scholarly journals The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

2019 ◽  
Author(s):  
Kai Xu
Keyword(s):  
Two Dimensions ◽  
Empirical Formulas ◽  
Von Neumann ◽  
Upper Limit ◽  
Edge Distribution ◽  
Cell Topology ◽  
Neighbor Relationship ◽  
New Formula ◽  
Geometric Formula ◽  
Ellipse Packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the new formula of the Aboav-Weaire’s law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.

scholarly journals The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

2019 ◽  
Author(s):  
Kai Xu
Keyword(s):  
Two Dimensions ◽  
Empirical Formulas ◽  
Von Neumann ◽  
Upper Limit ◽  
Edge Distribution ◽  
Cell Topology ◽  
Neighbor Relationship ◽  
New Formula ◽  
Geometric Formula ◽  
Ellipse Packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the new formula of the Aboav-Weaire’s law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.

scholarly journals The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

2019 ◽  
Author(s):  
Kai Xu
Keyword(s):  
Voronoi Diagrams ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Empirical Formulas ◽  
Von Neumann ◽  
Second Moment ◽  
Upper Limit ◽  
Cell Topology ◽  
Geometric Formula ◽  
Ellipse Packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams and analyzed the cell topology based on ellipse packing and then given the improved formulas. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the improved formula of Aboav-Weaire’s law.

Simulation of Ideal Grain Growth Using the Multi-Phase-Field Model

2007 ◽  
Vol 558-559 ◽  
pp. 1177-1181 ◽  
Author(s):  
Philippe Schaffnit ◽  
Markus Apel ◽  
Ingo Steinbach
Keyword(s):  
Grain Size ◽  
Grain Growth ◽  
Phase Field ◽  
Large Scale ◽  
Field Model ◽  
Three Dimensional ◽  
Phase Field Model ◽  
Two Dimensions ◽  
Von Neumann ◽  
Average Grain Size

The kinetics and topology of ideal grain growth were simulated using the phase-field model. Large scale phase-field simulations were carried out where ten thousands grains evolved into a few hundreds without allowing coalescence of grains. The implementation was first validated in two-dimensions by checking the conformance with square-root evolution of the average grain size and the von Neumann-Mullins law. Afterwards three-dimensional simulations were performed which also showed fair agreement with the law describing the evolution of the mean grain size against time and with the results of S. Hilgenfeld et al. in 'An Accurate von Neumann's Law for Three-Dimensional Foams', Phys. Rev. Letters, 86(12)/2685, March 2001. Finally the steady state grain size distribution was investigated and compared to the Hillert theory.

Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field

1986 ◽  
Vol 403 (1824) ◽  
pp. 135-166 ◽  
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Periodic Function ◽  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Effective Hamiltonian ◽  
Hall Conductance ◽  
Von Neumann ◽  
Wannier Functions ◽  
Bloch Electrons

The problem of Bloch electrons in a magnetic field in two dimensions can be reduced to a one-dimensional problem with a Hamiltonian Ĥ that is a periodic function of x ^ and p ^ . Wannier functions can be defined for the sub-bands of the spectrum of this effective Hamiltonian. When the Chern class (quantized Hall conductance integer) of the sub-band is zero, the Weyl-Wigner formalism can be used to represent these Wannier functions by a von Neumann lattice. It is shown how this von Neumann lattice of Wannier functions can be defined for irrational as well as rational magnetic fields. An important benefit from using the Weyl-Wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective Hamiltonian in phase space. It is shown how the Wannier functions can be defined so that their Wigner functions have the point symmetries of the effective Hamiltonian. An example of how these results can prove useful is given: if we take matrix elements of the Hamiltonian between the Wannier states of a sub-band, we derive a new effective Hamiltonian describing this sub-band, which is again a periodic function of coordinate and momentum operators. Since, by projecting onto a sub-band, we have also reduced the number of degrees of freedom, this operation is a renormalization group transformation. It is shown that the symmetry of the new effective Hamil­tonian in phase space is the same as that of the original one. This preservation of symmetry helps to explain some unusual properties of the spectrum when the Hamiltonian has fourfold symmetry.

scholarly journals SHIP-GENERATED WAVES OVER A COMPLEX BATHYMETRY

2018 ◽  
pp. 35
Author(s):  
Björn Almström ◽  
Magnus Larson ◽  
Lars Granath ◽  
Hans Hanson
Keyword(s):  
Numerical Models ◽  
Water Levels ◽  
Empirical Formulas ◽  
Wave Regime ◽  
Ship Waves ◽  
Ship Wave ◽  
Energy Environment ◽  
Length Width ◽  
Stockholm Archipelago ◽  
New Formula

Problems related to shipping have increased worldwide during the last decades as a result of more traffic travel-ling at higher speeds and using larger vessels. When ships move in a restricted fairway they generate primary (drawdown) and secondary (transverse and divergent) waves (Bertram 2000) that often cause adverse impact to adjacent shores. An example of this is the Furusund fairway in Sweden, which since the 1980’s has experienced increased traffic and larger ships. This has resulted in a loss of natural fine sediment habitats along the shores as well as structural damages to piers and jetties (Granath 2015). Furusund is an important fairway into Stockholm, the capital of Sweden, and is located about 25 km north of the city within the Stockholm archipelago. It is mainly trafficked by large ferries (length/width/draft: 200x30x7m). The wind-wave regime in the fairway can be described as a low-energy environment, due to the short fetches and no swell. Hence, ship waves have a significant impact on the shores in terms of bed and bank erosion. This study aims at determining the primary ship wave characteristics and their relationship to ship properties and bathymetric conditions in the Furusund fairway. Measured water levels were collected for this purpose during three months at three locations. Existing empirical formulas for drawdown are evaluated based on the measurements and compared with a new formula derived for the specific fairway. The results are used for designing nature-based protection against ship-generated waves along the shores and to validate analytical and numerical models that can be employed for ship wave generation and propagation.

scholarly journals The bulk-edge correspondence in three simple cases

2019 ◽  
Vol 32 (03) ◽  
pp. 2030003 ◽  
Author(s):  
Jacob Shapiro
Keyword(s):  
Time Reversal ◽  
Topological Insulators ◽  
Two Dimensions ◽  
Symmetry Classes ◽  
New Formula

We present examples in three symmetry classes of topological insulators in one or two dimensions where the proof of the bulk-edge correspondence is particularly simple. This serves to illustrate the mechanism behind the bulk-edge principle without the overhead of the more general proofs which are available. We also give a new formula for the [Formula: see text]-index of our time-reversal invariant systems inspired by Moore and Balents.

A New Formula to Calculate the Fundamental Frequency of a Multistory Building by Approximate Numerical Method

2000 ◽  
Author(s):  
George Weiss
Keyword(s):  
Fundamental Frequency ◽  
Limited Range ◽  
Maximum Height ◽  
Lateral Loads ◽  
Empirical Formulas ◽  
Building Frames ◽  
Fundamental Frequencies ◽  
Multistory Building ◽  
Vibration Problems ◽  
New Formula

Abstract A new formula has been developed in this study, to calculate the fundamental frequency of a multistory building. This formula is derived from Dunkerley’s estimate, complies with the requirements of Uniform Building Code (UBC) and the Building Officials Code Administrators (BOCA) Code, and accounts for multistory building frames of different type and size. A coefficient Γ, function of moments of inertia and lengths of column/girder was introduced in the formula, in order to obtain a relative simple algebraic equation. To verify the validity of the new formula a thorough investigation has been conducted. Fundamental frequencies and periods have been calculated for different building frames using the new formula (Formula 1). A formula for buildings with rigid floors, Formula 2, has been derived from Formula 1. Formula 2 has been used to calculate the fundamental frequency for the same type and size of buildings used in the calculations with Formula 1, considered with rigid floors. In addition, three different empirical formulas have been used to calculate the fundamental period for the same buildings. Two of these formulas are recommended in different versions of the codes. For the same type and size of buildings the fundamental frequencies and periods have been determined by computer. Comparing with the computer values, formulas 1 and 2 give the best results and can be used in preliminary design, in vibration problems or to determine seismic lateral loads for buildings with a maximum height of 160/65 ft. The empirical formulas show large differences and have to be used with caution, only in the limited range imposed by UBC and BOCA.

ENTANGLEMENT IN ONE-AND TWO-DIMENSIONAL ANDERSON MODELS WITH LONG-RANGE CORRELATED-DISORDER

2009 ◽  
Vol 07 (05) ◽  
pp. 959-968
Author(s):  
Z. Z. GUO ◽  
Z. G. XUAN ◽  
Y. S. ZHANG ◽  
XIAOWEI WU
Keyword(s):  
Long Range ◽  
Ground State ◽  
Two Dimensions ◽  
Von Neumann Entropy ◽  
Two Dimensional ◽  
Correlation Effects ◽  
Von Neumann ◽  
Range Correlation ◽  
The Difference ◽  
Anderson Models

The ground state entanglement in one- and two-dimensional Anderson models are studied with consideration of the long-range correlation effects and using the measures of concurrence and von Neumann entropy. We compare the effects of the long-range power-law correlation for the on-site energies on entanglement with the uncorrelated cases. We demonstrate the existence of the band structure of the entanglement. The intraband and interband jumping phenomena of the entanglement are also reported and explained to as the localization-delocalization transition of the system. We also demonstrated the difference between the results of one- and two-dimensions. Our results show that the correlation of the on-site energies increases the entanglement.

scholarly journals On Using an Approximate Noncentral t-Distribution in Determining a One-Side Upper Limit for Future Sample Relative Reproducibility Standard Deviations

2007 ◽  
Vol 90 (2) ◽  
pp. 575-581 ◽  
Author(s):  
Foster D McClure ◽  
Jung K Lee
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Degrees Of Freedom ◽  
Normal Approximation ◽  
Upper Limit ◽  
Standard Deviations ◽  
New Formula ◽  
T Distribution ◽  
Future Sample

Abstract For future sample relative reproducibility standard deviations (RSDR), collaboratively obtained under a completely randomized model (CRM), a new formula for determining a one-tailed 100p% upper limit (p) for such RSDR values was developed based on an approximate noncentral t-distribution with degrees of freedom obtained using Satterthwaite's adjustment. The accuracy of p was assessed by comparing p and its probability levels with similar values associated with a Monte Carlo simulation and with those obtained using another formula (p) that was developed for the same purpose but based on a normal approximation.

scholarly journals New formula proposal for the determination of variable speed pumps efficiency

RBRH ◽  
2018 ◽  
Vol 23 (0) ◽  
Author(s):  
José Nilton de Abreu Costa ◽  
Marco Aurélio Holanda de Castro ◽  
Luís Henrique Magalhães Costa ◽  
João Marcelo Costa Barbosa
Keyword(s):  
Rotation Speed ◽  
Operating Conditions ◽  
The Other ◽  
Energy Costs ◽  
Variable Speed ◽  
Empirical Formulas ◽  
Speed Reduction ◽  
Power And Energy ◽  
New Formula

ABSTRACT The use of pumps which work with variable rotation speed consists of an appropriate tactics for reduction of energy costs in the pump systems. In case of change in speed, the pump starts to work in a new flow pattern, head and efficiency values. Two empirical formulas can be found in the literature in order to estimate the final efficiency of a pump in this situation. This article aims at assessing the accuracy of these formulas and, next, to present a new proposal for the efficiency estimate in the new operating conditions. 50 pumps have been chosen and, for each of them, the above-mentioned formulas have been assessed in three procedures of speed reduction. The analysis of these equations’ accuracy has been accomplished by means of the comparison between the efficiencies estimated by them with the efficiencies calculated with the aid of the affinity laws. The results indicated that, in 95 of the 150 analysis procedures accomplished, the formula proposed presented estimates which have shown to be more precise than the other two formulas. It has also been possible to verify that the average of error detected through the procedure of the new formula (1.60%) has been lower than the average of the other two (1.93% and 2.22%). In light of this fact, the use of the equation here presented is considered advantageous since, from their more accurate efficiency estimates, the calculations for power and energy are also more precise.

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