Analytical solutions of the one-dimensional advection equation and two-dimensional or three-dimensional dispersion equation

1990 ◽  
Vol 26 (7) ◽  
pp. 1475-1482 ◽  
Author(s):  
Feike J. Leij
Keyword(s):  
Dispersion Equation ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Advection Equation ◽  
Two Dimensional ◽  
One Dimensional ◽  
The One

Preliminary Ship Design Using One and Two-Dimensional Models

1999 ◽  
Vol 36 (02) ◽  
pp. 102-112
Author(s):  
Michael D. A. Mackney ◽  
Carl T. F. Ross
Keyword(s):  
Finite Element ◽  
Dimensional Analysis ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Finite Element ◽  
Dimensional Models ◽  
Support Conditions ◽  
The One ◽  
Three Dimensional Finite Element

Computational studies of hull-superstructure interaction were carried out using one-, two-and three-dimensional finite element analyses. Simplification of the original three-dimensional cases to one- and two-dimensional ones was undertaken to reduce the data preparation and computer solution times in an extensive parametric study. Both the one- and two-dimensional models were evaluated from numerical and experimental studies of the three-dimensional arrangements of hull and superstructure. One-dimensional analysis used a simple beam finite element with appropriately changed sections properties at stations where superstructures existed. Two-dimensional analysis used a four node, first order quadrilateral, isoparametric plane elasticity finite element, with a corresponding increase in the grid domain where the superstructure existed. Changes in the thickness property reflected deck stiffness. This model was essentially a multi-flanged beam with the shear webs representing the hull and superstructure sides, and the flanges representing the decks One-dimensional models consistently and uniformly underestimated the three-dimensional behaviour, but were fast to create and run. Two-dimensional models were also consistent in their assessment, and considerably closer in predicting the actual behaviours. These models took longer to create than the one-dimensional, but ran in very much less time than the refined three-dimensional finite element models Parametric insights were accomplished quickly and effectively with the simplest model and processor, but two-dimensional analyses achieved closer absolute measure of the displacement behaviours. Although only static analysis with simple loading and support conditions were presented, it is believed that similar benefits would be found for other loadings and support conditions. Other engineering components and structures may benefit from similarly judged simplification using one- and two-dimensional models to reduce the time and cost of preliminary design.

A novel one-dimensional silver cylinder stabilized by mixed 2-mercaptobenzoic acid and ethylenediamine ligands

2012 ◽  
Vol 68 (8) ◽  
pp. m229-m232
Author(s):  
Di Sun ◽  
Zhi-Hao Yan
Keyword(s):  
Three Dimensional ◽  
Van Der Waals Interactions ◽  
Two Dimensional ◽  
Binding Modes ◽  
Supramolecular Framework ◽  
One Pot ◽  
Tetrahedral Geometry ◽  
Coordination Environment ◽  
One Dimensional ◽  
The One

A novel infinite one-dimensional silver cylinder, namely poly[μ-ethylenediamine-μ5-(2-sulfanidylbenzoato)-μ4-(2-sulfanidylbenzoato)-tetrasilver(I)], [Ag4(C7H4O2S)2(C2H8N2)]n, has been synthesized by one-pot reaction of equivalent molar silver nitrate and 2-mercaptobenzoic acid (H2mba) in the presence of ethylenediamine (eda). One Ag atom is located in an AgS2NO four-coordinated tetrahedral geometry, two other Ag atoms are in an AgS2O three-coordinated T-shaped geometry and the fourth Ag atom is in an AgSNO coordination environment. The two mba ligands show two different binding modes. The μ2-N:N′-eda ligand, acting as a bridge, combines with mba ligands to extend the AgIions into a one-dimensional silver cylinder incorporating abundant Ag...Ag interactions ranging from 2.9298 (11) to 3.2165 (13) Å. Interchain N—H...O hydrogen bonds extend the one-dimensional cylinder into an undulating two-dimensional sheet, which is further packed into a three-dimensional supramolecular framework by van der Waals interactions; no π–π interactions were observed in the crystal structure.

Combined Effects of 2D and 3D Inhomogeneities on the Dynamic Susceptibility of Superlattices

2010 ◽  
Vol 168-169 ◽  
pp. 97-100
Author(s):  
V.A. Ignatchenko ◽  
D.S. Tsikalov
Keyword(s):  
Density Of States ◽  
Three Dimensional ◽  
Wave Number ◽  
Dynamic Susceptibility ◽  
Combined Effects ◽  
Two Dimensional ◽  
Simultaneous Presence ◽  
One Dimensional ◽  
The One ◽  
2D And 3D

The dynamic susceptibility and the one-dimensional density of states (DOS) of an initially sinusoidal superlattice (SL) with simultaneous presence of two-dimensional (2D) phase inhomogeneities that simulate the deformations of the interfaces between the SL’s layers and three-dimensional (3D) amplitude inhomogeneities of the layer material of the SL were investigated. An analytical expression for the averaged Green’s function of the sinusoidal SL with 2D phase inhomogeneities was obtained in the Bourret approximation. It was shown that the effect of increasing asymmetry of heights of the dynamic susceptibility peaks at the edge of the Brillouin zone of the SL, which was found in [6] at increasing the rms fluctuations of 2D inhomogeneities, also takes place at increasing the correlation wave number of such inhomogeneities. It was also shown that the increase of the rms fluctuations of 3D amplitude inhomogeneities in the superlattice with 2D phase inhomogeneities leads to the suppression of the asymmetry effect and to the decrease of the depth of the DOS gap.

Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata

1997 ◽  
Vol 07 (07) ◽  
pp. 1451-1496 ◽  
Author(s):  
André Barbé
Keyword(s):  
Cellular Automata ◽  
Three Dimensional ◽  
Coarse Graining ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Complex Order ◽  
Simple Order ◽  
Problems And Solutions ◽  
The One

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).

GENERALIZATIONS OF THE CHUA EQUATIONS

1992 ◽  
Vol 02 (04) ◽  
pp. 889-909 ◽  
Author(s):  
RAY BROWN
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Type Ii ◽  
Two Dimensional ◽  
Chua’S Circuit ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Chua's Circuit ◽  
The One ◽  
Block Approach

In this paper we present two generalizations of the equations governing Chua’s circuit. In order to obtain the first generalization we simplify Chua’s equations by replacing the piecewise-linear term with a signum function. The resulting simplified system produces a double scroll similar to the one observed experimentally in Chua’s circuit. What is significant about this simplified system is that it can be reduced to what we shall call a two-dimensional single scroll, and from the two-dimensional single scroll we are able to derive a one-dimensional map. This entire derivation is carried out analytically, in contrast to the one-dimensional map analysis that has been carried out for the Lorenz equations which is based on axioms. After we have carried out our analysis for this simplified version of Chua’s equations, we use these equations as a guide to the construction of the first generalization to be presented in this paper. We call this a type-I generalization of Chua’s equations. The generalization consists in using a two-dimensional autonomous flow as a component in a three-dimensional autonomous flow in such a way that the resulting equations will have double scroll attractors similar to those observed experimentally in Chua’s circuit. The value of this generalization is that: (1) it provides a building block approach to the construction of chaotic circuits from simpler two-dimensional components which are not chaotic by themselves. In so doing it provides an insight into how chaotic systems can be built up from simple nonchaotic parts; (2) it illustrates a precise relationship between three-dimensional flows and one-dimensional maps. Of particular significance in this regard is a recent paper of Misiurewicz [1993], which analytically connects the two-dimensional single scroll to the class of unimodal maps, thus providing a framework within which a theory linking unimodal maps to three-dimensional flows may be possible. The second generalization is suggested by considering three-dimensional flows whose only nonlinearities are sigmoid, sgn, or piecewise-linear functions. Clearly, such flows are a generalization of the Chua equations. We call these equations type-II generalization Chua equations. The significance of this direction of investigation is that attractors similar to the Lorenz and Rössler attractors can be produced from type-II generalized Chua equations in a building block approach using only piecewise-linear vector fields. As a result we have a method of producing the Lorenz and Rössler dynamics in a circuit without the use of multipliers. This suggests that the type-II generalized Chua equations are in some sense fundamental in that the dynamics of the three most important autonomous three-dimensional differential equations producing chaos are seen as variations of a single class of equations whose nonlinearities are generalizations of the Chua diode.

One- and two-dimensional CdIIcoordination polymers incorporating organophosphinate ligands

2014 ◽  
Vol 70 (11) ◽  
pp. 1069-1074 ◽  
Author(s):  
Jeffrey A. Rood ◽  
Steven Boyer ◽  
Allen G. Oliver
Keyword(s):  
Three Dimensional ◽  
Dimensional Structure ◽  
Cadmium Nitrate ◽  
Two Dimensional ◽  
X Ray Diffraction ◽  
One Dimensional ◽  
Phenyl Rings ◽  
Ft Ir ◽  
Ir Analysis ◽  
The One

Reaction of cadmium nitrate with diphenylphosphinic acid in dimethylformamide solvent yielded the one-dimensional coordination polymercatena-poly[[bis(dimethylformamide-κO)cadmium(II)]-bis(μ-diphenylphosphinato-κ2O:O′)], [Cd(C12H10O2P)2(C3H7NO)2]n, (I). Addition of 4,4′-bipyridine to the synthesis afforded a two-dimensional extended structure, poly[[(μ-4,4′-bipyridine-κ2N:N′)bis(μ-diphenylphosphinato-κ2O:O′)cadmium(II)] dimethylformamide monosolvate], {[Cd(C12H10O2P)2(C10H8N2)]·C3H7NO}n, (II). In (II), the 4,4′-bipyridine molecules link the CdIIcenters in the crystallographicadirection, while the phosphinate ligands link the CdIIcenters in the crystallographicbdirection to complete a two-dimensional sheet structure. Consideration of additional π–π interactions of the phenyl rings in (II) produces a three-dimensional structure with channels that encapsulate dimethylformamide molecules as solvent of crystallization. Both compounds were characterized by single-crystal X-ray diffraction and FT–IR analysis.

scholarly journals In situsingle-crystal to single-crystal (SCSC) transformation of the one-dimensional polymercatena-poly[[diaqua(sulfato)copper(II)]-μ2-glycine] into the two-dimensional polymer poly[μ2-glycine-μ4-sulfato-copper(II)]

2014 ◽  
Vol 70 (11) ◽  
pp. 1057-1063 ◽  
Author(s):  
Helen Stoeckli-Evans ◽  
Olha Sereda ◽  
Antonia Neels ◽  
Sebastien Oguey ◽  
Catherine Ionescu ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Coordination Polymer ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Apical Position ◽  
Coordination Environment ◽  
One Dimensional ◽  
Carboxylate Groups ◽  
The One

The one-dimensional coordination polymercatena-poly[diaqua(sulfato-κO)copper(II)]-μ2-glycine-κ2O:O′], [Cu(SO4)(C2H5NO2)(H2O)2]n, (I), was synthesized by slow evaporation under vacuum of a saturated aqueous equimolar mixture of copper(II) sulfate and glycine. On heating the same blue crystal of this complex to 435 K in an oven, its aspect changed to a very pale blue and crystal structure analysis indicated that it had transformed into the two-dimensional coordination polymer poly[(μ2-glycine-κ2O:O′)(μ4-sulfato-κ4O:O′:O′′:O′′)copper(II)], [Cu(SO4)(C2H5NO2)]n, (II). In (I), the CuIIcation has a pentacoordinate square-pyramidal coordination environment. It is coordinated by two water molecules and two O atoms of bridging glycine carboxylate groups in the basal plane, and by a sulfate O atom in the apical position. In complex (II), the CuIIcation has an octahedral coordination environment. It is coordinated by four sulfate O atoms, one of which bridges two CuIIcations, and two O atoms of bridging glycine carboxylate groups. In the crystal structure of (I), the one-dimensional polymers, extending along [001], are linkedviaN—H...O, O—H...O and bifurcated N—H...O,O hydrogen bonds, forming a three-dimensional framework. In the crystal structure of (II), the two-dimensional networks are linkedviabifurcated N—H...O,O hydrogen bonds involving the sulfate O atoms, forming a three-dimensional framework. In the crystal structures of both compounds, there are C—H...O hydrogen bonds present, which reinforce the three-dimensional frameworks.

The coordination chemistry of two symmetric fluorene-based organic ligands with cuprous chloride

2013 ◽  
Vol 69 (12) ◽  
pp. 1488-1493 ◽  
Author(s):  
Yan-Fei Liu ◽  
Chao-Wei Zhao ◽  
Jian-Ping Ma ◽  
Qi-Kui Liu ◽  
Yu-Bin Dong
Keyword(s):  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Helical Chain ◽  
Cuprous Chloride ◽  
Dimensional Chain ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Network ◽  
The Difference ◽  
The One

Two novel symmetric fluorene-based ligands, namely, 2,7-bis(1H-imidazol-1-yl)-9,9-dimethyl-9H-fluorene [L1 or (I), C21H18N4] and 2,7-bis(1H-imidazol-1-yl)-9,9-dipropyl-9H-fluorene (L2), have been used to construct the coordination polymerscatena-poly[[dichloridodicopper(I)(Cu—Cu)]-μ-2,7-bis(1H-imidazol-1-yl)-9,9-dimethyl-9H-fluorene], [Cu2Cl2(C21H18N4)]n, (II), andcatena-poly[[tetra-μ2-chlorido-tetracopper(I)]-bis[μ-2,7-bis(1H-imidazol-1-yl)-9,9-dipropyl-9H-fluorene]], [Cu4Cl4(C25H26N4)2]n, (III). There are three types of C—H...N hydrogen bonds in (I), resulting a two-dimensional network in theabplane, including a chiral helical chain along thebaxis. Compounds (II) and (III) are related one-dimensional polymers. In both, CuIatoms connect the symmetric ligands (L1 orL2) into a one-dimensional chain. In (II), the {[CuICl2]−} unit, acting as a co-anion, adheres to the one-dimensional chain through a weak Cu...Cu interaction. However, in (III), the {[CuI2Cl4]2−} unit links two different chains into a one-dimensional rope-ladder-type chain. In addition, there are C—H...Cl hydrogen bonds and π–π interactions in the extended structures of (II) and (III), the difference is that the chains in (II) are linked into a two-dimensional network while the chains in (III) are stacked into a three-dimensional framework.

Synthons and design in metal phosphates and oxalates with open architectures

2001 ◽  
Vol 57 (1) ◽  
pp. 1-12 ◽  
Author(s):  
C. N. R. Rao ◽  
Srinivasan Natarajan ◽  
Amitava Choudhury ◽  
S. Neeraj ◽  
R. Vaidhyanathan
Keyword(s):  
Linear Chain ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Ladder Structure ◽  
One Dimensional ◽  
Metal Phosphates ◽  
Open Framework ◽  
Framework Materials ◽  
Layer Structures ◽  
The One

We briefly describe the structures of open-framework metal phosphates with different dimensionalities, such as the one-dimensional linear-chain and ladder structures, two-dimensional layer structures and three-dimensional structures with channels. We demonstrate the role of the zero-dimensional four-membered ring monomer and of the one-dimensional ladder structure as the starting building units or synthons involved in the formation of the complex architectures. Thus, we show how the one-dimensional ladder structure transforms to two- and three-dimensional structures under mild conditions. The two-dimensional layer structures also transform to three-dimensional structures, while the zero-dimensional monomer transforms to layered and three-dimensional structures under ordinary reaction conditions. These transformations provide an insight into the possible pathways involved in the building up of the complex structures of metal phosphates. The isolation of amine phosphates during the hydrothermal synthesis of metal phosphates and also the facile reactions between amine phosphates and metal ions to yield a variety of open-framework materials have thrown light on the mechanism of formation and design of these structures. The existence of a hierarchy of open-framework metal oxalates and their ready formation by employing amine oxalates as intermediates provides additional support to the observations made earlier with regard to the phosphates.

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