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Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
One approach to simplifying the computational-geometry problem is to use the classical marching cubes algorithm [261,291]. This algorithm is commonly used to construct isosurfaces of field quantities. In this approach, each vertex of the hexahedron is identified as either interior or exterior to the given domain. For a hexahedron, there are 28=256 $ 2^8=256 $ possible cases, which can be reduced to 15 base configurations by rotation and symmetry operations. These 15 base configurations are shown in Figure 11.9. With this categorization of vertices, the edges of the hexahedron that intersect the surface of the domain are then known. The actual intersection points within an edge can be determined by linear interpolation if a scalar field is assigned to each vertex, such as a signed distance function. A polyhedral element can then be constructed from this triangulated surface within each hexahedron as shown in Figure 11.10. In practice, the created polyhedra may be non-convex. Also, tolerancing issues due to small edges are a primary concern, and topology issues may require disambiguation techniques [283,401]. These techniques are extensible to adaptively refined (quadtree in 2D or octree in 3D) overlay meshes [358,424].
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Published in Luis Liz-Marzán, Colloidal Synthesis of Plasmonic Nanometals, 2020
Javier Reguera, Judith Langer, Dorleta Jimenez de Aberasturi, Luis M. Liz-Marzán
Nanocubes (NCs) are nanostructures composed of six symmetric square faces, each three of them meeting at each vertex (regular hexahedron). In general, for AgNCs it has been easier to obtain higher quality and uniformity than for AuNCs.29 A standard method to prepare AgNCs comprises the reduction of AgNO3 with ethylene glycol, which is used as both solvent and reducing agent. By adding HCl to the reaction mixture, single-crystal AgNCs are obtained, at the expense of longer reaction times. Xia et al.29 presented a sulfide-mediated protocol, with reaction times below 15 min, by including a trace amount of Na2S to the synthesis. In the presence of ethylene glycol and AgNO3, Ag2S nanocrystallites were produced, which then catalyse the reduction of AgNO3. Because of the fast reduction rate, the formation of twinned Ag seeds was limited and the cube shape further promoted. The authors claimed that PVP selectively binds onto {100} Ag facets, thereby facilitating the formation of NPs with cubic shape. The synthesis of monodisperse AuNCs has however been challenging, and key parameters such as reproducibility and fine size control still require further optimization.30 The most common method is based on seed-mediated growth, in combination with alkyltrimethyl ammonium halide surfactants such as CTAC, CTAB or CPC (cetylpyridinium chloride). The final shape is thus directed by kinetic control, reducing agent (ascorbic acid) concentration and selective absorption ofhalide ions on certain facets. In addition, crystal habits of the seeds play a key role on the final shape, single-crystal seeds being required for cubic structures.
Natural gas hydrate
Published in Jon Steinar Gudmundsson, Flow Assurance Solids in Oil and Gas Production, 2017
Crystals are three-dimensional structures. Cubes have six sides (also called faces) and triangular pyramids four sides. The cube form is also called hexahedron and the pyramid form tetrahedron. Collectively, the various forms are called polyhedra (singular, polyhedron). A polyhedron with 12-faces is called dodecahedron. Each of the 12 faces is shaped like a pentagon. One of the crystal cages that can trap small gas molecules is shaped like a dodecahedron, shown in Figure 5.1.
Geometry in Our Three-Dimensional World
Published in Technometrics, 2023
Chapter 6 is a joy to read and discusses, among other things, Platonic and Archimedean solids. These solids are polyhedral and are the natural spatial equivalents of plane objects such as triangles, squares, and other polygons. The fact that there are only five Platonic solids can be easily established thanks to Euler’s characteristics formula. Using the concept of duality, it is shown that a dodecahedron (with 20 faces and 12 vertices) is the dual of an icosahedron (with 12 faces and 20 vertices). A hexahedron (aka a cube) is shown as the dual of an octahedron. A tetrahedron, however, is self-dual and is without a Platonic partner. Somewhat less popular but equally interesting Kepler-Poinsot solids have also been discussed in considerable detail.
Analysis of unreinforced and reinforced tubular T-joints structures with open source finite element software
Published in Mechanics of Advanced Materials and Structures, 2023
Sina Saberi, Nicholas Fantuzzi
As it is listed in Table 4 the number of tetrahedral elements is much higher than hexahedral elements. Approximately one hexahedral element corresponds to six tetrahedral elements [24]. This fact demonstrates that tetrahedral element simulations were computationally expensive compared to hexahedral element simulations.