Magnetic Field Reconstruction for a Realistic Multi-Point, Multi-Scale Spacecraft Observatory

Future in situ space plasma investigations will likely involve spatially distributed observatories comprised of multiple spacecraft, beyond the four and five spacecraft configurations currently in operation. Inferring the magnetic field structure across the observatory, and not simply at the observation points, is a necessary step towards characterizing fundamental plasma processes using these unique multi-point, multi-scale data sets. We propose improvements upon the classic first-order reconstruction method, as well as a second-order method, utilizing magnetometer measurements from a realistic nine-spacecraft observatory. The improved first-order method, which averages over select ensembles of four spacecraft, reconstructs the magnetic field associated with simple current sheets and numerical simulations of turbulence accurately over larger volumes compared to second-order methods or first-order methods using a single regular tetrahedron. Using this averaging method on data sets with fewer than nine measurement points, the volume of accurate reconstruction compared to a known magnetic vector field improves approximately linearly with the number of measurement points.

Design and analysis of three-dimensional printing of a porous titanium scaffold

Objective Mechanic strength, pore morphology and size are key factors for the three-dimensional (3D) printing of porous titanium scaffolds, therefore, developing optimal structure for the 3D printed titanium scaffold to fill bone defects in knee joints is instructive and important. Methods Structural models of titanium scaffolds with fifteen different pore unit were designed with 3D printing computer software; five different scaffold shapes were designed: imitation diamond-60°, imitation diamond-90°, imitation diamond-120°, regular tetrahedron and regular hexahedron. Each structural shape was evaluated with three pore sizes (400, 600 and 800 μm), and fifteen types of cylindrical models (size: 20 mm; height: 20 mm). Autodesk Inventor software was used to determine the strength and safety of the models by simulating simple strength acting on the knee joints. We analyzed the data and found suitable models for the design of 3D printing of porous titanium scaffolds. Results Fifteen different types of pore unit structural models were evaluated under positive pressure and lateral pressure; the compressive strength reduced when the pore size increased. Under torsional pressure, the strengths of the imitation diamond structure were similar when the pore size increased, and the strengths of the regular tetrahedron and regular hexahedron structures reduced when the pore size increased. In each case, the compressive strength of the regular hexahedron structure was highest, that of the regular tetrahedron was second highest, and that of the imitation diamond structure was relatively low. Fifteen types of cylindrical models under a set force were evaluated, and the sequence of comprehensive compressive strength, from strong to weak was: regular hexahedron > regular tetrahedron > imitation diamond-120° > imitation diamond-90° > imitation diamond-60°. The compressive strength of cylinder models was higher when the pore size was smaller. Conclusion The pore size and pore morphology were important factors influencing the compressive strength. The strength of each structure reduced when the pore size (400, 600 and 800 μm) increased. The models of regular hexahedron, regular tetrahedron and imitation diamond-120°appeared to meet the conditions of large pore sizes and high compressive strength.

Design and Analysis of Three-Dimensional Printing of A Porous Titanium Scaffold

Objective To develop suitable structural designs for the three-dimensional (3-D) printing of a porous titanium scaffold to fill bone defects in knee joints. Pore diameter and mechanic strength are key factors for the 3-D printing of porous titanium scaffolds. Methods Fifteen different pore unit structural models of titanium scaffolds were designed with 3-D printing computer software; five different scaffold shapes were designed: imitation diamond-60°, imitation diamond-90°, imitation diamond-120°, regular tetrahedron and regular hexahedron. Each structural shape was evaluated with three pore diameters 400μm, 600μm and 800μm, and fifteen types of cylindrical models(diameter: 20mm; height: 20mm). Autodesk Inventor software was used determine the strength and safety of the models by simulating simple strength acting on the knee joints. We analyzed the data and found suitable models for 3-D printing of porous titanium scaffolds. Results Fifteen different types of pore unit structural models were evaluated under positive pressure; the compressive strength was lower when the pore diameter(400μm, 600μm and 800μm) was larger, except for the regular tetrahedron structure. Under lateral pressure, the compressive strength was also lower when the pore diameter(400μm, 600μm and 800μm) was larger. Under torsional pressure, the strength of the imitation diamond structure was similar when the pore diameter(400μm, 600μm and 800μm) was larger, and the strengths of the regular tetrahedron and regular hexahedron structures were lower when the pore diameter(400μm, 600μm and 800μm) was larger. In each case, the compressive strength of the regular hexahedron structure was highest, that of the regular tetrahedron was second highest, and that of the imitation diamond structure was relatively low. Fifteen types of cylindrical models under a set force were evaluated, and the sequence of comprehensive compressive strength, from strong to weak was: regular hexahedron> regular tetrahedron> imitation diamond-120°> imitation diamond-90°> imitation diamond-60°. The compressive strength of cylinder models was higher when the pore diameter was smaller. Conclusion The compressive strength differed among titanium scaffolds with different pore structures. The pore diameter and shapes of the pore structure were important factors influencing the compressive strength. The models of regular hexahedron, regular tetrahedron and imitation diamond-120°appeared to meet the conditions of large pore diameters and high compressive strength. The strength of each structure was lower when the pore diameter(400μm, 600μm and 800μm) was larger.

Covering a Regular Tetrahedron with Diminished Copies

Let T be a unit regular tetrahedron. A diminished copy of T is the image of T under a homothety with positive ratio smaller than 1. Let m be a positive integer and let γm(T) be the smallest positive number r such that T can be covered by m translates of rT. Zong gave the results of γ4(T) = 3/4and γ5(T) = 9/13. However, the values of γ6(T) , γ7(T) and γ8(T) were not given then. In this article we give the upper bounds of γ6(T), γ7(T) and γ8(T).

Renormalization group theory of the generalized multi-vertex sine-Gordon model

We investigate the renormalization group theory of the generalized multi-vertex sine-Gordon model by employing the dimensional regularization method and also the Wilson renormalization group method. The vertex interaction is given by $\cos(k_j\cdot \phi)$, where $k_j$ ($j=1,2,\ldots,M$) are momentum vectors and $\phi$ is an $N$-component scalar field. The beta functions are calculated for the sine-Gordon model with multiple cosine interactions. The second-order correction in the renormalization procedure is given by the two-point scattering amplitude for tachyon scattering. We show that new vertex interaction with the momentum vector $k_{\ell}$ is generated from two vertex interactions with vectors $k_i$ and $k_j$ when $k_i$ and $k_j$ meet the condition $k_{\ell}=k_i\pm k_j$, called the triangle condition. A further condition $k_i\cdot k_j=\pm 1/2$ is required within the dimensional regularization method. The renormalization group equations form a set of closed equations when $\{k_j\}$ form an equilateral triangle for $N=2$ or a regular tetrahedron for $N=3$. The Wilsonian renormalization group method gives qualitatively the same result for beta functions.

CONHECIMENTOS GEOMÉTRICOS E ALGÉBRICOS DO TETRAEDRO FRACTAL 3D

A fractal is a figure that has a unique characteristic that will be present in the entire domain of the figure. There are several different types of fractals, some of which are constructed from a simple figure such as a triangle ofplane geometry or a tetrahedron of spatial geometry. From the initial construction of a two-dimensional fractal starting with an equilateral triangle and using Napoleon's Theorem, in this article, we present a construction of a new three-dimensional fractal using ideas similar to Napoleon's Theorem in a tetrahedron. Using concepts of plane and spatial geometry, this fractal can be built from a regular tetrahedron, and from the midpoints of its edges a new tetrahedron with a 1/2 ratio side is built in relation to the initial tetrahedron. After this construction, the characteristics of the infinite application fractal are studied, such as the sum of the surface areas and the total volume of the formed figure.

Elasticity of Anisotropic Low-Density Lattice Materials

Computational first-order homogenization theory is used for the elastic analysis of generally anisotropic lattice materials within classical continuum mechanics. The computational model is tailored for structural one-dimensional (1D) elements, which considerably reduces the computational cost comparing to previously developed models based on solid elements. The effective elastic behavior of lattice materials is derived consistently with several homogenization approaches including strain- and stress-based methods together with volume and surface averaging. Comparing the homogenization based on the Hill–Mandel Lemma and constitutive approach, a shear correction factor is also introduced. In contrast to prior studies that are usually limited to a specific class of lattice materials such as lattices with cubic symmetry or similarly situated joints, this computational tool is applicable for the analysis of any planar or spatial stretching- and bending-dominated lattices with arbitrary topology and anisotropy. Having derived the elasticity of the lattice, the homogenization is then complemented by the symmetry identification based on the monoclinic distance function. This step is essential for lattices with non-apparent symmetry. Using the computational model, nine different spatial anisotropic lattices are studied among which four are fully characterized for the first time, i.e., non-regular tetrahedron (with trigonal symmetry), rhombicuboctahedron type a (with cubic symmetry), rhombicuboctahedron type b (with transverse isotropy), and double-pyramid dodecahedron (with tetragonal symmetry).

Necessary condition for the existence of a simple closed geodesic on a regular tetrahedron in the spherical space

In the spherical space the curvature of the tetrahedron’s faces equals 1, and the curvature of the whole tetrahedron is concentrated into its vertices and faces. The intrinsic geometry of this tetrahedron depends on the value α of faces angle, where π/3 < α ⩽ 2π/3. The simple (without points of self-intersection) closed geodesic has the type (p,q) on a tetrahedron, if this geodesic has p points on each of two opposite edges of the tetrahedron, q points on each of another two opposite edges, and (p+q) points on each edges of the third pair of opposite one. For any coprime integers (p,q), we present the number αp, q (π/3 < αp, q < 2π/3) such that, on a regular tetrahedron in the spherical space with the faces angle of value α > αp, q, there is no simple closed geodesic of type (p,q)