Numerical simulations of the steady Navier–Stokes equations using adaptive meshing schemes

In this paper, we consider an adaptive meshing scheme for solution of the steady incompressible Navier–Stokes equations by finite element discretization. The mesh refinement and optimization are performed based on an algorithm that combines the so‐called conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) and residual‐type local a posteriori error estimators. Numerical experiments in the two‐dimensional space for various examples are presented with quadratic finite elements used for the velocity field and linear finite elements for the pressure. The results show that our meshing scheme can equally distribute the errors over all elements in some optimal way and keep the triangles very well shaped as well at all levels of refinement. In addition, the convergence rates achieved are close to the best obtainable. Extension of this approach to three‐dimensional cases is also discussed and the main challenge is the efficient implementation of three‐dimensional CfCVDT generation that is still under development. Copyright © 2007 John Wiley & Sons, Ltd.     

Two novel methods for vibration diagnosis to characterize non-linear response

Poincaré maps have been proved to be a valuable tool in the analysis of non-linear dynamical systems, which usually reduce a continuous phase flow into a two-dimensional discrete map. However, they may be inconvenient for reflecting some characteristics of the system response. In this paper, two novel methods, using the period sampling peak-to-peak value (PSP) diagram and the modified Poincaré map, are presented for characterizing different types of non-linear response. These two methods take advantage of some parameters of the response, such as the peak-to-peak value within an exterior excitation period and the mean value of the displacement. In the PSP diagram method, a two-dimensional graph is plotted by taking the peak-to-peak value as ordinate and the sequential periodically sampling number as abscissa. On the other hand, the modified Poincaré map takes the mean value of the velocity within an exterior excitation period as ordinate and the relevant mean value of the displacement as abscissa. The non-linear responses of a Duffing system, a pendulum with circular motion support and an oscillating circuit are studied by these methods. We also studied the intermittent chaos of the Lorenz system by the PSP diagram method. The PSP diagram is a set of mapping points, which form: a straight line for a one-period response; multi-straight lines for a multi-period response; orderly periodic curves for a quasi-period response; long lines interrupted by transitoriness confusion points for intermittent chaos; and totally out-of-order points for chaos. The figures for the modified Poincaré maps for the period, multi-period, quasi-period responses and chaos are almost identical to those for the Poincaré maps, but the modified maps take more sampling points and can reflect the mean values of the responses. Some numerical results are given based on these methods to show their efficiency in distinguishing different non-linear responses.     

Cations of halogenated methanes: adiabatic ionization energies, potential energy surfaces, and ion fragment appearance energies

The DFT-B3LYP and G3X model chemistry were used to predict the cation structures and energetics of fluorinated, chlorinated, and brominated methanes. Ion–complex structures between methylene cations and HX (X = F, Cl, Br) were found for all H-containing cations, and [CHF–FH]⁺, [CF₂–FH]⁺, [CCl₂–ClH]⁺, and [CCl₂–FH]⁺ structures are more stable than their normal tetravalent structures. Several cations should also be better described as ion–complex structures between methyl cations and halogen atoms, e.g., [CF₃–Br]⁺. Transition states connecting normal and ion–complex structures were also located, and potential energy diagrams were constructed for decomposition of methane cations and to predict the fragmentation pathways. The G3X energies were used to predict the adiabatic ionization energies (IE_as) and ion fragment appearance energies (AEs) from methanes. Many of the experimental AEs correspond to the energies of transition states instead of the thermodynamic dissociation limits. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

On graphs determining links with maximal number of components via medial construction

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K₁ by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.     

Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88-91] and the second author [P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n+3 facets, Electron. J. Combin. 14 (2007), R69, 36 pp.], this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.     

Investigation of the Quasi Binary System BaAu–BaPt

Phase equilibria in the system BaAu–BaPt have been investigated by X‐ray powder diffraction. Depending on composition, three structure types occur, the FeB type for BaAu, and NiAs for BaPt, while the CrB type of structure is adopted in between. The homogeneity range for the CrB type of structure was established to extend from BaPt_0.15Au_0.85 to BaPt_0.90Au_0.10. The respective lattice parameters vary linearly, in accordance with Vegard's law. The crystal structure of the new CrB type compounds have been confirmed by X‐ray powder diffraction for the solid solution range, and by single crystal X‐ray diffraction exemplary for the composition BaAu_0.5Pt_0.5 (Cmcm; a = 4.3915(5) Å; b = 11.9149(12) Å; c = 4.7920(5) Å; Z = 4). BaAu was also established by single crystal structure determination (Pnma; a = 8.3220(10) Å; b = 4.9252(10) Å; c = 6.3844(10) Å; Z = 4) to complete the results. According to ESCA measurements BaAu_0.5Pt_0.5 and BaAu can be formulated as [Ba²⁺·0.5e^?]·[Au^?_0.5·Pt^2?_0.5] and [Ba²⁺·e^?]·[Au^?], respectively.     

Nitrous oxide: Saturation properties and the phase diagram

The experimental values of the coordinates of the triple point and of the critical point of nitrous oxide registered in the literature were assessed and those judged as most reliable have been selected. Empirical equations have been found for the vapour pressure, sublimation and fusion curves. The virial coefficients and saturation properties as functions of temperature along the equilibrium curves are described by reduced equations. They were used in arriving at the molar enthalpies at the triple point and the normal boiling temperature. Equations for the sublimation and fusion curves resulting from the exactly integrated Clapeyron equation compare favourably with the results from the empirical treatment and the experimental data.     

Influence of methanol on the phase behavior of nonionic fluorinated surfactant: Relation to the structure of mesoporous silica materials

We have investigated the effect of methanol addition on the R^F₈(EO)₉ and R^F₇(EO)₈ surfactant-based systems. While upon the addition of methanol the L₁ micellar phase grows, the direct hexagonal (H₁) and the lamellar (L_α) liquid crystals progressively melt with the increase of alcohol content. Phase behavior and SAXS measurements proved that methanol molecules interact with the oxyethylene units of the surfactant. This involves a folding up of the hydrophobic chains in the liquid crystal phases. Moreover, for the R^F₇(EO)₈ surfactant, the cloud point curve is shifted to high temperatures upon addition of alcohol. Starting from these systems, we have prepared mesoporous materials. Results show that due to the hydrogen bonds between the alcohol and the EO groups, the hexagonal structure of the mesostructured silica obtained from R^F₈(EO)₉ is lost when the content of CH₃OH is increased. In contrast, for the compounds prepared from the R^F₇(EO)₈-based system, the pore ordering occurs in the presence of alcohol. This phenomenon has been related to the moving of the cloud point curve toward high temperatures with the addition of methanol. Our study reveals also that under our conditions the methanol released during the hydrolysis of the silica precursor does not affect the self-assembly mechanism in a positive or negative way.     

Thermodynamic analysis of LiF-BeF2 and KF-BeF2 melts by a structural model

An earlier structural model for binary silicate melts and glasses is extended to MF-BeF₂ (M = Li, K) systems. The evaluation of the thermodynamic properties as well as the phase diagrams for the binary LiF-BeF₂ system and the integral enthalpy of mixing of the KF-BeF₂ system are carried out with this model. This thermodynamic model is based on the assumption that each alkali fluoride produces the depolymerization of BeF₂ network with a characteristic free energy change. A least squares optimization program permits all available thermodynamic and phase diagram data to be optimized simultaneously. In this manner, data for these binary systems have been analysed and represented with a small number of parameters. The model predicts the chain-length distribution of polymeric ions, even though these are not explicitly treated as structural units of the model. The calculated fluoride polyanion chain-length distribution for the LiF-BeF₂ system is in quantitative agreement with the predictions reported in the literature.     

Theoretical Study on the Excited-state Intramolecular Hydrogen Abstraction Reactions of Butanal

The excited-state intramolecular hydrogen abstraction reactions of butanal have been investigated using the CAS-MP2/6-311＋G^＊//CASSCF/6-31G^＊ methods. Calculated results show that the hydrogen transfer induced fluorescence quenching of the n,π^＊-excited state of covalent butanal with three paths： （1） The first path corresponds to direct S0-react reconstitution, which involves the first S1 decay by partial hydrogen atom transfer. （2） The second stepwise mechanism can be viewed as a full hydrogen atom transfer followed by a partial hydrogen atom back transfer, electron transfer （near S1/S0 or S0-TS） and finally a proton transfer to S0-react. （3） On the triplet surface, the surface crossing to the singlet state would be clearly much efficient at the T1/S0 region due to the large SOC value of 8.3 cm^-1. The S0-react decay route from T1/S0 was studied with an intrinsic reaction coordinate （IRC） calculation at the CASSCF level, resulting in the S0-React minimum.