Central Finite Difference Method Research Articles

A Comprehensive Numerical Model of Steady-State Negative Corona for Coaxial Cylindrical Electrode in Atmosphere Air

Corona numerical model to predict the distributions of space charge and electric field is an important aspect in coaxial cylindrical electrode. Compared with the existing steady-state negative corona models, electron diffusion, space charge recombination, photoionization, and secondary electron emission mechanism are simultaneously considered in a comprehensive negative corona model. The 1-D hydrodynamic equations are calculated by the fourth-order Runge–Kutta method and the central finite difference method. The effects of these four terms in the conversation equations are analyzed quantitatively. In addition, the influence of corona current and conductor radius on the distribution of space charge and electric field and voltage–current characteristics has been discussed. The results from the comprehensive model show that under the high corona current, the comprehensive scheme is superior to the simplified model and the ion flow model in the accuracy.

Eringen’s small length scale coefficient for vibration of axially loaded nonlocal Euler beams with elastic end restraints

AbstractThis paper presents the calibration of Eringen’s small length scale coefficient

Influence of Prandtl number on heat transfer of a flat vertical plate

Liquid metals, such as sodium (Na), lead (Pb), and lead-bismuth (Pb-Bi) eutectic (e), are considered as potential coolants for the fast spectrum nuclear reactors of the next generation. So the main objective of this paper is to study the heat transfer and fluid flow characteristics of liquid metal coolants flowing over a nuclear fuel element having uniform volumetric energy generation. Stream function vorticity formulation method was used to solve the full Navier Stokes equations governing the flow. The energy equation was solved using central finite difference method. For the two-dimensional steady state heat conduction and stream-function equation, the discretisation was done in the form suitable to solve using 'line-by-line Gauss-Seidel' solution technique whereas the discretisation of vorticity transport and energy equations was done using Alternating Direction Implicit (ADI) scheme. After discretisation the systems of algebraic equations were solved using 'Thomas algorithm'. The complete work was done by writing a well-validated indigenous computer code using C-language. The parameters considered for the study were: aspect ratio of fuel element, Ar, conduction-convection parameter Ncc, total energy generation parameter Qt, and flow Reynolds number ReH. The results obtained can be used to minimise the maximum temperature in the fuel element (hot spots) and prevent its melting.

A fourth order accuracy summation-by-parts finite difference scheme for acoustic reverse time migration in boundary-conforming grids

The fourth order accuracy finite difference scheme is known advantageous in reducing memory and improving efficiency. Summation-by-parts finite difference operator is a natural way for wavefield simulation in complicated domains containing surface topography and irregular interfaces. The application of summation-by-parts method guarantees the stability of numerical approximation for heterogeneous media on curvilinear grids. This paper extends the second order summation-by-parts finite difference method to the fourth order case for the discretization of acoustic wave equation and perfect matched layer in boundary-conforming grids. In particular, the implementation of the fourth order method for wavefield simulation and reverse time migration in complicated domains can significantly improve the efficiency and decrease the storage. The elliptic method is applied for boundary-conforming grid generation in complicated domains. Under such grids, the two-dimensional acoustic wave equation in second order displacement formulation is compactly reformulated for forward modeling and reverse time migration, and the symmetric and compact form of perfectly matched layers expressed in a curvilinear coordinate system are applied to suppress artificial reflections. The discretizations of the acoustic wave equation and perfectly matched layer formula are fourth and second order accuracy in space and time respectively, where the spatial discretization satisfies the principle of summation-by-parts and is stable. Numerical experiments are presented to compare the accuracy of the second with fourth order summation-by-parts finite difference methods and to evaluate the efficiency of reverse time migration by using these two methods. As well, comparisons are performed between the fourth order accuracy summation-by-parts finite difference method and central finite difference method to illustrate the stability superiority of summation-by-parts operators.

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.

A three-dimensional mathematical model for the anode of a direct ethanol fuel cell

In this paper, we develop a mathematical model to analyze a direct ethanol fuel cell (DEFC). The three-dimensional model is able to predict the flow on all layers of the fuel cell and allow a better analysis of physical and chemical phenomena that occur inside it. In addition, the calculation of mole fraction of species allows one to observe that the diffusion layer has great influence on mass transfer of fuel between the input channel and the catalyst layer. Numerical simulation of reactive flow was made based on the central finite difference method. The equations were integrated in time using the simplified Runge-Kutta multistage scheme. The results obtained are in agreement with the experimental data found in the literature, for different feed concentrations of ethanol and for different operating temperatures of the cell. In this way, the paper contributes to the development of a model for direct ethanol fuel cells, taking into account all losses overpotentials at the anode and the cathode, providing a better understanding of the physical and chemical behavior inside the cell, and on the conversion of chemical energy into electrical energy.

Non-Hydrostatic Model for Solitary Waves Passing Through a Porous Structure

The non-hydrostatic depth-integrated model we developed to study solitary waves passing undisturbed in shape through a porous structure, involves hydrodynamic pressure. The equations are nonlinear, diffusive, and weakly dispersive wave equation for describing solitary wave propagation in a porous medium. We solve the equation numerically using a staggered finite volume with a predictor-corrector method. To demonstrate our non-hydrostatic scheme’s performance, we implement our scheme for simulating solitary waves over a flat bottom in a free region to examine the balance between dispersion and nonlinearity. Our computed waves travel undisturbed in shape as expected. Furthermore, the numerical scheme is used to simulate the solitary waves pass through a porous structure. Results agree well with results of a central finite difference method in space and a fourth-order Runge-Kutta integration technique in time for the Boussinesq model. When we quantitatively compare the wave amplitude reduction from our numerical results to experimental data, we find satisfactory agreement for the wave transmission coefficient.

Embedded Crack Identification in Beam-Column Structures Under Axial Load Using an Efficient Static Data Based Indicator

A triangular model base on an investigation which has done by Sinha et al. has been developed for evaluating embedded crack localization in beam-column structures. In the assessment of this member’s behavior, the effects of displacement slope are necessary. In order to propose a crack localization method for embedded crack, an efficient static data based indicator is proposed for this crack in Euler-Bernoulli beam-columns under axial load effect. A finite element procedure is implemented for calculating the Static responses. Then, base on a central finite difference method, the slope and curvatures of horizontal displacements are evaluated. For this purpose, a simply supported beam-column and a two-span beam-column are considered and two different scenarios base on the damage of one element (single damage) and multiple elements (multiple damages) by considering the noise have been assessed. The numerical results have shown that this crack localization method has considerable accurate.

Electrokinetic and aspect ratio effects on secondary flow of viscoelastic fluids in rectangular microchannels

The secondary flow of PTT fluids in rectangular cross-sectional plane of microchannels under combined effects of electroosmotic and pressure driving forces is the subject of the present study. Employing second-order central finite difference method in a very refined grid network, we investigate the effect of electrokinetic and geometric parameters on the pattern, strength and the average of the secondary flow. In this regard, we try to illustrate the deformations of recirculating vortices due to change in the dimensionless Debye–Huckel and zeta potential parameters as well as channel aspect ratio. We demonstrate that, in the presence of thick electric double layers, significant alteration occurs in the secondary flow pattern by transition from favorable to adverse pressure gradients. Moreover, it is found that for polymer-electrolyte solutions with large Debye lengths, the secondary flow pattern and the shape of vortices are generally dependent upon the width-to-height ratio of the channel cross section. Also, the inspections of strength and average of secondary flow reveal that the sensitivity of these quantities with respect to the electrokinetic, geometric and rheological parameters increases by increasing the absolute value of velocity scale ratio. In this regard, utilizing the curve fitting of the results, several empirical expressions are presented for the strength and average of the secondary flow under various parametric conditions. The obtained relations with the other predictions for secondary flow are of high practical importance when dealing with the design of microfluidic devices that manipulate viscoelastic fluids.

An immersed boundary method for direct and large eddy simulation of stratified flows in complex geometry

A sharp-interface Immersed Boundary Method (IBM) is developed to simulate density-stratified turbulent flows in complex geometry using a Cartesian grid. The basic numerical scheme corresponds to a central second-order finite difference method, third-order Runge–Kutta integration in time for the advective terms and an alternating direction implicit (ADI) scheme for the viscous and diffusive terms. The solver developed here allows for both direct numerical simulation (DNS) and large eddy simulation (LES) approaches. Methods to enhance the mass conservation and numerical stability of the solver to simulate high Reynolds number flows are discussed. Convergence with second-order accuracy is demonstrated in flow past a cylinder. The solver is validated against past laboratory and numerical results in flow past a sphere, and in channel flow with and without stratification. Since topographically generated internal waves are believed to result in a substantial fraction of turbulent mixing in the ocean, we are motivated to examine oscillating tidal flow over a triangular obstacle to assess the ability of this computational model to represent nonlinear internal waves and turbulence. Results in laboratory-scale (order of few meters) simulations show that the wave energy flux, mean flow properties and turbulent kinetic energy agree well with our previous results obtained using a body-fitted grid (BFG). The deviation of IBM results from BFG results is found to increase with increasing nonlinearity in the wave field that is associated with either increasing steepness of the topography relative to the internal wave propagation angle or with the amplitude of the oscillatory forcing. LES is performed on a large scale ridge, of the order of few kilometers in length, that has the same geometrical shape and same non-dimensional values for the governing flow and environmental parameters as the laboratory-scale topography, but significantly larger Reynolds number. A non-linear drag law is utilized in the large-scale application to parameterize turbulent losses due to bottom friction at high Reynolds number. The large scale problem exhibits qualitatively similar behavior to the laboratory scale problem with some differences: slightly larger intensification of the boundary flow and somewhat higher non-dimensional values for the energy fluxed away by the internal wave field. The phasing of wave breaking and turbulence exhibits little difference between small-scale and large-scale obstacles as long as the important non-dimensional parameters are kept the same. We conclude that IBM is a viable approach to the simulation of internal waves and turbulence in high Reynolds number stratified flows over topography.

Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs

AbstractThis paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

Investigation of Fluid-structure Interaction by Explicit Central Finite Difference Methods

Fluid-structure interaction (FSI) occurs when the dynamic water hammer forces; cause vibrations in the pipe wall. FSI in pipe systems due to Poisson and junction coupling has been the center of attention in recent years. It causes fluctuations in pressure heads and vibrations in the pipe wall. The governing equations of this phenomenon include a system of first order hyperbolic partial differential equations (PDEs) in terms of hydraulic and structural quantities. In the present paper, a two-step variant of the Lax-Friedrichs (LxF) method, and a method based on the Nessyahu-Tadmor (NT) are used to simulate FSI in a reservoir-pipe-valve system. The computational results are compared with those of the Method of Characteristics (MOC), Godunov's scheme and also with the exact solution of linear hyperbolic four-equation system to verify the proposed numerical solution. The results reveal that the proposed LxF and NT schemes can predict discontinuous in fluid pressure with an acceptable order of accuracy. The independency of time and space steps allows for setting different spatial grid size with a unique time step, thus increasing the accuracy with respect to the conventional MOC. In these schemes, no Riemann problems were solved and hence field-by-field decompositions were avoided which led to reduced run times compared with Godunov scheme.

Multi-scale Coupling between Monte Carlo Molecular Simulation and Darcy-Scale Flow in Porous Media

In this work, an efficient coupling between Monte Carlo (MC) molecular simulation and Darcy- scale flow in porous media is presented. The cell centered finite difference method with non- uniform rectangular mesh were used to discretize the simulation domain and solve the governing equations. To speed up the MC simulations, we implemented a recently developed scheme that quickly generates MC Markov chains out of pre-computed ones, based on the reweighting and reconstruction algorithm. This method astonishingly reduces the required computational times by MC simulations from hours to seconds. To demonstrate the strength of the proposed coupling in terms of computational time efficiency and numerical accuracy in fluid properties, various numerical experiments covering different compressible single-phase flow scenarios were conducted. The novelty in the introduced scheme is in allowing an efficient coupling of the molecular scale and the Darcy's one in reservoir simulators. This leads to an accurate description of thermodynamic behavior of the simulated reservoir fluids; consequently enhancing the confidence in the flow predictions in porous media.

Modelling of quantum confinement in optical nanostructures

An efficient sensitivity analysis approach for quantum nanostructures is proposed. The imaginary time propagation method (ITP) is utilized to solve the time dependent Schrödinger equation (TDSE). Using this method, an extraction of all the modes and their sensitivity with respect to all the design parameters have been performed with minimal computational effort. The sensitivity analysis is done using the adjoint variable method (AVM) and results are comparable to those obtained using central finite difference method (CFD) applied directly on the response level.

The BFGS method for estimating the interface temperature and convection coefficient in ultrasonic welding

An inverse heat transfer problem is investigated in the present study by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to predict the unknown time-dependent heat generation at the weld interface and convection heat transfer coefficient during an ultrasonic metal welding process based on the knowledge of temperature measurements taken on the horn. With known temperature data at some locations on the horn, the inverse solution was rapidly obtained by solving nonlinear direct problem, Central Finite Difference and Simple Step Method. The proposed method which did not need solving adjoint and sensitivity problem revealed the characteristics of high efficiency, lower iterations for a computational algorithm and high accuracy for estimating values even when measurement error was considered. Besides, a comparison of the BFGS method with some previous methods (i.e. CGM, SCGM) was established. These results show that an excellent estimation on interfacial heat generation (or temperature), as well as a convection heat transfer coefficient, can be simultaneously obtained in this study. The current methodology will provide a useful tool to optimize welding conditions in ultrasonic welding.

Transient Response of Cross Flow Heat Exchangers Subjected to Simultaneous Temperature and Flow Perturbations

This paper compares the transient thermal performance between counter and parallel cross flow heat exchangers subjected to time varying inlet mass flow rates and inlet temperatures that hasn’t been previously discussed in the available literature. Specifically the transient performance of 2 pass and 3 pass cross flow heat exchangers is discussed in this paper. In the present study the energy balance equations for the hot and cold fluids and the heat exchanger wall were solved using an implicit central finite difference method. Representative values of NTU were considered, and the NTU’s of the heat exchanger were assumed to be uniformly distributed among the heat exchanger passes. Other physically significant parameters such as the capacity rate ratio and the convection heat transfer resistance ratio were systematically varied. A detailed summary based on the observations has been presented.

Damage identification in plate-like structure using mode shape derivatives

In this paper, a new damage indicator based on modal data, such as mode shapes and its derivatives, is presented for damage identification in plate-like structures. The proposed indicator is determined using modal analysis information extracted from a finite element code in MATLAB. After obtaining the mode shapes, the slope and curvature of the plate in each mode are calculated based on central finite difference methods. A numerical example with and without noise is considered to evaluate the exact location of different damage scenarios. In order to validate the proposed indicator for structural damage detection, the obtained results have been compared with another study which was based on experimental data. Moreover, in order to better assess the performance of the proposed indicator, a comparison has been made between the proposed indicator and two well-known indicators found in the literature. The results indicate that the proposed damaged indicator is able to detect precisely the location of single and multiple damage cases having different characteristics in plate-like structures.

Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments

Peridynamics is a non-local continuum mechanics formulation that can handle spatial discontinuities as the governing equations are integro-differential equations which do not involve gradients such as strains and deformation rates. This paper employs bond-based peridynamics. Cellular Automata is a local computational method which, in its rectangular variant on interior domains, is mathematically equivalent to the central difference finite difference method. However, cellular automata does not require the derivation of the governing partial differential equations and provides for common boundary conditions based on physical reasoning. Both methodologies are used to solve a half-space subjected to a normal load, known as Lamb’s Problem. The results are compared with theoretical solution from classical elasticity and experimental results. This paper is used to validate our implementation of these methods.

Convergence Analysis of the Standard Central Finite Difference Method for Poisson Equation

We consider the standard central finite difference method for solving the Poisson equation with the Dirichlet boundary condition. This scheme is well known to produce second order accurate solutions. From numerous tests, its numerical gradient was reported to be also second order accurate, but the observation has not been proved yet except for few specific domains. In this work, we first introduce a refined error estimate near the boundary and a discrete version of the divergence theorem. Applying the divergence theorem with the estimate, we prove the second order accuracy of the numerical gradient in arbitrary smooth domains.

A Parametric Study of Turbulent Flow Past a Circular Cylinder Using Large Eddy Simulation

Flow over a circular cylinder at a Reynolds number of 3900 is investigated using large eddy simulations (LES) to assess the affect of four numerical parameters on the resulting flow-field. These parameters are subgrid scale (SGS) turbulence models, wall models, discretization of the advective terms in the governing equations, and grid resolution. A finite volume method is employed to solve the incompressible Navier–Stokes equations (NSE) on a structured grid. Results are compared to the experiments of Ong and Wallace (1996, “The Velocity Field of the Turbulent Very Near Wake of a Circular Cylinder,” Exp. Fluids, 20(6), pp. 441–453) and Lourenco and Shih (1993, “Characteristics of the Plane Turbulent Near Wake of a Circular Cylinder: A Particle Image Velocimetry Study,” private communication (taken from Ref. [2]); and the numerical results of Beaudan and Moin (1994, “Numerical Experiments on the Flow Past a Circular Cylinder at Sub-Critical Reynolds Number,” Technical Report No. TF-62), Kravchenko and Moin (2000, “Numerical Studies of Flow Over a Circular Cylinder at ReD = 3900,” Phys. Fluids, 12(2), pp. 403–417), and Breuer (1998, “Numerical and Modelling Influences on Large Eddy Simulations for the Flow Past a Circular Cylinder,” Int. J. Heat Fluid Flow, 19(5), pp. 512–521). It is concluded that the effect of the SGS models is not significant; results with and without a wall model are inconsistent; nondissipative discretization schemes, such as central finite difference methods, are preferred over dissipative methods, such as upwind finite difference methods; and it is necessary to properly resolve the boundary layer in the vicinity of the cylinder in order to accurately model the complex flow phenomena in the cylinder wake. These conclusions are based on the analysis of bulk flow parameters and the distribution of mean and fluctuating quantities throughout the domain. In general, results show good agreement with the experimental and numerical data used for comparison.