Bi-Dbar-Approach for a Coupled Shifted Nonlocal Dispersionless System

We propose a Bi-Dbar approach and apply it to the extended coupled shifted nonlocal dispersionless system. We introduce the nonlocal reduction to solve the coupled shifted nonlocal dispersionless system. Since no enough constraint conditions can be found to curb the norming contants in the Dbar data, the “solutions” obtained by the Dbar dressing method, in general, do not admit the coupled shifted nonlocal dispersionless system. In the Bi-Dbar approach to the extended coupled shifted nonlocal dispersionless system, the norming constants are free. The constraint conditions on the norming constants are determined by the general nonlocal reduction, and the solutions of the coupled shifted nonlocal dispersionless system are derived

Experimental and Thermodynamic Investigation of RE-Mg-Zn (RE=Sc, Y, La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) Systems

Lors d’une “optimisation” / modélisation thermodynamique, des paramètres du modèle ajustables sont affinés à partir de toutes les données thermodynamiques et d’équilibres de phases disponibles de façon à obtenir un ensemble d’équations du modèle dépendant de la température et de la composition. A partir de ces équations du modèle, toutes les propriétés thermodynamiques et tous les diagrammes de phases peuvent être calculés par minimisation de l’énergie de Gibbs à l’aide d’un logiciel tel que FactSage. En général, l’optimisation d’un système ternaire commence par l’optimisation des trois sous-systèmes binaires. Les paramètres binaires du modèle sont ensuite utilisés pour estimer les propriétés des phases ternaires, et ces estimations sont ensuite améliorées par l’introduction de paramètres ternaires si nécessaire de façon à reproduire les données ternaires disponibles. Tous les systèmes binaires Mg-Zn et Mg-RE (où RE = terre rare) ont déjà été optimisés auparavant et les paramètres du modèle pour ces systèmes sont disponibles dans le logiciel FactSage. Dans le présent projet, tous les systèmes binaires RE-Zn et la plupart des systèmes ternaires RE-Mg-Zn sont optimisés.En premier lieu, toutes les données thermodynamiques et de diagrammes de phases disponibles pour les systèmes Re-Zn (Sc, Y, La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) ont été rassemblées et évaluées de manière critique. Les terres rares ont des propriétés très similaires. Les diagrammes de phases de tous les systèmes Re-Zn sont très semblables. Les tendances observées dans les propriétés des systèmes terre rare (RE)-Zn pour toute la séquence des terres rares ont été exploitées pour estimer les données manquantes et pour vérifier la cohérence des données existantes. Le Modèle de Miedema est également utilisé dans le présent projet pour évaluer l’enthalpie de mélange des phases liquides. A partir de toutes les données disponibles, une évaluation critique et une optimisation thermodynamiques de ces systèmes a été effectuée et des paramètres du modèle pour les propriétés thermodynamiques de toutes les phases ont été obtenus. En second lieu, des expériences de diffraction de neutrons (DN) in-situ ont été réalisées pour des échantillons choisis dans les systèmes Ce-Mg-Zn et Nd-Mg-Zn de façon à identifier les phases et les températures de transition. Grâce à la grande capacité de pénétration des neutrons, des échantillons de grande taille (10-20 grammes) peuvent être utilisés dans le présent travail, conduisant à un meilleur contrôle de la composition et à une résistance à l’oxydation accrue. Des informations plus précises à propos des relations de phases et du comportement en transformation découlent des présentes expériences de DN car celles-ci sont réalisées in-situ à des températures élevées. Toutes les données expérimentales de DN sont utilisées pour valider et affiner le modèle thermodynamique. Finalement, toutes les données de diagrammes de phases pour les systèmes RE-Mg-Zn ont été évaluées de manière critique et tous les systèmes ternaires RE-Mg-Zn (à l’exception de Sc-Mg-Zn, Pm-Mg-Zn, Eu-Mg-Zn et Yb-Mg-Zn) ont été optimisés à partir des systèmes binaires Mg-Zn, Mg-RE et RE-Zn. Les tendances et régularités observées sont utilisées à nouveau lors de l’optimisation des systèmes ternaires. Comme on pouvait s’y attendre, tous les systèmes RE-Mg-Zn sont liés de près. Les systèmes Ce-Mg-Zn et Nd-Mg-Zn sont optimisés de façon critique en prenant en compte les nouvelles données de DN. L’optimisation thermodynamique de tous les autres systèmes RE-Mg-Zn est grandement facilitée par l’optimisation simultanée des systèmes Ce-Mg-Zn et Nd-Mg-Zn. Il faut noter que le Modèle Quasichimique Modifié (MQM) est utilisé dans le présent projet pour décrire la phase liquide. Puisque l’ordre à courte distance est pris en compte par ce modèle, on s’attend à une meilleure description de la phase liquide. Le présent projet a pour but de construire une banque de données thermodynamiques la plus complète et la plus précise possible pour les systèmes RE-Mg-Zn. Les chercheurs s’intéressant aux alliages de Mg pourront en bénéficier. ---------- In a thermodynamic “optimization/modeling”, adjustable model parameters are refined based on all available thermodynamic and phase-equilibrium data in order to obtain one set of model equations as functions of temperature and composition. From the model equations, all the thermodynamic properties and phase diagrams can be back-calculated by Gibbs energy minimization using software such as FactSage. Generally, in the optimization of a ternary system one begins by optimizing the three binary sub-systems. The binary model parameters are then used to estimate the properties of the ternary phases, and these estimates are then refined by introducing ternary model parameters where required to reproduce available ternary data. All binary Mg-Zn and Mg-RE (where RE = rare earth) systems have already been optimized and the model parameters of these systems are readily available in the FactSage software. In the present project, all binary RE-Zn and most of the ternary RE-Mg-Zn systems are optimized.Firstly, all available phase diagram and thermodynamic data for the RE-Zn (Sc, Y, La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) systems have been collected and critically assessed. The rare earth elements have very similar properties. The phase diagrams of all RE-Zn systems are very similar. Trends in the properties of the rare earth (RE)-Zn systems as one traverses the rare earth series have been exploited for purposes of estimating missing data and for checking existing data for consistency. The Miedema Model is also used in the present project to estimate the enthalpy of mixing of the liquid phases. Based on all available data, critical thermodynamic evaluation and optimization of these systems have been carried out and model parameters for the thermodynamic properties of all phases have been obtained.Secondly, in-situ neutron diffraction (ND) experiments have been performed on selected samples in the Ce-Mg-Zn and Nd-Mg-Zn systems to identify phases and transition temperatures. Due to the high penetrating power of neutrons, large samples (10-20 grams) are used in the present work, leading to better control of composition and increased resistance to oxidation. More accurate information about phase relationships and transformation behavior can also be expected from the present ND experiments because they are performed in-situ at high temperatures. All the ND experimental data are used to validate and refine the thermodynamic modelling.Finally, all phase diagram data for the RE-Mg-Zn systems have been critically assessed and all ternary RE-Mg-Zn (excluding Sc-Mg-Zn, Pm-Mg-Zn, Eu-Mg-Zn and Yb-Mg-Zn) systems have been optimized, based on the binary Mg-Zn, Mg-RE and RE-Zn systems. Observed trends and regularities are again used in the optimization of the ternary systems. As expected, all RE-Mg-Zn systems are similarly closely related. The Ce-Mg-Zn and Nd-Mg-Zn systems are critically optimized taking into account the new ND data. Thermodynamic optimization of all other RE-Mg-Zn systems is greatly aided by simultaneous optimization of the Ce-Mg-Zn and Nd-Mg-Zn systems.It should be noted that the Modified Quasichemical Model (MQM) is used in the present project to describe the liquid phase. Since short range ordering (SRO) is taken into account by this model, better description of the liquid phase is expected.The present project is aimed at building the most complete and accurate thermodynamic database for the RE-Mg-Zn systems from which all investigators of Mg alloys can benefit

Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we introduce a hierarchy of integrable higher order equations of Camassa-Holm (CH) type, that is, we present infinitely many nonlinear equations depending on inertia operators which generalize the standard momentum operator A2=∂xx−1 appearing in the Camassa-Holm equation mt=−mxu−2mux, m=A2(u). Our higher order CH-type equations are integrable in the sense that they possess an infinite number of local conservation laws, quadratic pseudo-potentials, and zero curvature formulations. We focus mainly on the fifth order CH-type equation and we show that it admits {\em pseudo-peakons}, this is, bounded solutions with differentiable first derivative and continuous and bounded second derivative, but whose higher order derivatives blow up. Furthermore, we investigate the Cauchy problem of our fifth order equation on the real line and prove local well-posedness for initial conditions u0∈Hs(R), s\u3e7/2. In addition, we discuss conditions for global well-posedness in H4(R) as well as conditions causing local solutions to blow up in a finite time. We finish our paper with some comments on the geometric content of our equations of CH-type

Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we discuss integrable higher order equations {\em of Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically integrable", that is, they describe one-parametric families of pseudo-spherical surfaces, in a sense explained in Section 1, and they are integrable in the sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many local conservation laws. The major focus of the present paper is on a specific fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is, weak bounded solutions with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. Furthermore, we investigate the Cauchy problem of this fifth order CH-type equation on the real line and prove local well-posedness under the initial conditions $u_0 \in H^s(\mathbb{R})$, $s > 7/2$. In addition, we study conditions for global well-posedness in $H^4(\mathbb{R})$ as well as conditions causing local solutions to blow up in a finite time. We conclude our paper with some comments on the geometric content of the high order CH-type equations.Comment: 6 figures; 32 page

Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model

This paper devotes to present analysiswork on the fifth order Camassa-Holm (FOCH) modelwhich recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data

Multiple Periodic Vibrations of Auxetic Honeycomb Sandwich Plate with 1:2 Internal Resonance

In this paper, we focus on the multiple periodic vibration behaviors of an auxetic honeycomb sandwich plate subjected to in-plane and transverse excitations. Nonlinear equation of motion for the plate is derived based on the third-order shear deformation theory and von Kármán type nonlinear geometric assumptions. The Melnikov method is extended to detect the bifurcation and multiple periodic vibrations of the plate under 1:2 internal resonance. The effects of transverse excitation on nonlinear vibration behaviors are discussed in detail. Evolution laws and waveforms of multiple periodic vibrations are obtained to analyze the energy transfer process between the first two order modes. Even quite small transverse excitation can cause periodic vibration in the system, and there can be at most three periodic orbits in certain bifurcation regions. The periodic orbits are classified into two families by tracing their sources. The study provides the possibility for the classification study on generation mechanism of system complexity and energy transfers between different modes

Numerical and experimental investigation of aerodynamic noise from automotive cooling fan module

The numerical investigation of automotive cooling fan module aerodynamic noise based on Computational Fluid Dynamics (CFD) / Computational Aero Acoustics (CAA) hybrid method is taken. Considering the influence of fan shroud, an aerodynamic steady simulation is made at first, and then Large Eddy Simulation (LES) with the Smagorinsky model is applied to capture the unsteady pressure data on fan surface. Based on Lowson formula, a prediction of aerodynamic noise is made by Boundary Element Method (BEM). Finally, the prediction results are compared with the experimental results, which show that the acoustic response is with a strong dipole character. Sound pressure level (SPL) at receive points increased with air flow rate. SPL at outlet was higher than inlet. Tonal noise was the major component of the aerodynamic noise. Broadband noise was relatively lower and distributed evenly. Predicted results is consistent with the experimental results, which validates the numerical prediction method. This method can provide a reference to acoustic optimization

Rogue peakon, well-posedness, ill-posedness and blow-up phenomenon for an integrable Camassa-Holm type equation

In this paper, we study an integrable Camassa-Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution - called {\sf rogue peakon}, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space $B_{p,r}^{s}$ with $1\leq p,r\leq\infty$, $s>\max \left\{1+1/p,3/2\right\}$ or $B_{2,1}^{3/2}$, and then prove the ill-posedness of the solution in $B_{2,\infty}^{3/2}$. Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if $m_0(x)=u_0-u_{0xx}\geq(\not\equiv) 0$, then the corresponding solution exists globally, meanwhile, if $m_0(x)\leq(\not\equiv) 0$, then the corresponding solution blows up in a finite time.Comment: 23 pages, 6 figure