Finite Difference Numerical Method Research Articles

Study on the Mechanical Behavior of Excavation and Support Construction of Shallow Tunnel

In this paper, the mechanical behavior of excavation and support construction of Weishe tunnel, which is a section of the Yangwu expressway, is studied quantitatively using 3D finite difference numerical simulation method. A sequential excavation method is used and the results show that the vault settlement occurs mainly on the phase of upper bench excavation. The convergences of upper and lower sidewalls occur mainly on the phase of lower bench excavation. During the construction, the surrounding rock pressure in the vault and sidewall of the tunnel decrease. Axial force of anchor reaches the maximum value after the finish of second lining. However, the surrounding rock pressure and internal force of steel arch reach the maximum value after completing the upper bench excavation, and then become as smaller as half of the peak value during the lower bench excavation.

Effects of tool eccentralization on cement-bond-log measurements: Numerical and experimental results

The cement-bond log (CBL) is a conventional and widely used cement quality evaluation technology for vertical wells. With the increase in horizontal wells around the world, the existing cement evaluation technologies are not appropriate. We have explored the possibilities of utilizing CBL in horizontal wells through investigating the effects of a noncentralized tool on CBL measurements. The parallel finite-difference numerical simulation method and experiments in calibration wells were adopted in the study. The numerical and experimental results matched very well, and indicated that the CBL amplitude decreases linearly with increasing tool eccentricity in a well with free pipe (i.e., a cased but uncemented well). For a standard pipe with a diameter of 5.5 in (139.7 mm) and a thickness of 7.72 mm, an eccentricity of [Formula: see text] (17% of the maximum eccentricity) could cause the CBL amplitude to be reduced by about 20%. The numerical simulations of CBL in wells with fluid channels in the cement showed that tool eccentralization could either increase or reduce the CBL amplitude relative to a centered tool, depending on the channel azimuth relative to eccentered direction. To explain this phenomenon, we investigated numerically the polarizations of casing waves in a well with free pipe and in a well with a fluid channel, and casing waves at higher frequencies in a well with free pipe. The relationship between the CBL amplitude and the percentage of cemented area for a borehole-centered tool was also studied. Our results provided some insights into understanding CBL measurements in horizontal wells.

Integrated Wide-Band Power Divider/Combiner at Plasmonic Interfaces

We propose a wide-band power divider/combiner configuration to facilitate light access and manipulation in planar nano-plasmonic devices. Light is incident from a standard silicon nanowire (negligible propagation losses) to be accessed by a nano-plasmonic circuit providing subwavelength confinement. The structure circumvents the inherent losses associated with long distance light propagation in nano-plasmonic splitters as coupling and splitting are performed at the same interface with small splitting losses. Plasmonic slot waveguides (PSWs) are utilized for subwavelength confinement. Two PSWs placed orthogonally to a silicon nanowire are utilized for the power division/combining functionality. The power splitter has been analyzed by the finite-difference time-domain numerical method. The proposed device is of compact size and wide-band operation. The relative displacement of the silicon nanowire from the PSWs is optimized. A splitting of 34%, over most of the wavelength spectrum from 0.8-2.5 μm , is demonstrated.

Beam Manipulating via an Array of Nanoslits Modified by Perpendicular Cuts and Bumps

We report the observation of focusing and deflection phenomena by employing a novel technique to perform phase front profile design in nanoslit-based planar plasmonic lenses and beam deflectors. Introducing perpendicular cuts and bumps to the perforated nanoslits on a thin metallic film is utilized to change the effective depth of the nanoslits which provide the possibility of manipulating the phase front profile based on the propagation property of the surface plasmon polaritons in the metal–insulator–metal waveguides. Using the dispersive finite-difference time-domain numerical method, simulations are conducted to explore the beam focusing and deflection phenomena, and the performance parameters of the lens and beam deflector include the focal length, full-width half-maximum, depth of focus, the efficiency of focusing, and the deflection angle. The whole structure is formed on a planar thin film which is convenient for miniaturization and high density integration besides that it can be fabricated by well-known techniques such as focused ion beam milling.

Application of the two-step Lax and Wendroff FCT and the CE-SE method to flow transport in wall-flow monoliths

Gas dynamic codes are computational tools applied to the analysis of air management in internal combustion engines. The governing equations in one-dimensional elements are approached assuming compressible unsteady non-homentropic flow and are commonly solved applying finite difference numerical methods. These techniques can also be applied to the calculation of flow transport in complex systems such as wall-flow monoliths. These elements are characterized by alternatively plugged channels with porous walls. It filters the particulates when the flow goes through the wall from the inlet to the outlet channels. Therefore, this process couples the solution of every pair of inlet and outlet channels. In this study, the adaptation of the two-step Lax and Wendroff method and the space-time Conservation Element and Solution Element method is performed to be applied in the solution of flow transport in wall-flow monolith channels. The influence on the prediction ability is analysed by a shock-tube test and experimental data obtained under impulsive flow conditions.

Mathematical modeling of heat transfer processes of coal waste combustion in a chamber of automated energy generating complex

The automated energy generating complex allows obtaining heat energy from waste coal-water slurry fuel (WCF) that is a mixture of fine coal particles from coal enrichment wastes with water. The mixture is blown into the swirl chamber under the pressure through the special sprayers. The received heat energy is used in different ways. One of the important issues is to estimate the heat losses through the walls of this chamber. In this paper we solved the boundary problem of mathematical physics to estimate the temperature fields in the walls of the swirl chamber. The obtained solution allows us to estimate the heat losses through the walls of the swirl chamber. The task of the mathematical physics has been solved by a numerical finite-difference method. The method for solving this problem can be used in the calculation of temperature fields and evaluation of heat losses in other thermal power units.

Stability of coal pillar in gob-side entry driving under unstable overlying strata and its coupling support control technique

Considering the situation that it is difficult to control the stability of narrow coal pillar in gob-side entry driving under unstable overlying strata, the finite difference numerical simulation method was adopted to analyze the inner stress distribution and its evolution regularity, as well as the deformation characteristics of narrow coal pillar in gob-side entry driving, in the whole process from entry driving of last working face to the present working face mining. A new method of narrow coal pillar control based on the triune coupling support technique (TCST), which includes that high-strength prestressed thread steel bolt is used to strain the coal on the goaf side, and that short bolt to control the integrity of global displacement zone in coal pillar on the entry side, and that long grouting cable to fix anchor point to constrain the bed separation between global displacement zone and fixed zone, is thereby generated and applied to the field production. The result indicates that after entry excavating along the gob under unstable overlying strata, the supporting structure left on the gob side of narrow coal pillar is basically invalid to maintain the coal-pillar stability, and the large deformation of the pillar on the gob side is evident. Except for the significant dynamic pressure appearing in the coal mining of last working face and overlying strata stabilizing process, the stress variation inside the coal pillar in other stages are rather steady, however, the stress expansion is obvious and the coal pillar continues to deform. Once the gob-side entry driving is completed, a global displacement zone on the entry side appears in the shallow part of the pillar, whereas, a relatively steady fixed zone staying almost still in gob-side entry driving and present working face mining is found in the deep part of the pillar. The application of TCST can not only avoid the failure of pillar supporting structure, but exert the supporting capacity of the bolting structure left in the pillar of last sublevel entry, thus to jointly maintain the stability of coal pillar.

Simulation and modelling of charge transport in dye-sensitized solar cells based on carbon nano-tube electrodes

For a better understanding of the mechanisms of dye-sensitized solar cells (DSSCs), based on carbon nano-tube (CNT) electrodes, a phenomenological model is proposed. For modelling purposes, the meso-scopic porous CNT electrode is considered as a homogeneous nano-crystalline structure with thickness L. The CNT electrode is covered with light-absorbing dye molecules, and interpenetrated by the tri-iodide (I−/I3−) redox couple. A simulation platform, designed to study coupled charge transport in such cells, is presented here. The work aims at formulating a mathematical model that describes charge transfer and charge transport within the porous CNT window electrode. The model is based on a pseudo-homogeneous active layer using drift–diffusion transport equations for free electron and ion transport. Based on solving the continuity equation for electrons, the model uses the numerical finite difference method. The numerical solution of the continuity equation produces current–voltage curves that fit the diode equation with an ideality factor of unity. The calculated current–voltage (J–V) characteristics of the illuminated idealized DSSCs (100 mW cm−2, AM1.5), and the different series resistances of the transparent conductor oxide (TCO) layer were introduced into the idealized simulated photo J–V characteristics. The results obtained are presented and discussed in this paper. Thus, for a series resistance of 4 Ω of the TCO layer, the conversion efficiency (η) was 7.49% for the CNT-based cell, compared with 6.11% for the TiO2-based cell. Two recombination kinetic models are used, the electron transport kinetics within the nano-structured CNT film, or the electron transfer rate across the CNT–electrolyte interface. The simulations indicate that both electron and ion transport properties should be considered when modelling CNT-based DSSCs and other similar systems. Unlike conventional polycrystalline solar cells which exhibit carrier recombination, which limits their efficiency, the CNT matrix (in CNT-based cells) serves as the conductor for majority carriers and prevents recombination. This is because of special conductivity and visible–near-infrared transparency of the CNT. Charge transfer mechanisms within the porous CNT matrix and at the semiconductor–dye–electrolyte interfaces are described in this paper.

Free Convection of Non-Newtonian Nanofluids in Porous media with Gyrotactic Microorganisms

The free convection of non-Newtonian nanofluids along a vertical plate in porous medium is investigated numerically. It is assumed that the medium contains gyrotactic microorganisms along with nanoparticles and the plate is subjected to prescribed temperature, concentration of nanoparticles and density of motile microorganisms. It is further assumed that the plate is impermeable. The governing partial differential equations are reduced to nonlinear ordinary differential equations using similarity transformations. The nonlinear ordinary differential equations are then solved by a finite difference numerical method. The effects of controlling parameters on several dimensionless quantities and numbers of our interest are investigated. The numerical results are compared with the published data and an excellent agreement has been found. It is found that nanofluid and bioconvection parameters have strong effects on local Nusselt, Sherwood and density numbers.

Self-focusing of femtosecond surface plasmon polaritons

We study the propagation of femtosecond pulses in nonlinear metal-dielectric plasmonic waveguiding structures by employing the finite-difference time-domain numerical method. Self-focusing of plasmon pulses is observed for defocusing Kerr-like nonlinearity of the dielectric medium due to normal dispersion. We compare the nonlinear propagation of plasmon pulses along a single metal-dielectric interface with the propagation within a metal-dielectric-metal slot waveguide and observe that nonlinear effects are more pronounced for the single surface where longer propagation length may compensate for lower field confinement.

Investigation of effective EM parameter based on impedance simulation

In order to calculate the equivalent EM parameter of mixed medium, in this paper, we proposes a new method of impedance simulation based on an equivalent circuit. First, the relationship between electromagnetic parameters and capacitance, inductance of material is set up, and then the model that can describe real composite accurately is established. The equivalent EM parameter could be obtained by calculating the equivalent impedance of mixed medium. Comparing this method with the classical formulae and finite-difference numerical method, it is proved that it can be used to calculate the equivalent EM parameter of material. In addition, introducing thin films to simulate the factor such as surface effect, the result of calculation would match with experimental results perfectly. This method has more advantages than classical formulae and numerical method.

A 2-D Barakat-Clark finite difference numerical method and its comparison for water quality simulation modeling

This paper provides an advance numerical algorithm to solve both ordinary and partial differential equations of surface water quality models. It uses finite difference methods and structures explicit or implicit or other process forms to solve the water quality model. This study also considers the stability of solutions to obtain more accurate results among those numerical algorithms. Water quality modeling commonly manifests itself in ordinary and partial differential equations in a realistic world. This study has applied numerical solutions to simulate the changing process of water quality in one and two dimensional spaces or in multiple dimensional spaces. The solutions of these analytical methods are provided in this paper to attest the justifiability of these numerical methods. It demonstrates that the 2-dimensional Barakat-Clark numerical method can be a highly efficient tool in obtaining approximate results of ordinary and partial differential equations, which may prove difficult in finding the accurate solution by using conventional methods. At the same time, the stability analysis corroborated the convergence of those numerical solutions. This study is the first attempt to compare the multiple numerical methods with the 2-D Barakat-Clark method in the water quality modeling process. The results clearly suggest that the Barakat-Clark method is better in reflecting the accuracy of the water quality modeling with stability for hydrological systems.

2-D Numerical Multimedia Environmental Analysis System (NMEAS) for pollution and risk assessment

All pollution issues involve potential impacts on the surrounding interconnected air, water, and soil (i.e., multimedia) environment. Effectively addressing a wide range of multimedia pollution problems is of crucial importance to major socioeconomic sectors. This study presents an attempt to develop a new Numerical Multimedia Environmental Analysis System (NMEAS), which incorporates numerical analyses of pollutants fate and transport in the multimedia environment. Specifically, three different approaches were evaluated for solving two dimensional coupled pollutants advection–diffusion–reaction equations using experimental data from the literature: (a) NMEAS/FEM (numerical finite element method); (b) NMEAS/FDM (numerical finite difference method); and, (c) NMEAS/analytical method. Preliminary validation of the proposed NMEAS method has been conducted through a 2-D case study. In the case study implementing the NMEAS/FEM solution is found to be better than that of NMEAS/FDM, especially at low concentrations of pollutants; and both pollutant fluxes across and distribution in the interconnected compartments are simulated in both numerical spatial and temporal scale for complex multimedia environment. The new NMEAS will be an innovative risk assessment tool aiming to formulate effective strategies of managing environmental impacts on multiple compartments resulting from pollution emissions.

Numerical Study of Hydrogen Trapping: Application to an API 5L X60 Steel

A numerical finite difference method is developed here to solve the diffusion equation for hydrogen in presence of trapping sites. A feature of our software is that an optimization of diffusion and trapping parameters is achieved via a non linear least squares fit. On the other hand, we have demonstrated that usual electrochemical hydrogen permeation tests are enough to assess hydrogen free energies of trapping in the range of −35 kJ/mol to −70 kJ/mol. These conclusions are obtained by assuming the presence of saturable traps in local equilibrium with hydrogen and are validated by means of simulated permeation and degassing transients. In addition, we check our model performing electrochemical hydrogen permeation tests at 30°C, 50°C, and 70°C, on an API 5L X60 as received steel state to study its trapping and diffusion properties considering only one type of trapping site. The binding energies (ΔG) and the trap densities (N) are determined by fitting the theoretical model to the experimental permeation data. The steel presents a high density of weak traps, |ΔG|<35 KJ/mol, namely, N=1.4×10−5 mol cm−3. Strong trapping sites which alter the shape of the permeation transient are also detected; their ΔG values ranged from 57 to 72 KJ/mol.

بررسی مقایسه‌ای فرآیند انتقال جرم بر اساس روش عددی تفاضل محدود و نتایج آزمایشگاهی برای خشک شدن محصول خربزه درختی

In this research, Papaya slices with dimensions of 0.5×2×5 cm3 were dried at 45 °C using a cabinet dryer in which drying air velocity and relative humidity were 0.9 m/s and 30%, respectively. The moisture diffusion coefficient of Papaya was determined in this drying condition. Mass transfer equation with its boundary conditions was solved based on finite difference method. Finally, coefficient of determination and goodness of fitting between the gained theoretical model by solving mass transfer equation, and experimental data were obtained R2=0.996 and RMSE=0.00115. Therefore, finite difference numerical method showed a suitable correlation with low error into the experimental data for solving of mass transfer equation.

Dissolution of Weak Acids Under Laminar Flow and Rotating Disk Hydrodynamic Conditions: Application of a Comprehensive Convection–Diffusion–Migration–Reaction Transport Model

A steady-state mass transfer model that incorporates convection, diffusion, ionic migration, and ionization reaction processes was extended to describe the dissolution of weak acids under laminar flow and a rotating disk hydrodynamics. The model accurately predicted the experimental dissolution rates of benzoic acid, 2-naphthoic acid, and naproxen in unbuffered and monoprotic buffers within the physiological pH range for both hydrodynamic systems. Simulations at various flow rates indicated a cube root dependency of dissolution rate on the flow rate for a given bulk pH value for the laminar hydrodynamic system, as proposed earlier by Shah and Nelson (1975. J Pharm Sci 64(9):1518-1520) for neutral compounds. The model has limitations in its ability to accurately predict the dissolution of weak acids under certain conditions that imposed steep concentration gradients, such as high pH values, and for polyprotic buffer systems that caused the numerical solution to be unstable, suggesting that alternative numerical techniques may be required to obtain a stable numerical solution at all conditions. The model presents many advantages, most notably the ability to successfully predict the complex process under physiological conditions without simplifying assumptions, and therefore accurately representing the system in a comprehensive manner.

A collocated method for the incompressible Navier–Stokes equations inspired by the Box scheme

We present a new finite-difference numerical method to solve the incompressible Navier–Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity–pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier–Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.

Calculation of transient heat transfer through the envelope of an underground cavern using Z-transfer coefficient method

A Z-transfer function method is proposed to calculate transient heat flow through the envelope of an underground cavern in this paper. The envelope of an underground cavern has great heat capacity and high thermal inertia. In order to obtain the correct Z-transfer coefficients, indoor air temperature is expressed as a superposition of mean annual temperature, annual temperature wave and daily temperature wave. Corresponding to the three types of temperatures, three heat flows are calculated separately, and then are superposed. The thermal zones formed by annual temperature wave and daily temperature wave are different; thus, the ratio of amplitude frequency characteristics of the transfer functions is proposed to define the boundaries of the two different thermal systems. After transforming an underground cavern into an equivalent cylinder, the Z-transfer function coefficients of two thermal systems are calculated. Case study shows that heat fluxes at interior surface calculated by Z-transfer coefficient method have good agreement with those by numerical finite difference method, and the deviation between the results from two methods is only 1% of that from numerical method. However, the computation time of Z-transfer coefficient method is less than 1% of that of numerical method.

Nonlinear Analysis of Large Deflection of Elastically-Bossed Sensor Plate under Initial Tension

The nonlinear problem of large deflection of an elastically-bossed layered plate under pretension due to lateral load is studied. The approach follows von Karman plate theory for large deflection for a symmetrically layered isotropic case. The thus derived nonlinear governing equations are solved using a numerical finite difference method with the aid of an iteration scheme. For a nearly monolithic plate with a thin boss, the obtained solutions correlate well with those available in literature for a single-layered flat plate, thus validates the presented approach. For three layered symmetric plates made of typical silicon based materials, various initial tension and lateral pressure are implemented. The results indicate that, edge behavior may appear at both the boss edge and the clamped end of the plate, thereby revealing severe variations for the structural responses. Varying the central boss size and relative thickness may have a sensible influence upon the behavior of the bossed layered plate. Furthermore, lateral pressure appears to have a sensible effect upon the nonlinear behavior of the bossed layered plate. For a relatively large initial tension, however, the pretension effect dominates, yielding a total membrane behavior for the bossed plate, regardless of the size of the center boss, except in the vicinity of the clamped edge.

Free Convection Boundary Layer Flow from a Heated Upward Facing Horizontal Flat Plate Embedded in a Porous Medium Filled by a Nanofluid with Convective Boundary Condition

The steady laminar incompressible free convective flow of a nanofluid over a permeable upward facing horizontal plate located in porous medium taking into account the thermal convective boundary condition is studied numerically. The nanofluid model used involves the effect of Brownian motion and the thermophoresis. Using similarity transformations the continuity, the momentum, the energy, and the nanoparticle volume fraction equations are transformed into a set of coupled similarity equations, before being solved numerically, by an implicit finite difference numerical method. Our analysis reveals that for a true similarity solution, the convective heat transfer coefficient related with the hot fluid and the mass transfer velocity must be proportional to x−2/3, where x is the horizontal distance along the plate from the origin. Effects of the various parameters on the dimensionless longitudinal velocity, the temperature, the nanoparticle volume fraction, as well as on the rate of heat transfer and the rate of nanoparticle volume fraction have been presented graphically and discussed. It is found that Lewis number, the Brownian motion, and the convective heat transfer parameters increase the heat transfer rate whilst the thermophoresis decreases the heat transfer rate. It is also found that Lewis number and the convective heat transfer parameter enhance the nanoparticle volume fraction rate whilst the thermophoresis parameter decreases nanoparticle volume fraction rate. A very good agreement is found between numerical results of the present article for special case and published results. This close agreement supports the validity of our analysis and the accuracy of the numerical computations.