scholarly journals Number and twistedness of strands in weavings on regular convex polyhedra

2014 ◽  
Vol 470 (2162) ◽  
pp. 20130608 ◽  
Author(s):  
F. Kovács
Keyword(s):  
Graph Theory ◽  
Structural Engineering ◽  
Closed Surface ◽  
Convex Polyhedra ◽  
Recursive Method ◽  
Simple Method ◽  
Linking Number ◽  
Regular Convex ◽  
Weaving Pattern ◽  
Symmetry Operations

This paper deals with two- and threefold weavings on Platonic polyhedral surfaces. Depending on the skewness of the weaving pattern with respect to the edges of the polyhedra, different numbers of closed strands are necessary in a complete weaving. The problem is present in basketry but can be addressed from the aspect of pure geometry (geodesics), graph theory (central circuits of 4-valent graphs) and even structural engineering (fastenings on a closed surface). Numbers of these strands are found to have a periodicity and symmetry, and, in some cases, this number can be predicted directly from the skewness of weaving. In this paper (i) a simple recursive method using symmetry operations is given to find the number of strands of cubic, octahedral and icosahedral weavings for cases where generic symmetry arguments fail; (ii) another simple method is presented to decide whether or not a single closed strand can run along the underlying Platonic without a turn (i.e. the linking number of the two edges of a strand is zero, and so the loop can be stretched to a circle without being twisted); and (iii) the linking number of individual strands in an alternate ‘check’ weaving pattern is determined.

Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
Vinjamuri Venkata Kamesh ◽  
Kuchibhotla Mallikarjuna Rao ◽  
Annambhotla Balaji Srinivasa Rao
Keyword(s):  
High Velocity ◽  
Mechanical Energy ◽  
Gear Train ◽  
Recursive Method ◽  
Gear Pair ◽  
Simple Method ◽  
Epicyclic Gear ◽  
Thick Line ◽  
Gear Trains ◽  
Isomorphic Graphs

Epicyclic gear trains (EGTs) are used in the mechanical energy transmission systems where high velocity ratios are needed in a compact space. It is necessary to eliminate duplicate structures in the initial stages of enumeration. In this paper, a novel and simple method is proposed using a parameter, Vertex Incidence Polynomial (VIP), to synthesize epicyclic gear trains up to six links eliminating all isomorphic gear trains. Each epicyclic gear train is represented as a graph by denoting gear pair with thick line and transfer pair with thin line. All the permissible graphs of epicyclic gear trains from the fundamental principles are generated by the recursive method. Isomorphic graphs are identified by calculating VIP. Another parameter “Rotation Index” (RI) is proposed to detect rotational isomorphism. It is found that there are six nonisomorphic rotation graphs for five-link one degree-of-freedom (1-DOF) and 26 graphs for six-link 1-DOF EGTs from which all the nonisomorphic displacement graphs can be derived by adding the transfer vertices for each combination. The proposed method proved to be successful in clustering all the isomorphic structures into a group, which in turn checked for rotational isomorphism. This method is very easy to understand and allows performing isomorphism test in epicyclic gear trains.

Evaluation of coupled partial models in structural engineering using graph theory and sensitivity analysis

2011 ◽  
Vol 33 (12) ◽  
pp. 3726-3736 ◽  
Author(s):  
Holger Keitel ◽  
Ghada Karaki ◽  
Tom Lahmer ◽  
Susanne Nikulla ◽  
Volkmar Zabel
Keyword(s):  
Sensitivity Analysis ◽  
Graph Theory ◽  
Structural Engineering ◽  
Partial Models

scholarly journals Wind Field Synthesis for Animating Wind-induced Vibration

2011 ◽  
Vol 10 (1) ◽  
pp. 53-60 ◽  
Author(s):  
Oyundolgor Khorloo ◽  
Zorig Gunjee ◽  
Batjargal Sosorbaram ◽  
Norishige Chiba
Keyword(s):  
Wind Field ◽  
Structural Engineering ◽  
Computational Cost ◽  
Three Dimensional ◽  
Wind Fields ◽  
Simple Method ◽  
Turbulent Wind ◽  
Key Characteristics ◽  
Induced Motion ◽  
Wind Induced Vibration

We present a simple method to generate three-dimensional frozen and non-frozen turbulent wind fields for use in the animation of wind-induced motion. Our approach uses 1/f^_ noise to match the characteristics of natural wind. By employing a noise-based approach, the complexity as well as computational cost is reduced. Additionally, by considering key characteristics of actual wind that are applied in the structural engineering field, our proposed method is able to produce plausible results in outdoor wind field simulations. In this paper, we describe the implementation results of our proposed method and compare them with other existing approaches used to construct turbulent wind fields. The implementation and visualization are carried out for both two- and three-dimensional scenarios and compared with the results of other well-known methods.

Wright-Fisher Geometry

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Normal Form ◽  
Euclidean Space ◽  
Convex Polyhedron ◽  
Convex Polyhedra ◽  
Linearly Independent ◽  
Regular Convex ◽  
Normal Vectors ◽  
Manifolds With Corners ◽  
Natural Domains ◽  
Manifold With Corners

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural domains of definition for generalized Kimura diffusions: polyhedra in Euclidean space or, more generally, abstract manifolds with corners. Amongst the convex polyhedra, the chapter distinguishes the subclass of regular convex polyhedra P. P is a regular convex polyhedron if it is convex and if near any corner, P is the intersection of no more than N half-spaces with corresponding normal vectors that are linearly independent. These definitions establish that any regular convex polyhedron is a manifold with corners. The chapter concludes by defining the general class of elliptic Kimura operators on a manifold with corners P and shows that there is a local normal form for any operator L in this class.

scholarly journals A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

2016 ◽  
Vol 20 (3) ◽  
pp. 733-753 ◽  
Author(s):  
J. Thomas Beale ◽  
Wenjun Ying ◽  
Jason R. Wilson
Keyword(s):  
Singular Integrals ◽  
Quadrature Rule ◽  
Accurate Method ◽  
Closed Surface ◽  
Special Treatment ◽  
Ewald Summation ◽  
Simple Method ◽  
Nearly Singular Integrals ◽  
Grid Points ◽  
Improved Accuracy

AbstractWe present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

A Method for Ultimate Strength Assessment of Plates in Combined Stresses

2015 ◽  
Author(s):  
Shengming Zhang ◽  
Lei Jiang
Keyword(s):  
Ultimate Strength ◽  
Structural Engineering ◽  
Fe Analysis ◽  
Strength Analysis ◽  
Element Analysis ◽  
Simple Method ◽  
Strength Assessment ◽  
Industry Standards ◽  
Non Linear ◽  
Offshore Industry

It is becoming a normal practice in structural engineering, including ships and offshore industry, to perform non-linear finite element analysis to assess the structure’s capacity for design or evaluation purposes. However, experience has shown that the quality and accuracy of the non-linear FE analysis results are highly dependent on the person’s skills and analysis procedures used. In some cases, the results could lead to a wrong conclusion. Simplified method with good accuracy is still a preferred approach in design by the industry because it is simple and easy to use. However, even the method for the simple model, a plate under combined loads, has not been addressed completely because of its complexity. This paper has thus developed a simple method for the ultimate strength analysis of square plates under combined longitudinal and transverse compressive stresses. The method is fully validated with a systematic non-linear FE analysis results. The paper has also compared the methods from the industry standards BS5400 and DIN18800. Analysis examples are also provided in the paper for reference and discussions.

Convex Polyhedra with Regular Faces

1966 ◽  
Vol 18 ◽  
pp. 169-200 ◽  
Author(s):  
Norman W. Johnson
Keyword(s):  
The Other ◽  
Convex Polyhedra ◽  
Regular Polygons ◽  
Platonic Solids ◽  
Geometric Figures ◽  
Regular Polyhedra ◽  
The Subject ◽  
Symmetry Operations

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.

scholarly journals The Platonic solids and fundamental tests of quantum mechanics

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 293 ◽  
Author(s):  
Armin Tavakoli ◽  
Nicolas Gisin
Keyword(s):  
Quantum Mechanics ◽  
Natural Philosophy ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Convex Polyhedra ◽  
Classical Antiquity ◽  
Platonic Solids ◽  
Mathematical Beauty ◽  
Regular Convex ◽  
And Mathematics

The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millennia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the European scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.

scholarly journals Toric Geometry of the Regular Convex Polyhedra

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Fiammetta Battaglia ◽  
Elisa Prato
Keyword(s):  
Convex Polyhedra ◽  
Toric Geometry ◽  
Regular Tetrahedron ◽  
Regular Octahedron ◽  
Regular Dodecahedron ◽  
Regular Convex ◽  
Regular Icosahedron

We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry.

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