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.     

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal (k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.     

An unconditionally stable time-domain discretization on cartesian meshes for the simulation of nonuniform magnetized cold plasma

In this paper an unconditionally stable, spatially and temporally implicit time-domain discretization for nonuniform magnetized cold plasma is developed. The discrete dispersion relation is free of spurious solutions and approximates the continuous dispersion relation for well-resolved wavelengths and frequencies (kΔ ? π, ωΔ_t ? π). For a specific choice of parameters, the discrete dispersion relation approximates the continuous dispersion relation for all wavelengths and frequencies up to the Nyquist limit. A few examples, amongst them one involving mode conversion, illustrate the new method.     

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.     

A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss–Lobatto–Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge–Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400–5000 and flows around an impulsively started cylinder for Re = 550–9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

Tunable characteristics of one dimensional magnetic photonic crystal composed with single-negative materials

In this paper, we present the study of dispersion and transmittance characteristics of one dimensional magnetic photonic crystal composed by single negative indexed materials. For this structure, we have considered magnetic negative (MNG) with ? = 1 and μ < 0 and electric negative (ENG) with ? < 0 and μ = 4. We used simple transfer matrix method and Bloch's theorem for its analytical explanations. Analyzing transmittance characteristics of the proposed structure, we obtain the tunneling of certain frequency range where as the dispersion characteristic shows total forbidden for the same range for TM mode. The tunable property is found inside the band structure due to zero-?, zero-μ and magnetic behavior of the material. To identify zero-? and zero-μ of the structure, we have calculated the dispersion and the transmittance of the magnetic structure of MNG–ENG on different angles of incidence and thickness of layers.     

Dissociation of dimethyl ether at high temperatures

The decomposition of dimethyl ether (CH₃OCH₃) has been investigated behind incident shock waves in a diaphragmless shock tube using laser schlieren densitometry, LS (T = 1500–2450 K, P = 57 ± 4, 125 ± 5 and 253 ± 12 Torr). The LS density gradient profiles were simulated and excellent agreement was found between the simulations and experimental profiles. Rate coefficients for CH₃OCH₃ → CH₃O + CH₃ were obtained. They showed strong fall-off, and at the lower end of the experimental temperature range are close to the low pressure limit. First order rate coefficient expressions were determined over 1500 < T < 2450 K. k_57Torr = (3.10 ± 1.0) × 10⁷⁹T^?19.03 exp(?54417/T) s^?1, k_125Torr = (1.12 ± 0.3) × 10⁸³T^?19.94 exp(?55554/T) s^?1and k_253Torr = (1.02 ± 0.3) × 10⁷³T^?17.09 exp(?51500/T) s^?1. The effect of a roaming channel for decomposition of dimethyl ether was assessed and the best agreement was obtained with 1% dissociation of DME via the roaming path.     

GPU-based single-cluster algorithm for the simulation of the Ising model

We present the GPU calculation with the common unified device architecture (CUDA) for the Wolff single-cluster algorithm of the Ising model. Proposing an algorithm for a quasi-block synchronization, we realize the Wolff single-cluster Monte Carlo simulation with CUDA. We perform parallel computations for the newly added spins in the growing cluster. As a result, the GPU calculation speed for the two-dimensional Ising model at the critical temperature with the linear size L = 4096 is 5.60 times as fast as the calculation speed on a current CPU core. For the three-dimensional Ising model with the linear size L = 256, the GPU calculation speed is 7.90 times as fast as the CPU calculation speed. The idea of quasi-block synchronization can be used not only in the cluster algorithm but also in many fields where the synchronization of all threads is required.     

Shock tube/laser absorption studies of the decomposition of methyl formate

Reaction rate coefficients for the major high-temperature methyl formate (MF, CH₃OCHO) decomposition pathways, MF → CH₃OH + CO (1), MF → CH₂O + CH₂O (2), and MF → CH₄ + CO₂ (3), were directly measured in a shock tube using laser absorption of CO (4.6 μm), CH₂O (306 nm) and CH₄ (3.4 μm). Experimental conditions ranged from 1202 to 1607 K and 1.36 to 1.72 atm, with mixtures varying in initial fuel concentration from 0.1% to 3% MF diluted in argon. The decomposition rate coefficients were determined by monitoring the formation rate of each target species immediately behind the reflected shock waves and modeling the species time-histories with a detailed kinetic mechanism [12]. The three measured rate coefficients can be well-described using two-parameter Arrhenius expressions over the temperature range in the present study: k₁ = 1.1 × 10¹³ exp(?29556/T, K) s^?1, k₂ = 2.6 × 10¹² exp(?32052/T, K) s^?1, and k₃ = 4.4 × 10¹¹ exp(?29 078/T, K) s^?1, all thought to be near their high-pressure limits. Uncertainties in the k₁, k₂ and k₃ measurements were estimated to be ±25%, ±35%, and ±40%, respectively. We believe that these are the first direct high-temperature rate measurements for MF decomposition and all are in excellent agreement with the Dooley et al. [12] mechanism. In addition, by also monitoring methanol (CH₃OH) and MF concentration histories using a tunable CO₂ gas laser operating at 9.67 and 9.23 μm, respectively, all the major oxygen-carrying molecules were quantitatively detected in the reaction system. An oxygen balance analysis during MF decomposition shows that the multi-wavelength laser absorption strategy used in this study was able to track more than 97% of the initial oxygen atoms in the fuel.     

On the rate constants of OH + HO2 and HO2 + HO2: A comprehensive study of H2O2 thermal decomposition using multi-species laser absorption

Hydrogen peroxide (H₂O₂) and hydroperoxy (HO₂) reactions present in the H₂O₂ thermal decomposition system are important in combustion kinetics. H₂O₂ thermal decomposition has been studied behind reflected shock waves using H₂O and OH diagnostics in previous studies (Hong et al. (2009) [9] and Hong et al. (2010) [6,8]) to determine the rate constants of two major reactions: H₂O₂ + M → 2OH + M (k₁) and OH + H₂O₂ → H₂O + HO₂ (k₂). With the addition of a third diagnostic for HO₂ at 227 nm, the H₂O₂ thermal decomposition system can be comprehensively characterized for the first time. Specifically, the rate constants of two remaining major reactions in the system, OH + HO₂ → H₂O + O₂ (k₃) and HO₂ + HO₂ → H₂O₂ + O₂ (k₄) can be determined with high-fidelity.No strong temperature dependency was found between 1072 and 1283 K for the rate constant of OH + HO₂ → H₂O + O₂, which can be expressed by the combination of two Arrhenius forms: k₃ = 7.0 × 10¹² exp(550/T) + 4.5 × 10¹⁴ exp(?5500/T) [cm³ mol^?1 s^?1]. The rate constants of reaction HO₂ + HO₂ → H₂O₂ + O₂ determined agree very well with those reported by Kappel et al. (2002) [5]; the recommendation therefore remains unchanged: k₄ = 1.0 × 10¹⁴ exp(?5556/T) + 1.9 × 10¹¹⁺exp(709/T) [cm³ mol^?1 s^?1]. All the tests were performed near 1.7 atm.     

High-spatial-resolution monitoring of strong magnetic field using Rb vapor nanometric-thin cell

We have implemented the so-called λ-Zeeman technique (LZT) to investigate individual hyperfine transitions between Zeeman sublevels of the Rb atoms in a strong external magnetic field B in the range of 2500 ? 5000 G (recently it was established that LZT is very convenient for the range of 10 ? 2500 G). Atoms are confined in a nanometric thin cell (NTC) with the thickness L = λ, where λ is the resonant wavelength 794 nm for Rb D₁ line. Narrow velocity selective optical pumping (VSOP) resonances in the transmission spectrum of the NTC are split into several components in a magnetic field with the frequency positions and transition probabilities depending on the B-field. Possible applications are described, such as magnetometers with nanometric local spatial resolution and tunable atomic frequency references.     

Multi-species time-history measurements during high-temperature acetone and 2-butanone pyrolysis

High-temperature acetone and 2-butanone pyrolysis studies were conducted behind reflected shock waves using five species time-history measurements (ketone, CO, CH₃, CH₄ and C₂H₄). Experimental conditions covered temperatures of 1100–1600 K at 1.6 atm, for mixtures of 0.25–1.5% ketone in argon. During acetone pyrolysis, the CO concentration time-history was found to be strongly sensitive to the acetone dissociation rate constant k₁ (CH₃COCH₃ → CH₃ + CH₃CO), and this could be directly determined from the CO time-histories, yielding k₁(1.6 atm) = 2.46 × 10¹⁴ exp(?69.3 [kcal/mol]/RT) s^?1 with an uncertainty of ±25%. This rate constant is in good agreement with previous shock tube studies from Sato and Hidaka (2000) [3] and Saxena et al. (2009) [4] (within 30%) at temperatures above 1450 K, but is at least three times faster than the evaluation from Sato and Hidaka at temperatures below 1250 K. Using this revised k₁ value with the recent mechanism of Pichon et al. (2009) [5], the simulated profiles during acetone pyrolysis show excellent agreement with all five species time-history measurements. Similarly, the overall 2-butanone decomposition rate constant k_tot was inferred from measured 2-butanone time-histories, yielding k_tot(1.5 atm) = 6.08 × 10¹³ exp(?63.1 [kcal/mol]/RT) s^?1 with an uncertainty of ±35%. This rate constant is approximately 30% faster than that proposed by Serinyel et al. (2010) [11] at 1119 K, and approximately 100% faster at 1412 K. Using the measured 2-butanone and CO time-histories and an O-atom balance analysis, a missing removal pathway for methyl ketene was identified. The rate constant for the decomposition of methyl ketene was assumed to be the same as the value for the ketene decomposition reaction. Using the revised k_tot value and adding the methyl ketene decomposition reaction to the Serinyel et al. mechanism, the simulated profiles during 2-butanone pyrolysis show good agreement with the measurements for all five species.     

Characterization of stimulated Brillouin scattering in small core microstructured chalcogenide fiber

We report a full modal characterization of the stimulated Brillouin scattering (SBS) properties in small core As₂Se₃-based chalcogenide photonic crystal fibers (PCFs). Our results include the calculation of Brillouin gain spectrum (BGS), Brillouin gain coefficient (g_B), Brillouin frequency shift (BFS) by taking into account the contribution of the higher-order acoustic modes. We show that for a highly nonlinear PCF having a 2-μm hole-to-hole pitch and a 0.5-μm hole diameter, a Brillouin gain coefficient g_B = 5.91 10^? 9 m.W^? 1 is obtained around the acoustic frequency of 8.19 GHz, which is more than 340 times larger than that of the same PCF made with silica glass. We demonstrate that the BGS of small core PCF structures show strong SBS and multipeaked behavior, with a presence of a second peak, when decreasing the core diameter which is to be attributed to the higher-order acoustic modes. We designed small core PCFs with tailored Brillouin response for a wide range of applications.     

Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation

Many applications lead to large systems of linear equations with dense matrices. Direct matrix-vector products become prohibitive, since the computational cost increases quadratically with the size of the problem. By exploiting specific kernel properties fast algorithms can be constructed.A directional multilevel algorithm for translation-invariant oscillatory kernels of the type K(x, y) = G(x ? y)e^?k∣x?y∣, with G(x ? y) being any smooth kernel, will be presented. We will first present a general approach to build fast multipole methods (FMMs) based on Chebyshev interpolation and the adaptive cross approximation (ACA) for smooth kernels. The Chebyshev interpolation is used to transfer information up and down the levels of the FMM. The scheme is further accelerated by compressing the information stored at Chebyshev interpolation points using ACA and QR decompositions. This leads to a nearly optimal computational cost with a small pre-processing time due to the low computational cost of ACA. This approach is in particular faster than performing singular value decompositions.This does not address the difficulties associated with the oscillatory nature of K. For that purpose, we consider the following modification of the kernel K^u = K(x, y)e^??ku·(x?y), where u is a unit vector (see Brandt [1]). We proved that the kernel K^u can be interpolated efficiently when x ? y lies in a cone of direction u. This result is used to construct an FMM for the kernel K.Theoretical error bounds will be presented to control the error in the computation as well as the computational cost of the method. The paper ends with the presentation of 2D and 3D numerical convergence studies, and computational cost benchmarks.     

Reduction of thin-film ceria on Pt(111) by supported Pd nanoparticles probed with resonant photoemission

Local defects present in CeO_2 ? x films result in a mixture of Ce³⁺ and Ce⁴⁺ oxidation states. Previous studies of the Ce 3d region with XPS have shown that depositing metal nanoparticles on ceria films causes further reduction, with an increase in Ce³⁺ concentration. Here, we compare the use of XPS and resonant photoemission spectroscopy (RESPES) to estimate the concentration of Ce³⁺ and Ce⁴⁺ in CeO_2 ? x films grown on Pt (111), and the variation of this concentration as a function of Pd deposition. Due to the nature of the electronic structure of CeO_2 ? x, resonant peaks are observed for the 4d–4f transitions when the photon energy matches the resonant energy; (hν = 121.0 eV) for Ce³⁺ and (hν = 124.5 eV) for Ce⁴⁺. This results in two discrete resonant photoemission peaks in valence band spectra. The ratio of the difference of these peaks with off-resonance scans gives an indication of the relative contribution of Ce³⁺. Results from RESPES indicate reduction of CeO_2 ? x on deposition of Pd, confirming earlier findings from XPS studies.     

Densely mapping the phase diagram of cuprate superconductors using a spatial composition spread approach

A novel spatial composition spread approach was used successfully to deposit a 52-member library of La_2?xSr_xCuO₄ (0 ? x ? 0.18) using magnetron sputtering combined with physical masking techniques. Two homemade targets of La₂CuO₄ and La_1.82Sr_0.18CuO₄ were sputtered at a power of 41 W RF and 42 W DC, respectively, in a process gas of 15 mTorr argon. The libraries were sputtered onto LaSrAlO₄ (0 0 1), SrTiO₃ (1 0 0) and MgO (1 0 0) substrates through a 52-slot shadow mask for which a ?20 V substrate bias was applied to prevent resputtering. The resulting amorphous films were post-annealed (800 °C for 1 h then at 950 °C for 2 h) in a tube sealed with oxygen gas. Wavelength Dispersive Spectroscopy (WDS) analysis revealed the expected linear variation of Sr content from 0 to 0.18 with an approximate change of 0.003 per library member. Transport measurements revealed superconducting transitions as well as changes in the quasiparticle scattering rate. These transitions and scattering rate changes were mapped to produce the T-hole concentration phase diagram.     

Interface capacitance of nano-patterned electrodes

By employing numerical solutions of the Poisson–Boltzmann equation we have studied the interface capacitance of flat electrodes with stripes of different potentials of zero charge ?_pzc. The results depend on the ratio of the width of the stripes l to the dielectric screening length in the electrolyte, the Debye length d_Debye, as well as on the difference Δ?_pzc in relation k_BT/e. As expected, the capacitance of a striped surface has its minimum at the mean potential of the surface if l/d_Debye << 1 and displays two minima if l/d_Debye >> 1. An unexpected result is that for Δ?_pzc ? 0.2V, the transition between the two extreme cases does not occur when l ? d_Debye, but rather when l > 10d_Debye. As a consequence, a single minimum in the capacitance is observed for dilute electrolytes even for 100 nm wide stripes. The capacitance at the minimum is however higher than for homogeneous surfaces. Furthermore, the potential at the minimum deviates significantly from the potential of zero mean charge on the surface if l > 3d_Debye and Δ?_pzc is larger than about 4k_BT/e. The capacitance of stepped, partially reconstructed Au(11n) surfaces is discussed as an example. Consequences for Parsons–Zobel-plots of the capacitances of inhomogeneous surfaces are likewise discussed.     

Sub-THz continuous wave generation scheme using high-order harmonics modulated lightwave

We propose a sub-THz continuous wave (CW) generation scheme using a high-order harmonics modulated lightwave (HML) to reduce an electronic dependency of a conventional double sideband suppressed carrier (DSB-SC) scheme. The electronic dependency should be overcome to increase frequency tunability of the conventional DSB-SC scheme. This is because the frequency of a local oscillator (LO), f_LO, should be one-half frequency of the frequency of a desired sub-THz CW in the conventional DSB-SC scheme. The proposed scheme is formed by adding an optical feedback loop to the conventional DSB-SC scheme. In order to verify our proposed scheme, a 120 GHz CW is generated using the LO with f_LO = 20 GHz. Based on our experimental results, we have found that the frequency of the LO can be reduced by our proposed scheme up to one-sixth (20 GHz) of 120 GHz. The 120 GHz CW generated by the proposed scheme has 52 dB higher photomixed output power with narrow spectral linewidth than that of the 120 GHz CW generated by the conventional DSB-SC scheme using the LO with f_LO = 20 GHz. Consequently, our proposed scheme can be helpful to reduce the electronic dependency of the conventional DSB-SC scheme.