Discrete Levels Research Articles

An energy-stable and conservative numerical method for multicomponent Maxwell–Stefan model with rock compressibility

Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this paper, taking molar densities as the primary unknowns rather than the pressure and molar fractions, we propose an alternative formulation of multicomponent Maxwell–Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation has a distinct feature that it follows an energy dissipation law, and namely, it is consistent with the second law of thermodynamics. Additionally, the formulation obeys the famous Onsager's reciprocal principle. An efficient energy-stable numerical scheme is constructed using the stabilized energy factorization approach for the Helmholtz free energy density and certain carefully designed formulations involving explicit and implicit mixed treatments for the coupling between molar densities, pressure, and porosity. We rigorously prove that the scheme inherits the energy dissipation law at the discrete level. The fully discrete scheme has the ability to ensure the mass conservation law for each component as well as preserve the Onsager's reciprocal principle. Numerical tests are conducted to verify our theories, and in particular, to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theories.

Energy stable finite element strategy for simulating spreading, sliding and rolling flow dynamics of viscoelastic droplets

Moving mesh finite element method (FEM) for simulating dynamics of viscoelastic droplets on inclined surfaces is proposed. Viscoelasticity is incorporated into the governing system employing the Oldroyd–B constitutive model. The supporting (inclined) surface is allowed to have non–homogeneous properties incorporated into the mathematical model through the generalized Navier boundary conditions (GNBC). The droplet motion (sliding and/or rolling) is handled by employing arbitrary Lagrangian Eulerian (ALE) framework. Energy balance is derived from the governing system and it is a starting point in the derivation of the numerical scheme. The overall numerical strategy is designed in such a way that a counterpart of the (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system which, in turn, guarantees the stability of the scheme in the energy norm. The framework proposed is very general and encapsulates both two and three dimensional scenarios. Numerical studies in this work focus primarily on the three dimensional scenarios since such are significantly more challenging and seem to be much scarcer in the literature. The newly proposed numerical strategy is validated on several examples to confirm the theoretical predictions. The role of viscoelasticity in the overall droplet dynamics is briefly investigated and the behaviors of Newtonian and non–Newtonian droplets are compared.

A fully discrete finite element method for a constrained transport model of the incompressible MHD equations

In this paper, we propose and analyze a fully discrete finite element method for a constrained transport (CT) model of the incompressible magnetohydrodynamic (MHD) equations. The spatial discretization is based on mixed finite elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, the magnetic field and magnetic vector potential are discretized by H(curl)-conforming edge element. The time marching is combining a backward Euler scheme and some subtle implicit-explicit treatments for nonlinear and coupling terms. With these treatments, the fully discrete scheme is linear in the implementation and the computation of the magnetic vector potential is decoupled from the whole coupled system. The most attractive feature of this scheme that it can yield the exactly divergence-free magnetic field and current density on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic field and magnetic vector potential are further demonstrated under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.

Spin-dependent resonant Andreev tunneling in hybrid ferromagnetic metal–magnetic quantum dot–superconductor nanostructures

There exists a variety of theoretical proposals to transform states induced by magnetic nanoparticles inside a superconducting gap into Majorana fermion states. The main challenge in this route is a conclusive proof and undoubted distinguishing between topologically trivial subgap Andreev bound states and topologically nontrivial magnetically polarized Majorana bound states. This motivated us to investigate a nonequilibrium electrons tunneling through a ferromagnetic normal metal–magnetic quantum dot–s-wave superconductor (F-mQD-SC) nanostructure, where the mQD’s discrete levels are spin splitted. By using the Keldysh Green’s function method, the expressions for a tunnel current and probability of the Andreev reflection (AR) versus energy are derived and studied. We find that the system’s resonant ARs conductance exhibits different kinds of peaks depending on a spin splitting of the mQD levels, the spin polarization magnitude of the F-lead current, the gate voltage, and an external magnetic field magnitude. The nanostructure’s conductance versus a bias voltage exhibits extra peaks which at some combination of its parameters can mimic ones expected for Majorana modes in a topological superconducting state. The distinguishing transport characteristics of a F-mQD-SC nanoscale structure being in non-topological state are discussed. We suggest that the results obtained can provide helpful clarification for understanding recent experiments in superconductor–ferromagnet hybrid nanostructures with topologically protected excitations.

Entropy of electron gas in semiconductor nanowires

The thermodynamic potential and entropy of the electron gas of a semiconductor nanowire are obtained. When calculating the entropy of an electron gas with a nonparabolic energy dispersion, the Nelson method was used. The entropy of the electron gas with the value of the chemical potential of the smaller line, which is equal to or greater than the energy of an arbitrary discrete energy level, is studied. An analytical expression for the entropy is obtained in the case when the chemical potential of the electron gas is equal to the energy of the discrete level. The temperature dependences of the entropy of the electron gas of semiconductor nanowires on the degree of nonparabolicity of the energy dispersion law are found.

Characterization of silicification and dissolution zones by integrating borehole image logs and core samples: a case study of a well from the Brazilian pre-salt

Precise knowledge of the spatial distribution patterns of non-matrix porosity zones and the establishment of the geological factors controlling their evolution are crucial for building more accurate carbonate reservoir models and improving hydrocarbon production. The occurrence of intervals affected by significant carbonate dissolution may result in drilling fluid loss and time-consuming drawbacks during well construction. Vug or cave-rich reservoirs may exhibit excess permeability and extremely high initial flow rates. Similar situations have been reported in exploration activities in the Brazilian pre-salt plays, where evidence of dissolution and other burial diagenetic processes, such as severe silicification and dolomitization, are common. In this study, we investigate evidence of major post-depositional changes in the lacustrine carbonate reservoirs of the Barra Velha Formation, which comprises the most prolific hydrocarbon play in Brazil. Using a comprehensive database consisting of both core samples and well-log data from a vertical well in the Santos Basin, we have characterized, at multiple scales, reservoir zones affected by silicification and carbonate dissolution. In addition, we have performed a petrophysical evaluation of the reservoir to understand the impact of such processes on porosity and permeability development. The results suggested an intimate relationship between silicification and dissolution processes, which can be associated with late fluid percolation under a deep burial flow regime. The occurrence of silicified and vuggy beds, associated with specific zones and lithofacies, indicates an important degree of stratigraphic control on fluid percolation and lateral migration. Furthermore, the presence of fractures at discrete stratigraphic levels has preferentially influenced the development of high-permeability zones, including metric-scale fracture-related conduits. This study contributes to the general knowledge of carbonate reservoirs affected by silicification and dissolution while providing support for the recognition of such processes in partially- or non-cored wells.

Experimental investigation of electron-mercury atom collisions: insights into atomic interaction dynamics

This experiment aims to validate the Bohr’s model, which posits the existence of discrete energy levels within Hg and Ne atoms, through the Frank-Hertz experimental methodology. Utilizing two specialized tubes, one containing Neon gas and the other Mercury gas, electrons are accelerated to collide with the gas atoms. As the applied voltage is gradually increased, a distinct min-max pattern emerges on the current-voltage graph, with the equidistant minima and maxima, indicating the quantization of atomic energy levels, a phenomenon predicted by Bohr’s model. The experimentally determined distance between these peaks correlates with the energy required to excite the respective gas atoms to a higher energy state. The measured excitation energies – 16.58±2.36 eV for neon and 4.90±0.12 eV for mercury – align closely with the widely accepted energy levels for these atoms. These findings not only corroborate the quantized nature of atomic properties but also reinforce the validity of Bohr’s atomic model.

A Physical Description of the Variability in Single‐ReRAM Devices and Hardware‐Based Neuronal Networks

Resistive switching devices are candidates to be used as weight for the future hardware realization of neuronal networks. Herein, these resistive switches base on oxygen vacancy migration and can be toggled between different resistance states. Internal processes lead to a variability over time in these states. To understand this, a dynamical model is developed, which links the internal physical mechanism to the current response. Thereby, the electronic current is calculated by tunneling processes over oxygen vacancies. The model allows to understand the physical transport properties and the current–voltage dependence. Integrating ionic hopping processes clarifies the variability in the electronic conduction. The derived model does not require empirical input parameters and only depends on the properties of the materials. This allows to use literature values and independently compare them to electrical measurements. The model predicts the experiments correctly. Thereupon, the physical properties of the conduction are investigated with respect to the transport over discrete energy levels, which change under the hopping of oxygen vacancies. Afterwards, the current level and variability are simulated for increasing oxygen vacancy concentrations to investigate analog switching, which is required for neuronal network layers. Finally, the variability is tested with multiple interacting cells in a neuronal layer.

Model-free idealization: Adaptive integrated approach for idealization of ion-channel currents

Single-channel electrophysiological recordings provide insights into transmembrane ion permeation and channel gating mechanisms. The first step in the analysis of the recorded currents involves an "idealization" process, in which noisy raw data are classified into two discrete levels corresponding to the open and closed states of channels. This provides valuable information on the gating kinetics of ion channels. However, the idealization step is often challenging in cases of currents with poor signal-to-noise ratios and baseline drifts, especially when the gating model of the target channel is not identified. We report herein on a highly robust model-free idealization method for achieving this goal. The algorithm, called adaptive integrated approach for idealization of ion-channel currents (AI2), is composed of Kalman filter and Gaussian mixture model clustering and functions without user input. AI2 automatically determines the noise reduction setting based on the degree of separation between the open and closed levels. We validated the method on pseudo-channel-current datasets that contain either computed or experimentally recorded noise. We also investigated the relationship between the noise reduction parameter of the Kalman filter and the cutoff frequency of the low-pass filter. The AI2 algorithm was then tested on actual experimental data for biological channels including gramicidin A, a voltage-gated sodium channel, and other unidentified channels. We compared the idealization results with those obtained by the conventional methods, including the 50%-threshold-crossing method.

ATP-Dependent Mismatch Recognition in DNA Replication Mismatch Repair

Mismatch repair is a critical step in DNA replication that occurs after base selection and proofreading, significantly increasing fidelity. However, the mechanism of mismatch recognition has not been established for any repair enzyme. Speculations in this area mainly focus on exploiting thermodynamic equilibrium and free energy. Nevertheless, non-equilibrium processes may play a more significant role in enhancing mismatch recognition accuracy by utilizing adenosine triphosphate (ATP). This study aimed to investigate this possibility. Considering our limited knowledge of actual mismatch repair enzymes, we proposed a hypothetical enzyme that operates as a quantum system with three discrete energy levels. When the enzyme is raised to its highest energy level, a quantum transition occurs, leading to one of two low-energy levels representing potential recognition outcomes: a correct match or a mismatch. The probabilities of the two outcomes are exponentially different, determined by the energy gap between the two low energy levels. By flipping the energy gap, discrimination between mismatches and correct matches can be achieved. Within a framework that combines quantum mechanics with thermodynamics, we established a relationship between energy cost and the recognition error.

Energy-preserving splitting finite element method for nonlinear stochastic space-fractional wave equations with multiplicative noise

It is well-known that numerical methods that preserve physical quantities of original system often yield physically reasonable results and possess better numerical stability in long-time simulations. In this paper, we propose a methodology of constructing efficient fully-discrete method preserving averaged energy evolution law for nonlinear stochastic space-fractional wave equations with multiplicative noise. Specifically, the space variable is firstly discretized by Galerkin finite element method, and then the splitting technique and averaged vector field method are employed for the time integration of the resulting spatial semi-discrete scheme. We prove that the proposed fully-discrete method preserves the averaged energy evolution law in the discrete level. Numerical experiments are given to verify the theoretical result.

An efficient energy conserving semi-Lagrangian kinetic scheme for the Vlasov-Ampère system

In this paper, we present a novel kinetic scheme termed the Energy Conserving Semi-Lagrangian (ECSL) for the Vlasov-Ampère system. The novelty of the ECSL is that it retains the efficiency of the explicit scheme while simultaneously maintaining energy conservation and unconditional stability properties of the implicit scheme without relying on nonlinear iteration. The proposed ECSL method includes two main ingredients: the conservative Semi-Lagrangian (CSL) scheme and a novel field solver. The CSL scheme is utilized for the phase space discretization of the Vlasov equation, which enables the scheme to conserve mass exactly and remove the Courant-Friedrichs-Lewy restriction. The novel field solver is proposed by coupling the Ampère equation and the moments of the Vlasov equation in a semi-implicit way, allowing for an explicit and efficient calculation of the electric field. The combination of the novel field solver and CSL ensures that ECSL scheme conserves total energy and mass on the fully discrete level, regardless of spatial resolution and time step size. Moreover, the ECSL scheme still provides reliable solution even when its spatial and temporal resolution are insufficient to fully resolve the Debye length and plasma period, making it a promising tool for multiscale and lengthy simulations. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and conservation properties of the proposed method.

High-order supplementary variable methods for thermodynamically consistent partial differential equations

In recent years, there has been growing interest in developing numerical methods to retain consistency between the partial differential equation system and its deduced equations (e.g., the thermodynamically consistent model and its energy law) after numerical approximations. A supplementary variable method (SVM) of second-order was recently introduced in [1,2] to preserve thermodynamic consistency for thermodynamically consistent models. In this paper, we present a new class of high-order supplementary variable methods. These methods consist of two parts: firstly, the original model is discretized using a high-order prediction method, and secondly, the SVM reformulated system is approximated by a high-order correction method. The newly developed schemes are rigorously proven to preserve high-order accuracy with increased stability through the use of a stabilizer. Additionally, the schemes inherently maintain thermodynamic consistency for thermodynamically consistent models, retaining the original energy law at the discrete level. To assess the high-order supplementary variable methods, we apply them to two thermodynamically consistent partial differential equation systems: the functionalized Cahn–Hilliard model and the ternary Cahn–Hilliard model. We then provide several benchmark examples to demonstrate the accuracy, stability, and efficiency of the newly developed high-order numerical methods.

How to Cross an Energy Barrier at Zero Kelvin without Tunneling Effect.

This Letter deals with the broad class of magnetic systems having a single or collective spin S with an energy barrier, such as rare-earth elements and their compounds, single molecule magnets with uniaxial anisotropy, and more generally any other anisotropic quantum system made of single or multiple objects with discrete energy levels. Till now, the reversal of the magnetization of such systems at zero kelvin required making use of quantum tunneling with a significant transverse field or transverse anisotropy term, at resonance. Here, we show that another very simple method exists. It simply consists in the application of a particular sequence of electromagnetic radiations in the ranges of optical or microwave frequencies, depending on the characteristics of the system (spin and anisotropy values for magnetic systems). This produces oscillations of the Rabi type that pass above the barrier, thus extending these oscillations between the two energy wells with mixtures of all the 2S+1 states. In addition to its basic character, this approach opens up new directions of research in quantum information with possible breakthroughs in the current use of multiple quantum bits.

Damage detection in ship hull structures under operational variability through strain sensing

Recently, growing concern for the reliability of ship structures has emerged among stakeholders in the maritime sector. In the transition towards predictive maintenance, Structural Health Monitoring (SHM) has emerged as a viable option. SHM systems seek to infer the existence or level of structural degradation using response data from in-situ sensors. In this work, damage detection in marine stiffened panels is treated. Out-of-plane deflections of the panel’s plate elements act as the damage case of interest, due to being potential triggers for buckling collapse. Strain response data are employed as the damage-sensitive features. Two different approaches are employed for damage detection. In the first, the problem is cast as one of multi-class classification, where the different classes correspond to discrete deflection levels, and a detection theory-based classifier is employed. In the second, the problem is treated as one of probabilistic regression, where strain readings are mapped to the probability distribution of the deflection magnitude. Both methods are implemented on a realistic stiffened panel geometry subjected to operational variability, based on an existing vessel, while all the employed data is obtained through Finite Element (FE) simulations. Results indicate that the proposed architecture can effectively provide uncertainty-informed predictions of out-of-plane deflection levels.

An energy stable discontinuous Galerkin time-domain finite element method in optics and photonics

In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell’s equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with Qk-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy.

Enhanced Femtosecond Nonlinearities and Multiphoton Absorptions in Discrete Bands of Porphyrins.

Optical nonlinearities of discrete absorption energy levels of one of the typical heterocyclic aromatic molecules, free-base porphyrins, have been probed over a broad spectral region (400-1600 nm) utilizing intense femtosecond pulses. A wide range of strong one- and multiphoton-induced nonlinear absorptions of both the blue-end Soret (B) band (au → b1g) and red-end orbital mixing split quasi-allowed Q-bands (Qx(0,0; 0,1), Qy(0,0; 0,1), au → eg) are critically probed and reported. During the resonant excitation within B- (400 nm) and Q-bands (600-750 nm), the nonlinear absorption has become predominant by the saturation of absorption (SA) of the one-photon absorption (1PA) process due to ground-state bleaching. At nonresonant wavelengths, it is dominated by the reverse saturation of absorption (RSA), involving various nonlinear processes of two-, three-, and four-photon (2PA, 3PA, and 4PA) absorptions, either to B- or Q-bands (1100-1600 nm). The laser intensity-dependent nonresonant (2PA, 800 nm) excitations for the prominent B-band show a distinct cross-over from SA to RSA, contributed by the excited-state absorption (ESA) utilizing a three-photon induced (3PA) process, whereas resonant (1PA, 400 nm) excitation reveals a systematic strong SA process. Both wavelength- and intensity-dependent nonlinear refractive index studies exhibit positive electronic Kerr-based self-focusing effects, with prominent contributions of nonlinear absorption and higher-order effects. The spectrally discrete, highly intense laser probing of individual energy bands and the consequent variety of nonlinearities can be broadly generalized for many free-base porphyrins and metalloporphyrins. The present studies provide a strong foundation and new insight into the broad categories of macrocycles, such as porphyrins and phthalocyanines, for myriad applications in nonlinear optics and bio/optophotonics.

Revenue implications of choosing discrete bid levels in a Japanese–English auction

We consider the set-up of a Japanese–English auction with exogenously fixed discrete bid levels for a specific game (the wallet game with two bidders, following Gonçalves and Ray in Econ Lett 159:177–179, 2017). We show that in this auction, partition equilibria exist that may be separating or pooling. We illustrate separating and pooling equilibria in games with two and three discrete bid levels and compare the revenues of the seller from these equilibria to find the optimal bid levels for these cases.

Three-Phase Motor Inverter and Current Sensing GaN Power IC.

A three-phase GaN-based motor inverter IC with three integrated phase current mirror sensors (sense-FETs or sense-HEMTs, 1200:1 ratio), a temperature sensor, and an amplifier is presented and experimentally operated. The three low-side currents are read out by virtual grounding transimpedance amplifiers. A modified summed DC current readout circuit using only one amplifier is also discussed. During continuous 24 V motor operation with space-vector pulse width modulation (SVPWM), the sensor signal is measured and a bidirectional measurement capability is verified. The measured risetime of the sensor signal is 51 ns, indicating around 7 MHz bandwidth (without intentional optimization for high bandwidth). The IC is operated up to 32 V on DC-biased semi-floating substrate to limit negative static back-gating of the high-side transistors to around -7% of the DC-link voltage. Analysis of the capacitive coupling from the three switch-nodes to the substrate is calculated for SVPWM based on capacitance measurement, resulting in four discrete semi-floating substrate voltage levels, which is experimentally verified. Integrated advanced power converter topologies with sensors improve the power density of power electronics applications, such as for low-voltage motor drive.

Contact and control: Engagement and influence among women of color state agency heads

AbstractWhile scholarship is delving into the representation of women and people of color within the U.S. bureaucracy, less is known about these individuals' discretion within their roles. Using an intersectionality framework, this article explores the experiences of women of color in leadership roles within state government agencies, their communication with elected superiors, and perceptions of their own policy discretion and communication compared with White men and women, and men of color. Leveraging the American State Administrators Project, we find that women of color have less communication with elected officials and their staff but also perceive that they have less discretionary control over their departments. This article highlights that while women of color are increasing in leadership roles within state bureaucracies, disparities exist between women of color and their colleagues. We call attention to the disproportionate levels of oversight and discretion perceived by women of color.