in the supercooled region. Three states, corresponding to the low density amorphous ice, high density amorphous ice, and very high density amorphous ice phases are chosen for the analyses. These states are located close to the liquid-liquid coexistence lines already shown to exist for the considered model. Thermodynamic and structural quantities are calculated, in order to characterize the properties of the three phases. The results point out the increasing relevance of the interstitial neighbors, which clearly appear in going from the low to the very high density amorphous phases. The interstitial neighbors are found to be, at the same time, also distant neighbors along the hydrogen bonded network of the molecules. The role of these interstitial neighbors has been discussed in connection with the interpretation of recent neutron scattering measurements. The structural properties of the systems are characterized by looking at the angular distribution of neighboring molecules, volume and face area distribution of the Voronoi polyhedra, and order parameters. The cumulative analysis of all the corresponding results confirms the assumption that a close similarity between the structural arrangement of molecules in the three explored amorphous phases and that of the ice polymorphs I(h), III, and VI exists.

Stiffness and strength of tridimensional periodic lattices

This paper presents a method for the linear analysis of the stiffness and strength of open and closed cell lattices with arbitrary topology. The method hinges on a multiscale approach that separates the analysis of the lattice in two scales. At the macroscopic level, the lattice is considered as a uniform material; at the microscopic scale, on the other hand, the cell microstructure is modelled in detail by means of an in-house finite element solver. The method allows determine the macroscopic stiffness, the internal forces in the edges and walls of the lattice, as well as the global periodic buckling loads, along with their buckling modes. Four cube-based lattices and nine cell topologies derived by Archimedean polyhedra are studied. Several of them are characterized here for the first time with a particular attention on the role that the cell wall plays on the stiffness and strength properties. The method, automated in a computational routine, has been used to develop material property charts that help to gain insight into the performance of the lattices under investigation. © 2012 Elsevier B.V.