Finite Difference Numerical Method Research Articles

Analytical modeling of the thermal behavior of a thin lubricant film under nonlinear conditions

Lubrication is an important phenomenon in a wide field of industry such as automotive, aerospace, mechanical transmission systems and many others. The viscosity of fluid is a determining factor in the thermal behavior of lubricant and solid surfaces in friction. In practice the viscosity varies strongly as a function of local pressure and temperature. In this study we are interested in the effect of temperature on the viscosity and the thermal behavior of the lubricant. We solve the dynamic and energy equations under nonlinear conditions considering that the viscosity decreases following an exponential law of the temperature as it is known in the literature, ? = ?0 e-? (T-T0). The analytical solution is compared to a numerical modeling using a finite difference methods. The results show an excellent agreement. We analyse the effect of the viscosity coefficient, ?, on the velocity and the temperature in the thin lubricant film.

Об одном разностном методе решения задачи Трикоми для уравнения Лаврентьева-Бицадзе

Получена априорная оценка решения задачи Трикоми для уравнения Лаврентьева-Бицадзе, из которой следует единственность регулярного решения. Предложен разностный метод нахождения приближенного решения исследуемой задачи. Получена априорная оценка решения разностной задачи, из которой следует сходимость построенной разностной схемы со вторым порядком точности.

Effect of bainite in microstructure on hydrogen diffusion and trapping behavior of ferritic steel used for sour service application

Abstract

Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems

Transient behavior of three-dimensional semiconductor device with heat conduction is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an elliptic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two concentrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

Transient Heat and Mass Transfer Flow through Salt Water in an Ocean by Inclined Angle

Abstract In the present computational study, the inclined angle effect of unsteady heat and mass transfer flow through salt water in an ocean was studied. The governing equations together with continuity, momentum, salinity and temperature were developed using the boundary layer approximation. Cartesian coordinate system was introduced to interpret the physical model where x-axis chosen along the direction of salt water flow and y-axis is inclined to x-axis. Two angle of inclination was considered such as 90° and 120°. The time dependent governing equations under the initial and boundary conditions were than transformed into the dimensionless form. A numerical solution approach so-called explicit finite difference method (EFDM) was employed to solve the obtained dimensionless equations. Different physical parameter was found in the model such as Prandtl number, Modified Prandtl number, Grashof number, Heat source parameter and Soret number. A stability and convergence analysis was developed in this study to describe the aspects of the finite difference scheme and this analysis is significant due to accuracy of the EFDM approach. The convergence criteria were observed to be in terms of dimensionless parameter as Pr ≥ 0.0128 and Ps ≥ 0.016. The distributions of the temperature and salinity profiles of salt water flow over different time steps were investigated for the effect of different dimensionless parameters and shown graphically.

Effects of ionized impurity and interface roughness scatterings on the electron mobility in InAs/GaSb type II superlattices at low temperatures

The in-plane electron mobility has been calculated in InAs/GaSb type-II superlattices (SLs) at low temperatures. The interface roughness scattering and ionized impurity scattering are investigated as the dominant scattering mechanisms in limiting the electron transport at low temperatures. For this purpose, the band structures and wave functions of electrons in such SLs are calculated by solving the K.P Hamiltonian using the numerical Finite Difference method. The scattering rates have been obtained for different temperatures and structural parameters. We show that the scattering rates are high in thin-layer SLs and the mobility rises as the temperature increases in low-temperature regime.

Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

Natural Convection of Non-Newtonian Fluids in a Square Cavity with a Localized Heat Source

This study investigates the natural convection cooling of a heat source placed on the bottom of a cavity filled with non-Newtonian power-law fluids. The heat source produces a uniform heat flux. The remaining parts of the bottom wall of the cavity are insulated, and the vertical, and top walls are kept at a low temperature. The governing equations for the power-law fluid flow are solved with the numerical finite difference method based on the control volume formulation and SIMPLE algorithm. The study investigates the effects of relevant parameters such as the Rayleigh number, the power-law index and the heat source length and heat source location on the thermal performance of the cavity. The results show that as the Rayleigh number increases, particularly when n<1, the thermal performance of the cavity is improved. Furthermore, using non-Newtonian shear thinning fluids, especially at higher Rayleigh numbers, improves the cooling performance of the heat source. The results also show that, depending on the Rayleigh number and the power-law index, the length and position of the heat source have significant effects on the thermal performance of the cavity.

A Mesoscale Three-Dimensional Moisture Transport Model in Concrete

Concrete was considered as a three-component composite material consisting of mortar matrix, coarse aggregates and interfacial transition zones (ITZ) at the mesoscale level. Based on the random accumulation model of spherical aggregates, a mesoscale geometric model was developed after discretizing the mortar matrix and ITZ into mesh elements using Voronoi diagram method. Combined with the third boundary condition, a mesoscale three-dimensional model to simulate the moisture transport process in concrete exposed to atmosphere environment was then developed using finite difference numerical method, where the transport of liquid water and water vapor were considered as permeation and diffusion respectively. Moreover, the model was verified and then applied to investigate the influence of ITZ on the distribution of relative humidity in concrete. The results indicated that the moisture transport process was overall accelerated since blocking effect of coarse aggregates was partially counteracted by the existence of ITZ.

Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects

The present study is one kind of numerical application to a thermal extrusion manufacturing processing system energy conversion problem by using some improved parameters control method. Combined electrical MHD Ohmic dissipation forced and free convection of an incompressible Maxwell fluid on a stagnation point heat and mass transfer energy conversion problem have been studied. The governing equations are solved by an analysis similarity transformation method and an improved numerical finite difference method. The above two methods have been used to analyze present problem which is provided a different method to deal with the similar thermal system energy conversion problems by using parameter control method. The combination thermal system numerical solutions of the flow velocity field, temperature field, mass transfer and heat conduction had been produced out as functions of the viscoelastic number (E), Prandtl number (Pr) and buoyancy parameters (Gc, Gt), etc. The effects of related importance parameters have also been discussed in detail. The results are shown that it will be produced greater heat transfer effects with larger values of viscoelastic number, Prandtl number, free convection parameters, electric parameter (E1), heat source/sink (AL) and conduction-convection number (Ncc). At last, it can be obtained a higher efficiency thermal extrusion system.

Proposal of a micromagnetic standard problem for ferromagnetic resonance simulations

Nowadays, micromagnetic simulations are a common tool for studying a wide range of different magnetic phenomena, including the ferromagnetic resonance. A technique for evaluating reliability and validity of different micromagnetic simulation tools is the simulation of proposed standard problems. We propose a new standard problem by providing a detailed specification and analysis of a sufficiently simple problem. By analyzing the magnetization dynamics in a thin permalloy square sample, triggered by a well defined excitation, we obtain the ferromagnetic resonance spectrum and identify the resonance modes via Fourier transform. Simulations are performed using both finite difference and finite element numerical methods, with OOMMF and Nmag simulators, respectively. We report the effects of initial conditions and simulation parameters on the character of the observed resonance modes for this standard problem. We provide detailed instructions and code to assist in using the results for evaluation of new simulator tools, and to help with numerical calculation of ferromagnetic resonance spectra and modes in general.

Plasmonic nanodipole antenna array with extra arms for sensing applications

In this paper, a periodic array of plasmonic nanodipole antennas with extra arms for sensing applications has been proposed. The extra arms provide more hot spots in the sensing regime, and so the electric field intensity is enhanced in comparison to the simple nanodipole. The optical responses of both the simple nanodipole and the proposed nanodipole with the extra arms, which include the transmission, the reflection, and the absorption spectra of the system, have been studied. It has been shown that the sensitivity of the proposed plasmonic nanodipole antennas array with the extra arms is much higher than the simple one. Moreover, by replacing the Si substrate with SiO2, the sensitivity increases up to 1420 and 1350 nm/RIU at the wavelengths of 1310 and 1550 nm, respectively, which are very suitable for sensing applications. The simulations are based on the finite-difference time-domain numerical method.

Direct Calculation of Permeability by High-Accurate Finite Difference and Numerical Integration Methods

AbstractVelocity of fluid flow in underground porous media is 6~12 orders of magnitudes lower than that in pipelines. If numerical errors are not carefully controlled in this kind of simulations, high distortion of the final results may occur [1–4]. To fit the high accuracy demands of fluid flow simulations in porous media, traditional finite difference methods and numerical integration methods are discussed and corresponding high-accurate methods are developed. When applied to the direct calculation of full-tensor permeability for underground flow, the high-accurate finite difference method is confirmed to have numerical error as low as 10–5% while the high-accurate numerical integration method has numerical error around 0%. Thus, the approach combining the high-accurate finite difference and numerical integration methods is a reliable way to efficiently determine the characteristics of general full-tensor permeability such as maximum and minimum permeability components, principal direction and anisotropic ratio.

Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products

In this paper, using the eigenvalues and eigenvectors of symmetric block diagonal matrices with infinite dimension and numerical method of finite difference, a closed-form solution for exact solution of Laplace equation is presented. The method of this paper has applications in different states of boundary conditions like Neumann, Dirichlet, and other mixed boundary conditions. Using the method of this paper, a mathematical model for the exact solution of the Poisson equation is derived. Since these equations have many applications in engineering problems, in each part of this paper, examples, like water seepage problem through the soil and torsion of prismatic bars, are presented. Finally, a method is provided for torsion problem of prismatic bars with non-circular and non-rectangular cross-sections utilizing conformal mapping.

Numerical Finite Difference Method of Electric Field Generated to determine the SAR Generated in a Microstrip Antenna Type Applied to a WBAN

&#61472;This paper gives an alternative method to determine the Specific Absorption Rate (SAR), amount of electromagnetic energy absorbed by the human body, using the numerical finite difference method with MATLAB software, from the electric fields generated by a type patch antenna applied to a Wireless Body Sensor Network (WBAN). These networks consist of bodily wireless sensors distributed inside or outside the human body for measuring physiological parameters. The research begins by defining electrical characteristics within the human body, to finally present and calculate the specific absorption rate SAR.

Earth-Air Heat Exchanger thermal performance in Egyptian conditions: Experimental results, mathematical model, and Computational Fluid Dynamics simulation

In this paper, the thermal performance of an Earth-Air Heat Exchanger (EAHE) used for heating and cooling purposes is investigated under Egyptian weather conditions. The soil temperature profile and the temperature distribution of flowing air through horizontal Earth-Air Heat Exchanger (EAHE) is experimentally studied. Also, a mathematical model based on unsteady, one-dimensional and quasi-state is developed for energy conservation equation. Moreover, an explicit finite difference numerical method is used to solve the developed mathematical model with the help of MATLAB code. Finally, three-dimensional, steady and double precision Computational Fluid Dynamics (CFD) ANSYS Fluent simulation model is established to predict the air and soil temperature. Whereas, the standard κ-∈ model is applied to simulate the turbulence kinetic energy of the flowing fluid. The mathematically developed model and CFD simulation result validated against experimental results. Good agreement is achieved with an average error and correlation coefficient of 2.09, 97% and 3.3 and 95.5% for CFD simulation and mathematical model respectively. The CFD model is used in a parametric investigation. A parametric study carried out to explore the impact of different parameters such as pipe diameter, pipe material, pipe space, pipe length and flowing fluid velocity. The results show that some of these parameters have noticeable results in air temperature. Whereas, the pipe diameter increases the air temperature decreases. The outlet air temperature declines from 20.4°C to 18.7°C as the pipe diameter expands from 2 to 3in. Furthermore, as pipe length increases, outlet air temperature enhances. The temperature changes from 19.7 to 19.9°C as the pipe length elongates from 5.45m to 7m. A bit change occurs in outlet air temperature from 19.7°C to 19.8°C when pipe space changes from 0.2 to 0.5m. Moreover, three different pipe materials such as PVC, steel and copper are implied. The outlet air temperature was 19.7°C in PVC pipe and 19.8, 19.8°C for steel and copper respectively. So the conclusion is that the change in outlet air temperature for various pipe material is neglected compared with their prices. Finally, the effect of fluid velocity was investigated. Therefore, the outlet air temperature declines from 20.4°C to 19.2°C as air accelerates from 1 to 3m/s.

Numerical models for ground deformation and gravity changes during volcanic unrest: simulating the hydrothermal system dynamics of a restless caldera

Abstract. Ground deformation and gravity changes in restless calderas during periods of unrest can signal an impending eruption and thus must be correctly interpreted for hazard evaluation. It is critical to differentiate variation of geophysical observables related to volume and pressure changes induced by magma migration from shallow hydrothermal activity associated with hot fluids of magmatic origin rising from depth. In this paper we present a numerical model to evaluate the thermo-poroelastic response of the hydrothermal system in a caldera setting by simulating pore pressure and thermal expansion associated with deep injection of hot fluids (water and carbon dioxide). Hydrothermal fluid circulation is simulated using TOUGH2, a multicomponent multiphase simulator of fluid flows in porous media. Changes in pore pressure and temperature are then evaluated and fed into a thermo-poroelastic model (one-way coupling), which is based on a finite-difference numerical method designed for axi-symmetric problems in unbounded domains.Informed by constraints available for the Campi Flegrei caldera (Italy), a series of simulations assess the influence of fluid injection rates and mechanical properties on the hydrothermal system, uplift and gravity. Heterogeneities in hydrological and mechanical properties associated with the presence of ring faults are a key determinant of the fluid flow pattern and consequently the geophysical observables. Peaks (in absolute value) of uplift and gravity change profiles computed at the ground surface are located close to injection points (namely at the centre of the model and fault areas). Temporal evolution of the ground deformation indicates that the contribution of thermal effects to the total uplift is almost negligible with respect to the pore pressure contribution during the first years of the unrest, but increases in time and becomes dominant after a long period of the simulation. After a transient increase over the first years of unrest, gravity changes become negative and decrease monotonically towards a steady-state value.Since the physics of the investigated hydrothermal system is similar to any fluid-filled reservoir, such as oil fields or CO2 reservoirs produced by sequestration, the generic formulation of the model will allow it to be employed in monitoring and interpretation of deformation and gravity data associated with other geophysical hazards that pose a risk to human activity.

An Equivalent Circuit Model for Graphene-Based Terahertz Antenna Using the PEEC Method

The electromagnetic (EM) characterization of graphene under general EM environments is becoming of interest in the engineering and scientific research fields. However, its numerical modeling process is extremely cost prohibitive due to the huge contrast between its thickness and other dimensions. In this work, for the first time, the EM features of graphene are characterized by a circuit model through the partial element equivalent circuit (PEEC) method. The atomically thick graphene is equivalently replaced by an impedance boundary condition. After incorporating the PEEC method, a novel surface conductivity circuit model is derived for graphene. A physical resistor and inductor are added into the conventional PEEC cell due to the dispersive conductivity property of graphene. The proposed novel method significantly reduces the memory and CPU time consumption for general graphene structures when compared with standard numerical finite element method (FEM) or finite difference (FD) methods, where 3-D meshing is unavoidable. This model also transforms the surface conductivity of graphene into a vivid circuit, and physical properties of the material can be conveniently obtained, such as radiation, scattering, and resistance properties, when compared with method of moments (MOM). In addition, the radiation and scattering calculation by MOM entail the cumbersome steps of defining a bounding surface and implementing a multidimensional integrand, while in PEEC, these complications are entirely bypassed by the concise vector–matrix–vector product (VMVP) formulas. To validate the introduced algorithm, various numerical examples are presented and compared with existing references.

Hencky bar-chain model for buckling analysis of non-uniform columns

The Hencky bar-chain model is developed for the buckling analysis of non-uniform columns. While the end rotational spring stiffnesses of the Hencky bar-chain are uniquely calibrated by the finite difference column model, the internal rotational spring stiffnesses may be defined in various ways for non-uniform columns. This paper studies three different ways in specifying the internal rotational spring stiffnesses and examine their effects on the buckling loads for various shapes of non-uniform columns. The developed Hencky bar-chain model facilitates buckling analysis and optimization of non-uniform columns as well as providing a physical interpretation to the numerical finite difference method.

Mathematical modeling and validation of the carburizing of low carbon steels

This paper shows the mathematical modeling of heat and mass transfer in transient state of cylindrical bars of low carbon steel subjected to carburizing process. The model solution for the two phenomena was performed using a one-dimensional analysis in the radius direction, using the numerical method of finite differences; also a sensitivity analysis by varying the coefficient of convective heat transfer (h) is performed. The modeling results show that this carburization steel is strongly dependent on h. These results suggest that if it can increase the value of h in this kind of process could reduce the time of process for this heat treatment. Additionally, an experimental procedure was established by carburization of a steel AISI SAE 1010, which develops cementing solid phase and the specimen steel and micrographic hardness profiles obtained from samples of the specimen analysis was performed, to determine the penetration depth of the carbon and validate this result over the values obtained by the computer model.