Value Of Scale Parameter Research Articles

Mixed FEM implementation of three-point bending of the beam with an edge crack within strain gradient elasticity theory

This paper considers the problem of symmetrical three-point bending of a prismatic beam with an edge crack. The solution is obtained by the mixed finite element method within the simplified Toupin–Mindlin strain gradient elasticity theory. A mixed variational formulation of the boundary value problem for displacements–strains–stresses and their gradients is applied, simplifying the choice of approximating functions. The concept of energy balance is adopted to calculate the energy release rate with a virtual increase in crack length. The increment of the potential energy of an elastic body is determined by accounting for the strain and stress gradient contribution. Numerical calculations were performed using a quasi-uniform triangular mesh of the cross-type. The mesh refinement was applied in the vicinity of the crack tip, at the concentrated support, and the point of application of the transverse force, and uniform mesh partitioning was utilized in the rest of the beam. The fine-mesh analysis was carried out on the successively condensed meshes in the stress concentration domain for different values of the length scale parameter. The crack opening displacements and the distribution of strains and Cauchy stresses for various values of the length scale parameter are presented. An increase in this parameter increases the stiffness of the crack, which leads to a decrease in the crack opening displacements and a smooth closure of its faces at the crack tip. In addition, accounting for the scale parameter reduces the calculated values of strains and stresses near the crack tip. Based on the energy balance criterion, local fracture parameters such as the release rate of elastic energy at the crack tip and the stress intensity factor are determined for different values of the mesh step. The numerical calculations indicate the convergence of the obtained approximations. The main feature of solutions, which includes the strain gradient contribution, is the decrease in the values of the calculated parameters associated with the fracture energy compared to the classical elasticity theory.

Evaluation of reliability and characteristics measure using exponential decay distribution of non‐series parallel network

AbstractThe purpose of this paper is to analyze the reliability and characteristics of the non‐series‐parallel network. The illustrative non‐series parallel networks, that is, the Twin T‐Network and the network with six components of resistor and capacitor, have been converted into a series of parallel network combinations by use of the minimal path technique. The stochastic variables related to component failure rate have been demonstrated using exponential decay distribution. We obtain the reliability, mean time to failure, and sensitivity with the use of the exponential decay distribution. The reliability and its attributes are examined by assuming different values of scale and shape parameters at different points of time. Using Owen's methodology, we obtained minimal signature, tail signature, expected time, expected cost, and Barlow‐Proschan index were determined. The performance of all reliability characteristics has been observed graphically for arbitrary values of parameters. A real‐world example is used to introduce the point.

Water retention curves of sandy soils obtained from direct measurements, particle size distribution, and infiltration experiments

AbstractAccurate information about soil water retention curves (SWRCs) of sands is essential for evaluating groundwater recharge and vulnerability to contamination in many shallow sandy aquifers which are widespread on post glacial areas in Northern Europe and North America. Pedotransfer functions (PTFs) allow to estimate SWRC from basic physical characteristics of soils, such as textural composition. However, in the case of clean sands which are dominated by a single textural fraction, PTFs should be based on more detailed information given by the particle size distribution. In this study we evaluated three parametric PTFs, which estimate parameters of the van Genuchten SWRC based on empirical correlations to the parameters of soil particle size distribution, and five semi‐physical PTFs, which derive the pore size distribution from particle size distribution. PTFs were compared to SWRCs fitted to the results of drainage experiments on sandy soil samples from six locations in Gdańsk region (northern Poland). Although in all samples the content of silt and clay fractions was low (<3.5%), the differences in actual content of fines strongly influenced the shape of SWRC. In contrast, the amount of gravel fraction (varying from 1% to 35%) did not have significant effect on SWRC. Semi‐physical PTFs were found to be more accurate than parametric PTFs. The best overall performance was shown by the semi‐physical Chang and Cheng PTF. Among the parametric PTFs the best accuracy was obtained with the Schaap and Bouten method. However, all considered functions showed limited accuracy in higher suction range. Additionally, infiltration experiments were performed on four sites. SWRCs were obtained from ring infiltrometer tests using the Beerkan estimation of soil transfer parameters (BEST) method and from the tension infiltrometer (TI) tests using numerical solution of the inverse problem based on the Richards equation. In almost all cases the wetting SWRCs were characterized by higher values of the pressure scaling parameter α compared to SWRCs measured in drainage experiments, which is consistent with the well‐known phenomenon of hysteresis in soils. However, the BEST method resulted in significantly higher α and hydraulic conductivity Ks than TI, probably due to activation of the largest soil pores during ponded infiltration.

A derivative-free phase-field theory for capturing local buckling induced damage in architected plates

Architected plates are commonly used to enhance the load-bearing capacity and stability of structures such as ship hulls and aircraft components. Local buckling-induced damage in these plates are critical events that occur when the plate is subjected to external transverse loading. Performing the analysis to accurately capture this damage and estimation of local buckling load requires explicit 3D- modeling of the architected plate, which is computationally expensive. We address this issue by proposing a derivative-free phase-field theory to capture local buckling-induced damage in architected plates. Here the architected plate is modeled using the shear deformable plate theory and the internal web-core structure is preserved in terms of length scale parameters. The numerical value of the length scale parameter depends on the shape of the webcore structure so that any complicated shape of the webcore is handled based on this parameter. This approach makes the analysis simple to predict the damage behavior with a lesser number of degrees of freedom. The extra energy required to attenuate the zero energy mode-induced oscillations in the solution is estimated from the in-plane buckling analysis of the architected plate. To showcase the efficiency, simulations are conducted based on the proposed approach with different loading cases. The local buckling load and the induced damage behavior of the webcore are compared with the 3D-finite element solution obtained from ABAQUS. The comparison shows that the damage variable is an equivalent estimator of out-of-plane stretch of the webcore in predicting the buckling behavior of the webcore with reduced computational cost and time.

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

Influence of CO2 laser surface treatment of basalt fibers on the mechanical properties of epoxy/basalt composites

AbstractIn fiber reinforced composite materials, the interfacial strength between the fibers and the matrix plays a key role in controlling the stress transfer and damage mechanisms of the composite. In this study, CO2 laser surface treatment of the fibers was investigated as a potential sustainable substitute for conventional chemical treatments, that can be costly and have negative environmental effects. The influence of the laser treatment on basalt fiber fabric was comprehensively investigated. The fibers were subjected to different laser power levels and characterized from a morphological and mechanical point of view. From optical and scanning electron microscopy, it was observed that the treated fibers manifested increased surface roughness along with spots of fused and bonded fibers. Individual treated fibers exhibited improved tensile properties with increased values of scale parameter (by about 21%) in the case of a laser power equal to 1.04 W/mm2, and no substantial changes in Young's modulus. The treated fibers were subsequently used in the preparation of epoxy‐based microcomposites, and microdebonding tests revealed an increase in the interfacial shear strength (IFSS) up to 8%. Therefore, this work proved that a laser surface treatment of basalt fibers is a valid alternative to conventional fiber surface modification to enhance the mechanical compatibility between fibers and matrix, and therefore to improve the mechanical performances of basalt fiber composites.Highlights Failure in composites due to weak interfacial adhesion with epoxy. CO2 laser treatment of basalt fibers to enhance interfacial adhesion. Treated fibers exhibit improved tensile properties. Treated fibers manifested improved interfacial shear strength (IFSS, +8%).

Mapping burned areas in Thailand using Sentinel-2 imagery and OBIA techniques

Monitoring burned areas in Thailand and other tropical countries during the post-harvest season is becoming increasingly important. High-resolution remote sensing data from Sentinel-2 satellites, which have a short revisit time, is ideal for accurately and efficiently mapping burned regions. However, automating the mapping of agriculture residual on a national scale is challenging due to the volume of information and level of detail involved. In this study, a Sentinel-2A Level-1C Multispectral Instrument image (MSI) from February 27, 2018 was combined with object-based image analysis (OBIA) algorithms to identify burned areas in Mae Chaem, Chom Thong, Hod, Mae Sariang, and Mae La Noi Districts in Chiang Mai, Thailand. OBIA techniques were used to classify forest, agricultural, water bodies, newly burned, and old burned regions. The segmentation scale parameter value of 50 was obtained using only the original Sentinel-2A band in red, green, blue, near infrared (NIR), and Normalized Difference Vegetation Index (NDVI). The accuracy of the produced maps was assessed using an existing burned area dataset, and the burned area identified through OBIA was found to be 85.2% accurate compared to 500 random burned points from the dataset. These results suggest that the combination of OBIA and Sentinel-2A with a 10 m spatial resolution is very effective and promising for the process of burned area mapping.

Techno-economic assessment and wind energy potential of Nagad in Djibouti

The use of small scaled horizontal and vertical axis wind turbines in urban installation is increasing over the world. However, in Djibouti, the latter is still in the development phase. The paper presents a techno-economical analysis and wind energy potential for the period of five years (2015-2019) in Nagad based on actual measured wind speed data collected every 10 min at 10 m height. The energy pattern factor method has been used to estimate the Weibull parameters. With this method, the mathematical complexity is reduced with a minimization of the error at any heights and locations when calculating the wind power density. At 50 m height, the shape parameter showed a small variation for different periods. The scale parameter values of 7.78 m/s and 4.8 m/s were obtained in the hot and cold seasons, respectively. The results showed that the Nagad site is suitable for wind power development. According to the economic viability, RX30, Vestas V20, Enercon, Nordex N27, and Vestas V44 wind turbines are recommended for the Nagad site due to their low energy price ranging from 0.05$/kWh to 0.31$/kWh. This is 2-6 times cheaper than the average local tariff of electricity in Djibouti.

Estimation of Weibull Parameters for Wind Energy Application in Al Shihabi City

Wind energy is environmentally friendly and low-cost energy compared to other renewable energy technologies and is easily exploited by different users. Al-Shihabi ia a city with high potential wind resources. The aim of the research is to assess the mean winds and Weibull parameters such as shape parameter and scale parameter. The assessment helps in understanding the wind behavior for the application of the wind energy project. Field data for year 2019 were collected at an interval of 10 minutes at altitudes of 50 and 100 meters. Then the data was analyzed using the statistical Weibull distribution function. The annual average of monthly means at 50 m and 100 m heights are 5.04 m/s and 5.56 m/s, respectively. While the annual average of monthly maximum winds at 50 m and 100 m heights were 13.65 m/s and 15.05 m/s. The larger values were in summer months. The mean wind speed and maximum wind speed values suggest that Al-Shihabi City is a promising site to install a wind farm. Estimating Weibull distribution parameters showed for 50 m and 100 heights that the annual average of monthly values of shape parameter and scale parameter were 5.6840, and 2.4035, respectively. At 100 m height, It was found that the annual average of monthly values of shape parameter, scale parameter were 6.2634 and 2.3215, respectively. It can also be noted that the values of parameters are higher in summer than in winter. It was found that the Weibull distribution is applicable to describe the probability density of wind speed.

Characterization of Raindrop Size Distribution over Santa Clara Valley

This study presents a year-long (January 2019–April 2020) analysis of the Z–R relationship and drop size distribution (DSD) scaling parameters for size, concentration, and shape of rain events over Santa Clara Valley, CA. External influences were analyzed based on synoptic variability and seasons. For the former, 850 hPa winds were separated into groups based on direction and magnitude. Results show that greater drop size, lower concentration, and larger shape parameters for spring, while winter and fall showed smaller drop sizes, higher concentrations, and smaller shape parameters. For synoptic variability, southeasterly-to-southwesterly flow was associated with larger drop sizes, larger concentrations, and smaller shape parameters relative to northwesterly flow. Differences in the DSD scaling parameter values and Z–R relationship were also observed between strong and weak low-level flow. The results of this study suggest that it is beneficial to derive specific microphysical relationships based on seasons and different synoptic events to improve radar rain rate retrieval algorithms using the Z–R relationship.

A novel [formula omitted] control for T-S fuzzy systems with transform matching membership functions

This article studies H∞ optimization issue of Takagi-Sugeno (T-S) fuzzy systems with an improved transform matching membership functions (MFs) approach. Different from existing transform matching MFs control method, a polynomial disassembling strategy is first proposed for T-S fuzzy systems. Then, combining with parameterized linear matrix inequality (LMI) technique which can remove the inequality constraints between the MFs in systems and fuzzy controllers, sufficient conditions are presented to maintain asymptotic stability and desired H∞ performance for studied plants. Afterwards, a novel MFs online optimization algorithm is proposed for the first time to automatically adjust the values of scaling and bias parameters so as to realize better H∞ performance. In contrast to the traditional control using improved matching MFs, the practical behavior of disturbance attenuation index is reduced efficiently. Additionally, the proposed MFs online optimization algorithm is capable of sustaining the desired H∞ performance when the disturbance channels exist modeling errors, i.e., the practical disturbance attenuation index γ is still less than the preset value. For guaranteeing the convergence of cost function, sufficient condition is obtained based upon Lyapunov stability theory. Finally, two demonstrative simulations are presented to confirm the advantages and benefits of the novel MFs online optimization control strategy.

Wind and solar resource complementarity and its viability in wind/PV hybrid energy systems in Machakos, Kenya

Integration of intermittent renewable energy resources provides the potential to mitigate the impact of the variability of independent sources. The intermittent nature creates stability, reliability and power quality problems in power grids. Wind and solar energies are the most viable resources whose complementarity could be deployed in the development of hybrid renewable energy systems to enhance their performance. In this study, wind-solar resource complementarity is investigated to establish its viability in hybrid energy systems in Machakos, a rural-urban town whose geographical location is 1°31′S, 37,016′E in the Eastern region of Kenya. The study findings could be fundamental in energy planning and developments in the area to provide more flexible and dependable hybrid energy systems which would supply energy services under constantly changing conditions. Ground measurements included PVGIS simulation and experimental tools installed at a height of 20 m. Weibull's distribution model and energy potential of wind resource are investigated to characterize wind resource for energy generation. The Wind distribution revealed a positively skewed profile with scale (c) and shape (k) parameter values of 2.68 m/s (≈ 3.0 m/s) and 1.9 at a mean wind speed of 2.47 m/s, respectively. The findings revealed a wind power density of 17 W/m2 at an average wind speed of 3.0 m/s. Further, an annual solar insolation of 2130 kWh/m2 with monthly average of 178 kWh/m2 translating to daily insolation of 5.8 kWh/m2 at an installation capacity of 1 kWp is reported. The months with minimum and maximum solar insolation were July and March which recorded monthly averages of 145 kWh/m2 and 213 kWh/m2 translating to a daily average of 4.7 kWh/m2 and 6.9 kWh/m2, respectively. The understanding gained from this study could be useful to the renewable energy research community and can be extended to stakeholders in PV and wind energy systems for micro grids and utility applications.

Wind Energy Analysis, Utilizations, And Economic Viability For Different Applications In Iraq

Thirty years of wind data, for 22 stations and a experimental locations inside Iraq, were used to calculate the potential of wind energy in Iraq. A full statistical analysis has been carried out for five selected stations which involved hourly wind speed, monthly average speed, annual values of shape and scale parameters of normal and weibull distribution. Frequency distribution, speed distribution, with their individual and cumulative probability pattern, are analyzed. It can be seen that 24% of the country enjoys monthly wind speed greater than 4 m/sec. Therefore, wind energy is sufficient for water pumping or electricity supply for remote areas in Iraq. Here we examine the viability of utilizing wind energy to supplement solar energy by comparing the supply and demand for different utilizations. The economic viability of wind energy visa vis traditional energy resources is also discussed.

A Minimum Discrepancy Method for Weibull Distribution Parameter Estimation

The most applied principles for parameter estimation are maximum likelihood and least square error. This paper presents a new principle with regard to the parameter estimation of the three-parameter Weibull distribution. By transforming the cumulative distribution function, constructed is a mapping from the value of the random variable and its corresponding cumulative distribution probability to the scale parameter. The scale parameter estimated by such a mapping is the random variable value and the corresponding cumulative distribution probability dependent when the shape parameter and/ or location parameter applied in the mapping is subject to error. Given a set of random variable values or a set of sample values, a larger error in the shape/location parameter brings about larger differences between the scale parameter values obtained with the individual random variable values or sample values, respectively. Based on such a causal relationship between the discrepancy and the shape/ location parameter value applied in the mapping relation, a new parameter estimation method is proposed. For the Weibull distribution parameter estimation according to a set of sample values, the right shape parameter and location parameter are those minimizing the discrepancy between the scale parameter values obtained with the individual sample values, respectively. Case studies demonstrate that the proposed method outperforms the maximum likelihood method and the Weibull plot-based least squares method.

Free vibration analysis of a nanoscale FG-CNTRCs sandwich beam with flexible core: Implementing an extended high order approach

The present work uses an extended high-order technique to examine free vibration analysis of Nanoscale CNTRCs sandwich beams. In this study, face sheets are made of PMMA reinforced with single-wall carbon nanotubes, and the honeycomb structure is selected for the flexible core. First-order shear deformation theory is utilized for the skins. The so-called high-order sandwich panel theory (HSAPT) is used for the core to consider the transverse shear strains in theoretical formulations. Then, skins and cores are subjected to the modified couple stress theory (MCST). For obtaining equations and related boundary conditions, the Hamilton principle is used. Additionally, three forms of beam support are examined: simply supported on both sides (SS), simply supported on one side and clamped supported on the other (SC), and clamped supported on both sides (CC). The material properties in skins are simulated to be graded in the thickness direction. The shooting method, as the numerical solution, is adopted to solve the governing differential equations to determine the natural frequencies of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams in which boundary conditions are converted into initial conditions. Compared to other techniques, this method is a reliable way to solve complicated equations and boundary conditions; therefore, its applicability, convergence, and accuracy are verified by comparing obtained results with those existing ones in the literature. Furthermore, the influences of distribution pattern and volume fraction of carbon nanotubes, boundary conditions, the size dependency parameter, core thickness, and slenderness ratio on vibration responses are discussed. It is concluded that natural frequencies rise with an increase in skin thicknesses and length scale parameter values while decreasing when beams’ length and core thicknesses increase.

An asymptotic consistent couple stress theory for the three‐dimensional free vibration analysis of functionally graded microplates resting on an elastic medium

Based on the three‐dimensional (3D) consistent couple stress theory (CCST), we develop an asymptotic theory for the free vibration analysis of functionally graded (FG) microplates resting on an elastic medium. The material properties of the FG microplate are assumed to obey a power‐law distribution of the volume fractions of the constituents in the thickness direction, for which the effective material properties are estimated using the rule of mixtures. The interactions between the FG microplate and its foundation medium are simulated using a Winkler–Pasternak foundation. Carrying out the asymptotic expansion method, we obtain recursive sets of motion equations for various order problems. The CCST‐based classical plate theory is derived as a first‐order approximation of the 3D CCST. The asymptotic theory is shown to converge rapidly and to be in excellent agreement with the exact solutions for FG macroplates reported in the literature when the value of the material length scale parameter is assigned to be zero.

Energy, exergy, economic, environmental (4E) and frequency distribution analysis of train wind gust with real-time data for energy harvesting

The wind gust velocity of trains are above the cut in speed of wind turbines. Multiple cases studies estimate the available wind energy and potential electrical output with numerical and computational models. These gust velocities are dynamic nature. This work collects real time data of wind gust using data acquisition, conducted 4E and Weibull frequency distribution analysis. The acquired data is further used as a velocity signal to Simulink and wind emulator wind energy harvesting systems. This distinguishes in producing benchmarking results when compared with numerical and computational models. From data interpretation and analysis, the wind gust are non-uniform and gust velocity ranges from 2.3 to 7.1 m s−1 is recorded with a Weibull scale parameter value(A) of 5.54 m s−1. The maximum power available for harvesting is after considering Betz limit is 159.6 W, whilst Simulink and emulator energy harvesting systems produces 126.4 W and 123.08 W with a maximum exergy efficiency of 49.38 and 49.14%. The estimated wind energy available for 1KM range with wind energy systems on both side of traction poles is about 3.3 KW/KM. The compared environmental and economic analysis reconfirms the feasibility of the proposed model with capacity factor 5.74%. Other findings are the corresponding variation in output with respect to dynamic-wind velocities is limited due to inertia and stored kinetic energy of system, the role of location, weather statistics and influence of tail winds in shaping wind gust velocity is also adjudged as crucial factors.

Impact of Technology Change on Productivity Difference between Tissue Culture and Non-Tissue Culture Pointed Gourd in Gujarat

This paper attempted an economic analysis of tissue-culture-pointed and non-tissue-culture-pointed gourd apart from studying the constraints the tissue-culture growing farmers faced. The coefficient of determination (R ) was 0.629 and 0.56 for the estimated 2 functions of tissue-culture, and non-tissue-culture technology, respectively. It revealed that the contribution of technological change (tissue-culture technology) to the total per hectare output was 26.41 per cent. The contribution of the neutral technological (shift in values of scale parameters) component in the productivity difference was positive (56.39 per cent). In contrast, the nonneutral technological (change in values of parameters) component contributed negatively (-29.98 per cent) to the total difference in the output. Age had a negative and significant impact, while family size had a positive and significant effect on adopting tissueculture technology. The analysis revealed that the tissue-culture technology in pointed gourd cultivation could significantly enhance the farmers' income.

On the static, vibration, and transient responses of micro-plates made of materials with different microstructures

In the present paper, for the first time, theoretical analysis for static, dynamic, and transient responses of micro-plates made of different materials were comprehensively performed. The constitutive relations of analyzed materials satisfy assumptions of the modified couple stress theory. Additionally, values of length scale parameters of studied materials are taken from well-established experimental results presented in the literature. A generalized model of micro-plates made from Cu, Ni, Ti, and Epoxy satisfies assumptions of Reissner–Mindlin plate theory. The formulated generalized boundary value problem is solved by analytical and the radial basis function finite difference (RBF-FD) methods. A convergence study is performed to show the advantages and correctness of the obtained numerical solution. The meshless numerical method is easy to implement and it provides accurate results that are in excellent agreement with analytical solutions. Effects of material parameters, boundary conditions, and assumptions of classical and the modified couple stress theory on the static bending, free vibration and transient responses of micro-plates are presented and discussed.

A gradient electromechanical theory for thin dielectric curved beams considering direct and converse flexoelectric effects

This work presents the development of a unified gradient electromechanical theory for thin flexoelectric beams considering both direct and converse flexoelectric effects. The two-way coupled electromechanical theory is developed starting from 3D variationals formulation by considering an electric field-strain-based free energy function. The formulation incorporates mechanical as well as electrical size effects. The coupled 3D theory is specialized to isotropic materials, and a 1D beam theory for composite flexoelectric curved beams is derived using the classical Kirchhoff assumptions. The beam theory is solved using a novel \(C^2\) continuous finite element framework for different loading and boundary conditions. Our finite element results are verified with analytical solutions for a simply supported flexoelectric beam operating in both actuator and sensor modes. The results are also compared with existing literature for the special case of a passive micro-beam. Our computational framework is subsequently used to perform various parametric studies to analyze the effect of electrical and mechanical length scale parameters, geometric parameters like the radius of curvature, flexoelectric layer thickness etc., on the response of the beam. Also, contribution of converse flexoelectricity in the overall response of the flexoelectric beam is compared with that of the direct effect. Our simulation results predict that the converse effect is significant (\(\approx \) 10–25% of the direct effect) for a wide range of thickness and length scale parameter values. It is also observed that the effective electromechanical coupling coefficient, calculated in terms of the voltage developed across the flexoelectric layer thickness, is higher in flexoelectric materials compared to piezoelectric materials at smaller length scales (thickness of the order of a few microns). Our simulation results also agree well with the trends observed in recent experiments [1].