Order Central Finite Difference Research Articles

Weighted pseudo [formula omitted]-almost periodic sequence solution and guaranteed cost control for discrete-time and discrete-space stochastic inertial neural networks

This article considers the dual hybrid influences of both discrete spatial diffusions and discrete time in a stochastic inertial neural networks via the methods of exponential Euler difference and second order central finite difference. Based on a non-decomposed constant variation formula, and the theories of semi-flow and metric dynamical systems, the existence of a unique weighted pseudo θ-almost periodic sequence solution is addressed to the discrete space and time stochastic inertial neural networks. Further, a guaranteed cost controller is designed to complete a global exponential stabilization for this discrete networks by establishing a framework of drive, response and error networks. Meanwhile, global exponential stability in the sense of mean square is achieved as well. This discussion is pioneering in considering discrete spatial diffusions in inertial neural networks and offers the studying bases for future researches.

Ion temperature profile importance in collisional sheath modelling

A plasma fluid model is being developed for the simulation of a direct current plasma discharge simulation including the sheath regions. The code uses a second order centered finite difference scheme and time integration is done by strong stability preserving third order Runge-Kutta method. The separation of scalar and vectorial quantities in two different grids gives stable results. After validation by comparison with theoretical ion sheath profiles, a one dimensional direct current argon discharge was simulated and compared to 1D3v particle-in-cell simulation results. It is shown that the inclusion of a non constant ion temperature profile is mandatory in fluid models in order to recover correct increase of ion velocity in sheaths and thus to simulate direct current (DC) discharges where collisions are not negligible in the sheaths.

Dynamic Measurements with the Bicone Interfacial Shear Rheometer: The Effects of the Numerical Implementation of the Interfacial Boundary Condition

The increasing popularity of flow field-based data analysis (FFBDA) techniques has a paradigmatic example in the routines already developed for the rotational oscillating bicone bob interfacial shear rheometer. Such routines use a second order centered finite difference (SOCFD) discretization scheme, in both the vertical and radial coordinates, for the velocity field in the bulk fluid subphase and a first order forward finite difference (FOFFD) scheme in the vertical coordinate for the velocity field at the air/water interface. Such a mixture of schemes causes non-smooth flow fields at the interface that can be tackled by appropriately devising a SOCFD scheme for the vertical coordinate at the interface using a line of “phantom” nodes that merely serve to adequately merge the Navier–Stokes equations and the Boussinesq–Scriven boundary condition at the interface. Here we report on a detailed analysis of the quantitative improvements of such a scheme over the previous one by comparing the structure of the flow fields at and close to the interface, the differences in the interfacial and bulk drag torques on the bicone bob, and the differences in the torque/displacement complex amplitude ratio.

A structured adaptive mesh refinement strategy with a sharp interface direct-forcing immersed boundary method for moving boundary problems

We develop a versatile and accurate structured adaptive mesh refinement (S-AMR) strategy with a moving least square sharp-direct forcing immersed boundary method (IBM) for incompressible fluid-structure interaction (FSI) simulations. The computational grid consists of several nested blocks in different refinement levels. While blocks with the coarsest grid cover the entire computational domain, the computational domain is locally refined at the location of solid boundary (moving or fixed) by bisecting selected blocks in every coordinate direction. The grid topology and data structure is managed by an extended version of Afivo toolkit (Teunissen and Ebert, 2018), where a novel technique is introduced for conservative data transfer between the coarser and the finer blocks, particularly in velocity transformation for which the mass conservation plays a crucial role. In the present study, the continuity and Navier-Stokes equations for incompressible flows are spatially discretized with a second order central finite difference method using a collocated arrangement and are time-integrated using a semi-implicit second order fractional step method, although the proposed S-AMR strategy can be used with different discretization schemes. An IBM using a moving least square approach is utilized to impose boundary conditions. To handle FSI problems, all the governing equations for the dynamics of fluid and structure are simultaneously advanced in time by a predictor-corrector strategy. Several test cases of increasing complexity are solved in order to demonstrate the robustness and accuracy of the proposed method as well as its capability in simulation-driven mesh adaptivity.

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.

A Comparison Study of HO-CFD and DSC-RSK with Small Computational Bandwidths for Solving Some Classes of Boundary-Value and Eigenvalue Problems

This paper conducts a comparison analysis of high order central finite difference (HO-CFD) method and discrete singular convolution-regularized Shannon kernel (DSC-RSK) scheme with small computational bandwidths for solving some classes of boundary-value and eigenvalue problems. Second-, fourth- and sixth-order partial differential equations are taken into account. New strategies to generate parameters [Formula: see text] in DSC-RSK are proposed to ensure minimum errors for each case, and the influence of parameters [Formula: see text] with more decimal places is analyzed. Apart from the existing matched interface and boundary (MIB) scheme, a new double-parameter MIB scheme is also proposed. The influence of small computational bandwidths is discussed in detail. Numerical results by using HO-CFD and DSC-RSK are presented and compared to illustrate the performance of both methods in small bandwidth limit. Some remarkable conclusions have been drawn at the end of this study.

A NUMERICAL SCHEME FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS OF CONVECTION-DIFFUSION TYPE ON AN ADAPTIVE GRID

In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

SEMI-ANALYTICAL SOLUTION OF THE HEAT CONDUCTION IN A PLATE WITH HEAT GENERATION

In the present work, a formulation for the solution of the two-dimensional steady state heat conduction with heat generation is presented. The classical integral transform technique (CITT) is used to solve the problem in a semi- analytical manner. CITT deals with expansions of the sought solution in terms of infinite orthogonal basis of eigenfunctions, keeping the solution process always within a continuous domain. For the particular problem, the resulting system is composed of a set of uncoupled differential equations which can be solved analytically. However, a truncation error is involved since the infinite series must be truncated to obtain numerical results. For comparison and validation purposes, the second order central finite difference method (FDM) is also implemented. The convergence analysis showed that CITT has a greater performance having no difficulties to obtain accurate results with very few terms in the solution summation. The FDM had convergence troubles specially for the positions near the center and for high concentration of heat generation in the center of the plate.

On boundary conditions for buckling and vibration of nonlocal beams

In this paper, we argue that the boundary conditions for Eringen's nonlocal beam theory have to take on the discrete forms similar to those for the first order central finite difference beam model for beam buckling and vibration problems. The latter finite difference beam model has been found to be analogous to the physical Hencky bar-chain and therefore these two models can be regarded as one discrete beam model. Based on the phenomenological similarities between this discrete beam model and the Eringen's nonlocal beam theory, one may calibrate the Eringen's small length scale coefficient e0 which is expected to be a constant appropriate to each material. When using the classical continuous nonlocal boundary conditions, we find that e0 is not a constant but it depends on the cases of boundary conditions (such as pinned-pinned, clamped-clamped, clamped-pinned ends). However, this conundrum may be resolved if we use the discrete boundary conditions instead. By adopting the discrete boundary conditions, analytical solutions for e0 may be obtained and the e0 values finally converge to 0.289 for buckling problems and 0.408 for free vibration problems regardless of the cases of boundary conditions.

Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation

This paper is concerned with the development of the Hencky bar-chain model (HBM) for the buckling and vibration analyses of non-uniform beams resting on partial variable elastic foundation. The HBM allows analysts to obtain beam buckling and vibration solutions by solving a set of algebraic equations instead of a differential equation. To get the HBM, we resort to the first order central finite difference model (FDM) because both discrete models have phenomenological similarities. Based on bending moment, shear force and deflection equivalence between the two discrete beam models, the expressions for the internal spring stiffness, the end spring stiffness and the elastic foundation stiffness for the HBM are obtained. Some example problems are considered to demonstrate the simplicity of the HBM for buckling and vibration analyses of non-uniform beams on partial variable elastic foundation by taking the number of segments of the HBM to infinity. The effects of the foundation length and stiffness on the buckling load and natural frequencies are also discussed. Naturally, the HBM can be used to obtain solutions for articulated beams or beam-like structures with repetitive cells as well.

Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem are treated by the simplified inverse Lax–Wendroff procedure. For the fully discrete case, a third order explicit Runge–Kutta method is used as an example for the analysis. Stability is analyzed by both the Gustafsson, Kreiss and Sundstrom theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

Numerical comparison of viscosity models on mixed convection in double lid-driven cavity utilized CuO-water nanofluid

Laminar incompressible mixed-convective heat transfer in two-dimensional lid-driven cavity, filled with nanofluid CuO-water, is studied numerically. Eight different viscosity models are compared to investigate the enhancement in the heat transfer and the increase in the average Nusselt number. The point of view of each model essentially differs in terms of whether it takes various parameters such as temperature effects, Brownian motion of the nanoparticles, the radii of aggregated particles, and the volume-fraction of nanoparticles into account or not. The governing stream-vorticity equations are solved using a second order central finite difference scheme, coupled to the conservation of mass and energy. The main sensitive parameters of interest to investigate the viscosity models are chosen as volume fraction of the nanoparticles ?, and Richardson number Ri. The performance study of the viscosity models and the interpretation of the corresponding results of velocity components are done in a different range of ? and Ri. <br><br><font color="red"><b> This article has been corrected. Link to the correction <u><a href="http://dx.doi.org/10.2298/TSCI160418084E">10.2298/TSCI160418084E</a><u></b></font>

Conjugate Heat Transfer in a Porous Cavity Heated by a Triangular Thick Wall

The conjugate natural convection-conduction heat transfer in a square domain composed of a cavity heated by a triangular solid wall is studied under steady state condition. The vertical and horizontal walls of the triangular solid are kept isothermal and at the same hot temperature T h . The other boundaries surrounding the porous cavity are kept adiabatic except the right vertical wall, where it is kept isothermally at the lower temperature T c . Equations governing the heat transfer in the triangular wall and heat and fluid flow, based on the Darcy model, in the fluid-saturated porous medium together with the derived relation of the interface temperature are solved numerically using the second order central differences finite difference scheme with the successive over relaxation (SOR) method. The investigated parameters are the Rayleigh number Ra (100-1000), solid to fluid saturated porous medium thermal conductivity ratio Kr (0.1–10), and the triangular wall thickness D (0.05-1). The results are presented in the conventional form; contours of streamlines and isotherms and the local and average Nusselt numbers. An uncommon behavior of the heat transfer in the porous medium with the triangular wall thickness D is observed and accounted.

Assessing the Performance of a Three Dimensional Hybrid Central-WENO Finite Difference scheme with Computation of a Sonic Injector in Supersonic Cross Flow

AbstractA hybridization of a high order WENO-Zfinite difference scheme and a high order central finite difference method for computation of the two-dimensional Euler equations first presented in [B. Costa and W. S. Don, J. Comput. Appl. Math., 204(2) (2007)] is extended to three-dimensions and for parallel computation. The Hybrid scheme switches dynamically from a WENO-Zscheme to a central scheme at any grid location and time instance if the flow is sufficiently smooth and vice versa if the flow is exhibiting sharp shock-type phenomena. The smoothness of the flow is determined by a high order multi-resolution analysis. The method is tested on a benchmark sonic flow injection in supersonic cross flow. Increase of the order of the method reduces the numerical dissipation of the underlying schemes, which is shown to improve the resolution of small dynamic vortical scales. Shocks are captured sharply in an essentially non-oscillatory manner via the high order shock-capturing WENO-Zscheme. Computations of the injector flow with a WENO-Zscheme only and with the Hybrid scheme are in very close agreement. Thirty percent of grid points require a computationally expensive WENO-Zscheme for high-resolution capturing of shocks, whereas the remainder of grid points may be solved with the computationally more affordable central scheme. The computational cost of the Hybrid scheme can be up to a factor of one and a half lower as compared to computations with a WENO-Zscheme only for the sonic injector benchmark.

Energy extrapolation schemes for adaptive multi-scale molecular dynamics simulations

This paper evaluates simple schemes to extrapolate potential energy values using the set of energies and forces extracted from a molecular dynamics trajectory. In general, such a scheme affords the maximum amount of information about a molecular system at minimal computational cost. More specifically, schemes like this are very important in the field of adaptive multi-scale molecular dynamics simulations. In this field, often the computation of potential energy values at certain trajectory points is not required for the simulation itself, but solely for the a posteriori analysis of the simulation data. Extrapolating the values at these points from the available data can save considerable computational time. A set of extrapolation schemes are employed based on Taylor series and central finite difference approximations. The schemes are first tested on the trajectories of molecular systems of varying sizes, obtained at MM and QM level using velocity-Verlet integration with standard simulation time steps. Remarkably good accuracy was obtained with some of the approximations, while the failure of others can be explained in terms of the distinct features of a molecular dynamics trajectory. We have found that, for a Taylor expansion of the potential energy, both a first and a second order truncation exhibit errors that grow with system size. In contrast, the second order central finite difference approximation displays an accuracy that is independent of the size of the system, while giving a very good estimate of the energy, and costing as little as a first order truncation of the Taylor series. A fourth order central finite difference approximation requires more input data, which is not always available in adaptive multi-scale simulations. Furthermore, this approximation gives errors of similar magnitude or larger than its second order counterpart, at standard simulation time steps. This leads to the conclusion that a second order central finite difference approximation is the optimal choice for energy extrapolation from molecular dynamics trajectories. This finding is confirmed in a final application to the analysis of an adaptive multi-scale simulation.

Application of residuals from regression of experimental mode shapes to locate multiple crack damage in a simply supported reinforced concrete beam

The objective of the study was to propose an effective, simple and reliable technique to determine the location of triple-crack damage in a simply-supported reinforced concrete beam using the method of mode shape regression. The study required simply supported finite element reinforced concrete beam models to be constructed with one as control and another as the test beam with predetermined triple-cracks along the length of the beam. The technique necessitated the performance of linear and Eigen analyses on the control beam, and nonlinear analysis on the beam with triple-cracks. Residuals obtained from regression of the mode shape using the Chebyshev series rational on the modal frequencies and transformation and application into the fourth order centered finite divided difference formula were used. The use of the regressed mode shapes for the reinforced concrete beam model showed large residuals around the areas of the crack damage. The results showed that the method was successful in determining the locations of the triple cracks and was comparable with other techniques proposed by other researchers in terms of its simplicity and reliability.

Honeycomb damage detection in a reinforced concrete beam using frequency mode shape regression

The objective of the study was to propose a technique to determine the location and severity of honeycomb damage in a reinforced concrete beam using frequency mode shape regression focusing on minimal data. Simply supported reinforced concrete beams were constructed with two different volumes of honeycomb materials implanted at mid-span of the beams. The technique necessitated the performance of linear and eigen analyses on a control beam, and nonlinear analysis on the test beams with damage. Local stiffness indicators obtained from regression using the Levenberg–Marquardt algorithm and residuals from regression of the frequency mode shapes using the Chebyshev series rational and transformation and application into the fourth order centered finite divided difference formula were utilized. Both approaches were successful in determining the location of the honeycomb damage. The Chebyshev series rational method was unable to determine the severity of damage.

Robust particle image velocimetry using gradient method with upstream difference and downstream difference

In the first part of this paper, sub-pixel displacement detection methods in particle image velocimetry (PIV) measurements were discussed. Some schemes for calculating the spatial gradient in a gradient method, which is one of sub-pixel displacement detection methods, were evaluated. It was found that the second order central finite difference scheme tends to underestimate the gradient of the tracer-particle’s intensity. Consequently it leads to an overestimation of sub-pixel displacements. A first-frame upstream and second-frame downstream finite difference scheme was then proposed and evaluated for one-dimensional flow measurements. Furthermore, the proposed difference scheme was extended to apply to two-dimensional flows. It was shown that taking the direction of the sub-pixel correction into account and using fewer adjacent pixels are important for PIV images consisting of small tracer particle images. Finally, a robust procedure for the execution of the gradient method was proposed, which can eliminate the effect of noises. It was found that the proposed scheme is not only highly accurate but is highly robust to noise.

Study on direct measurement method of vorticity from particle images

To overcome the shortcomings of conventional methods for vorticity measurement, a new direct measurement of vorticity (DMV) method extracting vorticity from particle images was proposed. Based on the theory of fluid flow, two matched particle patterns were extracted from particle images in the DMV method. The pattern vorticity was determined from the average angular displacement of rotation between the two matched particle patterns. The method was applied on standard particle images, and was compared with the second and third order central finite difference methods. Results show that the accuracy of DMV method is independent of the spatial resolution of the sampling, and the uncertainty errors in the velocity measurement are not propagated into the vorticity. The method is applicable for measuring vorticity of a stronger rotational flow. The time interval of image sampling should be shortened to increase the measurement ranges for higher shearing distortion flows.

High order Hybrid central—WENO finite difference scheme for conservation laws

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.