Central Finite Difference Research Articles

Numerical Analysis of Transient Heat Transfer in Radial Porous Moving Fin with Temperature Dependent Thermal Properties

In this article, a time dependent partial differential equation is used to model the nonlinear boundary value problem describing heat transfer through a radial porous moving fin with rectangular profile. The study is performed by applying a numerical solver in MATLAB (pdepe), which is a centered finite difference scheme. The thermal conductivity and fin surface emissivity are linearly dependent on temperature while the heat transfer coefficient is given by power law function of temperature. The effects of thermo-physical parameters, such as the Peclet number, surface emissivity coefficient, power index of heat transfer coefficient, convective-conductive parameter, radiative-conductive parameter and non-dimensional ambient temperature on temperature are studied.

A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term

In this paper, a fitted mesh numerical method on Shishkin mesh is proposed to solve a weakly coupled system of two singularly perturbed reaction-diffusion equations containing equal diffusion parameters with discontinuous source terms. This method uses the standard centred finite difference scheme constructed on piecewise-uniform Shishkin mesh with the average of the source terms on either side of the point of discontinuity and then the problem is solved by an iterative procedure. An error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to confirm the theoretical results and compares well with the existing results.

A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow over a Linearly Stretching Surface

Present contribution is concerned with the construction and application of a numerical method for the fluid flow problem over a linearly stretching surface with the modification of standard Gradient descent Algorithm to solve the resulted difference equation. The flow problem is constructed using continuity, and Navier Stoke equations and these PDEs are further converted into boundary value problem by applying suitable similarity transformations. A central finite difference method is proposed that gives third-order accuracy using three grid points. The stability conditions of the present proposed method using a Gauss-Seidel iterative procedure is found using VonNeumann stability criteria and order of the finite difference method is proved by applying the Taylor series on the discretised equation. The comparison of the presently modified optimisation algorithm with the Gauss-Seidel iterative method and standard Newton’s method in optimisation is also made. It can be concluded that the presently modified optimisation Algorithm takes a few iterations to converge with a small value of the parameter contained in it compared with the standard descent algorithm that may take millions of iterations to converge. The present modification of the steepest descent method converges faster than Gauss-Seidel method and standard steepest descent method, and it may also overcome the deficiency of singular hessian arise in Newton’s method for some of the cases that may arise in optimisation problem(s).

A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term

In this paper, a fitted mesh numerical method on Shishkin mesh is proposed to solve a weakly coupled system of two singularly perturbed reaction-diffusion equations containing equal diffusion parameters with discontinuous source terms. This method uses the standard centred finite difference scheme constructed on piecewise-uniform Shishkin mesh with the average of the source terms on either side of the point of discontinuity and then the problem is solved by an iterative procedure. An error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to confirm the theoretical results and compares well with the existing results.

SIMULASI PERAMBATAN GELOMBANG ELEKTROMAGNETIK PADA KRISTAL FOTONIK 2D MENGGUNAKAN METODE FINITE DIFFERENCE FREQUENCY DOMAIN

Numerical simulations on two-dimensional photonic crystals with defects were analyzed using the Finite Difference Frequency Domain (FDFD). This approach consists of Maxwell's formulation that uses Central Finite Difference to place fields and materials at discrete points of the Yee grid, so that the matrix wave equation is obtained in the form of column vectors. Absorbent boundary conditions use Perfectly Matched Layer (PML) with fictitious magnetic conductivity to shed incoming waves at the edge of the domain calculation. Photonic crystals can be assumed to be a periodic lattice of dielectric material that produces the phenomenon of photonic band gap (PBG). The results of FDFD simulations are compared with the literature with a difference of 0.056. This small difference value means that this method is good enough to analyze PBG phenomena. For point defects and the accumulation of electromagnetic waves, linear defects are investigated and analyzed with spectral responses. Insertion of defects in photonic crystals will produce a photonic pass band (PPB). The simulation results show that PPB depends on the angle of arrival vector, material permittivity, and width of the defect structure.

Vertical Discretization for a Nonhydrostatic Atmospheric Model Based on High-Order Spectral Elements

AbstractThis study addresses the treatment of vertical discretization for a high-order, spectral element model of a nonhydrostatic atmosphere in which the governing equations of the model are separated into horizontal and vertical components by introducing a coordinate transformation, so that one can use different orders and types of approximations in both directions. The vertical terms of the decoupled governing equations are discretized using finite elements based on either Lagrange or basis-spline polynomial functions in the sigma coordinate, while maintaining the high-order spectral elements for the discretization of the horizontal terms. This leads to the fact that the high-order model of spectral elements with a nonuniform grid, interpolated within an element, can be easily accommodated with existing physical parameterizations. Idealized tests are performed to compare the accuracy and efficiency of the vertical discretization methods, in addition to the central finite differences, with those of the standard high-order spectral element approach. Our results show, through all the test cases, that the finite element with the cubic basis-spline function is more accurate than the other vertical discretization methods at moderate computational cost. Furthermore, grid dependency studies in the tests with and without orography indicate that the convergence rate of the vertical discretization methods is lower than the expected level of discretization accuracy, especially in the Schär mountain test, which yields approximately first-order convergence.

Adjoint-Based Design Optimization of Nonlinear Switched Reluctance Motors

This work investigates the application of the adjoint variable method (AVM) to switched reluctance motors (SRMs). A MATLAB toolbox developed by the authors estimates the sensitivities of the required objective function with respect to different geometric design parameters using at most one adjoint simulation. In this work, the AVM evaluates the sensitivities of the x and y components of the magnetic flux density, the phase flux linkage, and the electromagnetic torque of switched reluctance motors with respect to teeth height, yoke thickness, teeth pole arc angle, and teeth taper angle of both stator and rotor. The nonlinearity of the motor magnetic material is taken into consideration. The estimated sensitivities using AVM are compared with those obtained using the more accurate but time intensive central finite differences (CFD). An interior-point optimization algorithm utilizes the sensitivities of the electromagnetic torque of an SRM to maximize the motor static torque profile. Structural mapping technique is used to control the geometric design parameters through the optimization process.

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

SummaryIn this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection‐diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid‐point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second‐order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.

Higher Order Fitted Operator Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems

In this paper, we consider singularly perturbed parabolic convection-diffusion initial boundary value problems with two small positive parameters to construct higher order fitted operator finite difference method. At the beginning, we discretize the solution domain in time direction to approximate the derivative with respect to time and considering average levels for other terms that yields two point boundary value problems which covers two time level. Then, full discretization of the solution domain followed by the derivatives in two point boundary value problem are replaced by central finite difference approximation, introducing and determining the value of fitting parameter ended at system of equations that can be solved by tri-diagonal solver. To improve accuracy of the solution with corresponding higher orders of convergence, we applying Richardson extrapolation method that accelerates second order to fourth order convergent. Stability and consistency of the proposed method have been established very well to assure the convergence of the method. Finally, validate by considering test examples and then produce numerical results to care the theoretical results and to establish its effectiveness. Generally, the formulated method is stable, consistent and gives more accurate numerical solution than some methods existing in the literature for solving singularly perturbed parabolic convection- diffusion initial boundary value problems with two small positive parameters.

An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type

In this work we are interested in the numerical approximation of the solutions to 1D semilinear parabolic singularly perturbed systems of reaction-diffusion type, in the general case where the diffusion parameters for each equation can have different orders of magnitude. The numerical method combines the classical central finite differences scheme to discretize in space and a linearized fractional implicit Euler method together with a splitting by components technique to integrate in time. In this way, only tridiagonal linear systems must be solved to compute the numerical solution; consequently, the computational cost of the algorithm is considerably less than that of classical schemes. If the spatial discretization is defined on appropriate nonuniform meshes, the method is uniformly convergent of first order in time and almost second order in space. Numerical results for some test problems are presented which corroborate in practice the uniform convergence and the efficiency of the algorithm.

An adaptive grid-based curvelet optimized solution for nonlinear Schrödinger equation

In this work, a dynamically adaptive curvelet technique has been developed for solving nonlinear Schrödinger equation (NLS). Central finite difference method is used for approximating the one- and two-dimensional differential operators and radial-basis functions (RBFs) are utilized for approximating the differential operators on the sphere. The grid on which the equation is solved, is obtained using curvelets. For test problems 1 and [Formula: see text] (1d & 2d problems) considered in this paper, the computational time carried out by the proposed technique is analyzed with the computational time carried out by the finite difference technique. Moreover, the problem on the sphere has been considered, for which the computational time carried out by the RBF collocation technique is analyzed with the computational time carried out by the proposed technique. It is found that the developed technique performs better in terms of computational time, for example, on sphere, computational effort reduces by four times using the proposed method.

Three-Dimensional Dynamic Stress and Vibration Analyses of Thick Singular-Kernel Fractional-Order Viscoelastic Annular Rotating Discs Under Nonuniform Loads

In this paper, the dynamic stress and radial/lateral vibration of circular/annular discs made of fractional-order viscoelastic materials under nonuniform mechanical loads are investigated for the first time, utilizing the exact 3D theory of elasticity, rather than the plate theories. The governing equations of motion of the disc are derived based on the Kelvin–Voigt fractional viscoelastic model. To solve these equations, the spatial partial and the time ordinary derivatives are replaced by adequate central, backward or forward finite difference expressions. Then the resulting Caputo-type time-dependent system of the coupled integro-differential governing equations of the fractional-order is solved by a novel numerical procedure. Namely, a time-marching procedure is employed to extract the time histories of the responses, in the space-time domain for various time and spatial distributions. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed. In this regard, effects of the fractional-order of the constitutive law, viscoelastic parameters, material rigidity, distribution and time variation patterns of the nonuniform distributed transverse loads, and boundary conditions on the distributions of the displacement and stress components are investigated.

Numerical Solutions of Unsteady Advection-Diffusion Equations by Using EG Iteration with Wave Variable Transformation

The primary goal of this paper is to investigate the effectiveness of the 4-point Explicit Group (4-point EG) iterative method for solving one-dimensional unsteady advection-diffusion problems via similarity transform. By using this transformation approach, the proposed problem can be reduced into the corresponding two-point boundary volume problem. By imposing the second-order central finite difference discretization scheme, then the corresponding approximation equation can be derived to construct a system of linear equations. Having a large linear system, the 4-point EG iterative method has been used to solve the generated system of linear equations. The formulation of the 4-point EG method is also derived. Some numerical experiments are conducted that to verify the 4-point EG method is more effective than the Gauss-Seidel (GS) method.

Numerical methods for Bogoliubov-de Gennes excitations of Bose-Einstein condensates

In this paper, we study the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs) describing the elementary excitation of Bose-Einstein condensates around the mean field ground state, which is governed by the Gross-Pitaevskii equation (GPE). Derived analytical properties of BdGEs can serve as benchmark tests for numerical algorithms and three numerical methods are proposed to solve the BdGEs, including sine-spectral method, central finite difference method and compact finite difference method. Extensive numerical tests are provided to validate the algorithms and confirm that the sine-spectral method has spectral accuracy in spatial discretization, while the central finite difference method and the compact finite difference method are second-order and fourth-order accurate, respectively. Finally, sine-spectral method is extended to study elementary excitations under the optical lattice potential and solve the BdGEs around the first excited states of the GPE. The numerical experiments demonstrate the efficiency and accuracy of the proposed methods for solving BdGEs.

Comparison of Derivative Estimation Methods in Optimal Control Using Direct Collocation

A study is conducted to evaluate four derivative estimation methods when solving a large sparse nonlinear programming problem that arises from the approximation of an optimal control problem using a direct collocation method. In particular, the Taylor series-based finite difference, bicomplex-step, and hyper-dual derivative estimation methods are evaluated and compared alongside a well-known automatic differentiation method. The performance of each derivative estimation method is assessed based on the number of iterations, the computation time per iteration, and the total computation time required to solve the nonlinear programming problem. The efficiency of each of the four derivative estimation methods is compared by solving three benchmark optimal control problems. It is found that, although central finite differencing is typically more efficient per iteration than either the hyper-dual or the bicomplex-step method, the latter two methods have significantly lower overall computation times due to the fact that fewer iterations are required by the nonlinear programming problem when compared with central finite differencing. Furthermore, although the bicomplex-step and hyper-dual methods are similar in performance, the hyper-dual method is significantly easier to implement. Moreover, the automatic differentiation method is found to be substantially less computationally efficient than any of the three Taylor series-based methods. The results of this study show that the hyper-dual method offers several benefits over the other three methods: both in terms of computational efficiency and ease of implementation.

A conservative diffuse interface method for two-phase flows with provable boundedness properties

Central finite difference schemes have long been avoided in the context of two-phase flows for the advection of the phase indicator function due to numerical overshoots and undershoots associated with their dispersion errors. We will show however, for an incompressible flow, in the context of a specific diffuse interface model, one can maintain the boundedness of the phase field while also taking advantage of the low cost and ease of implementation of central differences to construct a non-dissipative discretization scheme for the advective terms. This is made possible by combining the advection and reinitialization steps of a conservative level set scheme introduced by Olsson and Kreiss (2005) [1] to form a phase field equation similar to that of Chiu and Lin (2011) [24]. Instead of resorting to specialized upwind methods as in these articles, we prove that the boundedness of the phase field is guaranteed for certain choices of the free parameters (ϵ and γ) for a specific central difference scheme that we propose. The proposed discretely conservative and bounded phase field equation, which is free of any reinitialization or mass redistribution, possesses desirable properties that can be leveraged in the coupled finite difference discretization of the two-phase momentum equation. Additionally, as compared to the state-of-the-art conservative and bounded two-phase flow methods, the proposed method boasts competitive accuracy-vs-cost trade-off, small memory requirements, ease of implementation, providing a viable alternative for realistic two-phase flow calculations. Given that the proposed method uses cheap, local numerical stencils with uniform loading throughout the computational domain, its implementation is expected to achieve superior parallel scalability compared to the commonly employed two-phase flow methods such as level set and VOF.

Crack detection in a shaft using finite difference technique

Fatigue cracks arise in a rotor shaft due to cyclic loading. A growing crack in a machine element may lead to a calamitous failure of the machinery. Identification of crack at an early stage supports in safe and economic running of machinery. The appearance of crack in a shaft reduces local stiffness and causes a slope discontinuity in the shaft elastic line during vibration. Detection of slope discontinuity reveals the presence of crack in a shaft. However, in the presence of measurement noise, identification of slope discontinuity is difficult. In the present work, finite element model of a cracked simply supported shaft is considered for the simulation work. A transverse surface crack is considered for the cracked model. The shaft is assumed to be non-rotating and an external sinusoidal forcing is used to excite the shaft. To imitate actual experimentation, Gaussian noise is added in the simulated cracked shaft response. A dominant peak is observed at the crack position, which is obtained by using the central finite difference approximation on the simulated response of simply supported shaft. The method is used for the localization of the cracks present in the shaft. It is based on detecting the slope discontinuity in the elastic line of the shaft. The finite difference method based evaluation is also implemented for varying crack depth ratios at different signal to noise ratio (SNR).

Convergence and stability of a BSLM for advection-diffusion models with Dirichlet boundary conditions

In this paper, we present a concrete analysis of the convergence and stability of a backward semi-Lagrangian method for a non-linear advection-diffusion equation with the Dirichlet boundary conditions. The total time derivative and the diffusion term are discretized by BDF2 and the second-order central finite difference, respectively, together with the local Lagrangian interpolation. The Cauchy problem for characteristic curves is resolved by an error correction method based on a quadratic polygon. Under the mesh restriction ▵t=O(▵x1/2) between the temporal step size △t and the spatial grid size △x, it turns out that the proposed method has the convergence order O(▵t2+▵x2+▵xp+1/▵t) in the sense of the discrete H1-norm, where p is the degree of an interpolation polynomial. Further, the unconditional stability of the method is established. Numerical tests are presented to support the theoretical analyses.

Algebra of Fuzzy Random Variable and its Application to Science and Engineering Problem- Radon Transport from Soil into Building

The theme of this paper is to bring out the mathematical details of fuzzy random variable and its application to model radon transport from soil into buildings. The physical processes that influence radon transport are advection and diffusion. Parameters associated with the governing partial differential equation describing radon transport from soil into buildings are radon diffusion coefficient and advective velocity of radon in air. Both the parameters are imprecise. Imprecision of these parameters is addressed as fuzzy random variable due to presence of inherent randomness within their fuzziness. Handling fuzziness within randomness is an innovative concept of computation to quantify the uncertainty of any physical system. Paper presents the mathematical structure (algebra) of fuzzy random variables. The concept of fuzzy randomness is implemented in developing radon transport model. Numerical solution of radon transport model with fuzzy random parameters is obtained by explicit forward time central space finite difference method. Support, uncertainty index, possibility, necessity and credibility of the spatial and temporal behaviour of the random behaviour of radon concentration are also computed to explore the concept and role of the fuzzy randomness in decision making issue

Finite difference approach and successive over relaxation (SOR) method for MHD micropolar fluid with Maxwell–Cattaneo law and porous medium

This analysis considered the flow of micropolar fluid to describe the micro-rotational influence of fluid particles. The oscillatory motion of stretchable disk is responsible for the fluid flow. The impacts of magnetic field and porous medium are incorporated in a momentum equation. The Maxwell–Cattaneo law of material-invariant version having damped-hyperbolic equation is proposed in order to overcome the limitations of Fourier’s law. The model of three-dimensional flow problem is then normalized by adopting suitable similarity variables. The resulting normalized system of partial differential equations is solved in a numerical way using first and second order central and backward finite difference approximations by converting the semi-infinite domain into the finite one. The resulting expressions are solved iteratively using the successive over relaxation parameter method. The impacts of various dimensionless parameters are elaborated through graphs and tables. Higher values of porosity and magnetic parameters reduced the radial velocity curves. Micropolar parameters, namely vortex and spin gradient viscosity parameters, enhanced the microrotational tangential velocity profiles. Radial shear stresses and frictional torque are decreased due to larger values of the vortex viscosity parameter. Couple stresses along radial and tangential directions are enhanced for increasing values of porosity parameter. The three-dimensional and two-dimensional flow phenomenon is also sketched. The numerical results for the limiting scenario are compared for various physical parameters to validate the numerical scheme.