High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics

We present a class of hybridizable discontinuous Galerkin (HDG) methods for the numerical simulation of wave phenomena in acoustics and elastodynamics. The methods are fully implicit and high-order accurate in both space and time, yet computationally attractive owing to their following distinctive features. First, they reduce the globally coupled unknowns to the approximate trace of the velocity, which is defined on the element faces and single-valued, thereby leading to a significant saving in the computational cost. In addition, all the approximate variables (including the approximate velocity and gradient) converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k ? 0 are used to represent the numerical solution and when the time-stepping method is accurate with order k + 1. When the time-stepping method is of order k + 2, superconvergence properties allows us, by means of local postprocessing, to obtain better, yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k + 2 in the L²-norm when k ? 1 for both acoustics and elastodynamics. Extensive numerical results are provided to illustrate these distinctive features.     

An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier–Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure–velocity formulation of the incompressible Navier–Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k?0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H (div)-conforming, and converges with order k + 2 for k ? 1 and with order 1 for k = 0 in the L²-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.     

Second-order optical nonlinearities from χ gratings induced by holographic all-optical poling

The paper presents the second-order optical nonlinearities from χ⁽²⁾ gratings induced by holographic all-optical poling for azobenzene polymer. Second harmonic (SH) signal along the directions with two different vectors was measured. One is strong SH signal diffracted in the same direction as 2ω writing beam with wave vector k_2ω and the other is weak SH signal diffracted in the direction of wave vector of 4k_ω - k_2ω + Δk where k_ω is wave vector of ω beam and Δk is the wave vector mismatch whose vector is parallel to k_ω. The latter signal was used as a tool to monitor the formation of holographic χ⁽²⁾ gratings in real-time because it has off-axis wave vector different from both k_ω and k_2ω. The increase of 2ω intensity on poling process led to the large increase of second-order optical nonlinearity. The real-time monitoring showed that it also gave the large relaxation of second-order optical nonlinearity on poling process. The increase of 2ω (532 nm) energy enhanced the increase of local heating, which led to easier alignment of azobenzene chromophore and also larger relaxation of aligned chromophore.     

Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics

We present hybridizable discontinuous Galerkin methods for solving steady and time-dependent partial differential equations (PDEs) in continuum mechanics. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the numerical flux to provide stability and consistency; and a global jump condition that enforces the continuity of the numerical flux to arrive at a global weak formulation in terms of the numerical trace. The HDG methods are fully implicit, high-order accurate and endowed with several unique features which distinguish themselves from other discontinuous Galerkin methods. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, they provide, for smooth viscous-dominated problems, approximations of all the variables which converge with the optimal order of k + 1 in the L²-norm. Third, they possess some superconvergence properties that allow us to define inexpensive element-by-element postprocessing procedures to compute a new approximate solution which may converge with higher order than the original solution. And fourth, they allow for a novel and systematic way for imposing boundary conditions for the total stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the methods. In addition, they possess other interesting properties for specific problems. Their approximate solution can be postprocessed to yield an exactly divergence-free and H(div)-conforming velocity field for incompressible flows. They do not exhibit volumetric locking for nearly incompressible solids. We provide extensive numerical results to illustrate their distinct characteristics and compare their performance with that of continuous Galerkin methods.     

Nature of electric and magnetic dipoles gleaned from the Poynting theorem and the Lorentz force law of classical electrodynamics

Starting with the most general form of Maxwell's macroscopic equations in which the free charge and free current densities, ρ_free and J_free, as well as the densities of polarization and magnetization, P and M, are arbitrary functions of space and time, we compare and contrast two versions of the Poynting vector, namely, S = μ_o^− 1E × B and S = E × H. Here E is the electric field, H is the magnetic field, B is the magnetic induction, and μ_o is the permeability of free space. We argue that the identification of one or the other of these Poynting vectors with the rate of flow of electromagnetic energy is intimately tied to the nature of magnetic dipoles and the way in which these dipoles exchange energy with the electromagnetic field. In addition, the manifest nature of both electric and magnetic dipoles in their interactions with the electromagnetic field has consequences for the Lorentz law of force. If the conventional identification of magnetic dipoles with Amperian current loops is extended beyond Maxwell's macroscopic equations to the domain where energy, force, torque, momentum, and angular momentum are active participants, it will be shown that “hidden energy” and “hidden momentum” become inescapable consequences of such identification with Amperian current loops. Hidden energy and hidden momentum can be avoided, however, if we adopt S = E × H as the true Poynting vector, and also accept a generalized version of the Lorentz force law. We conclude that the identification of magnetic dipoles with Amperian current loops, while certainly acceptable within the confines of Maxwell's macroscopic equations, is inadequate and leads to complications when considering energy, force, torque, momentum, and angular momentum in electromagnetic systems that involve the interaction of fields and matter.     

Propagation regimes of intense circularly polarized laser beam in magnetoactive plasma

This paper presents an analytical and numerical investigation of an intense circularly polarized wave propagating along the static magnetic field parallel to oscillating magnetic field in magnetoactive plasma. In the relativistic regime such a magnetic field is created by pulse itself. The authors have studied different regimes of propagation with relativistic electron mass effect for magnetized plasma. An appropriate expression for dielectric tensor in relativistic magnetoactive plasma has been evaluated under paraxial theory. Two modes of propagation as extraordinary and ordinary exist; because of the relativistic effect, ultra-strong magnetic fields are generated which significantly influence the propagation of laser beam in plasma. The nature of propagation is characterized through the critical-divider curves in the normalized beam width with power plane For given values of normalized density (ω_p/ω) and magnetic field (ω_c/ω) the regions are namely steady divergence (SD), oscillatory divergence (OD) and self-focusing (SF). Numerical computations are performed for typical parameters of relativistic laser-plasma interaction: magnetic field B = 10-100 MG; intensity I = 10¹⁶ to 10²⁰ W/cm²; laser frequency ω = 1.1 × 10¹⁵ s⁻¹; cyclotron frequency ω_c = 1.7 × 10¹³ s⁻¹; electron density n_e = 2.18 × 10²⁰ cm⁻³. From the calculations, we confirm that a circularly polarized wave can propagate in different regimes for both the modes, and explicitly indicating enhancement in wave propagation, beam focusing/self-guiding and penetration of E-mode in presence of magnetic field.     

Negative refraction at various crystal interfaces

In an anisotropic positive index media the tangential components of Poynting vector S and that of wavevector k can have the opposite signs under certain conditions. The phenomenon is called as the anomalous negative refraction. The negative refraction phenomenon at planar interfaces between isotropic medium-crystal and crystal-crystal media has been analyzed. The crystals can be both uniaxial axis and biaxial axis and the azimuth of optical axis can be arbitrary. The negative refraction phenomenon under various azimuth of optical axis is analyzed. The phenomena can be very evident if the optical axis angle is arranged appropriately. The optimal conditions are obtained. For strong anisotropy crystal, the maximum incident angle that yields the negative refraction and the bending angle could be very large. For the interface of biaxial crystal the anisotropy parameter of crystal u₁ = n_z/n_x plays a dominant role in effect on negative refraction.     

The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes

We extend the mimetic finite difference (MFD) method to the numerical treatment of magnetostatic fields problems in mixed div–curl form for the divergence-free magnetic vector potential. To accomplish this task, we introduce three sets of degrees of freedom that are attached to the vertices, the edges, and the faces of the mesh, and two discrete operators mimicking the curl and the gradient operator of the differential setting. Then, we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div–curl variational formulation of the magnetostatic equations. This construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the MFD method. We also discuss the linear algebraic form of the resulting MFD scheme, its practical implementation, and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by Hyman and Shashkov to the case of unstructured polyhedral meshes. The accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem.     

Two new upwind difference schemes for a coupled system of convection–diffusion equations arising from the steady MHD duct flow problems

In this paper, we develop two new upwind difference schemes for solving a coupled system of convection–diffusion equations arising from the steady incompressible MHD duct flow problem with a transverse magnetic field at high Hartmann numbers. Such an MHD duct flow is convection-dominated and its solution may exhibit localized phenomena such as boundary layers, namely, narrow boundary regions where the solution changes rapidly. Most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy. In contrast, the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability, and they are capable of resolving high gradients near the layer regions without refining the grid. The accuracy of the first new upwind scheme is O(h + k) and the second one improves the accuracy to O(ε²(h + k) + ε(h² + k²) + (h³ + k³)), where 0 < ε ? 1/M ? 1 and M is the high Hartmann number. Numerical examples are provided to illustrate the performance of the newly proposed upwind difference schemes.     

Low temperature thermodynamical properties of ErCu2Si2

ErCu₂Si₂ crystallises in the tetragonal ThCr₂Si₂-type crystal structure. In this paper results of magnetometric, electrical transport, specific heat as well as neutron diffraction are reported. Results of electrical resistivity and specific heat measurements performed at low temperature yield existence of magnetic ordering roughly at 1.3 K. These results are in concert with neutron diffraction measurements, which reveal simple antiferromagnetic ordering between 0.47 and 1.00 K. At temperatures ranging from 1.00 up to 1.50 K an additional incommensurate magnetic structure was observed. The propagation vector k=(0;0;0.074) was proposed to describe magnetic reflections within the amplitude modulated magnetic structure. Basing on specific heat studies the crystal field levels splitting scheme and magnetic entropy were calculated.     

Magnetic structure of TmCu2Ge2

TmCu₂Ge₂ compound crystallizes in the tetragonal ThCr₂Si₂-type crystal structure. The neutron diffraction reveals the presence of an incommensurate antiferromagnetic order below T_N=2.5 K. The Tm magnetic moment of 5.0(1) μ_B at 0.47 K is parallel to the c-axis. The order is described by the propagation vector k=[k_x, k_x, 0], where k_x=0.117(3). The increase of the values of the components k_x near the Néel temperature is observed.     

Metal ion-molecule kinetics at combustion temperatures: The reaction of Ca with O2

The high-temperature photochemistry (HTP) technique, previously used for reactions of neutral species, has been adapted to the study of atomic metal ion-molecule reactions. Ca⁺ ions were generated by 193 nm multi-photon photolysis of calcium acetyl acetonate and its pyrolysis fragments. The relative ion concentrations were monitored by laser-induced fluorescence at 393.4 nm. Ar was used as the bath gas. The data for the Ca⁺ + O₂ + M → CaO₂⁺ + M association reaction (1) are fitted by k₁(907-1425 K) = 3.5 × 10⁻³² exp(+3161 K/T) cm⁶ molecule⁻² s⁻¹. Combining with an approximate k₁(296 K) value in the literature leads to k₁(296-1425 K) = 5.8 × 10⁻²² (T/K)^−2.9 exp(−601 K/T) cm⁶ molecule⁻² s⁻¹. Over much of the observed temperature range reaction (1) has much smaller rate coefficients than the corresponding neutral Ca association reaction. Reaction (1) is shown to behave very similarly to the O₂ association reaction with neutral K atoms, with which Ca⁺ is iso-electronic. This suggests that the initial step is ion-pair complex formation of the superoxide Ca²⁺(O₂)⁻, which is also consistent with results from density functional calculations. The k₁ values are rationalized via Troe’s unimolecular formalism, which leads to good accord with the experiments.     

Synthesis and luminescent properties of two Schiff-base boron complexes

Schiff bases N,N′-o-phenylenebis (salicylideneimine) (H₂L¹), N,N′-p-phenylenebis (salicylideneimine) (H₂L²) and their corresponding boron complexes (BF₂)₂L¹, (BF₂)₂L² were synthesized, respectively. The two boron complexes have been characterized by ¹H NMR, mass spectrometry and elemental analysis, while the luminescent properties of them were investigated with UV-VIS spectroscopy and photoluminescence spectroscopy. Then the three-layer devices [ITO/NPB (60 nm)/(BF₂)₂L¹ (50 nm)/Alq3 (20 nm)/LiF (1 nm)/Al (200 nm)] (device I) and [ITO/NPB (60 nm)/(BF₂)₂L² (50 nm)/Alq3 (20 nm)/LiF (1 nm)/Al (200 nm)] (device II) were fabricated by vacuum deposition. These two devices both exhibited blue green emission at 500 nm, but showed different luminances and efficiencies.     

Electron-phonon interaction in magnesium: From the monolayer to the Mg(0 0 0 1) surface

We report ab initio study of the electron-phonon coupling in a free standing magnesium monolayer and at the Mg(0 0 0 1) surface. The calculations were carried out using a linear-response approach in the plane-wave pseudopotential representation. Eliashberg spectral function α²F(ω) averaged over electron states at the Fermi surface is presented for the monolayer while for the Mg(0 0 0 1) surface, we compute the electron-phonon spectral function α²F_k,i(ω) for surface states at the and points.     

Nucleation of Instability of the Meissner State of 3-Dimensional Superconductors

This paper concerns a nonlinear partial differential system in a 3-dimensional domain involving the operator curl², which is a simplified model used to examine nucleation of instability of the Meissner state of a superconductor as the applied magnetic field reaches the superheating field. We derive a priori C ^2+α estimates for a weak solution H, the curl of the magnetic potential, and determine the location of the maximal points of |curlH| which correspond to the nucleation of instability of the Meissner state. We show that, if the penetration length is small, the solution exhibits a boundary layer. If the applied magnetic field is homogeneous, |curlH| is maximal around the points on the boundary where the applied field is tangential to the surface.     

Magnetic structure of the Mn5Si3-type Er5Ge3 compound

The Er₅Ge₃ compound (Mn₅Si₃-type, hP16, P6₃/mcm) at 4 K shows magnetic ordering of the antiferromagnetic type. Its magnetic structure consists of sine modulated collinear magnetic moments of Er that are parallel to the c axis (with a propagation vector k=[0 0 ±0.3]). This corresponds to the magnetic unit cell (a a 10c), the values of the magnetic moment of the Er atoms being, as a general formula, M_z≈M₀ cos [2π(Z–1/4)(1–kZ)], with M₀=9.2(2) μ_B at 4 K.     

Higher order differential lift equations over manifolds

LetM be a differentiable manifold modeled on a Banach space overK=R or C. LetT ^k(M) be thekth iterated tangential extension ofM, and letk ^M be thekth Bowman (=restricted tangential) extension ofM. It is shown that there is an embedding ϕ_k:k ^{→T k}(M), and that such embeddings constitute a natural transformation of functors. LetQ be a subset/submanifold inT ^k(M), and letV:Q→T(Q) be a differentiable vector field. CallV k-suitable if everyK-curveg inQ satisfyingg′=V° g has the formg=f ^[k], wheref ^[k] denotes thekth iterated differential lift of aK-curvef inM. It is shown thatV isk-suitable if and only if: (a) , where is a subset/submanifold ink ^M, and (b) , where isk-suitable relative to restricted tangentialK-curve liftsf ^(k). Interpretive consequences for motion problems are discussed.     

Ab initio chemical kinetics for the reactions of HNCN with O(P) and O2

The kinetics and mechanisms of the reactions of cyanomidyl radical (HNCN) with oxygen atoms and molecules have been investigated by ab initio calculations with rate constant prediction. The doublet and quartet state potential energy surfaces (PESs) of the two reactions have been calculated by single-point calculations at the CCSD(T)/6-311+G(3df, 2p) level based on geometries optimized at the CCSD/6-311++G(d, p) level. The rate constants for various product channels of the two reactions in the temperature range of 300-3000 K are predicted by variational transition state and RRKM theories. The predicted total rate constants of the O(³P) + HNCN reaction at 760 Torr Ar pressure can be represented by the expressions k_total (O + HNCN) = 3.12 × 10⁻¹⁰ × T^−0.05 exp (−37/T) cm³ molecule⁻¹ s⁻¹ at T = 300-3000 K. The branching ratios of primary channels of the O(³P) + HNCN are predicted: k₁ for producing the NO + CNH accounts for 0.72-0.64, k₂ + k₉ for producing the ³NH + NCO accounts for 0.27-0.32, and k₆ for producing the CN + HNO accounts for 0.01-0.07 in the temperature range studied. Meanwhile, the predicted total rate constants of the O₂ + HNCN reaction at 760 Torr Ar pressure can be represented by the expression, k_total(O₂ + HNCN) = 2.10 × 10⁻¹⁶ × T^1.28exp (−12200/T) cm³ molecule⁻¹ s⁻¹ at T = 300-3000 K. The predicted branching ratio for k₁₁ + k₁₃ producing HO₂ + ³NCN as the primary products accounts for 0.98-1.00 in the temperature range studied.     

Intermolecular photoinduced electron-transfer processes between C60 and aniline derivatives in benzonitrile

The quenching behavior of the triplets of C₆₀ by various aniline derivatives (1a-d and 2a-e) was investigated by means of laser flash photolysis in benzonitrile at 293 K. Electron transfer process was proposed to be the main mechanism because of the direct detection of radical ions of aniline derivatives and C₆₀ in time-resolved transient absorption spectra. The quenching rate constants (k_q) of by different substrates determined at 740 nm approach or reach the diffusion-controlled limit. DFT method was employed to calculate the unknown oxidation potentials of substrates in solution. With these E_ox values, free energy changes (ΔG) were obtained through Rehm-Weller equation. Dependence of observed quenching rate constants on the free energy changes further indicates the photoinduced reactions between ³C₆₀^* and substrates proceed through an electron transfer mechanism. Obtained k_q values for the aniline derivatives are impacted obviously by ground-state configurations and the kinds substituents quantified by Hammett σ constant. Good correlation between log k_q and σ values conforms to the empirical Hammett equation. A more negative ρ value (−3.356) was gained for anilines (2a-e) than that of N,N-dimethylanilines (1a-d) (−1.382), which suggests a more susceptible reactivity for the former substrates. Charge density distribution of reaction center “N” originated from quantum calculation supports this suggestion. In addition, a relationship between quenching rate constants and solvent viscosity was gained from C₆₀/dimethyl-p-toluidine system in altered mixtures of acetonitrile and toluene.     

Easy axis reorientation and magneto-crystalline anisotropic resistance of tensile strained (Ga,Mn)As films

We present a study of magnetic anisotropy by using magneto-transport and direct magnetization measurements on tensile strained (Ga,Mn)As films. The magnetic easy axis of the films is in-plane at low temperatures, while the easy axis flips to out-of-plane when temperature is raised or hole concentration is increased. This easy axis reorientation is explained qualitatively in a simple physical picture by Zener’s p-d model. In addition, the magneto-crystalline anisotropic resistance was also investigated experimentally and theoretically based on the single magnetic domain model. The dependence of sheet resistance on the angle between the magnetic field and [1 0 0] direction was measured. It is found that the magnetization vector M in the single-domain state deviates from the external magnetic field H direction at low magnetic field, while for high magnetic field, M continuously moves following the field direction, which leads to different resistivity function behaviors.