Application Of Finite Difference Method Research Articles

Application of finite difference method to 2D heat conduction in young hydrating mass concretes structures.

Finite Difference Method is a numerical approximation scheme usually applied in solving differential equations derived from the knowledge of the behaviour of engineering systems. Due to the unsteady state nature of cement hydration and heat conduction in concretes, Crank Nicholson implicit approach was adopted. Concrete block of size 1.10 m x 1.10 m x 1.10 m, cast with mix ratio 1:3:6, probed thermocouples and digital thermometer were used to verify the numerical computational analysis. The temperature profile depicted a relatively hot core, with peak temperature values occurring at 24 hours of placement. Implicit finite difference method was utilized in determining time dependent temperature profile within mass concretes at early ages. The paper could provide a guide towards proactive decision making in terms of controlling thermal cracks.

Application of Finite Difference Method in Simulating 2D Partial Dam-break Flow with an Obstacle

Application of Finite Difference Method in Simulating 2D Partial Dam-break Flow with an Obstacle

Application of finite difference method for solving problem of seismic resistance of underground pipelines

The paper analyzes the dynamic response of an underground main pipe under the action of a longitudinal wave propagating in soil along the pipe. The article assumes that the elastic pipe is of a finite length, and a linear viscoelastic model of the “pipe-soil” system interaction is considered. The problem is solved numerically using the explicit scheme of the finite difference method. A longitudinal wave in the soil is taken as a traveling sine wave. The article presents a comparative analysis of the results for certain values of elastic and viscous interaction coefficients, propagation velocity, and pulse duration. Under elastic interaction of the “pipe-soil” system, the reflection of the wave propagating in the underground pipeline at the boundaries of the pipeline coincides with the propagating wave in soil, leading to an increase in the maximum deformation of the underground pipeline, the value of which can double. The viscosity coefficient of interaction at the “pipe-soil” system contact leads to attenuation of the wavefront in the underground pipeline. For soils with values of viscous interaction coefficient of more than 100 kN·s/m2, this may lead to complete attenuation of the bursts at the wavefront in the pipeline. The choice of the step ratio in coordinate and time equal to the wave propagation velocity in the pipeline allows for obtaining results that coincide with the exact solution.

Application of finite difference method in solving a second- and fourth-order PDE blending denoising model

Application of finite difference method in solving a second- and fourth-order PDE blending denoising model

The application of finite difference method on 2-D heat conductivity problem

The finite difference method is one of the numerical methods that is often used to solve partial differential equations arose in the real world physical problems. The method is approximated by Taylor series. The study considers the FDM method to calculate the heat diffusion in any point in a rectangular domain. The results show that, it has a good level of accuracy with various values of error.

Application of Finite Difference Method in Image Reconstruction of Electrical Capacitance Tomography

Electrical capacitance tomography is to invert the medium distribution inside the object by using a limited set of measured capacitance values. In order to realize image reconstruction, the sensor sensitive field is solved by finite difference method under the condition of known sensor distribution and internal permittivity distribution of pipeline, and satisfactory results are obtained. According to the calculated sensitive field, the distribution image of the fluid on the fault plane is reconstructed by using the back-projection algorithm. Back-projection algorithm has the character of simple structure and fast speed of reconstructing images. However, the number of projections obtained by the capacitive tomography system is much smaller than the number of image pixels and the soft field characteristics of the imaging system make it difficult to accurately image, so only an approximate phase distribution image can be reconstructed.

Application of Finite Difference Method in Multifractal Techniques to Identify Pb–Zn Anomalies: a Case Study from Khomein, Iran

Separation of geochemical anomalies from background is an important and essential issue in the exploration of mineral deposits. Statistical methods are used to separation of anomaly from background. Fractal/multifractal and singularity methods are very suitable for the separation of weak anomalies. For interpolation, various methods could be used such as most distance-weighting methods, kriging, spline interpolation, interpolating polynomials, and finite difference methods. This paper addressed application of multifractal techniques together with finite difference procedure to detect the anomalous regions. Fractal models such as concentration–area, spectrum–area, and singularity index model were selected to identify Pb–Zn anomalies. In this research, 170 samples from 33 boreholes were taken in the Haft-Savaran area and analyzed for determination of Pb–Zn anomaly. Singularity method was a suitable method for separating Pb and Zn anomalies. Also, the results showed that the geochemical anomaly has occurred along NE–SW trend and is consistent with the sulfide mineralization process and the faults of the region.

About mechanical behaviour of metal matrix composites under shock wave loading

In this work the results of studying of the mechanical behaviour of metal-ceramic composites with an aluminium matrix on meso-scale level are submitted. Computer simulation of mechanical reaction of representative volume of composite material, considered as ensemble of the interacting structural elements (ceramic particles and metal matrix), is used for studying mechanisms of deformation on meso-scale level of metal-ceramic composites under the loading by shock waves. The mechanical behaviour of aluminium matrix is described by the model of the damaged elastic-plastic medium. The model of the damaged brittle solid is used for ceramics. The absence of macroscopic and mesoscopic voids and cracks in the material before loading is assumed. Ideal adhesion of ceramic material to metal is supposed at the initial condition. The problem is solved in 2D statement with application of finite-difference method. Results of numerical simulation of the mechanical behaviour of metal-ceramic composites under shock wave loading have shown the formation of non-stationary and essentially non-uniform fields of stresses and strains at the meso-scale level. The generation of a dissipative structure on the meso-scale level of the composites under shock wave loading was revealed in the simulations.

Primjena metode konačnih razlika u određivanju utjecaja statičkog posmičnog naprezanja na likvefakciju tla

Primjena metode konačnih razlika u određivanju utjecaja statičkog posmičnog naprezanja na likvefakciju tla

Computer simulation of damage to metal-ceramic composites under shock loading

In this work the results of studying of the mechanical behaviour of metal-ceramic composites with an aluminium matrix on mesoscale level are submitted. Computer simulation of mechanical reaction of representative volume of composite material, considered as ensemble of the interacting structural elements (ceramic particles and metal matrix), is used for studying processes of the nucleation and growth of damages on mesoscale level of metal-ceramic composites under the loading by shock pulses. The sizes of representative volume give the possibilities to enter effective mechanical parameters of a composite material. The mechanical behaviour of aluminium matrix is described by the model of the damaged elastic-plastic medium. The model of the damaged brittle solid is used for ceramics. The problem is solved in 2D statement with application of finite-difference method. The results of computer simulation show that the local tensile stresses can appear in unloading wave and can cause damages in ceramic particles. In the region of composite where two opposing unloading waves interacts (spall zone) the calculation predicts the existence of cracked ceramic particles, local damages in metal matrix and existence of mesoscale and macro-scale cracks.

Application of Finite Difference Method for determining lunar regolith diurnal temperature distribution

This study was performed in order to verify viability of using finite difference method and proposed simple astrometrical model for modelling heat transfer in lunar regolith. The concept was examined by developing FD model of heat flow for upper 0,9 m of lunar regolith, and comparing obtained results with in situ measurements provided by Apollo 15 and 17 heat flow experiments. The model was based on FDM approximation of Fourier’s law for one dimensional transient heat flow. Both constant and temperature-dependent thermophysical properties of lunar regolith were obtained from in situ measurements. Thermal boundary conditions were assumed on in situ measurements and on remote sensing based analytical model. In order to approximate Sun's position at lunar sky, simple analytical astrometric model of lunar rotation was developed. Matlab 2012a was used to conduct the calculations. Stable solutions were obtained for latitudes between 0 and 80°. Satisfactory agreement between Apollo 15 and 17 in situ measurements and FDM modelling was observed. A conclusion was reached, that both FDM and proposed astrometrical model are to be successfully applied for modelling heat transfer in lunar regolith.

Application of finite difference method of lines on the heat equation

In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three‐dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.

A Difference-Wavelet Method for Solving Generalized Density Evolution Equation in Stochastic Structural Analysis

The probability density evolution method (PDEM) provides a feasible approach for the dynamic response analysis of nonlinear stochastic structures. The key step in this regard is to solve a generalized density evolution equation (GDEE) in order to establish the probability density function (PDF). Previously, a finite difference method (FDM) has often been resorted to solve the GDEE. However, one may encounter the problem of mesh sensitivity in the application of FDM to the PDEM. To this end, a novel difference-wavelet method that can improve the finite difference result by means of a nonlinear wavelet density estimation method is proposed in the present paper. By exploiting the multi-resolution property of wavelet functions and by choosing the optimal scale at each instant, it is expected that the bothering mesh sensitivity issue in finite difference method can be overcome to some extent and a better probability density result can be obtained. In order to verify the proposed method, a single-degree-of-freedom (SDOF) oscillator and an 8-story frame structure are investigated in detail. The results show the notable superiority of the proposed method to finite difference method.

Stability of curvilinear Euler-Bernoulli beams in temperature fields

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles.The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method.The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method.Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.

Application of finite difference method in the study of diffusion with chemical kinetics of first order

The mathematical modelling of diffusion of a bleaching agent into a porous material is studied in the present paper. Law of mass conservation was applied to analize the mass transfer of a reactant from the bulk into the external surface of a solid geometrically described as a flat plate. After diffusion of the reactant, surface reaction following kinetics of first order was considered to take place. The solution of the differential equation that described the process leaded to an equation that represents the concentration profile in function of distance, porosity and Thiele modulus. The case of interfacial mass resistance is also discused. In this case, finite difference method was used for the solution of the differential equation taking into account the respective boundary conditions. The profile of concentration can be obtained after numerical especification of Thiele modulus and Biot number.

有限差分法在太阳能电池性质仿真中的应用

有限差分法在太阳能电池性质仿真中的应用

Detecting milling deformation in 7075 aluminum alloy thin-walled plates using finite difference method

With the development of modern airspace technology, entire thin-walled structures have been widely and increasingly used. Given the complicated milling process and poor rigidity of thin-walled plates, deformation and surface errors are apparent in workpieces. Finite difference method (FDM) has been applied to some engineering fields for establishing mathematical models because of its simplicity and efficiency. However, FDM has not yet been introduced and employed in the aerospace field, especially in the prediction of deformations in thin-walled plates. A novel mathematical method is developed in this study to detect deformations in thin-walled plates given the actual milling situation. This method is based on FDM. Then, the deformation can be estimated under three different boundary conditions using a program written with program codes. Given the stiffness of fixtures, a deformation experiment was conducted under preload forces to compare FDM with finite element method (FEM). In this experiment, FDM is similar to FEM. Moreover, the deformation parameters are validated by comparative results. Most of the errors are within 10 %. The reason for these errors is determined to improve the reliability of the calculation. Thus, this research introduces an effective method for obtaining solutions to the deformation of thin-walled plates in the milling process. This work also expands the application of FDM in the engineering field.

FINITE DIFFERENCES APPROACH FOR CALCULATING ELASTIC LATERAL TORSIONAL BUCKLING MOMENT OF CANTILEVER I SECTIONS

In steel structures, beams are usually assembled so as to be under bending about their major axis in order to use the structural material economically. Two major problems take place in design of these steel beams. First problem is exceeding of yield stress at the extreme fiber of the section. External force magnitude for any loading case which causes exceeding of yield stress on a steel beam can be easily calculated. Second problem is loss of stability. In beams which are loaded so as to be under bending about their major axis an equilibrium state is also possible for a certain load magnitude, in which the beam is twisted and buckled about its weak axis. This case is called lateral torsional buckling. Depending on the loading case, yield stress of the structural material and slenderness of the beam, which is related to section properties and length of the element, lateral torsional buckling may occur before the extreme fiber of the section reaches to yield stress. In this case, instead of first yield moment, lateral torsional buckling moment should be considered in design of the beam. In this paper, application of finite differences method for determining elastic critical lateral torsional buckling moment of cantilever I sections which are loaded from shear center is presented. The results are compared with ABAQUS software and it is seen that results obtained from the presented 1D model and ABAQUS software coincide.

Application of Finite Difference Method and PSO Algorithm in Seismic Analysis of Narmab Earth Dam

In this paper the dynamic behavior of the Narmab earth dam (Iran), considering dam-foundation interaction, under normalized Manjil earthquake as input motion has been studied. In order to assess the effect of the dam heights and the foundation widths, in the finite difference model on the earthquake response, various dam-foundation coupled models are analyzed by FLAC as finite difference program for solving mining and geotechnical engineering problems. In this research, a fitness function has been defined based on NonLinear Energy Operator (NLEO). The dam heights and the foundation widths has been chosen optimally using particle swarm optimization (PSO) method. The simulation results indicate considerable differences in the seismic responses.

Numerical Solution of Axisymmetric Problem for an Elastic Footing

Based on formulas of elastic mechanics, the axisymmetric problem applied to an elastic footing is analyzed. Using Hankel transform and finite difference method, the arbitrary points’ displacement and stress solutions of the axisymmetric elastic footing problem is obtained. Compared with recursive substitution method and transfer matrix method, this method is simple because it avoids complex formulas and exponential function calculation. The half space model is more flexible to simulate an actual footing. The numerical solutions are compared with Boussinesq’s Equation and results show that finite difference method is applicable to solve axisymmetric elastic footing problem. But in the application of finite difference method, there are a few factors that could cause calculation errors. Effective measures are needed to obtain exact solutions.