Effect of non-uniform swelling on coal multiphysics during gas injection: The triangle approach

In current dual porosity/permeability models,there exists a fundamental assumption that the adsorption-induced swelling is distributed uniformly within the representative elementary volume (REV),irrespective of its internal structures and transient processes.However,both internal structures and transient processes can lead to the non-uniform swelling.In this study,we hypothesize that the non-uniform swelling is responsible for why coal permeability in experimental measurements is not only controlled by the effective stress but also is affected by the adsorption-induced swelling.We propose a concept of the swelling triangle composed of swelling paths to characterize the evolution of the non-uniform swelling and serve as a core link in coupled multiphysics.A swelling path is determined by a dimensionless volumetric ratio and a dimensionless swelling ratio.Different swelling paths have the same start and end point,and each swelling path represents a unique swelling case.The swelling path as the diagonal of the triangle represents the case of the uniform swelling while that as the two perpendicular boundaries represents the case of the localized swelling.The paths of all intermediate cases populate inside the triangle.The corresponding relations between the swelling path and the response of coal multiphysics are established by a non-uniform swelling coefficient.We define this method as the triangle approach and corresponding models as swelling path-based ones.The proposed concept and models are verified against a long-term experimental measurement of permeability and strains under constant effective stress.Our results demonstrate that during gas injection,coal multiphysics responses have a close dependence on the swelling path,and that in both future experiments and field predictions,this dependence must be considered.

Pressure transient characteristics of non-uniform conductivity fractured wells in viscoelasticity polymer flooding based on oil-water two-phase flow

Polymer flooding in fractured wells has been extensively applied in oilfields to enhance oil recovery.In contrast to water,polymer solution exhibits non-Newtonian and nonlinear behavior such as effects of shear thinning and shear thickening,polymer convection,diffusion,adsorption retention,inaccessible pore volume and reduced effective permeability.Meanwhile,the flux density and fracture conductivity along the hydraulic fracture are generally non-uniform due to the effects of pressure distribution,formation damage,and proppant breakage.In this paper,we present an oil-water two-phase flow model that captures these complex non-Newtonian and nonlinear behavior,and non-uniform fracture characteristics in fractured polymer flooding.The hydraulic fracture is firstly divided into two parts:high-conductivity fracture near the wellbore and low-conductivity fracture in the far-wellbore section.A hybrid grid system,including perpendicular bisection(PEBI)and Cartesian grid,is applied to discrete the partial differential flow equations,and the local grid refinement method is applied in the near-wellbore region to accurately calculate the pressure distribution and shear rate of polymer solution.The combination of polymer behavior characterizations and numerical flow simulations are applied,resulting in the calculation for the distribution of water saturation,polymer concentration and reservoir pressure.Compared with the polymer flooding well with uniform fracture conductivity,this non-uniform fracture conductivity model exhibits the larger pressure difference,and the shorter bilinear flow period due to the decrease of fracture flow ability in the far-wellbore section.The field case of the fall-off test demonstrates that the proposed method characterizes fracture characteristics more accurately,and yields fracture half-lengths that better match engineering reality,enabling a quantitative segmented characterization of the near-wellbore section with high fracture conductivity and the far-wellbore section with low fracture conductivity.The novelty of this paper is the analysis of pressure performances caused by the fracture dynamics and polymer rheology,as well as an analysis method that derives formation and fracture parameters based on the pressure and its derivative curves.

Image Non-Uniformity Correction in 3T Gd-EOB-DTPA-Enhanced Magnetic Resonance Imaging: Comparison among Different Software Versions

Background: Non-uniformity in signal intensity occurs commonly in magnetic resonance (MR) imaging, which may pose substantial problems when using a 3T scanner. Therefore, image non-uniformity correction is usually applied. Purpose: To compare the correction effects of the phased-array uniformity enhancement (PURE), a calibration-based image non-uniformity correction method, among three different software versions in 3T Gd-EOB-DTPA-enhanced MR imaging. Material and Methods: Hepatobiliary-phase images of a total of 120 patients who underwent Gd-EOB-DTPA-enhanced MR imaging on the same 3T scanner were analyzed retrospectively. Forty patients each were examined using three software versions (DV25, DV25.1, and DV26). The effects of PURE were compared by visual assessment, histogram analysis of liver signal intensity, evaluation of the spatial distribution of correction effects, and evaluation of quantitative indices of liver parenchymal enhancement. Results: The visual assessment indicated the highest uniformity of PURE-corrected images for DV26, followed by DV25 and DV25.1. Histogram analysis of corrected images demonstrated significantly larger variations in liver signal for DV25.1 than for the other two versions. Although PURE caused a relative increase in pixel values for central and lateral regions, such effects were weaker for DV25.1 than for the other two versions. In the evaluation of quantitative indices of liver parenchymal enhancement, the liver-to-muscle ratio (LMR) was significantly higher for the corrected images than for the uncorrected images, but the liver-to-spleen ratio (LSR) showed no significant differences. For corrected images, the LMR was significantly higher for DV25 and DV26 than for DV25.1, but the LSR showed no significant differences among the three versions. Conclusion: There were differences in the effects of PURE among the three software versions in 3T Gd-EOB-DTPA-enhanced MR imaging. Even if the non-uniformity correction method has the same brand name, correction effects may differ depending on the software version, and these differences may affect visual and quantitative evaluations.

An improved non-uniform fast Fourier transform method for radio imaging of coronal mass ejections

Radioheliographs can obtain solar images at high temporal and spatial resolution,with a high dynamic range.These are among the most important instruments for studying solar radio bursts,understanding solar eruption events,and conducting space weather forecasting.This study aims to explore the effective use of radioheliographs for solar observations,specifically for imaging coronal mass ejections(CME),to track their evolution and provide space weather warnings.We have developed an imaging simulation program based on the principle of aperture synthesis imaging,covering the entire data processing flow from antenna configuration to dirty map generation.For grid processing,we propose an improved non-uniform fast Fourier transform(NUFFT)method to provide superior image quality.Using simulated imaging of radio coronal mass ejections,we provide practical recommendations for the performance of radioheliographs.This study provides important support for the validation and calibration of radioheliograph data processing,and is expected to profoundly enhance our understanding of solar activities.

Numerical Simulation of the Impact of Urban Non-uniformity on Precipitation

To evaluate the influence of urban non-uniformity on precipitation, the area of a city was divided into three categories （commercial, high-density residential, and low-density residential） according to the building density data from Landsat satel- lites. Numerical simulations of three corresponding scenarios （urban non-uniformity, urban uniformity, and non-urban） were performed in Nanjing using the WRF model. The results demonstrate that the existence of the city results in more precip- itation, and that urban heterogeneity enhances this phenomenon. For the urban non-uniformity, uniformity, and non-urban experiments, the mean cumulative summer precipitation was 423.09 mm, 407.40 mm, and 389.67 mm, respectively. Urban non-uniformity has a significant effect on the amount of heavy rainfall in summer. The cumulative precipitation from heavy rain in the summer for the three numerical experiments was 278.2 mm, 250.6 mm, and 236.5 mm, respectively. In the non- uniformity experiments, the amount of precipitation between 1500 and 2200 （LST） increased significantly. Furthermore, the adoption of urban non-uniformity into the WRF model could improve the numerical simulation of summer rain and its daily variation.

Study on effect of heat transfer space non-uniformity of combustion chamber components on in-cylinder NOx emission formation in internal combustion engine

The components of combustion chamber (cylinder head-cylinder liner-piston assembly-oil film) were taken as a coupled body.Based on the three-dimensional heat transfer numerical simulation of the coupled body,a coupled three-dimensional calculation model for in-cylinder working process and the combustion chamber components was built with domain decomposition and boundary coupled method,which implements the coupled three-dimensional simulation of in-cylinder working process and the combustion chamber components.The model was applied in the influence investigation of the space non-uniformity in heat transfer among combustion chamber components on the generation of in-cylinder emissions:NOx.The results showed that the heat transfer space non-uniformity of combustion chamber components directly influences the formation of in-cylinder NOx.The main area being influenced was the accessory area on the wall,while the influence on the generation of NOx in the central area couold be omitted.

The coefficient of variation representing the non-uniformity of the strain field and its implication to detect earthquake precursor

Before the major earthquake or rock damage occurs,it is often accompanied by a sudden change in the degree of non-uniformity of the strain field.In order to find a stronger non-uniformity signal before the rock failure,the coefficient of variation(Cv)is examined and reformed in this study.We test the Cv calculation way of the"normal-abnormal"model proposed in the previous studies.Based on the analysis of the physical process of rock failure and its relationship to the shear strain field,we construct a new way to calculate the Cv value.The variation of shear strain field on rock sample with the increase of stress is obtained by the digital speckle correlation method(DSCM).The new Cv value calculation way is used to study the non-uniformity of the spatial distribution for the shear strain field.The results show that this Cv calculation way can get more obvious abnormal signals.When the number of observation points are limited,the specific distribution of points can increase the signal strength,which may provide reference for the research on precursor detection of earthquakes.

Non-Uniformity of Heavy-Ion Beam Irradiation on a Direct-Driven Pellet in Inertial Confinement Fusion

Heavy-ion-driven fusion （HIF） is a scheme to achieve inertial confinement fusion （ICF）. Investigation of the non-uniformity of heavy-ion beam （HIB） irradiation is one of the key issues for ICF driven by powerful heavy-ion beams. Ions in HIB impinge on the pellet surface and deposit their energy in a relatively deep and wide area. Therefore, the non-uniformity of HIB irradiation should be evaluated in the volume of the deposition area in the absorber layer. By using the OK1 code with some corrections, the non-uniformity of heavy-ion beam irradiation for the different ion beams on two kinds of targets were evaluated in 12-beam, 20-beam, 60-beam and 120-beam irradiation schemes. The root-mean-square （RMS） non-uniformity value becomes aRMS = 8.39% in an aluminum mono-layer pellet structure and aRMS = 6.53% in a lead-aluminum layer target for the 12-uranium-beam system. The RMS non-uniformity for the lead-aluminum layer target was lower than that for the mono-layer target. The RMS and peak-to-valley （PTV） non-uniformities are reduced with the increase in beam number, and low at the Bragg peak layer.

Robust and Reusable Fuzzy Extractors from Non-Uniform Learning with Errors Problem

Afuzzy extractor can extract an almost uniformrandom string from a noisy source with enough entropy such as biometric data.To reproduce an identical key from repeated readings of biometric data,the fuzzy extractor generates a helper data and a random string from biometric data and uses the helper data to reproduce the random string from the second reading.In 2013,Fuller et al.proposed a computational fuzzy extractor based on the learning with errors problem.Their construction,however,can tolerate a sub-linear fraction of errors and has an inefficient decoding algorithm,which causes the reproducing time to increase significantly.In 2016,Canetti et al.proposed a fuzzy extractor with inputs from low-entropy distributions based on a strong primitive,which is called digital locker.However,their construction necessitates an excessive amount of storage space for the helper data,which is stored in authentication server.Based on these observations,we propose a new efficient computational fuzzy extractorwith small size of helper data.Our scheme supports reusability and robustness,which are security notions that must be satisfied in order to use a fuzzy extractor as a secure authentication method in real life.Also,it conceals no information about the biometric data and thanks to the new decoding algorithm can tolerate linear errors.Based on the non-uniform learning with errors problem,we present a formal security proof for the proposed fuzzy extractor.Furthermore,we analyze the performance of our fuzzy extractor scheme and provide parameter sets that meet the security requirements.As a result of our implementation and analysis,we show that our scheme outperforms previous fuzzy extractor schemes in terms of the efficiency of the generation and reproduction algorithms,as well as the size of helper data.

Radial Basis Approximations Based BEMD for Enhancement of Non-Uniform Illumination Images

An image can be degraded due to many environmental factors like foggy or hazy weather,low light conditions,extra light conditions etc.Image captured under the poor light conditions is generally known as non-uniform illumination image.Non-uniform illumination hides some important information present in an image during the image capture Also,it degrades the visual quality of image which generates the need for enhancement of such images.Various techniques have been present in literature for the enhancement of such type of images.In this paper,a novel architecture has been proposed for enhancement of poor illumination images which uses radial basis approximations based BEMD(Bi-dimensional Empirical Mode Decomposition).The enhancement algorithm is applied on intensity and saturation components of image.Firstly,intensity component has been decomposed into various bi-dimensional intrinsic mode function and residue by using sifting algorithm.Secondly,some linear transformations techniques have been applied on various bidimensional intrinsic modes obtained and residue and further on joining the transformed modes with residue,enhanced intensity component is obtained.Saturation part of an image is then enhanced in accordance to the enhanced intensity component.Final enhanced image can be obtained by joining the hue,enhanced intensity and enhanced saturation parts of the given image.The proposed algorithm will not only give the visual pleasant image but maintains the naturalness of image also.

Simulation of Corrosion-Induced Cracking of Reinforced Concrete Based on Fracture Phase Field Method

Accurate simulation of the cracking process caused by rust expansion of reinforced concrete(RC)structures plays an intuitive role in revealing the corrosion-induced failure mechanism.Considering the quasi-brittle fracture of concrete,the fracture phase field driven by the compressive-shear term is constructed and added to the traditional brittle fracture phase field model.The rationality of the proposed model is verified by a mixed fracture example under a shear displacement load.Then,the extended fracture phase model is applied to simulate the corrosion-induced cracking process of RC.The cracking patterns caused by non-uniform corrosion expansion are discussed for RC specimens with homogeneous macroscopically or heterogeneous with different polygonal aggregate distributions at the mesoscopic scale.Then,the effects of the protective layer on the crack propagation trajectory and cracking resistance are investigated,illustrating that the cracking angle and cracking resistance increase with the increase of the protective layer thickness,consistent with the experimental observation.Finally,the corrosion-induced cracking process of concrete specimens with large and small spacing rebars is simulated,and the interaction of multiple corrosion cracking is easily influenced by the reinforcement spacing,which increases with the decrease of the steel bar interval.These conclusions play an important role in the design of engineering anti-corrosion measures.The fracture phase field model can provide strong support for the life assessment of RC structures.

Comparative Analysis of Energy Characteristics of Two Southwest Vortices in Sichuan Under Similar Circulation Backgrounds

Based on ERA5 reanalysis data,the present study analyzed the thermal energy development mechanism and kinetic energy conversion characteristics of two extreme rainstorm processes in relation to the shallow southwest vortex in the warm-sector during a“rain-generated vortex”process and the deep southwest vortex in a“vortex-generated rain”process.The findings were as follows:(1)During the extreme rainstorm on August 11,2020(hereinafter referred to as the“8·11”process),intense surface heating and a high-energy unstable environment were observed.The mesoscale convergence system triggered convection to produce heavy rainfall,and the release of latent condensation heat generated by the rainfall promoted the formation of a southwest vortex.The significant increase(decrease)in atmospheric diabatic heating and kinetic energy preceded the increase(decrease)in vorticity.By contrast,the extreme rainstorm on August 16,2020(hereinafter referred to as the“8·16”process)involved the generation of southwest vortex in a low-energy and highhumidity environment.The dynamic uplift of the southwest vortex triggered rainfall,and the release of condensation latent heat from rainfall further strengthened the development of the southwest vortex.The significant increase(decrease)in atmospheric diabatic heating and kinetic energy exhibited a delayed progression compared to the increase(decrease)in vorticity.(2)The heating effect around the southwest vortex region was non-uniform,and the heating intensity varied in different stages.In the“8·11”process,the heating effect was the strongest in the initial stage,but weakened during the vortex's development.On the contrary,the heating effect was initially weak in the“8·16”process,and intensified during the development stage.(3)The available potential energy of the“8·11”process significantly increased in kinetic energy converted from rotational and divergent winds through baroclinic action,and the divergent wind energy continued to convert into rotational wind energy.By contrast,the“8·16”process involved the conversion of rotational wind energy into divergent wind energy,which in turn converted kinetic energy back into available potential energy,thereby impeding the further development and maintenance of the southwest vortex.

Line Patterns Segmentation in Blurred Images Using Contrast Enhancement and Local Entropy Thresholding

Finger vein extraction and recognition hold significance in various applications due to the unique and reliable nature of finger vein patterns. While recently finger vein recognition has gained popularity, there are still challenges associated with extracting and processing finger vein patterns related to image quality, positioning and alignment, skin conditions, security concerns and processing techniques applied. In this paper, a method for robust segmentation of line patterns in strongly blurred images is presented and evaluated in vessel network extraction from infrared images of human fingers. In a four-step process: local normalization of brightness, image enhancement, segmentation and cleaning were involved. A novel image enhancement method was used to re-establish the line patterns from the brightness sum of the independent close-form solutions of the adopted optimization criterion derived in small windows. In the proposed method, the computational resources were reduced significantly compared to the solution derived when the whole image was processed. In the enhanced image, where the concave structures have been sufficiently emphasized, accurate detection of line patterns was obtained by local entropy thresholding. Typical segmentation errors appearing in the binary image were removed using morphological dilation with a line structuring element and morphological filtering with a majority filter to eliminate isolated blobs. The proposed method performs accurate detection of the vessel network in human finger infrared images, as the experimental results show, applied both in real and artificial images and can readily be applied in many image enhancement and segmentation applications.

An improved binarization algorithm of wood image defect segmentation based on non-uniform background

In this study,an image binarization optimization algorithm,based on local threshold algorithms,is proposed because global and traditional local threshold segmentation algorithms cannot effectively address the problems of nonuniform backgrounds of wood defect images.The proposed algorithm calculates the threshold by the mean,standard deviation and the extreme value of the window.The results indicate that this modified algorithm enhances the image segmentation for wood defect images on a complex background,which is much superior to the global threshold algorithm and the Bernsen algorithm,and slightly better than the Niblack algorithm and Sauvola algorithm.Compared with similar models,the algorithm proposed in this paper has higher segmentation accuracy,as high as 92.6%for wood defect images with a complex background.

Experimental Study on Occupant's Thermal Responses under the Non-uniform Conditions in Vehicle Cabin during the Heating Period

The existing investigations on thermal comfort mostly focus on the thermal environment conditions, especially of the air-flow field and the temperature distributions in vehicle cabin. Less attention appears to direct to the thermal comfort or thermal sensation of occupants, even to the relationship between thermal conditions and thermal sensation. In this paper, a series of experiments were designed and conducted for understanding the non-uniform conditions and the occupant＇s thermal responses in vehicle cabin during the heating period. To accurately assess the transient temperature distribution in cabin in common daily condition, the air temperature at a number of positions is measured in a full size vehicle cabin under natural winter environment in South China by using a discrete thermocouples network. The occupant body is divided into nine segments, the skin temperature at each segment and the occupant＇s local thermal sensation at the head, body, upper limb and lower limb are monitored continuously. The skin temperature is observed by using a discrete thermocouples network, and the local thermal sensation is evaluated by using a seven-point thermal comfort survey questionnaire proposed by American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc（ASHRAE） Standard. The relationship between the skin temperature and the thermal sensation is discussed and regressed by statistics method. The results show that the interior air temperature is highly non-uniform over the vehicle cabin. The locations where the occupants sit have a significant effect on the occupant＇s thermal responses, including the skin temperature and the thermal sensation. The skin temperaWa-e and thermal sensation are quite different between body segments due to the effect of non-uniform conditions, clothing resistance, and the human thermal regulating system. A quantitative relationship between the thermal sensation and the skin temperature at each body segment of occupant in real life traffic is presented. The investigation result indicates that the skin temperature is a robust index to evaluate the thermal sensation. Applying the skin temperature to designing and controlling parameters of the heating, ventilation and air conditioning（HVAC） system may benefit the thermal comfort and reducing energy consumption.

Turbulent Variance Characteristics of Temperature and Humidity over a Non-uniform Land Surface for an Agricultural Ecosystem in China

This paper describes the application of the variance method for flux estimation over a mixed agricultural region in China. Eddy covariance and flux variance measurements were conducted in a near-surface layer over a non-uniform land surface in the central plain of China from 7 June to 20 July 2002. During this period, the mean canopy height was about 0.50 m. The study site consisted of grass （10% of area）, beans （15%）, corn （15%） and rice （60%）. Under unstable conditions, the standard deviations of temperature and water vapor density （normalized by appropriate scaling parameters）, observed by a single instrument, followed the Monin-Obukhov similarity theory. The similarity constants for heat （CT） and water vapor （Cq） were 1.09 and 1.49, respectively. In comparison with direct measurements using eddy covariance techniques, the flux variance method, on average, underestimated sensible heat flux by 21% and latent heat flux by 24%, which may be attributed to the fact that the observed slight deviations （20% or 30% at most） of the similarity ＂constants＂ may be within the expected range of variation of a single instrument from the generally-valid relations.

Numerical study on non-uniform heat transfer deterioration of supercritical RP-3 aviation kerosene in a horizontal tube

The convective heat transfer of supercritical-pressure RP-3(Rocket Propellant 3)aviation kerosene in a horizontal circular tube has been numerically studied,focusing mainly on the non-uniform heat transfer deterioration along the circumferential direction.The governing equations of mass,momentum and energy have been solved using the pressure-based segregated solver based on the finite volume method.The re-normalization group(RNG)k-εturbulence model with an enhanced wall treatment was selected.Considering the heat conduction in the solid wall,the mechanism of heat transfer deterioration and the buoyancy effect on deteriorated heat transfer were discussed.The evolution of secondary flow was analyzed.Effects of the outer-wall heat flux,mass flux,pressure and tube thermal conductivity on heat transfer were investigated.Moreover,the buoyancy criterion and the heat transfer correlation were obtained.Results indicate that the poor flow performance of near-wall fluid causes the pseudo-film boiling,further leads to the heat transfer deterioration.The strong buoyancy has an effect of enhancing the heat transfer at the bottom of tube,and weakening the heat transfer at the top of tube,which results in the non-uniform inner-wall temperature and heat flux distributions.Decreasing the ratio of outer-wall heat flux and mass flux,increasing the pressure could weaken the heat transfer difference along the circumferential direction,while the effect of thermal conductivity of tube on the circumferential parameters distributions is more complicated.When the buoyancy criterion of(Grq/Grth)max≤0.8 is satisfied,the effect of buoyancy could be ignored.The new correlations work well for non-uniform heat transfer predictions.

A simple algorithm to improve the performance of the WENO scheme on non-uniform grids

This paper presents a simple approach for improving the performance of the weighted essentially nonoscillatory(WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifthorder WENO-JS(WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable.The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On nonuniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime,the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme.

Structural strength deterioration of coastal bridge piers considering non-uniform corrosion in marine environments

In the aggressive marine environment over a long-term service period,coastal bridges inevitably sustain corrosion-induced damage due to high sea salt and humidity.This paper investigates the strength reduction of coastal bridges,especially focusing on the effects of non-uniform corrosion along the height of bridge piers.First,the corrosion initiation time and the degradation of reinforcement and concrete are analyzed for bridge piers in marine environments.To investigate the various damage modes of the concrete cover,a discretization method with fiber cells is used for calculating time-dependent interaction diagrams of cross-sections of the bridge piers at the atmospheric zone and the splash and tidal zone under a combination of axial force and bending moment.Second,the shear strength of these aging structures is analyzed.Numerical simulation indicates that the strength of a concrete pier experiences dramatic reduction from corrosion initiation to the spalling of the concrete cover.Strength loss in the splash and tidal zone is more significant than in the atmospheric zone when structures’service time is assumed to be the same.

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.