Finite Difference Numerical Method Research Articles

Numerical Investigation by the Finite Difference Method of the Laser Hardening Process Applied to AISI-4340

This paper presents a numerical and experimental analysis study of the temperature distribution in a cylindrical specimen heat treated by laser and quenched in ambient temperature. The cylinder studied is made of AISI-4340 steel and has a diameter of 14.5-mm and a length of 50-mm. The temperature distribution is discretized by using a three-dimensional numerical finite difference method. The temperature gradient of the transformation of the microstructure is generated by a laser source Nd-YAG 3.0-kW manipulated using a robotic arm programmed to control the movements of the laser source in space and in time. The experimental measurement of surface temperature and air temperature in the vicinity of the specimen allows us to determine the values of the absorption coefficient and the coefficient of heat transfer by convection, which are essential data for a precise numerical prediction of the case depth. Despite an unsteady dynamic regime at the level of convective and radiation heat losses, the analysis of the averaged results of the temperature sensors shows a consistency with the results of microhardness measurements. The feasibility and effectiveness of the proposed approach lead to an accurate and reliable mathematical model able to predict the temperature distribution in a cylindrical workpiece heat treated by laser.

The effect of conductive baffles on natural convection in a power-law fluid-filled square cavity

This research involves the numerical study of the conjugate natural convection of a power-law fluid in a two-dimensional square cavity with a pair of conducting baffles. The lower wall of the cavity contains a heat source with constant heat flux, while a pair of conductive baffles is embedded on its upper wall. The left and right side walls and some parts of the lower wall of the cavity are thermally insulated and its upper wall is kept at a low temperature. The governing equations along with the corresponding boundary conditions are solved using the numerical finite difference method based on the control volume formulation and SIMPLE algorithm. The effects of pertinent parameters on flow and temperature fields and heat transfer rate are investigated. From the results of the numerical solution, increasing the Rayleigh number and baffles thermal conductivity enhance the thermal performance of the cavity. So that for $$ Ra = 10^{6} , $$ with an increase in the thermal conductivity ratio from $$ 1 $$ to $$ 100, $$ the average Nusselt number increases by about 14% for both cases $$ n = 0.6 $$ and $$ n = 1.2 $$ . As well as a decrease in power-law index, except in low Rayleigh numbers, increases the heat transfer rate. Based on the results, the increase of the average Nusselt number is about 283% for a power-law index reduction in the range of $$ 0.6 \le n \le 1.2 $$ at $$ Ra = 10^{6} $$ . Also, an increase of the length of the baffles deteriorates the thermal performance of the cavity at high Rayleigh numbers, while enhances it at low Rayleigh numbers.

Calculation of the Maximum Slope Angles of the Temperature Curve for the Single-Flow Non-Stationary Method of Deriving the Thermal Characteristics of Heat Transfer Surfaces

Results are presented for the solution of the problem of gas flow in a porous body with the presence of a temperature “step” in the range 1–100 of the non-dimensional heat-transfer coefficient, while accounting for longitudinal thermal conductivity. The problem was solved using the numerical finite difference method. Maximum slope angles were derived for temperature curves that can be used to obtain the thermal characteristics of high-compact heat transfer surfaces with the single-flow non-stationary method. Recommendations were provided regarding the range of applicability of the method, taking into account the new received results.

Forward modeling and inversion of tensor CSAMT in 3D anisotropic media

Tensor controlled-source audio-frequency magnetotellurics (CSAMT) can yield information about electric and magnetic fields owing to its multi-transmitter configuration compared with the common scalar CSAMT. The most current theories, numerical simulations, and inversion of tensor CSAMT are based on far-field measurements and the assumption that underground media have isotropic resistivity. We adopt a three-dimensional (3D) staggered-grid finite difference numerical simulation method to analyze the resistivity in axial anisotropic and isotropic media. We further adopt the limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) method to perform 3D tensor CSAMT axial anisotropic inversion. The inversion results suggest that when the underground structure is anisotropic, the isotropic inversion will introduce errors to the interpretation.

A novel stochastic energy analysis of a solar air heater: case study in solar radiation uncertainty

There is a growing recognition of the fact that solar energy utilization plans cannot be carried out without explicitly accounting for the uncertainty presented in the received solar irradiation. This may be expressed as an uncertainty quantification problem. A novel stochastic energy analysis is introduced to study the transient heat transfer problem of a typical flat plate solar air heater, based on the polynomial chaos expansion approach. The constructed model was equipped with the numerical finite difference method and the Galerkin projection scheme in the random space. The numerical model was verified against the available exact analytical solutions. The results of polynomial chaos method was compared to corresponding basic Monte Carlo sampling results. Finally, a case study with realistic solar irradiance data of Urmia, a cold climate city in Iran, was conducted for a typical solar air heater. Afterward, the outlet temperature of the air heater was tracked in a probabilistic framework, to find the reliable hours for extracting solar energy stably during a typical summery day. These hours were found between 11 am to 5 pm.The proposed approach could be highly worthwhile in the designing and contriving control plans taking into consideration the non-negligible uncertainty of solar radiation.

Closed-Form Solution of Large Deflected Cantilever Beam a gainst Follower Loading Using Complex Analysis

The literature reveals that the non-conservative deflection of an elastic cantilever beam caused by applying follower tip loading was investigated and solved by various numerical methods like: Runge Kutta, iterative shooting, finite element, finite difference, direct iterative and non-iterative numerical methods. This is due to the fact that the Euler–Bernoulli nonlinear differential equation governing the problem contains the “slope at the free end”, this slope however needs special numerical treatment. On the other hand, some of these methods fail to find numerical solutions for extremely large loading conditions. Hence, this paper is aimed to obtain a closed-form solution for solving the large deflection of a cantilever beam opposed to a concentrated point follower load at its free end. This closed-form solution when compared with other conventional numerical approaches is characterized by simplicity, stability and straightforwardness in getting the beam deflection and slopes even for extremely large loading conditions. The closed-form solution is obtained by applying complex analysis along with elliptic-integral approach. Very good results were obtained when the elastica of the beam compared with that of various numerical methods which are used in analyzing similar problem.

Brackish water desalination using electrodialysis: predictive mass transfer and concentration distribution model along the electrodialyzer.

This study employs theory and experimental data from a laboratory-scale electrodialyzer to predict sodium chloride (NaCl) mass transport and concentration distribution along the electrodialyzer as a function of feed concentration, feed flow rate, applied voltage, and pressure. Moreover, a model was developed to predict the ion removal as a function of driving forces through solving the complete Navier-Stokes, continuity, and steady state Nernst-Planck equations by the finite difference numerical method. The findings of the experiments confirmed that concentration distributions are nonlinear along both the dilute and concentrate compartments. The results also demonstrated that increases in pressure and feed flow rate have a negative effect on salt removal, linear and nonlinear for pressure and flow rate, respectively. In the investigated ranges, higher voltage increased salt removal at a constant feed concentration.

English

In this study, we developed a new numerical finite difference method for solving various diffusion convection equations. The method involves reduction of the diffusion convection equations to a system of algebraic equations. By solving the system of algebraic equations we obtain the problem approximate solutions. The study of the numerical accuracy of the method has shown that the method provides similar results to the known explicit finite difference method for solving diffusion convection equations, but with fewer numbers of iterations. Key words: Lines, multistep collocation, parabolic, Taylor’s polynomial.

Nonlinear size-dependent pull-in instability and stress analysis of thin plate actuator based on enhanced continuum theories including nonlinear effects and surface energy

In recent years, size dependent continuum theories have been commonly used to simulate material discontinuities in micro/nano-scales. In the present article, modified strain gradient theory (MSGT) and Gurtin–Murdoch surface elasticity, are used to investigate the size-dependent nonlinear pull-in instability and conduct a stress analysis of thin microplates. This paper modifies the classical equations and surface elasticity theory as a non-classical method which is able to account for the surface effect of nanomaterials with sufficient precision and lower calculation cost in comparison to atomic methods. The effects of van der Waals and Casimir forces are incorporated into the nonlinear governing equation of the system. A finite difference method (FDM) is employed to solve the nonlinear differential equation, and the pull-in parameters of the microplate are extracted. The effect of dispersion attractions and size-dependency on instability as well as the importance of the coupling between them is discussed. The most common types of thin-plates, including fully-clamped, simply-supported, and plates with mixed boundary conditions, are investigated. It is found that the significant difference between the pull-in instability parameters in the MSGT and the classical theory for thin microplates is merely due to the consideration of the size effect parameter in the modified strain gradient theory. The effect of surface/interface is also accounted for in the numerical FDM approach. Then, the stress analysis of the microplate is studied numerically by examining the surface/interface effect on the microplate with a square/rectangular section. The results obtained through this numerical method are plotted graphically.

Research on the forward modeling of controlled-source audio-frequency magnetotellurics in three-dimensional axial anisotropic media

Controlled-source audio-frequency magnetotellurics (CSAMT) has developed rapidly in recent years and are widely used in the area of mineral and oil resource exploration as well as other fields. The current theory, numerical simulation, and inversion research are based on the assumption that the underground media have resistivity isotropy. However a large number of rock and mineral physical property tests show the resistivity of underground media is generally anisotropic. With the increasing application of CSAMT, the demand for probe accuracy of practical exploration to complex targets continues to increase. The question of how to evaluate the influence of anisotropic resistivity to CSAMT response is becoming important. To meet the demand for CSAMT response research of resistivity anisotropic media, this paper examines the CSAMT electric equations, derives and realizes a three-dimensional (3D) staggered-grid finite difference numerical simulation method of CSAMT resistivity axial anisotropy. Through building a two-dimensional (2D) resistivity anisotropy geoelectric model, we validate the 3D computation result by comparing it to the result of controlled-source electromagnetic method (CSEM) resistivity anisotropy 2D finite element program. Through simulating a 3D resistivity axial anisotropy geoelectric model, we compare and analyze the responses of equatorial configuration, axial configuration, two oblique sources and tensor source. The research shows that the tensor source is suitable for CSAMT to recognize the anisotropic effect of underground structure.

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

Fluid infiltration into unsaturated soil is of vital significance from many perspectives. Mathematically, such transient infiltrations are described by Richards' equation; a nonlinear parabolic Partial Differential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for 1-D horizontal and vertical water infiltration are derived and compared to numerical finite difference method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2- and 3-D vertical water infiltration are derived for constant, no-flow, and sinusoidal boundary and initial conditions. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use different computational techniques

Optical absorption enhancement by inserting ZnO optical spacer in plasmonic organic solar cells

Optical absorption enhancement (AE) using coupled optical spacer and plasmonic effects in standard and inverted organic solar cells (OSCs) are demonstrated using the finite-difference time-domain numerical method. The influence of an added zinc oxide (ZnO) optical spacer layer inserted below the active layer in standard architecture is first theoretically investigated while the influence of varying the ZnO cathodic buffer layer thickness in inverted design is studied on AE. Then, the embedding of a square periodic array of core–shell silver–silica nanospheres (Ag@SiO2 NSs) at different positions in standard and inverted OSCs is performed while AE and short-circuit current density (Jsc) are calculated. As a result of previous combined effects, the optimized standard plasmonic OSCs present 15% and 79.45% enhancement in Jsc over the reference with and without ZnO optical spacer layer, respectively, and a 16% increase of AE when Ag@SiO2 NSs are placed on top of the PEDOT:PSS layer. Compared to the inverted OSC reference, the plasmonic OSCs present 26% and 27% enhancement in Jsc and AE, respectively, when the Ag@SiO2 NSs are located on top of the ZnO layer. Furthermore, the spatial position of these NSs in such OSCs is a key parameter for increasing light absorption via enhanced electromagnetic field distribution.

Stresses minimization in functionally graded cylinders using particle swarm optimization technique

In the present work, minimization of the induced stresses in functionally graded cylinders due to pressure loading is carried out considering the plane-stress geometrical conditions. The functionally graded cylinder is made up of layers each of which is assumed to be isotropic while the properties of each layer are different from those of the others. Also, the number of layers is varied up to ten while all layers have the same thickness. For the optimization methodology, the evolutionary particle swarm optimization technique is applied in order to solve the nonlinear optimization problem. The optimization variables are the volume fractions of the constituent materials that lead to the moduli of elasticity of the layers and therefore the modulus of elasticity of the multi-layer cylinder varies in the radial direction. Since failure may occur at the interface between any two successive layers, due to its weakness, the maximum hoop-stress-jump at the interfaces is considered as an extra objective along with the maximum hoop stress through the cylinder thickness. The optimization is attained by minimizing these two objectives through the evaluation of the proper optimization variables using the finite difference numerical method. The obtained stresses are compared to the nonoptimized ones in order to have some insight about the effectiveness of the proposed analyses. Based on the obtained numerical results, it is found that the stresses are considerably decreased by applying the proposed methodology. Eventually, failure probability can be decreased and/or the pressure capacity can be increased by applying the evaluated optimized parameters.

Tunable midinfrared wavelength selective structures based on resonator with antisymmetric parallel graphene pair

A parallel graphene layer pair arranged in an antisymmetric configuration coupled through a cavity resonator is proposed and analyzed by the analytical method and the numerical finite-difference time-domain method. The structure operates as a bandpass filter in the midinfrared region. The feature, as the result of the wavelength selective property of the cavity resonator, can be tuned by varying the length of the resonator, the lateral coupling distance between the graphene layers, the dielectric refractive index of material inside the resonator, and, the most interesting, the chemical potential of the graphene layers. The proposed structure can be promoted to power splitters and refractive index sensors by proper designs. Various power division ratios can be realized by changing the relative positions and/or the chemical potentials of the output waveguides. The investigated components can be utilized in the design of midinfrared nanoscale photonic integrated circuits.

Analytical solution for the dynamic model of tumbling mills

Optimisation of grinding circuits is invariably dependent on sound process models together with process simulators that can solve the process models accurately. Most of the process models are solved numerically because analytical solutions are not available, which can lead to errors in the results due to the numerical approximation of mathematical equations. Whiten [1], and Valery Jnr & Morrell [2,3] have developed a dynamic model with numerical simulation for autogenous and semi-autogenous mills, and validated the model with dynamic response of mills in terms of power draw, grinding charge level, slurry level and product size distribution to changes in feed rate, feed size, feed hardness and water addition [2,3]. In this work, an analytical solution for their dynamic model of tumbling mills has been developed based on the knowledge of solutions to the first-order nonhomogeneous linear differential equations. Two algorithms, Direct Single Time method (DST) and Direct Multiple Time method (DMT), were applied to obtain the analytical solutions respectively. It was found that analytical solutions are more accurate than the traditional finite difference numerical methods. However, the DST analytical method has a drawback of numerical instability due to the accumulation of round-off errors which are amplified by exponential functions, whilst the DMT method can provide stable solutions. To test the DMT analytical method, two cases of SAG mill dynamic operation were studied with both the traditional numerical method and the newly developed analytical method, further proving the robustness and feasibility of the analytical solutions.

A Python Program for Solving Schrödinger’s Equation in Undergraduate Physical Chemistry

In undergraduate physical chemistry, Schrodinger’s equation is solved for a variety of cases. In doing so, the energies and wave functions of the system can be interpreted to provide connections with the physical system being studied. Solving this equation by hand for a one-dimensional system is a manageable task, but it becomes time-consuming once students aim to make various changes and investigate the impact of those changes on the results. To address this challenge, numerical methods, such as the shooting and linear finite-difference methods, have been utilized to quickly solve Schrodinger’s equation. In this technology report, we use the Python programming environment and the three-point finite-difference numerical method to find the solutions and plot the results (wave functions or probability densities) for a particle in an infinite, finite, double finite, harmonic, Morse, or Kronig–Penney finite potential energy well. We believe that this technology report will educate undergraduates on the basic ...

РАСЧЁТ ПОЛЕЙ ДАВЛЕНИЙ И ТЕМПЕРАТУР В СМАЗОЧНОМ СЛОЕ КОНИЧЕСКИХ ГИДРОСТАТОДИНАМИЧЕСКИХ ПОДШИПНИКОВ

The paper reports the consideration of the procedure for the computation of pressure and temperature fields in a lubricating layer of a tapered hydrostatic dynamic bearing based on a joint solution of Reynolds equations, a balance of energies and a balance of expenses and also additional correlations for thermal-physical properties of lubricating material taking into account turbulence and a thermal effect. The solution of this system was carried out through a numerical method of finite differences. The theoretical and experimental investigations show that lubricating material in a tapered hydrostatic dynamic bearing ensures a sufficient carrying capacity. This work offers the use of results shown for the analysis of rotor systems on tapered bearings and it will be useful for scientific workers and engineers dealing with the problems of computation and design of slider bearings in highspeed systems.

Reducing Friction Drag on Flat Plates

There are in existence several methods of boundary layer control which have been developed for the purpose of reducing friction drag of streamlined bodies. These include such elements as acceleration of the boundary layer (blowing), injection of a different gas, cooling of the wall, etc. All of them are described in some detail in scientific literature. In this paper, a new method for the reduction of friction drag, which has been insufficiently studied. The proposed method is based on the use of a liquid film on the surface of the streamlined body. One solution to the problem of the flow over a flat semi-infinite plate set at an angle of deflection to the horizon, and having a thin liquid film on its surface by external airflow is presented. The film is formed at the plate in the form of raindrops. Liquid moves by gravity and friction on the outer surface of the film. Influence on the boundary layer raindrops and thermal effects are ignored. In general, the problem is conjugate, including the problem of a film flow (internal problem) and the problem of the boundary layer of incoming air (external problem). For the solution of the dual problem, a method of successive approximations was used. This method assumes the fact that the external and internal problems can be solved separately, either logically or iteratively. At each new approximation, the inner problem is solved with regard to the friction resulting from the external problem; the solution of which, in turn, takes into account the speed at the boundary surface, obtained from the previous approximation of the interior problem. Thus, the iterative process continues until the speed and, consequently, the friction at the phase interface change little from iteration to iteration. For the interior problem, the method of asymptotic expansion in a small parameter was used. This paper gives null and first approximation. The external problem was solved using the numerical method of finite differences. In this paper we present some results showing the reduction of friction drag on the flat plate.

Modeling and optimization of phenol hydrothermal oxidation in supercritical water

The hydrothermal oxidation is an efficient and clean way for wastewater treatment; it is also an energy self sufficient process under certain conditions, by the mean of the highly exothermic reaction of oxidation.In this present work, we studied the modeling of hydrothermal oxidation of phenol in polluted water using supercritical conditions. First we apply a simple mathematical model for tubular reactor, resulting from mass, momentum and energy balances, which is often difficult to solve in its general form. According to adequate simplifications, the profiles of temperature and concentration of chemical species are computed along the reactor. Finite differences numerical method was used to solve the obtained system of equations and a MathCAD program was developed for this purpose. The results are close to experimental works from the literature.To study the influence of some operating conditions on the efficiency of the treatment and the amount of heat generated by the reaction and which will be recovered to heat the inlet flows, we exploited the modeling results to perform a full factorial design of experiment (DOE). The results allow fixing the optimal conditions for the studied factors on the efficiency and energy recovery of the process treatment.

Symmetry of Gold Neutral Clusters Au3-20 and Normal Modes of Vibrations by using the Numerical Finite Difference Method with Density-Functional Tight-Binding(DFTB) Approach

The geometries and vibrational frequency of the most stable small AuN clusters with N = 3 to 20 are presented through global structure re-optimization Study. The finite-differentiation method has been implemented within the density-functional tight-binding (DFTB) approach. The desired set of system eigenfrequencies (3N-6) is obtained by a diagonalization of the symmetric positive semi definite Hessian matrix. We have observed the vibrational modes between 0.55 and 370.72 cm-1 in wavenumbers at T=0 for the small gold clusters. The effect of the range of interatomic forces was calculated, and even the very lower frequencies were occupied in some clusters.