Equivalent Gradient Research Articles

Influence of vibrational and chemical non-equilibrium on the velocity-gradient and the pressure-Hessian fields in compressible turbulence

The influence of vibrational and chemical non-equilibrium on the dynamics of velocity gradients and pressure-Hessian tensors is investigated in this study. Such non-equilibrium flows typically occur in high-speed compressible turbulence at elevated temperatures, as observed in reentry vehicles or hypersonic flights. In the first part of the study, we derive the exact evolution equations for the velocity gradients and pressure-Hessian tensors under vibrational and chemical non-equilibrium conditions. We identify the inertial, vibrational, and chemical mechanisms in this evolution equation. In the second part of this study, we focus on assessing the relative importance of these mechanisms across different simulation cases. In this work, we present direct numerical simulations of isotropic decaying turbulence, which consider both vibrational and chemical non-equilibrium effects. It is found that in the presence of chemical and vibrational non-equilibrium, (i) vibrational relaxation processes are expedited in terms of mean and fluctuating flow fields. (ii) Vortical fluctuations increase while dilatational fluctuations are suppressed. (iii) The relative strength of the pressure-Hessian tensor compared to the velocity gradient tensor is reduced. (iv) The explicit effects of vibrational mechanisms on the pressure-Hessian evolution equation are diminished, whereas chemical mechanisms substantially influence the dynamics compared to inertial mechanisms throughout the turbulence decay process. This study highlights the need for robust turbulence closure models for chemical mechanisms to accurately capture the effects of thermal non-equilibrium on the dynamics of velocity gradients in compressible flows.

Response of Soil Detachment Capacity to Hydrodynamic Characteristics Under Different Slope Gradients

The mechanism of soil detachment on steep slopes is obviously different from that on gentle slopes. However, the slope effect of soil detachment remains unclear. The objective of this study was to quantify the slope effect of soil detachment capacity at the varying hydrodynamic characteristics. In this study, the soil detachment capacity (Dc) on clay loam and hydrodynamic characteristics were measured by conducting the runoff scouring experiments at 10 slope gradients (1.7–57.7%) and 5 unit flow discharges (0.022–0.089 m2·min−1). The results showed that the relationships between Dc and hydrodynamic parameters were affected by slope gradient. Based on the optimal functional relationship, the hydrodynamic characteristics (flow velocity, flow shear stress, stream power, unit stream power, and unit energy) calculated by maximum and minimum Dc in this study changed by 19.91–95138.10%, and the Dc calculated by the maximum and minimum hydrodynamic characteristics could differ by up to nine orders of magnitude. Overall, the power function of hydrodynamic parameters was superior to the linear function in different slope gradients. The stream power was the best predictor for Dc compared with other hydrodynamic parameters. For all combinations of slope gradients, the adjusted coefficient of determination (Adj. R2) of the power relationship between Dc and stream power was 9.41–27.40% higher than it was between Dc and other hydrodynamic parameters. The coefficient and index of power function for different hydrodynamic parameters showed a trend change with increasing slope gradient, indicating that there was a slope effect on Dc. Further analysis found that Dc could be well predicted using a power combination equation of slope gradient, flow velocity, and flow depth (Adj. R2 = 0.96). This study helps to better understand the mechanism of soil detachment and emphasizes that the slope effect should be considered when establishing a soil detachment equation.

Application of leak-off test data in formation fracture gradient correlation for Niger Delta Basin natural gas wells development

Offshore drilling of gas wells in the Niger Delta is becoming very expensive and requires highly skilled manpower. Therefore, it is essential to accurately update the fracture pressure data, maintain adequate measurement while drilling, and ensure acceptable prediction with conventional empirical correlations while drilling operations are ongoing to avoid non-productive time (NPT). To effectively drill an offshore well and deliver it safely while protecting personnel, the environment, and equipment, drillers must ensure that the wellbore pressure is sufficient to balance the pore pressure without exceeding the fracture pressure anywhere along the open section of the well. Calculating and predicting the fracture pressure is critical for reducing NPT in offshore drilling and ensuring the safety of workers and goods on-site. There are numerous relationships for predicting the fracture pressures. However, they are primarily limited to onshore and shallow water fields, and as wells are drilled deeper in the Offshore Niger Delta field, a correlation (equation) based on offshore leak-off test (LOT) data is required to correctly estimate the crack pressure. Thus, this study employs a mathematical modelling technique to create an improved offshore fracture gradient equation from leak-off test data tailored to the needs of the offshore Niger Delta. The generated correlation (equation) was statistically examined, yielding a reliable coefficient of determination. It was tested again with the field cases, and the results were similar to the LOT results.

The Spatial Distribution of Lipophilic Cations in Gradient Copolymers Regulates Polymer-pDNA Complexation, Polyplex Aggregation, and Intracellular pDNA Delivery.

Here, we demonstrate that the spatial distribution of lipophilic cations governs the complexation pathways, serum stability, and biological performance of polymer-pDNA complexes (polyplexes). Previous research focused on block/statistical copolymers, whereas gradient copolymers, where the density of lipophilic cations diminishes (gradually or steeply) along polymer backbones, remain underexplored. We engineered gradient copolymers that combine the polyplex colloidal stability of block copolymers with the transfection efficiency of statistical copolymers. We synthesized length- and compositionally equivalent gradient copolymers (G1-G3) along with statistical (S) and block (B) copolymers of 2-(diisopropylamino)ethyl methacrylate and 2-hydroxyethyl methacrylate. We mapped how polymer microstructure governs pDNA loading per polyplex, pDNA conformational changes, and polymer-pDNA binding thermodynamics via static light scattering, circular dichroism spectroscopy, and isothermal titration calorimetry, respectively. While gradient steepness is a powerful design handle to improve polyplex physical properties, augment pDNA delivery capacity, and attenuate polycation-triggered hemolysis, microstructural contrasts did not elicit differences in complement activation.

Combination of two-fluid model and delayed equilibrium model for the critical flow in a slit

Accurately predicting the mass flux, pressure profile, and velocity profile of the critical flow in a slit is essential for analyzing the breaking process of the liquid phase and calculating the aerosol source term for leak-before-break (LBB) monitoring and Loss of Coolant Accident (LOCA) risk analysis. A new critical flow model combining Two-fluid Model (TFM) and Delayed Equilibrium Model (DEM) is built to get accurate profiles while avoiding the same phase velocity in DEM and the arbitrary critical flow criterion in TFM. The new model is verified using past experiments of the critical flow in a slit. It proves to be accurate in mass flux but not in critical pressure, with maximum relative errors of around 25% in mass flux and around 80% in critical pressure. The new model is optimized for higher accuracy in critical pressure. The empirical equation of saturated phase mass flow rate fraction gradient is optimized by conducting approximate pressure profile calculation and regression analysis. The maximum relative error decreases little while the ratio of critical pressure relative errors lying in the range of ±40% increases after optimization. In contrast, the difference in the average abstract relative error of pressure between original TFM-DEM and DEM is much larger, for the maximum relative error of mass flux and critical pressure are around 25% and 110%. The comparison between the original and optimized TFM-DEM proves that the new critical flow model is accurate in mass flux and can be optimized to raise pressure calculation accuracy. The comparison between the original TFM-DEM and DEM proves that the phase velocity difference is the major source of accuracy improvement in the pressure profile.

STCSNN: High energy efficiency spike-train level spiking neural networks with spatio-temporal conversion

Brain-inspired spiking neuron networks (SNNs) have attracted widespread research interest due to their low power features, high biological plausibility, and strong spatiotemporal information processing capability. Although adopting a surrogate gradient (SG) makes the non-differentiability SNN trainable, achieving comparable accuracy for ANNs and keeping low-power features simultaneously is still tricky. In this paper, we proposed an energy-efficient spike-train level spiking neural network with spatio-temporal conversion, which has low computational cost and high accuracy. In the STCSNN, spatio-temporal conversion blocks (STCBs) are proposed to keep the low power features of SNNs and improve accuracy. However, STCSNN cannot adopt backpropagation algorithms directly due to the non-differentiability nature of spike trains. We proposed a suitable learning rule for STCSNNs by deducing the equivalent gradient of STCB. We evaluate the proposed STCSNN on static and neuromorphic datasets, including Fashion-Mnist, Cifar10, Cifar100, TinyImageNet, and DVS-Cifar10. The experiment results show that our proposed STCSNN outperforms the state-of-the-art accuracy on nearly all datasets, using fewer time steps and being highly energy-efficient.

Understanding the effect of Prandtl number on momentum and scalar mixing rates in neutral and stably stratified flows using gradient field dynamics

Recently, direct numerical simulations (DNS) of stably stratified turbulence have shown that as the Prandtl number ( $Pr$ ) is increased from 1 to 7, the mean turbulent potential energy dissipation rate (TPE-DR) drops dramatically, while the mean turbulent kinetic energy dissipation rate (TKE-DR) increases significantly. Through an analysis of the equations governing the fluctuating velocity and density gradients we provide a mechanistic explanation for this surprising behaviour and test the predictions using DNS. We show that the mean density gradient gives rise to a mechanism that opposes the production of fluctuating density gradients, and this is connected to the emergence of ramp cliffs. The same term appears in the velocity gradient equation but with the opposite sign, and is the contribution from buoyancy. This term is ultimately the reason why the TPE-DR reduces while the TKE-DR increases with increasing $Pr$ . Our analysis also predicts that the effects of buoyancy on the smallest scales of the flow become stronger as $Pr$ is increased, and this is confirmed by our DNS data. A consequence of this is that the standard buoyancy Reynolds number does not correctly estimate the impact of buoyancy at the smallest scales when $Pr$ deviates from 1, and we derive a suitable alternative parameter. Finally, an analysis of the filtered gradient equations reveals that the mean density gradient term changes sign at sufficiently large scales, such that buoyancy acts as a source for velocity gradients at small scales, but as a sink at large scales.

A geometric integration approach to smooth optimization: foundations of the discrete gradient method

Abstract Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimization that provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $\text{O}(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak–Łojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients—the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh–Abe discrete gradient—as well as for a randomised Itoh–Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.

Comparative analysis of viscous dissipation effects on Prandtl fluid including contraction and relaxation phenomena

AbstractThis paper investigates the effects of magnetohydrodynamics and heat transfer on the peristaltic transport of Prandtl fluid in a vertical endoscopic tube. The reduction of the complexity of the equations governing the flow of Prandtl fluid entails the use of long wavelength and low Reynolds number approximations. These complex equations for the pressure gradient and velocity profile are handled analytically using the perturbation technique with convective boundary conditions, and the temperature and concentration profiles are carefully solved for the exact solution. The frictional forces and pressure rise are also simulated with numerical integration. The resulting formulas for velocity, temperature, concentration, pressure rise, and pressure gradient are graphed using the MATLAB and MATHMATICA software, and the effects of all the different physical parameters are investigated and assessed. The streamlines with five distinct wave types are sketched at the conclusion to show the phenomenon of trapping. It is investigated that the velocity profile rises due to buoyancy forces and falls due to the influence of magnetic forces.

Structural seismic responses prediction using the gradient-enhanced hybrid PINN

To rapidly and effectively assess the bridge seismic-resistant capability, it is essential to conduct efficient predictions of bridge seismic responses. Recently, physics informed neural network (PINN) has made great progress and utilized to solve differential equations in different fields. However, how to increase its accuracy and efficiency still remains an open challenge. In this work, a novel gradient-enhanced Fourth-Order Runge-Kutta PINN (gRK4-PINN), as a powerful hybrid PINN, is utilized to achieve this goal. As for gRK4-PINN, the physical information is not simply embedded into the loss function; instead, the RK4 method and the physical model is intricately integrated with the neural network. In addition, to improve the predictive performance, additional gradient equation is directly embedded in loss function. A large-span continuous girder high speed railway (CGHSR) bridge is adopted as numerical experiment to validate the fidelity of the proposed method. Results reveal that the Mean Absolute Error (MAE) of the predicting seismic responses is relatively small, whose value is below 0.014 in most of the time. These small MAE values indicate that the proposed gRK4-PINN performs well in predicting the seismic responses of the CGHSR bridge.

Higher-order multi-scale computational approach and its convergence for nonlocal gradient elasticity problems of composite materials

A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials.

Exact solutions of the Navier-Stokes equations in the Boussinesq approximation for describing the flow of binary fluids

An exact solution is proposed for describing creeping convective flows of binary fluids. The velocity field is quadratically nonlinear in two spatial coordinates (horizontal or longitudinal). The coefficients of the forms depend on the third coordinate (vertical or transverse) and time. The pressure field, the concentration field, and the mass force field are nonlinear forms of the fourth order. Linear differential equations of the heat conductivity type and gradient equations for describing hydrodynamic fields are given.

Lithostratigraphic Interpretation, Geotemperature Analysis, and Hydrocarbon Windows Determination in Seloken Field, Chad Basin, Northeastern Nigeria

The aim of this study is to interpret the lithostratigraphic units, analyze geotemperature and determine the hydrocarbon windows in the Seloken Field. Gamma-ray logs and formation tops were used to interpret the lithostratigraphic units. Corrected bottom hole temperature (BHT) data from the wells were used to calculate the geothermal gradients. A formular was derived from the geothermal gradient equation to analyze the geotemperature. Another formular was derived from the geotemperature equation to obtain the hydrocarbon windows. Five major lithostratigraphic units were interpreted. They are: Bima and Gongila Formations, Fika Shale, Kerri-Kerri and Chad Formations…The calculated geothermal gradients range from 3.4oC/100m to 4.0oC/100m, with an average of 3.65oC/100m and the standard deviation of 0.01oC/100m. The temperature of shale units of the formations at various depths calculated from the geotemperature equation range from 30.99oC to 202.45oC. This shows that some of the shale units are thermally mature to cook petroleum The oil window calculated from the derived geomathematical equation is between 1178.1m and 2457.95m, while the gas window is from 2457.95m to 5424.7m. The boundary between the two windows is at 2457.95m. The results above were used to generate a hypothetical model for the hydrocarbon windows. The model indicates that the shale units are within the diagenesis and catagenesis phases. This work provides mathematical methods of source rock thermal maturity evaluation to complement geochemical methods. The outcome is a very important tool, which can be applied to other fields and sedimentary basins in hydrocarbon exploration.

Constrained nonlinear amplitude-variation-with-offset inversion for reservoir dynamic changes estimation from time-lapse seismic data

We develop a novel elastic amplitude-variation-with-offset (AVO) inversion process to estimate the fluid saturation and effective pressure or variations in these properties from time-lapse seismic data sets. These changes occur in oil and gas reservoirs caused by fluid injection or hydrocarbon production that leads to changes in the elastic wave properties, reflectivity, and seismic response. Our method is based on a seismic forward model that consists of a linearized AVO equation and a rock-physics model. The AVO equation links the elastic wave properties to seismic reflection amplitudes, whereas the rock-physics model maps the saturation and pressure into seismic velocities and density. The inversion approach relies on the gradient descent technique to estimate the unknown variables by searching for the minimum of the least-squares misfit between the observed and modeled data. The first-order gradient equations of the least-squares data misfit function with respect to effective pressure and water saturation are derived by using the adjoint-state method and the chain rule. The optimization method used to minimize the misfit function and obtain the best optimal solution is a limited-memory quasi-Newton algorithm. This inversion process allows us to incorporate prior constraints by using the logistic function to map the model variables to a bounded range. To achieve a stable solution to ill-posed inversion problems, optimal regularization weights are applied. The application of the developed workflow on 1D synthetic and real well-log data from the Edvard Grieg oil field simulating various saturation-pressure conditions during production demonstrates the validity of the approach with different noise levels. The inversion is then applied to a 2D synthetic data set modeled from the reservoir model of the Smeaheia field, a potential site for a large-scale offshore CO2 storage field located in the North Sea. Our results illustrate that our inversion method efficiently and accurately estimates reservoir saturation and pressure variations.

Electrical performance estimation and comparative study of heterojunction strained and conventional gate all around nanosheet field effect transistors

Abstract In this paper, we propose a novel type of Gate All Around Nanosheet Field Effect Transistor (GAA NS FET) that incorporates source heterojunctions and strained channels and substrate. We compare its electrical characteristics with those of the Heterojunction Gate All Around Nanosheet Field Effect Transistor (Heterojunction GAA NS FET) and the Conventional Gate All Around Nanosheet Field Effect Transistor (Conventional GAA NS FET). We investigate the impact of electrostatic control on both DC and analog parameters such as gate capacitance (C gg), transconductance g m, and cut-off frequency (f T) for all three device types. In our Proposed GAA NS FET, we employ Germanium for the source and substrate regions, Silicon/Germanium/Silicon (Si/Ge/Si) for the channel, and Silicon for the drain region. The introduction of strain into the nanosheet and the use of a heterojunction structure significantly enhance device performance. Before utilizing a model to analyze a semiconductor device, it is crucial to accurately determine and elaborate on the model parameters. In this case, we solve the Density Gradient (DG) equation self-consistently to obtain the electrostatic potential for a given electron Fermi-level distribution, use the Shockley-Read-Hall (SRH) equation to estimate carrier generation, account for bandgap narrowing in transport behavior, and consider auger recombination. Our general results indicate a notable improvement in drain current, transconductance, and unity-gain frequency by approximately 42%, 53%, and 31%, respectively. This enhancement results in superior RF performance for the Proposed GAA NS FET compared to both the heterojunction GAA NS FET and the conventional GAA NS FET.

Stochastic characterisation of unstable lean hydrogen–air annular premixed flames

This study is devoted to the characterisation of lean premixed hydrogen–air laminar flames stabilised on an annular burner. The flames studied are chosen to lie close to the lean flammability limit, spanning a range of Damköhler and Reynolds numbers, and correspond to situations where instabilities arise. The experimental characterisation is achieved by examining the statistical moments of the flame hydroxyl radical (OH*) chemiluminescence and schlieren. The standard deviation and the skewness, that are needed to fully characterise the measured probability density functions, are shown to exhibit a non-monotonic behaviour with the distance from the burner surface, first increasing, then decreasing downstream the flame tip. An exponentially modified Gaussian distribution model is proposed to describe the skewed schlieren fluctuations probability density function. This model is closed by developing relations for the probability density function parameters based on the Damköhler and Reynolds numbers, and then used to describe the OH* stochastic behaviour. The relation between the distribution parameters and the transport equation of a reactive scalar gradient is addressed also.

Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation

Abstract Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p ∈ L 2 ⁢ ( Ω ) {p\in L^{2}(\Omega)} . The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.

Real-Time Belt Deviation Detection Method Based on Depth Edge Feature and Gradient Constraint.

Aiming at the problems of the poor recognition effect and low recognition rate of the existing methods in the process of belt deviation detection, this paper proposes a real-time belt deviation detection method. Firstly, ResNet18 combined with the attention mechanism module is used as a feature extraction network to enhance the features in the belt edge region and suppress the features in other regions. Then, the extracted features are used to predict the approximate locations of the belt edges using a classifier based on the contextual information on the fully connected layer. Next, the improved gradient equation is used as a structural loss in the model training stage to make the model prediction value closer to the target value. Then, the authors of this paper use the least squares method to fit the set of detected belt edge line points to obtain the accurate belt edge straight line. Finally, the deviation threshold is set according to the requirements of the safety production code, and the fitting results are compared with the threshold to achieve the belt deviation detection. Comparisons are made with four other methods: ultrafast structure-aware deep lane detection, end-to-end wireframe parsing, LSD, and the Hough transform. The results show that the proposed method is the fastest at 41 frames/sec; the accuracy is improved by 0.4%, 13.9%, 45.9%, and 78.8% compared to the other four methods; and the F1-score index is improved by 0.3%, 10.2%, 32.6%, and 72%, respectively, which meets the requirements of practical engineering applications. The proposed method can be used for intelligent monitoring and control in coal mines, logistics and transport industries, and other scenarios requiring belt transport.

Liouville Theorems for infinity Laplacian with gradient and KPP type equation

Liouville Theorems for infinity Laplacian with gradient and KPP type equation

Non-homogeneous Riemannian gradient equations for sum of squares of Bures–Wasserstein metric

We investigate Riemannian gradient equations on the open convex cone Pm of all m×m positive definite Hermitian matrices, for the function WA(X)=∑j=1nwjdW2(X,Aj)where dW denotes the Bures–Wasserstein metric. On the special case where the gradient of the weighted sum of squares of the Bures–Wasserstein metrics vanishes, its unique solution is known as the Wasserstein mean of A1,…,An. We discuss the existence and uniqueness of solutions for non-homogeneous Riemannian gradient equations at which vector fields are constant and congruence transformations. Furthermore, we establish some bounds for these solutions.