Gradient Fractionation Research Articles

Inversion in binary gas mixtures in rarefied flow conditions: Direct simulation Monte Carlo solution and comparison with the analytical solutions at free molecular regime

This paper analyzes the mixing of gases in a plane channel at rarefied conditions. The direct simulation Monte Carlo method is employed to simulate gas mixing in parallel mixers working at different Knudsen numbers and having different values of wall accommodation coefficient. Results show that the normal-to-wall component of the mole fraction gradient may have the same sign as the corresponding component of the diffusive mass flux vector near the diffuse solid walls in contrast to the predictions of Fick's law for continuum conditions. This non-continuum behavior, which is called “inversion” in the present study, will become more pronounced at higher Knudsen numbers, whereas it will become less evident for smaller wall accommodation coefficients. To confirm that the observed phenomenon is consistent with the basic physical laws governing the rarefied gas dynamics and it is not an artifact of the numerical method, a new analytical model based on the kinetic theory of gases is developed for the parallel mixers that have diffuse walls and are working in the free-molecular regime. Excellent agreement is observed between the analytical and direct simulation Monte Carlo results in the free molecular flow regime. Both methods predict the occurrence of inversion near the diffuse walls at highly rarefied flow conditions.

Mass transfer of the multicomponent free jet and one numerical implementation on a graphic processing unit platform

Jet flows are ubiquitous in nature and laboratories. For example, helium and argon jets into ambient air are commonly used to produce atmospheric plasma jets. The jets before and after discharge both are typical multicomponent flows. This study reports on the implementation of a numerical solver based on the graphic processing units to model multicomponent mixture gas flows. To this end, two- and three-dimensional Jacobian matrices for the multicomponent Euler equations were obtained to reconstruct numerical fluxes and build a total variation diminishing scheme. The two-dimensional early-stage mass transfer of gas jets was studied for three mixture gases: He/air, Ar/air, and N2/air to numerically verify the steady functioning of the numerical solver despite the high mass fraction gradients. The mass fraction distribution of helium in ambient air exhibited more complex substructures than those of argon and nitrogen in ambient air. Using the watershed method, self-generated bullets were extracted from the mass fraction distributions of the helium jet. The proposed numerical solver implemented in this study is expected to provide a better understanding of gas jet dynamics, and the Jacobian matrices can provide a mathematical foundation for constructing potential higher schemes.

Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity

Abstract We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.

Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groups

We establish a novel definition of the fractional gradient and divergence of order $$\alpha \in (0, 1)$$ through the use of Riesz potential on homogeneous Carnot groups. We introduce and investigate the distributional fractional Sobolev space and the space of fractional BV functions in this context. Additionally, we provide a definition of fractional Caccioppoli sets on homogeneous Carnot groups and demonstrate their blow-up property, using similar methods as outlined in [13].

Fault diagnosis of rolling bearing based on BP neural network with fractional order gradient descent

The health of rolling bearing is of great importance for the normal operation of rotating machinery. The fault diagnosis process can be roughly summarized as signal processing, feature extraction, and fault classification. In this paper, a novel feature extraction and fault diagnosis method with fractional order back-propagation neural network is put forward. The new sine cosine algorithm optimized variational mode decomposition is performed on vibration signals, and the fault feature vectors are selected and built by singular value decomposition. Inspired by the fractional order calculus, a fractional order back-propagation neural network is employed to realize fault classification. The capability of the developed fault diagnosis algorithm is comprehensively evaluated via benchmark bearing data. The experimental results demonstrate that the designed method substantially extracts bearing defect features, increases classification accuracy and efficiency, and also outperforms existing algorithms.

Fractional gradient descent algorithm for switching models using self-organizing maps: One set data or all the collected data

This paper proposes two fractional gradient descent algorithms for switching models. Each submodel is assigned a weight which can determine the identity of the submodel in each sampling instant. By using the fractional gradient descent algorithms, the parameters of each submodel can be obtained, and then the weights of all the submodels can be estimated based on the self-organizing maps method. These two algorithms can deal with different kinds of switching models on a case by case basis. In addition, compared with the traditional identification algorithms, the proposed methods have two advantages: (1) has faster convergence rates; (2) has less computational efforts. Simulation example demonstrates the effectiveness of the proposed methods.

The characteristics of flow patterns in the shell-and-plate heat exchanger

The flow path of the fluid in the passage of the shell-and-plate heat exchanger shows the characteristics of first dispersing and then gathering, which significantly affects the efficiency of the heat exchanger. Compared with the traditional rectangular plate heat exchanger (PHE), an in-depth understanding of the flow characteristics in SPHE is insufficient. The properties of the flows between the adjacent plates were numerically studied in the present study. The single-phase fluid passes rapidly between the staggered contact points, and there is noticeable pressure fluctuation at the entrance and exit. In addition, the flow diffusion and convergence cause noticeable differences in local velocity and pressure variation. For two-phase flows, three flow patterns are defined, and a subregional discussion on the flow pattern in the passage and their pressure gradient and void fraction are also provided. The void fraction is also introduced to determine the flow patterns, i.e., the dominant flow pattern is bubbly flow when α<0.4, 0.4≤α≤0.7 is for slug flow, and the primary type is film flow when α>0.7. By analyzing the pressure drop, an empirical equation of pressure drop prediction is proposed.

Magnitude and seasonal variation of N2O and CH4 emissions over a mixed agriculture-urban region

Inventory estimates of N2O and CH4 emissions disregard temporal and spatial variabilities, which hinders the search for effective local strategies to lower greenhouse gas emissions. We have quantified the emissions of N2O and CH4 in a mixed agriculture-urban region using two independent approaches, i.e., the vertical gradient method (VGM) and the radon-tracer method (RTM), compared the estimated annual fluxes with the EDGARv6.0 emissions, revealed the seasonal variations of the VGM fluxes, and inferred the sources that most likely cause the seasonal variations based on the footprint analysis even though our methods cannot attribute different sources. We show that the annual RTM estimates represented by the mode of lognormal fit for N2O and CH4 are 0.4 g m−2 yr−1 and 12 g m−2 yr−1, and the VGM estimates are 0.6 ± 0.3 g m−2 yr−1 and 13 ± 4 g m−2 yr−1, respectively. Furthermore, the average EDGARv6.0 emissions constrained by the VGM and the RTM footprints are 1.3 g m−2 yr−1 and 0.9 g m−2 yr−1 for N2O, and 21 g m−2 yr−1 and 18 g m−2 yr−1 for CH4. Compared to our estimated fluxes, EDGARv6.0 N2O and CH4 emissions are both overestimated; for N2O, it is mainly caused by an overestimation of the chemical industry's emission. Moreover, in contrast to EDGARv6.0′s nearly constant monthly emissions throughout the year, the VGM estimates of N2O and CH4 show seasonal variations with relatively high values from March to September, which is most likely caused by agricultural activities. Our study demonstrates that large nighttime vertical gradients of atmospheric N2O and CH4 mole fractions at a tall tower can be used to derive surface fluxes by the VGM; taken together with the RTM fluxes, both the annual means and the temporal variations of N2O and CH4 emissions can be constrained on a regional scale.

An optimized delayed equilibrium model for predicting the pressure profile of the critical flow in a slit

The mass flux and pressure profile of the critical flow in a slit are connected to the original diameter and concentration of radioactive aerosols in leak-before-break monitoring in nuclear power plants. The Delayed Equilibrium Model (DEM), although accurate for mass flux and pressure prediction of critical flow in a circular tube, suffers from underprediction of the pressure drop for critical flow in the slit. In this study, an optimized DEM considering the shape effect of flash boiling is proposed by changing the saturated phase mass flow rate fraction gradient and two-phase friction coefficient multiplier in the DEM. Two novel equations for the new coefficients, C4 and C5, are introduced in the optimization. The relative error of the critical flow pressure in the slit can be reduced using the optimized DEM. In addition, a sensitivity analysis of the original critical flow model is conducted and the mass flux and critical pressure of the critical flow model are found to be most sensitive to the inlet temperature. The effectiveness of the optimization proves that the shape effect of the velocity of achieving thermal equilibrium in flash boiling is critical for the critical flow in a slit.

Fractional Gradient Methods via ψ-Hilfer Derivative

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the ψ-Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. The convergence of the ψ-Hilfer continuous fractional gradient method was studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we developed an algorithm for the ψ-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and step size optimization, the ψ-Hilfer fractional gradient method showed better results in terms of speed and accuracy. Our results generalize previous works in the literature.

A Constitutive Model of Water-Triggered Shape Memory Hydrogels and Its Finite Element Implementation

AbstractShape memory hydrogel is a type of hydrogel whose shape can transform between a temporary shape and its initial shape when exposed to external stimuli, such as water, temperature, and pH. Over the last decade, shape memory hydrogels have gained increasing interest owing to their distinct properties; however, constitutive models to describe their shape memory mechanism are still lacking. In this paper, we propose a constitutive model for water-triggered shape memory hydrogels based on the transition between the sparse and dense phases. In the model, the shape memory process is identified using two internal variables: the frozen deformation gradient and dense phase volume fraction. To validate the model for describing shape memory effects, we implemented the model in the finite element method using a user-defined element (UEL) subroutine in ABAQUS. To verify the accuracy of the proposed UEL, we simulated the water-triggered shape memory effects in different recovery processes under different uniaxial loads. Furthermore, we investigated the water-triggered shape memory behavior of a self-bending bilayer structure and a four-arm gripper structure using both experiments and simulations. Good agreement was observed between the simulation and experimental results.

Uptake of oomycete RXLR effectors into host cells by clathrin-mediated endocytosis.

Filamentous (oomycete and fungal) plant pathogens deliver cytoplasmic effector proteins into host cells to facilitate disease. How RXLR effectors from the potato late blight pathogen Phytophthora infestans enter host cells is unknown. One possible route involves clathrin-mediated endocytosis (CME). Transient silencing of NbCHC, encoding clathrin heavy chain, or the endosome marker gene NbAra6 encoding a Rab GTPase in the model host Nicotiana benthamiana, attenuated P. infestans infection and reduced translocation of RXLR effector fusions from transgenic pathogen strains into host cells. By contrast, silencing PP1c isoforms, susceptibility factors not required for endocytosis, reduced infection but did not attenuate RXLR effector uptake. Endosome enrichment by ultracentrifugation and sucrose gradient fractionation revealed co-localization of RXLR effector Pi04314-RFP with clathrin-coated vesicles. Immunopurification of clathrin- and NbAra6-associated vesicles during infection showed that RXLR effectors Pi04314-RFP and AvrBlb1-RFP, but not apoplastic effector PiSCR74-RFP, were co-immunoprecipitated during infection with pathogen strains secreting these effectors. Tandem mass spectrometry analyses of proteins co-immunoprecipitated with NbAra6-GFP during infection revealed enrichment of host proteins associated with endocytic vesicles alongside multiple pathogen RXLR effectors, but not apoplastic effectors, including PiSCR74, which do not enter host cells. Our data show that the uptake of P. infestans RXLR effectors into plant cells occurs via CME.

Applications of fractional gradient descent method with adaptive momentum in BP neural networks

A novel fractional gradient descent method with adaptive momentum is presented in this paper to improve the convergence speed and stability for BP neural network training. The fractional Grünwald-Letnikov derivative is used for the fractional gradient. The coefficient of the momentum term is set as an adaptive variable, depending on the fractional gradient of the current step and the weight change of the previous step. We give a detailed convergence proof of the proposed method. Experiments on MNIST data sets and XOR problem demonstrate that the fractional gradient descent method with adaptive momentum term can effectively improve convergence speed, maintain stability of BP neural network training, help escape from local minimum points, and enlarge the selection range of the learning rate.

Phase‐field modeling of chemically reactive multi‐component and multi‐phase systems

AbstractThe present phase‐field approach is based on a mixture theory for multiple components and phases within the framework of non‐equilibrium thermodynamics of internal state variables. More specifically, diffuse interfaces are included in the state potential by terms depending on spatial gradients of the component mass fractions as well as order parameters representing different phases. Coupled field equations of generalized Cahn‐Hilliard and Allen‐Cahn type are derived directly from a local entropy balance, under the assumption of a localized Gibbs fundamental equation. Additionally, thermodynamically consistent kinetics for equilibrium reactions are formulated. The field equations are recast into a mixed variational formulation, which allows a discretization by finite elements with low‐order ansatz functions. The numerical implementation is discussed by means of a benchmark problem for reactive binary systems, of which the thermodynamic equilibrium solution is known.

Isolation of Betulinic Acid from Byttneria herbacea (Malvaceae)

During the phytochemical analysis, flavonoids, saponins, tannins, glycosides, alkaloids, and terpenoids were identified in Byttneria herbacea (Malvaceae) among other things. The goal of this research is to identify and characterize the plant's bioactive principle. It has a wide range of folk medicinal applications. The gradient fractionation method was used to isolate and characterize phytoconstituents from ethanol extract. Physical, chemical, and spectroscopic data were used to determine the structure of the isolated substance (IR, UV, 1HNMR, MS). The ethanol extract of the plant yielded a triterpenoid structure. Various spectroscopical approaches were used to identify the odorless, colorless chemical that solubilizes in ethyl acetate and has a melting point of 315°C - 318°C and an Rf value of 0.62 in the mobile phase toluene: ethyl acetate (85:15). As a result of this research, the isolated component has been identified as Betulinic acid, which is accountable for the plant's diverse pharmacological properties.

Constrained 2D inversion of radio-magnetotelluric and controlled-source audio-magnetotelluric data using high-resolution reflection seismic data: An example in groundwater surveying from Heby, Sweden

We advance a previously established method for 2D inversion of electromagnetic data, in which the smoothness constraints are locally reweighted according to the seismic envelope fractional gradients (SEFGs). As the first step of our modifications, seismic envelope values are edited and normalized. This results in the rejection of noise-contaminated parts, clipping the envelope outliers and increasing the reflectivity power of weak seismic signals. Second, we introduce a weighting matrix in the normalization process to incorporate prior information in the constrained inversion regarding the relative contrasts of electrical resistivity in the given parts of the model. Third, due to normalizing the SEFG, there is no need to search for an optimum stabilization factor to modify local smoothing weights, and hence, we set it to a constant value. Finally, an Occam inversion with additional Levenberg-Marquardt damping is used to mitigate possible artifacts in the resistivity model generated by reflection seismic constraints. In applying the proposed scheme to a synthetic example, interfaces of various geologic units are restored as sharp boundaries. In addition, artifacts generated in the original approach are effectively mitigated thanks to the applied normalization process. For the first time, we apply the method to radio-magnetotelluric (RMT) and controlled-source audio-magnetotelluric (CSAMT) field data acquired along a profile across a known aquifer in Heby, Sweden. Our inversion models compare favorably to previously presented results from seismic and geoelectric data. By modifying the smoothness weights based on SEFG, the depth to the bedrock is recovered well, being constrained by the corresponding interfaces in the seismic image. The resistivity models from seismically constrained inversions of RMT and CSAMT data reveal steeply dipping and possibly fractured bedrock underneath the valley-shaped aquifer in the area. This interpretation is verified by borehole logs.

An approach to elliptic equations with nonlinear gradient terms via a modulation framework

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] is the nonlinearity, [Formula: see text] the potential and [Formula: see text] is a forcing term. Some examples of nonlinearities dealt with are [Formula: see text], [Formula: see text] and [Formula: see text], covering large values of [Formula: see text], and particularly variational supercritical powers for [Formula: see text] and super-[Formula: see text] ones for [Formula: see text] (superquadratic if [Formula: see text]). Moreover, we are able to consider some exponential growths, [Formula: see text] belonging to certain classes of power series, or [Formula: see text] satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

Extending linear growth functionals to functions of bounded fractional variation

In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply the direct method. We therefore utilize the recently introduced spaces of bounded fractional variation and study the extension of the linear growth functional to these spaces through relaxation with respect to the weak* convergence. Our main result establishes an explicit representation for this relaxation, which includes an integral term accounting for the singular part of the fractional variation and features the quasiconvex envelope of the integrand. The role of quasiconvexity in this fractional framework is explained by a technique to switch between the fractional and classical settings. We complement the relaxation result with an existence theory for minimizers of the extended functional.

Coronary CT angiography-derived plaque characteristics and physiologic patterns for peri-procedural myocardial infarction and subsequent events.

Peri-procedural myocardial infarction (PMI) after percutaneous coronary intervention (PCI) has been shown to be associated with worse clinical outcomes. We aimed to investigate the value of coronary plaque characteristics and physiologic disease patterns (focal vs. diffuse) assessed by coronary computed tomography angiography (CTA) in predicting PMI and adverse events. Three hundred fifty-nine patients with normal pre-PCI high-sensitivity cardiac troponin T (hs-cTnT) underwent CTA before PCI were analysed. The high-risk plaque characteristics (HRPC) were assessed on CTA. The physiologic disease pattern was characterized using CTA fractional flow reserve-derived pullback pressure gradients (FFRCT PPG). PMI was defined as an increase in hs-cTnT to >5 times the upper limit of normal after PCI. The major adverse cardiovascular events (MACE) were a composite of cardiac death, spontaneous myocardial infarction, and target vessel revascularization. The presence of ≥3 HRPC in the target lesions [odds ratio (OR) 2.21, 95% confidence interval (CI) 1.29-3.80, P = 0.004] and low FFRCT PPG (OR 1.23, 95% CI 1.02-1.52, P = 0.028) were independent predictors of PMI. In a four-group classification according to HRPC and FFRCT PPG, patients with ≥3 HRPC and low FFRCT PPG had the highest risk of MACE (19.3%; overall P = 0.001). Moreover, the presence of ≥3 HRPC and low FFRCT PPG was an independent predictor of MACE and showed incremental prognostic value compared with a model with clinical risk factors alone [C index = 0.78 vs. 0.60, P = 0.005, net reclassification index = 0.21 (95% CI: 0.04-0.48), P = 0.020]. Coronary CTA can evaluate plaque characteristics and physiologic disease patterns simultaneously, which plays an important role for risk stratification before PCI.

Fractional Gradient Descent-Based Auxiliary Model Algorithm for FIR Models with Missing Data

This study proposes a fractional gradient descent (FGD) algorithm for FIR models with missing data. By using the auxiliary model method, the missing data can be obtained. Then, the FGD algorithm is applied to update the parameters of the FIR models. Because of the fractional term in the conventional GD algorithm, the convergence rates of the GD algorithm can be increased. In addition, to avoid the step-size calculation, an Aitken FGD-based auxiliary model algorithm is also introduced. The convergence analysis and simulation examples are provided to show the effectiveness of the proposed algorithms.