Finite-difference Numerical Calculations Research Articles

Numerical Study on Karst Treatment Technology of Wuhan Metro Line 6

Taking Wuhan Metro Line 6 karst crossing area as the research object, combined with on-site survey data, a reasonable karst treatment technical plan is proposed. The finite difference numerical calculation method was used to carry out numerical calculation research on the treatment effect of karst treatment technology in the study area. The research results show that after grouting reinforcement, the maximum settlement of tunnel segments is 19.5mm, which is 73% less than before reinforcement, and the treatment effect is better. The lining deformation is within the allowable deformation value after the reinforcement of the three rows of jet grouting piles. The maximum value of the stress of the jet grouting piles is lower than the design value of the shear strength of the high jetting walls, indicating that the treatment technology can play a good supporting role when completely collapsed Without destruction.

Application of fiber-reinforced concrete lining for fault-crossing tunnels in meizoseismal area to improving seismic performance

The fault-crossing tunnel in meizoseismal area is directly subjected to strong ground motion, which leads to the failure of the tunnel lining. In order to improve the seismic safety of tunnel, fiber-reinforced concrete is applied to tunnel lining in this article. Taking the section of Zhongyi tunnel crossing Wanlong fault as an example, seismic performance of fiber-reinforced concrete tunnel lining was studied by finite difference numerical calculation software FLAC3D. The seismic displacement, stress response, and side wall convergence of secondary lining structures which are plain concrete, steel fiber-reinforced concrete, and steel-basalt hybrid fiber-reinforced concrete were comparatively analyzed. Moreover, the safety factor of each lining structure was investigated with the present numerical model. With the obtained data, seismic performance of steel-basalt hybrid fiber-reinforced concrete secondary lining is better than that of steel fiber-reinforced concrete secondary lining. The results may provide references for seismic design of fault-crossing tunnels in meizoseismal area.

Numerical study on temperature field of the sandwich plate defects based on MATLAB

As for the defects of the plate whose length and width are more than 8 times larger than the thickness direction, the finite difference numerical calculations are carried out by using the software of MATLAB. The surface temperature of defects in different positions, different thickness defects, different heating heat flow and heating time, and defects under defects are analyzed correspondingly by the program. According to the temperature difference results of the flawless surface and the flawed surface, the feasibility of infrared thermal imaging to detect the defects of the sandwich plate is analyzed to ensure the processing quality of the sandwich plate.

Numerical Simulation and Experimental Analysis of Microstructure of Magnetorheological Fluid

Magnetorheological fluid is a new type of smart material that is sensitive to magnetic fields and has controllable performance. It is widely regarded for its unique magnetorheological effect and good rheological properties. For materials, the microstructure determines its macroscopic properties. In order to better study its macroscopic properties, it is necessary to have a more comprehensive understanding and deep understanding of its microstructure. In this paper, the magnetization process of magnetorheological fluid is analyzed from a microscopic point of view. Based on Newton’s second law, the dynamic model of particle motion is established. The magnetic force, repulsive force, and viscous resistance of magnetic particles are analyzed. The finite difference numerical calculation method is used. The velocity-Verlet algorithm simulates the static microstructure chaining process of the magnetorheological fluid and the dynamic chaining process under shear force under different influencing factors. At the same time, a static observation device and a shear observation device were developed to observe the microstructure chaining morphology of magnetorheological fluid under different influencing factors, and to study the dynamic chaining law of magnetorheological fluid under the action of a shear force. Therefore, a reasonable contrast index is established, and the numerical simulation results are compared with the experimental observation results.

Multiple Waves Propagate in Random Particulate Materials

For over 70 years it has been assumed that scalar wave propagation in (ensemble-averaged) random particulate materials can be characterized by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the best of the authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions.

Numerical Simulation Analysis of Isolation Pile-Steel Shotcrete Combined Support of Deep Foundation Pit Excavation

The rapid development of deep foundation pit engineering, has become an important part of the urbanization construction, which brings deep excavation support of geotechnical engineering problem research also became a major issue. This paper uses the international well-known geotechnical engineering numerical simulation software FLAC3D, through 3D finite difference numerical calculation and analysis, to better simulation calculation and analysis of deep foundation pit construction site condition, forecast after excavation of the deep foundation pit deformation displacement and dangerous position, analysis of deep foundation pit excavation process isolation pile - steel shotcrete combined support effect. Three dimensional numerical model analysis and calculation in deep foundation pit engineering design and construction scheme optimization with economy is convenient wait for a obvious advantages, can for deep foundation pit excavation of deep foundation pit support design and construction to provide effective basis.

Effects of composition and grain size on the interdiffusional behaviour in B2-RuAl intermetallic compound

Compositional as well as grain size effects on the interdiffusional behaviour in single-phase B2-RuAl intermetallic compound were studied in the temperature range between 1473 and 1773 K. Variations in the diffusion coefficient due to composition changes were investigated using a diffusion couple technique. Boltzmann–Matano analysis and numerical finite difference calculations were performed and the influence was determined to be a factor of five between the end compositions. Contribution of grain boundary diffusion on the measured coefficients at the lower temperatures has to be assumed due to grain sizes of about 5 μm. Using Hart's relation a range of grain boundary interdiffusion coefficients was suggested.

Characterization of an external electro-optic sampling probe: Influence of probe height on distortion of measured voltage pulses

We experimentally and theoretically investigate the distortion that an external electro-optic sampling tip imprints on a measured voltage pulse that propagates along a coplanar stripline. By using the electro-optic effect of the GaAs substrate on which the stripline is fabricated we obtain the undisturbed voltage pulse, which serves as a reference pulse. This allows us to quantitatively compare the voltage pulse measured with the external electro-optic sampling tip to the undisturbed voltage pulse. The sensitivity of the sampling measurement and the distortion of the voltage pulse are determined for different heights of the tip which is lifted off the stripline. The experimental data agree very well with numerical finite difference calculations. We find that small changes of the tip height already influence the measurements strongly. From the experiments we infer an optimum working height of the sampling tip of only 0.5 μm. Our investigations help to improve the accuracy of electro-optic sampling measurements.

Critical cooling rate for the glass formation of ferromagnetic Fe 40Ni 40P 14B 6 alloy

Bulk ferromagnetic amorphous Fe–Ni–P–B alloys in rod shape were formed by a rapid solidification technique. The largest amorphous specimen prepared had a diameter of ∼ 2.5 mm and the corresponding cooling rate for the glass formation of this alloy system in our experiment can be estimated to be around 492.4 K/s by the method of finite-difference numerical calculation. This value is on the same order of magnitude as the critical cooling rate R c of Fe 40Ni 40P 14B 6 alloy estimated by the method of constructing the continuous-cooling-transformation (CCT) curve. It is indicated that the heterophase impurities have been eliminated well in our experiment.

Numerical local-potential-averaging method for quantum mechanical simulations

A simple and accurate numerical local-potential-averaging (NLPA) method is presented for calculating both the eigenstates of the Schrödinger equation and the quantum transport properties based on the Wigner distribution function. This method can handle the abrupt discontinuities in the potential profile of nano-scale electronic devices quite successfully that the accuracy of the numerical finite difference calculations combined with this method is less sensitive to the size of grid spacing along the discretized coordinate and thus, we can maintain the same level of numerical accuracy of the conventional method with much sparser grid points which make numerical computations less intensive.

A polynomial version of the generator coordinate Dirac-Fock method.

A polynomial version of the Generator Coordinate Dirac-Fock (p-GCDF) method is introduced and applied to develop Adapted Gaussian Basis Sets (AGBS) for helium- and beryllium-like atomic species (He, Ne +8, Ar +16, Sn +48, Be, Ne +6, Ar +14, and Sn +46) and for Kr and Xe atoms. The Dirac-Fock-Coulomb and Dirac-Fock-Breit energies obtained with these basis sets are in excellent agreement with numerical finite-difference calculations. Moreover, the sizes of the AGBS generated here with the p-GCDF method are significantly smaller than the size of previous relativistic Gaussian basis sets.

Computing permeability of fault zones in eolian sandstone from outcrop measurements

The large-scale equivalent permeabilities of strike-slip faults in porous sandstone are computed from detailed field measurements. The faults, which occur in the Valley of Fire State Park, Nevada, were previously characterized, and the flow properties of their individual features were estimated. The faults formed from the shearing of joint zones and are composed of a core of fine-grain fault rock (gouge) and deformation bands and a peripheral damage zone of joints and sheared joints. High-resolution fault-zone maps and permeability data, estimated using image analysis calibrated to actual measurements, are incorporated into detailed finite difference numerical calculations to determine the permeability of regions of the fault zone. Faults with slips of magnitude 6, 14, and 150 m are considered. The computed fault-zone permeabilities are strongly anisotropic in all cases. Permeability enhancement of nearly 1 order of magnitude (relative to the host rock) is observed for the fault-parallel component in some regions. Fault-normal permeability, by contrast, may be 2 orders of magnitude less than the host rock permeability. The fault-normal permeability is a minimum for the fault with the highest slip. For a representative fault region, the fault-parallel component of permeability is highly sensitive to the fracture aperture, although the fault-normal permeability is insensitive. The procedures developed and applied in this article can be used for any type of fault for which detailed structural and permeability data are available or can be estimated.

Choked flow of low density gas in a narrow parallel-plate channel with adiabatic walls

The continuum and slip choked flows of low-density air through a narrow parallel-plate channel with adiabatic walls are investigated by means of finite-difference numerical calculation, experiment and analytical solution. In the numerical calculation based on two-dimensional compressible viscous boundary layer equations, the previous method is improved to achieve a higher precision of the choked state by decoupling the pressure term from momentum equation. The numerical results of discharge coefficient, pressure distribution and surface temperature distribution agree well with the experimental data. As the upstream tank pressure drops, the numerical results of friction coefficient and recovery factor approach the asymptotic values for the incompressible developed flow.

The role of the diffusion of oxygen and radiation on the spontaneous combustibility of a coal pile in confined storage

This paper deals with the spontaneous combustibility and ignition behavior of coal stored in confined spaces. The geometry, which views the stockpile to be sealed from all sides except the top (which is exposed to the ambient conditions), is of interest in the transport of coals in barges or in cars by rail. Semi-analytical solutions are developed with the assumption that the coal pile is isothermal. Finite-difference numerical calculations show that the steady-state temperature at the exposed surface from a nonisothermal model is approximately the same as that from the isothermal formulation. Based on this approximation, the maximum temperature in the nonisothermal stockpile is linked analytically to the temperature calculated from the isothermal formulation. This approximation also permits rapid evaluation of the critical behavior of the nonisothermal stockpile in terms of the isothermal formulation. Detailed numerical solutions are necessary, however, to evaluate the transient behavior of the nonisothermal stockpile. Results in previous literature suggest that diffusion in a one-dimensional stockpile can lead to a maximum temperature rise of ≈ 80 K. This conclusion is shown to be a consequence of the unrealistic boundary conditions imposed on the oxygen and energy balance equations—diffusion alone can lead to dangerous self-heating. It is also shown that inclusion of the radiation at the boundary is extremely important in determining the stability of the stockpile. Finally, the maximum temperature within the pile at steady state, as well as the safe transit time, are calculated for the transport of coal in barges. These calculations confirm the importance of bed compaction in enhancing the safety of the coal pile.

Adapted Gaussian basis sets for the relativistic closed-shell atoms from helium to barium generated with the generator coordinate Dirac-Fock method

The generator coordinate Dirac-Fock method for closed-shell atoms is applied to generate adapted Gaussian basis sets for the relativistic closed-shell atoms from He ( Z = 2) to Ba ( Z = 56). Our Dirac-Fock-Coulomb and Dirac-Fock-Breit energies, for all the atoms studied here (except for He and Mg), are better than the corresponding Dirac-Fock-Coulomb and Dirac-Fock-Breit energies attained with previous Gaussian basis sets. Except for the Mg, Si, Ar and Ca atoms our Dirac-Fock-Coulomb energies are lower than the corresponding ones obtained by numerical finite-difference calculations. A discussion on the generation of universal and adapted Gaussian basis sets is presented.

A comparative study between an integral equation approach and a finite difference formulation for heat diffusion with nonlinear boundary conditions

The nonsingular, nonlinear Volterra integral equation of the second kind for the surface temperature of a sphere object to a general boundary condition is solved numerically by a linear multistep procedure based on the composite formula. A linearly increasing step size along with an alternating combination of Simpson's rule together with the trapezoid rule is used to carry out the numerical integration. A two-phase integration sequence is devised to eliminate an excessive number of linear steps for convection and radiation parameters greater than one. A linear boundary condition for which an analytical solution exists is used to test the accuracy of the numerical method, whereas the Stefan-Boltzmann law is considered as a nonlinear example of some physical significance. A finite-difference numerical calculation is also performed to verify that the integral method is computationally less expansive than the finite-difference formulation.

Comparison of Root Water Uptake Models

Four water uptake models for calculating soil moisture movement in a soil-root system are described and compared. Two models by Gardner (1964) and Molz and Remson (1970) are mainly based on the hydraulic properties of unsaturated soil, the third model by Herkelrath et al. (1977) on the soil moisture, and the fourth model by Feddes et al. (1978) on the soil water matric potential. The models are incorporated into a finite difference numerical calculation of Richards' equation including the root water uptake term. It is shown that soil moisture profiles, which are calculated using each model, are in relatively good agreement with those in laboratory experiments which are carried out under the constant environmental conditions. Among the four models, the third model agrees best with the experimental results. It is also shown that the water uptake profile depends on not only the root density distribution but also the soil moisture profile in the root zone.

A numerical study of the non-absorbable effects on the falling liquid film absorption

A numerical study for the simultaneous heat and mass transfer in a falling liquid film absorption process with the presence of non-absorbable gases is presented. Water vapor mixed with air as the non-absorbables being absorbed into a falling smooth aqueous lithium chloride film flow was chosen as the model problem for the study. The finite difference numerical calculation was proceeded by marching downward from the top end, owing to the parabolic type energy and concentration equations for both liquid and gas phases. The results indicate that the local non-absorbable gas concentration is much higher at the gas-liquid interface than that in the ambient, hence the local vapor pressure is lowered there such that the absorption driving potential of the vapor pressure difference is reduced. The resulting reduction of the absorption rate due to the presence of the non-absorbables suggests that its effect must be carefully considered in the application of absorption heat pump design. The present study can provide some useful information for this purpose.

The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium

The quasi-stationary state approximation to mass transport and fluid-rock interaction provides a quantitative description of metasomatic processes over geologically significant time spans. The time evolution of a geochemical system within the quasi-stationary state approximation is represented by a sequence of stationary states. Each stationary state describes the fluid composition and rates of reacting minerals as a function of distance corresponding to a particular state of alteration of the host rock. Because time steps separating stationary states are not restricted by stability and accuracy requirements as apply to conventional finite difference algorithms, geologic time spans are attainable for complex geochemical systems. The approximation is valid if mineral reaction zone boundaries, surface area, porosity and permeability change slowly compared to the time required to establish a stationary state. The propagation in time of mineral alteration zones can be predicted without any additional assumptions of the functional relationship of the zone boundaries on time. Pure advective mass transport within the quasi-stationary state approximation is equivalent to a multiple reaction path description of combined fluid flow and chemical reaction based on a Lagrangian reference frame. The validity of the quasi-stationary state approximation is examined for reaction of the minerals calcite and quartz. An exact expression is obtained for the velocity and position of the dissolution front formed in a single component system for a linear kinetic rate law. Under certain conditions the velocity of the front is independent of the reaction rate constant and surface area and equal to the local equilibrium result. The mineral volume fraction profile depends linearly on the rate constant and surface area in the vicinity of the front. Numerical and analytical examples for the dissolution of quartz at 550°C and 1 kb are in excellent agreement with numerical finite difference calculations based on exact transient mass conservation equations. Finally, the quasi-stationary state approximation is applied to the hydrochemical weathering of K-feldspar resulting from infiltrating rainwater. Product minerals gibbsite, kaolinite and muscovite form alteration zones which propagate with time in response to the reduction in surface area and, eventually, complete reaction of the dissolving K-feldspar mineral grains. During the initial stages of alteration gibbsite forms directly from K-feldspar, but in later stages forms indirectly from kaolinite.

Laminar natural-convection flow transitions in tilted three-dimensional longitudinal rectangular enclosures

Numerical finite-difference calculations have been carried out to determine laminar flow transition and heat transfer characteristics in tilted three-dimensional rectangular differentially-heated longitudinal enclosures. The tilt angle at which heat transfer attains a local minimum corresponds to the transition from multicell to unicell flow structures. Numerical results have been found to agree very well with existing experimental data and observations. Simulation calculations have also been made to determine the effect of the lateral walls and it is found that the close proximity between the walls restrict flow development in that direction, thus reducing the overall heat transfer across the enclosure.