A flexible multiscale algorithm based on an improved smoothed particle hydrodynamics method for complex viscoelastic flows

Viscoelastic flows play an important role in numerous engineering fields,and the multiscale algorithms for simulating viscoelastic flows have received significant attention in order to deepen our understanding of the nonlinear dynamic behaviors of viscoelastic fluids.However,traditional grid-based multiscale methods are confined to simple viscoelastic flows with short relaxation time,and there is a lack of uniform multiscale scheme available for coupling different solvers in the simulations of viscoelastic fluids.In this paper,a universal multiscale method coupling an improved smoothed particle hydrodynamics(SPH)and multiscale universal interface(MUI)library is presented for viscoelastic flows.The proposed multiscale method builds on an improved SPH method and leverages the MUI library to facilitate the exchange of information among different solvers in the overlapping domain.We test the capability and flexibility of the presented multiscale method to deal with complex viscoelastic flows by solving different multiscale problems of viscoelastic flows.In the first example,the simulation of a viscoelastic Poiseuille flow is carried out by two coupled improved SPH methods with different spatial resolutions.The effects of exchanging different physical quantities on the numerical results in both the upper and lower domains are also investigated as well as the absolute errors in the overlapping domain.In the second example,the complex Wannier flow with different Weissenberg numbers is further simulated by two improved SPH methods and coupling the improved SPH method and the dissipative particle dynamics(DPD)method.The numerical results show that the physical quantities for viscoelastic flows obtained by the presented multiscale method are in consistence with those obtained by a single solver in the overlapping domain.Moreover,transferring different physical quantities has an important effect on the numerical results.

A multiscale adaptive framework based on convolutional neural network:Application to fluid catalytic cracking product yield prediction

Since chemical processes are highly non-linear and multiscale,it is vital to deeply mine the multiscale coupling relationships embedded in the massive process data for the prediction and anomaly tracing of crucial process parameters and production indicators.While the integrated method of adaptive signal decomposition combined with time series models could effectively predict process variables,it does have limitations in capturing the high-frequency detail of the operation state when applied to complex chemical processes.In light of this,a novel Multiscale Multi-radius Multi-step Convolutional Neural Network(Msrt Net)is proposed for mining spatiotemporal multiscale information.First,the industrial data from the Fluid Catalytic Cracking(FCC)process decomposition using Complete Ensemble Empirical Mode Decomposition with Adaptive Noise(CEEMDAN)extract the multi-energy scale information of the feature subset.Then,convolution kernels with varying stride and padding structures are established to decouple the long-period operation process information encapsulated within the multi-energy scale data.Finally,a reconciliation network is trained to reconstruct the multiscale prediction results and obtain the final output.Msrt Net is initially assessed for its capability to untangle the spatiotemporal multiscale relationships among variables in the Tennessee Eastman Process(TEP).Subsequently,the performance of Msrt Net is evaluated in predicting product yield for a 2.80×10^(6) t/a FCC unit,taking diesel and gasoline yield as examples.In conclusion,Msrt Net can decouple and effectively extract spatiotemporal multiscale information from chemical process data and achieve a approximately reduction of 30%in prediction error compared to other time-series models.Furthermore,its robustness and transferability underscore its promising potential for broader applications.

Multiscale analysis of fine slag from pulverized coal gasification in entrained-flow bed

Fine slag(FS)is an unavoidable by-product of coal gasification.FS,which is a simple heap of solid waste left in the open air,easily causes environmental pollution and has a low resource utilization rate,thereby restricting the development of energy-saving coal gasification technologies.The multiscale analysis of FS performed in this study indicates typical grain size distribution,composition,crystalline structure,and chemical bonding characteristics.The FS primarily contained inorganic and carbon components(dry bases)and exhibited a"three-peak distribution"of the grain size and regular spheroidal as well as irregular shapes.The irregular particles were mainly adsorbed onto the structure and had a dense distribution and multiple pores and folds.The carbon constituents were primarily amorphous in structure,with a certain degree of order and active sites.C 1s XPS spectrum indicated the presence of C–C and C–H bonds and numerous aromatic structures.The inorganic components,constituting 90%of the total sample,were primarily silicon,aluminum,iron,and calcium.The inorganic components contained Si–O-Si,Si–O–Al,Si–O,SO_(4)^(2−),and Fe–O bonds.Fe 2p XPS spectrum could be deconvoluted into Fe 2p_(1/2) and Fe 2p_(3/2) peaks and satellite peaks,while Fe existed mainly in the form of Fe(III).The findings of this study will be beneficial in resource utilization and formation mechanism of fine slag in future.

The 2022 Extreme Heatwave in Shanghai,Lower Reaches of the Yangtze River Valley:Combined Influences of Multiscale Variabilities

In the summer of 2022,China(especially the Yangtze River Valley,YRV)suffered its strongest heatwave(HW)event since 1961.In this study,we examined the influences of multiscale variabilities on the 2022 extreme HW in the lower reaches of the YRV,focusing on the city of Shanghai.We found that about 1/3 of the 2022 HW days in Shanghai can be attributed to the long-term warming trend of global warming.During mid-summer of 2022,an enhanced western Pacific subtropical high(WPSH)and anomalous double blockings over the Ural Mountains and Sea of Okhotsk,respectively,were associated with the persistently anomalous high pressure over the YRV,leading to the extreme HW.The Pacific Decadal Oscillation played a major role in the anomalous blocking pattern associated with the HW at the decadal time scale.Also,the positive phase of the Atlantic Multidecadal Oscillation may have contributed to regulating the formation of the double-blocking pattern.Anomalous warming of both the warm pool of the western Pacific and tropical North Atlantic at the interannual time scale may also have favored the persistency of the double blocking and the anomalously strong WPSH.At the subseasonal time scale,the anomalously frequent phases 2-5 of the canonical northward propagating variability of boreal summer intraseasonal oscillation associated with the anomalous propagation of a weak Madden-Julian Oscillation suppressed the convection over the YRV and also contributed to the HW.Therefore,the 2022 extreme HW originated from multiscale forcing including both the climate warming trend and air-sea interaction at multiple time scales.

Blade-Coated Porous 3D Carbon Composite Electrodes Coupled with Multiscale Interfaces for Highly Sensitive All-Paper Pressure Sensors

Flexible and wearable pressure sensors hold immense promise for health monitoring,covering disease detection and postoperative rehabilitation.Developing pressure sensors with high sensitivity,wide detection range,and cost-effectiveness is paramount.By leveraging paper for its sustainability,biocompatibility,and inherent porous structure,herein,a solution-processed all-paper resistive pressure sensor is designed with outstanding performance.A ternary composite paste,comprising a compressible 3D carbon skeleton,conductive polymer poly(3,4-ethylene dioxythiophene):poly(styrenesulfonate),and cohesive carbon nanotubes,is blade-coated on paper and naturally dried to form the porous composite electrode with hierachical micro-and nano-structured surface.Combined with screen-printed Cu electrodes in submillimeter finger widths on rough paper,this creates a multiscale hierarchical contact interface between electrodes,significantly enhancing sensitivity(1014 kPa-1)and expanding the detection range(up to 300 kPa)of as-resulted all-paper pressure sensor with low detection limit and power consumption.Its versatility ranges from subtle wrist pulses,robust finger taps,to large-area spatial force detection,highlighting its intricate submillimetermicrometer-nanometer hierarchical interface and nanometer porosity in the composite electrode.Ultimately,this all-paper resistive pressure sensor,with its superior sensing capabilities,large-scale fabrication potential,and cost-effectiveness,paves the way for next-generation wearable electronics,ushering in an era of advanced,sustainable technological solutions.

Fast compressed sensing spectral measurement with adaptive gradient multiscale resolution

We propose a fast,adaptive multiscale resolution spectral measurement method based on compressed sensing.The method can apply variable measurement resolution over the entire spectral range to reduce the measurement time by over 75%compared to a global high-resolution measurement.Mimicking the characteristics of the human retina system,the resolution distribution follows the principle of gradually decreasing.The system allows the spectral peaks of interest to be captured dynamically or to be specified a priori by a user.The system was tested by measuring single and dual spectral peaks,and the results of spectral peaks are consistent with those of global high-resolution measurements.

Multiscale Simulation of Microstructure Evolution during Preparation and Service Processes of Physical Vapor Deposited c-TiAlN Coatings

Physical Vapor Deposited(PVD)TiAlN coatings are extensively utilized as protective layers for cutting tools,renowned for their excellent comprehensive performance.To optimize quality control of TiAlN coatings for cutting tools,a multi-scale simulation approach is proposed that encompasses the microstructure evolution of coatings considering the entire preparation and service lifecycle of PVD TiAlN coatings.This scheme employs phase-field simulation to capture the essential microstructure of the PVD-prepared TiAlN coatings.Moreover,cutting simulation is used to determine the service temperature experienced during cutting processes at varying rates.Cahn-Hilliard modeling is finally utilized to consume the microstructure and service condition data to acquaint the microstructure evolution of TiAlN coatings throughout the cutting processes.This methodology effectively establishes a correlation between service temperature and its impact on the microstructure evolution of TiAlN coatings.It is expected that the present multi-scale numerical simulation approach will provide innovative strategies for assisting property design and lifespan prediction of TiAlN coatings.

Effect of Heat Treatment on Microstructure and Mechanical Properties of Multiscale SiC_p Hybrid Reinforced 6061 Aluminum Matrix Composites

The performance of solid solution aging treatment on aluminum matrix composites prepared by powder metallurgy and reinforced with 6061 aluminum alloy powder as matrix;meanwhile, nano silicon carbide particles(nm Si Cp), submicron silicon carbide particles(1 μm Si Cp) and Ti particles were studied. The Al/Si Cp composite powder was prepared by high-energy ball milling, and then cold-pressed, sintered, hotextruded, and then heat-treated with different solution temperatures and aging times for the extruded composites. Optical microscopy, scanning electron microscopy, energy dispersive X-ray spectroscopy(EDS), X-ray diffractometer(XRD) and extrusion testing were used to analyze and test the microstructure and mechanical properties of aluminum matrix composites. The results show that after the multi-stage solid solution at 530 ℃×2 h+535 ℃×2 h+540 ℃×2 h, the particles are mainly equiaxed grains and uniformly distributed. There is no reinforcement agglomeration, and the surface is dense and the insoluble phase is basically dissolved. In the matrix, the strengthening effect is good, and the hardness and compressive strength are 179.43 HV and 680.42 MPa, respectively. Under this solution process, when the aluminum matrix composites are aged at 170 ℃ for 10 h, the hardness and compressive strength can reach their peaks and increase to 195.82 HV and 721.48 MPa, respectively.

Infrasound Event Classification Fusion Model Based on Multiscale SE-CNN and BiLSTM

The classification of infrasound events has considerable importance in improving the capability to identify the types of natural disasters.The traditional infrasound classification mainly relies on machine learning algorithms after artificial feature extraction.However,guaranteeing the effectiveness of the extracted features is difficult.The current trend focuses on using a convolution neural network to automatically extract features for classification.This method can be used to extract signal spatial features automatically through a convolution kernel;however,infrasound signals contain not only spatial information but also temporal information when used as a time series.These extracted temporal features are also crucial.If only a convolution neural network is used,then the time dependence of the infrasound sequence will be missed.Using long short-term memory networks can compensate for the missing time-series features but induces spatial feature information loss of the infrasound signal.A multiscale squeeze excitation–convolution neural network–bidirectional long short-term memory network infrasound event classification fusion model is proposed in this study to address these problems.This model automatically extracted temporal and spatial features,adaptively selected features,and also realized the fusion of the two types of features.Experimental results showed that the classification accuracy of the model was more than 98%,thus verifying the effectiveness and superiority of the proposed model.

Multiscale confinement nitridation in molybdenum carbide for efficient hydrogen production

The molybdenum carbide(Mo_(2)C)has been regarded as one of the most cost-efficient and stable electrocatalyst for the hydrogen evolution reaction(HER)by the virtue of its Pt-like electronic structures.However,the inherent limitation of high density of empty valence band significantly reduces its catalytic reactivity by reason of strong hydrogen desorption resistance.Herein,we propose a multiscale confinement synthesis method to design the nitrogen-rich Mo_(2)C for modulating the band structure via decomposing the pre-coordination bonded polymer in a pressure-tight tube sealing system.Pre-bonded c/N-Mo in the coordination precursor constructs a micro-confinement space,enabling the homogeneous nitrogenization in-situ happened during the formation of Mo_(2)C.Simultaneously,the evolved gases from the precursor decomposition in tube sealing system establish a macro-confinement environment,preventing the lattice N escape and further endowing a continuous nitridation.Combining the multiscale confinement effects,the nitrogen-rich Mo2C displays as high as 25%N-Mo concentration in carbide lattice,leading to a satisfactory band structure.Accordingly,the constructed nitrogen-rich Mo_(2)C reveals an adorable catalytic activity for HER in both alkaline and acid solution.It is anticipated that the multiscale confinement synthesis strategy presents guideline for the rational design of electrocatalysts and beyond.

An Initial Perturbation Method for the Multiscale Singular Vector in Global Ensemble Prediction

Ensemble prediction is widely used to represent the uncertainty of single deterministic Numerical Weather Prediction(NWP) caused by errors in initial conditions(ICs). The traditional Singular Vector(SV) initial perturbation method tends only to capture synoptic scale initial uncertainty rather than mesoscale uncertainty in global ensemble prediction. To address this issue, a multiscale SV initial perturbation method based on the China Meteorological Administration Global Ensemble Prediction System(CMA-GEPS) is proposed to quantify multiscale initial uncertainty. The multiscale SV initial perturbation approach entails calculating multiscale SVs at different resolutions with multiple linearized physical processes to capture fast-growing perturbations from mesoscale to synoptic scale in target areas and combining these SVs by using a Gaussian sampling method with amplitude coefficients to generate initial perturbations. Following that, the energy norm,energy spectrum, and structure of multiscale SVs and their impact on GEPS are analyzed based on a batch experiment in different seasons. The results show that the multiscale SV initial perturbations can possess more energy and capture more mesoscale uncertainties than the traditional single-SV method. Meanwhile, multiscale SV initial perturbations can reflect the strongest dynamical instability in target areas. Their performances in global ensemble prediction when compared to single-scale SVs are shown to(i) improve the relationship between the ensemble spread and the root-mean-square error and(ii) provide a better probability forecast skill for atmospheric circulation during the late forecast period and for short-to medium-range precipitation. This study provides scientific evidence and application foundations for the design and development of a multiscale SV initial perturbation method for the GEPS.

Multiscale and Auto-Tuned Semi-Supervised Deep Subspace Clustering and Its Application in Brain Tumor Clustering

In this paper,we introduce a novel Multi-scale and Auto-tuned Semi-supervised Deep Subspace Clustering(MAS-DSC)algorithm,aimed at addressing the challenges of deep subspace clustering in high-dimensional real-world data,particularly in the field of medical imaging.Traditional deep subspace clustering algorithms,which are mostly unsupervised,are limited in their ability to effectively utilize the inherent prior knowledge in medical images.Our MAS-DSC algorithm incorporates a semi-supervised learning framework that uses a small amount of labeled data to guide the clustering process,thereby enhancing the discriminative power of the feature representations.Additionally,the multi-scale feature extraction mechanism is designed to adapt to the complexity of medical imaging data,resulting in more accurate clustering performance.To address the difficulty of hyperparameter selection in deep subspace clustering,this paper employs a Bayesian optimization algorithm for adaptive tuning of hyperparameters related to subspace clustering,prior knowledge constraints,and model loss weights.Extensive experiments on standard clustering datasets,including ORL,Coil20,and Coil100,validate the effectiveness of the MAS-DSC algorithm.The results show that with its multi-scale network structure and Bayesian hyperparameter optimization,MAS-DSC achieves excellent clustering results on these datasets.Furthermore,tests on a brain tumor dataset demonstrate the robustness of the algorithm and its ability to leverage prior knowledge for efficient feature extraction and enhanced clustering performance within a semi-supervised learning framework.

PSMFNet:Lightweight Partial Separation and Multiscale Fusion Network for Image Super-Resolution

The employment of deep convolutional neural networks has recently contributed to significant progress in single image super-resolution(SISR)research.However,the high computational demands of most SR techniques hinder their applicability to edge devices,despite their satisfactory reconstruction performance.These methods commonly use standard convolutions,which increase the convolutional operation cost of the model.In this paper,a lightweight Partial Separation and Multiscale Fusion Network(PSMFNet)is proposed to alleviate this problem.Specifically,this paper introduces partial convolution(PConv),which reduces the redundant convolution operations throughout the model by separating some of the features of an image while retaining features useful for image reconstruction.Additionally,it is worth noting that the existing methods have not fully utilized the rich feature information,leading to information loss,which reduces the ability to learn feature representations.Inspired by self-attention,this paper develops a multiscale feature fusion block(MFFB),which can better utilize the non-local features of an image.MFFB can learn long-range dependencies from the spatial dimension and extract features from the channel dimension,thereby obtaining more comprehensive and rich feature information.As the role of the MFFB is to capture rich global features,this paper further introduces an efficient inverted residual block(EIRB)to supplement the local feature extraction ability of PSMFNet.A comprehensive analysis of the experimental results shows that PSMFNet maintains a better performance with fewer parameters than the state-of-the-art models.

Automatic Road Tunnel Crack Inspection Based on Crack Area Sensing and Multiscale Semantic Segmentation

The detection of crack defects on the walls of road tunnels is a crucial step in the process of ensuring travel safetyand performing routine tunnel maintenance. The automatic and accurate detection of cracks on the surface of roadtunnels is the key to improving the maintenance efficiency of road tunnels. Machine vision technology combinedwith a deep neural network model is an effective means to realize the localization and identification of crackdefects on the surface of road tunnels.We propose a complete set of automatic inspection methods for identifyingcracks on the walls of road tunnels as a solution to the problem of difficulty in identifying cracks during manualmaintenance. First, a set of equipment applied to the real-time acquisition of high-definition images of walls inroad tunnels is designed. Images of walls in road tunnels are acquired based on the designed equipment, whereimages containing crack defects are manually identified and selected. Subsequently, the training and validationsets used to construct the crack inspection model are obtained based on the acquired images, whereas the regionscontaining cracks and the pixels of the cracks are finely labeled. After that, a crack area sensing module is designedbased on the proposed you only look once version 7 model combined with coordinate attention mechanism (CAYOLOV7) network to locate the crack regions in the road tunnel surface images. Only subimages containingcracks are acquired and sent to the multiscale semantic segmentation module for extraction of the pixels to whichthe cracks belong based on the DeepLab V3+ network. The precision and recall of the crack region localizationon the surface of a road tunnel based on our proposed method are 82.4% and 93.8%, respectively. Moreover, themean intersection over union (MIoU) and pixel accuracy (PA) values for achieving pixel-level detection accuracyare 76.84% and 78.29%, respectively. The experimental results on the dataset show that our proposed two-stagedetection method outperforms other state-of-the-art models in crack region localization and detection. Based onour proposedmethod, the images captured on the surface of a road tunnel can complete crack detection at a speed often frames/second, and the detection accuracy can reach 0.25 mm, which meets the requirements for maintenanceof an actual project. The designed CA-YOLO V7 network enables precise localization of the area to which a crackbelongs in images acquired under different environmental and lighting conditions in road tunnels. The improvedDeepLab V3+ network based on lightweighting is able to extract crack morphology in a given region more quicklywhile maintaining segmentation accuracy. The established model combines defect localization and segmentationmodels for the first time, realizing pixel-level defect localization and extraction on the surface of road tunnelsin complex environments, and is capable of determining the actual size of cracks based on the physical coordinatesystemafter camera calibration. The trainedmodelhas highaccuracy andcanbe extendedandapplied to embeddedcomputing devices for the assessment and repair of damaged areas in different types of road tunnels.

Multiscale Feature Fusion for Gesture Recognition Using Commodity Millimeter-Wave Radar

Gestures are one of the most natural and intuitive approach for human-computer interaction.Compared with traditional camera-based or wearable sensors-based solutions,gesture recognition using the millimeter wave radar has attracted growing attention for its characteristics of contact-free,privacy-preserving and less environmentdependence.Although there have been many recent studies on hand gesture recognition,the existing hand gesture recognition methods still have recognition accuracy and generalization ability shortcomings in shortrange applications.In this paper,we present a hand gesture recognition method named multiscale feature fusion(MSFF)to accurately identify micro hand gestures.In MSFF,not only the overall action recognition of the palm but also the subtle movements of the fingers are taken into account.Specifically,we adopt hand gesture multiangle Doppler-time and gesture trajectory range-angle map multi-feature fusion to comprehensively extract hand gesture features and fuse high-level deep neural networks to make it pay more attention to subtle finger movements.We evaluate the proposed method using data collected from 10 users and our proposed solution achieves an average recognition accuracy of 99.7%.Extensive experiments on a public mmWave gesture dataset demonstrate the superior effectiveness of the proposed system.

A Multiscale Reliability-Based Design Optimization Method for Carbon-Fiber-Reinforced Composite Drive Shafts

Carbon fiber composites,characterized by their high specific strength and low weight,are becoming increasingly crucial in automotive lightweighting.However,current research primarily emphasizes layer count and orientation,often neglecting the potential of microstructural design,constraints in the layup process,and performance reliability.This study,therefore,introduces a multiscale reliability-based design optimization method for carbon fiber-reinforced plastic(CFRP)drive shafts.Initially,parametric modeling of the microscale cell was performed,and its elastic performance parameters were predicted using two homogenization methods,examining the impact of fluctuations in microscale cell parameters on composite material performance.A finite element model of the CFRP drive shaft was then constructed,achieving parameter transfer between microscale and macroscale through Python programming.This enabled an investigation into the influence of both micro and macro design parameters on the CFRP drive shaft’s performance.The Multi-Objective Particle Swarm Optimization(MOPSO)algorithm was enhanced for particle generation and updating strategies,facilitating the resolution of multi-objective reliability optimization problems,including composite material layup process constraints.Case studies demonstrated that this approach leads to over 30%weight reduction in CFRP drive shafts compared to metallic counterparts while satisfying reliability requirements and offering insights for the lightweight design of other vehicle components.

Exponentially Convergent Multiscale Finite Element Method

We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

Multiscale Finite Element Method for Coupling Analysis of Heterogeneous Magneto-Electro-Elastic Structures in Thermal Environment

Magneto-electro-elastic (MEE) materials, a new type of composite intelligent materials, exhibit excellent multifield coupling effects. Due to the heterogeneity of the materials, it is challenging to use the traditional finite element method (FEM) for mechanical analysis. Additionally, the MEE materials are often in a complex service environment, especially under the influence of the thermal field with thermoelectric and thermomagnetic effects, which affect its mechanical properties. Therefore, this paper proposes the efficient multiscale computational method for the multifield coupling problem of heterogeneous MEE structures under the thermal environment. The method constructs a multi-physics field with numerical base functions (the displacement, electric potential, and magnetic potential multiscale base functions). It equates a single cell of heterogeneous MEE materials to a macroscopic unit and supplements the macroscopic model with a microscopic model. This allows the problem to be solved directly on a macroscopic scale. Finally, the numerical simulation results demonstrate that compared with the traditional FEM, the multiscale finite element method (MsFEM) can achieve the purpose of ensuring accuracy and reducing the degree of freedom, and significantly improving the calculation efficiency.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.