Alternating Direction Implicit Research Articles

Efficient computation of heat distribution of processed workpiece with coupling to bottom water vapor in water-jet guided laser machining.

In this paper, we present a novel approach for calculating the heat distribution within a processed workpiece subjected to laser irradiation while accounting for the influence of bottom water vapor. A comprehensive mathematical model is introduced and numerical techniques using difference approximation are employed. Initially, the three-dimensional heat equation, originally defined in the rectangular coordinate system, is transformed into a corresponding model within the cylindrical coordinate system, incorporating a nonlinear boundary condition to account for coupling effects. Subsequently, leveraging the axial symmetry of the heat distribution, the three-dimensional model is simplified into a two-dimensional one. This simplified model is solved using the alternating direction implicit scheme coupled with the Crank-Nicolson method. Moreover, we develop a high-precision numerical treatment for the nonlinear boundary condition within the cylindrical coordinate system. To validate our methodology, simulation experiments are conducted on three distinct samples. Our comparative results demonstrate the feasibility and efficiency of the proposed approach in the context of water-jet guided laser processing.

BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

A fractional-order image segmentation model with application to low-contrast and piecewise smooth images

In this paper, we propose a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, we use the fractional-order regularization to approximate the Hausdorff measure of the Mumford-Shah (MS) model. The existence and uniqueness of the solution is proved and the alternating direction implicit (ADI) scheme is used to find the solution of the modified MS model. In the second stage, a thresholding is used to induce the segmentation of the target. The superior performances of the proposed model are demonstrated by some comparative experimental results with several state-of-art methods.

A novel exploration of how localized magnetic field affects vortex generation of trihybrid nanofluids

Abstract Nanofluidics have better thermal properties than regular fluids, which makes them useful for heat transfer applications. This research investigated the complex dynamics of confined magnetic forces that influence the rotation of nanostructures and vortex formation in a tri-hybrid nanofluid (Ag, Al2O3, TiO2) flow regime. The study shows that the magnetic field can change the flow and heat transfer of nanofluidic, depending on its direction and strength. The study also provides insights into the complex physics of nanofluid flow and heat transfer, which can help design devices that use nanofluids more efficiently for cooling electronics, harvesting solar energy, and generating power from fuel cells. We used a single-phase model to model the nanofluids while the governing partial differential equations were solved numerically. An alternating-direction implicit approach has been employed to analyze the impact of confined magnetic fields on the nanofluid flow and thermal properties. Unlike previous studies that assumed uniform magnetic fields, we introduced multiple confined magnetic fields in the form of horizontal and vertical strips. Using our custom MATLAB codes, we systematically examined various parameters, including the magnetic field strength, number of strips and their position, and nanoparticle volume fraction, to assess their effects on nanofluid flow and thermal characteristics. Our findings revealed that the confined Lorentz force induced the spinning of tri-hybrid nanoparticles, resulting in a complicated vortex structure within the flow regime. In the absence of a magnetic field, a single symmetric vortex can be seen in the flow field. However, the introduction of magnetic sources stretches this vortex until it splits into two smaller, weaker vortices in the lower cavity, rotating clockwise or counterclockwise. Furthermore, the magnetic field strength significantly reduces both skin friction and the Nusselt number, while Reynolds numbers mainly affect the Nusselt number.

Uncovering the mystery of the vortex dynamics in a micropolar fluid with multiple magnetic field strips: A novel case study

From transportation to energy production, environmental protection, and medical advancements, vortices are known to influence everything. Therefore, a remarkable aspect of engineering and scientific research in these areas is the prediction, control, and optimization of such vortices. Motivated by these potential applications, we numerically study ‘how confined magnetic fields may give rise to new vortices in the micropolar flow inside a square cavity, by employing the Alternating Direction Implicit approach. Unlike most of the previous studies, we do not assume a uniform magnetic field throughout the flow domain, which is more realistic. Instead, we introduce several confined magnetic fields in the form of multiple horizontal and vertical strips. Further, we have applied the theory of micropolar fluids proposed by Eringen to model the micropolar fluid and visualize the flow patterns around the magnetic strips in the flow regime. Using our self-developed MATLAB codes, we examine how various parameters, such as the magnetic field strength, the number and position of the strips, and the micropolar parameter affect the flow and thermal properties of the micropolar fluid. The findings of this study will contribute to a better understanding of the vortex generation and heat transfer enhancement of micropolar fluids in lid-driven cavities under localized magnetic fields. The study reveals that both the Nusselt number (Nu) and skin friction (CfRe) coefficient decrease by 45% and 1% respectively with a magnetic number (Mn) for a fixed Reynolds number (Re). However, increasing the Reynolds number at a fixed magnetic number increases the Nusselt number (909%) and skin friction coefficient (7%). This means that the higher flow velocity translates into increased wall shear stress and heat transfer, which destabilizes the flow and may cause turbulence. This knowledge can be used to develop new and improved micropolar fluid-based technologies for a variety of applications, such as energy production, medical devices, and microfluidics.

Effect of solid obstacle and thermal conditions on convective flow and entropy generation of nanofluid filled in a cylindrical chamber

PurposeOne of the major challenges in the design of thermal equipment is to minimize the entropy production and enhance the thermal dissipation rate for improving energy efficiency of the devices. In several industrial applications, the structure of thermal device is cylindrical shape. In this regard, this paper aims to explore the impact of isothermal cylindrical solid block on nanofluid (Ag – H2O) convective flow and entropy generation in a cylindrical annular chamber subjected to different thermal conditions. Furthermore, the present study also addresses the structural impact of cylindrical solid block placed at the center of annular domain.Design/methodology/approachThe alternating direction implicit and successive over relaxation techniques are used in the current investigation to solve the coupled partial differential equations. Furthermore, estimation of average Nusselt number and total entropy generation involves integration and is achieved by Simpson and Trapezoidal’s rules, respectively. Mesh independence checks have been carried out to ensure the accuracy of numerical results.FindingsComputations have been performed to analyze the simultaneous multiple influences, such as different thermal conditions, size and aspect ratio of the hot obstacle, Rayleigh number and nanoparticle shape on buoyancy-driven nanoliquid movement, heat dissipation, irreversibility distribution, cup-mixing temperature and performance evaluation criteria in an annular chamber. The computational results reveal that the nanoparticle shape and obstacle size produce conducive situation for increasing system’s thermal efficiency. Furthermore, utilization of nonspherical shaped nanoparticles enhances the heat transfer rate with minimum entropy generation in the enclosure. Also, greater performance evaluation criteria has been noticed for larger obstacle for both uniform and nonuniform heating.Research limitations/implicationsThe current numerical investigation can be extended to further explore the thermal performance with different positions of solid obstacle, inclination angles, by applying Lorentz force, internal heat generation and so on numerically or experimentally.Originality/valueA pioneering numerical investigation on the structural influence of hot solid block on the convective nanofluid flow, energy transport and entropy production in an annular space has been analyzed. The results in the present study are novel, related to various modern industrial applications. These results could be used as a firsthand information for the design engineers to obtain highly efficient thermal systems.

Numerical investigation of nanofluid buoyant flow behavior and heat transfer characteristics in annular-shaped enclosure with internal baffle

PurposeNatural convection in finite enclosures is a common phenomenon in various thermal applications. To provide the thermal design guidelines, this study aims to numerically explore the potential of using internal baffles and nanofluids to either enhance or suppress heat transport in a vertical annulus. Furthermore, the annular-shaped enclosure is filled with aqueous-silver nanofluid and the effects of five distinct nanoparticle shapes are examined. In addition, the influence of baffle design parameters, including baffle position, thickness and length, is thoroughly analyzed.Design/methodology/approachThe finite difference method is used in conjunction with the alternating direction implicit and successive line over relaxation techniques to solve nonlinear and coupled partial differential equations. The single phase model is used for nanofluid which is considered as a homogeneous fluid with improved thermal properties. The independence tests are carried out for assessing the sufficiency of grid size and time step for obtaining results accurately.FindingsThe baffle dimension parameters and nanoparticle shape exhibit significant impact on the convective flow and heat transfer characteristics, leading to the following results: sphere- and blade-shaped nanoparticles demonstrate around 30% enhancement in the heat transport capability compared with platelet-shaped nanoparticles, which exhibit the least. When considering the baffle design parameter, either a decrease in the baffle length and thickness or an increase in baffle height leads to an improvement in heat transport rate. Consequently, a threefold increase in baffle height yields a 40% improvement in thermal performance.Originality/valueUnderstanding the impact of nanoparticle shapes and baffle design parameters on flow and thermal behavior will enable engineers to provide valuable insight on thermal management and overall system efficiency. Therefore, the current work focuses on exploring buoyant nanofluid flow and thermal mechanism in a baffled annular-shaped enclosure. Specifically, an internal baffle that exhibits conductive heat transfer through it is considered, and the impact of baffle dimensions (thickness, length and position) on the fluid flow behavior and thermal characteristics is investigated. In addition, the current study also addresses the influence of five distinct nanoparticle shapes (e.g. spherical, cylindrical, platelet, blade and brick) on the flow and thermal behavior in the baffled annular geometry. In addition to deepening the understanding of nanofluid behavior in a baffled vertical annulus, the current study contributes to the ongoing advancements in thermal applications by providing certain guidelines to design application-specific enclosures.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

ALTERNATING DIRECTION IMPLICIT METHOD FOR POISSON EQUATION WITH INTEGRAL CONDITIONS

In this paper, we investigate the convergence of the Peaceman-Rachford Alternating Direction Implicit method for the system of difference equations, approximating the two-dimensional elliptic equations in rectangular domain with nonlocal integral conditions. The main goal of the paper is the analysis of spectrum structure of difference eigenvalue problem with nonlocal conditions. The convergence of iterative method is proved in the case when the system of eigenvectors is complete. The main results are generalized for the system of difference equations, approximating the differential problem with truncation error O(h4).

$$H^{1}$$-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems

$$H^{1}$$-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems

Reduction of edge beading defects on rectangular substrates using a rotation about an axis perpendicular to the spin axis

Here, we consider the coating dynamics of a binary liquid on a rectangular substrate that both spins and rotates about an axis perpendicular to the spin axis. This rotation causes not only a force parallel to the substrate but a large normal force on the horizontal region of the substrate, known as elevated gravity. This force not only smooths out any irregularities in the film, it also causes an increase in the evaporation rate, resulting in a higher viscosity, which in turn decreases the thinning of the coating due to centrifugal forces. We derive the lubrication form of the governing equations for such a geometry that includes surface tension and body forces acting on the liquid due to the spinning and rotation of the substrate, as well as viscosity changes and surface tension gradients due to evaporation of the volatile component. The resultant coupled nonlinear differential equations are solved numerically using an efficient alternating direction implicit method. The numerical results indicate that the rotation can reduce edge beading defects for parameters typically employed by the spin coating industry. It is particularly effective for small average coating thicknesses, where it can almost eliminate edge beading. Surface tension gradients, which act in the post rotation dry phase, can also further affect edge beading, particularly in the corner regions.

Picosecond to microsecond dynamics of X-ray irradiated materials at MHz pulse repetition rate

Modern X-ray free-electron lasers (XFELs) produce intense femtosecond X-ray pulses able to cause significant damage to irradiated targets. Energetic photoelectrons created upon X-ray absorption, and Auger electrons emitted after relaxation of core-hole states trigger secondary electron cascades, which contribute to the increasing transient free electron density on femtosecond timescales. Further evolution may involve energy and particle diffusion, creation of point defects, and lattice heating. This long-timescale (up to a microsecond) X-ray-induced dynamics is discussed on the example of silicon in two-dimensional geometry. For modeling, we apply an extended Two-Temperature model with electron density dynamics, nTTM, which describes relaxation of an irradiated sample between two successive X-ray pulses, emitted from XFEL at MHz pulse repetition rate. It takes into account ambipolar carrier diffusion, electronic and atomic heat conduction, as well as electron-ion coupling. To solve the nTTM system of equations in two dimensions, we developed a dedicated finite-difference integration algorithm based on Alternating Direction Implicit method with an additional predictor-corrector scheme. We show first results obtained with the model and discuss its possible applications for XFEL optics, detectors, and for diagnostics tools. In particular, the model can estimate the timescale of material relaxation relevant for beam diagnostic applications during MHz operation of contemporary and future XFELs.

Physics‐based preconditioning of Jacobian‐free Newton–Krylov solver for Navier–Stokes equations using nodal integral method

AbstractThe nodal integral methods (NIMs) have found widespread use in the nuclear industry for neutron transport problems due to their high accuracy. However, despite considerable development, these methods have limited acceptability among the fluid flow community. One major drawback of these methods is the lack of robust and efficient nonlinear solvers for the algebraic equations resulting from discretization. Since its inception, several modifications have been made to make NIMs more agile, efficient, and accurate. Modified nodal integral method (MNIM) and modified MNIM (M2NIM) are the two most recent and efficient versions of the NIM for fluid flow problems. M2NIM modifies the MNIM by replacing the current time convective velocity with the previous time convective velocity, leading to faster convergence albeit with reduced accuracy. This work proposes a preconditioned Jacobian‐free Newton–Krylov approach for solving the Navier–Stokes equation using MNIM. The Krylov solvers do not generally work well without an appropriate preconditioner. Therefore, M2NIM is used here as a preconditioner to accelerate the solution of MNIM. Due to pressure–velocity coupling in the Navier–Stokes equation, developing a quality preconditioner for these equations needs significant effort. The momentum equation is solved using the time‐splitting alternate direction implicit method. The velocities obtained from the solution are then used to solve the pressure Poisson equation. The computational results for the Navier–Stokes equation are presented to underscore the advantages of the developed algorithm.

Numerical study of encapsulated nanoparticles enhanced phase change material in thermal energy cool storage packed bed system

A numerical study is performed to investigate the dynamic behavior of a packed bed containing spherical capsules filled with Al2O3 nanoparticles dispersed in pure water as an enhanced phase change material (NEPCM) that can be utilized in ice storage of air conditioning systems. The heat transfer fluid (coolant) employed in the current work is an aqueous solution of 30 vol% ethylene glycol. For energy conservation of the coolant and the NEPCM, coupled partial differential equations are used to create a two-dimensional axisymmetric time-dependent model. While the energy equation for the NEPCM is solved using a completely explicit scheme, the energy equation for the coolant is discretized using a finite difference method and solved using the Alternating Direction Implicit (ADI) technique. The Al2O3 nanoparticles are dispersed in PCM with a volume concentration range of 0–5 %. The coolant inlet temperature with a range of -5 °C to -15 °C and mass flow rate of range of 0.2 to 0.8 kg/s for a storage volume of 0.5 m3 are the parameters used in the current study. The effect of the nanoparticles volume concentration, coolant mass flow rate, and coolant inlet temperature are investigated. The timewise variation of the solidified volume fraction of the bed, the cool energy stored in the bed, the bed centerline temperature, and the coolant exit temperature are predicted for both charging and recovery modes under different operating conditions.

Entropy generation and natural convection analyses in a non-darcy porous square cavity with thermal radiation and viscous dissipation

This study investigates the phenomena of fluid flows, general heat transfer and entropy generation on transient natural convection in a square cavity occupied with a saturated non-Darcy porous medium with dissipation effects and thermal radiation. The governing equations for the fluid flows and heat transfer, with corresponding boundary and initial conditions in the porous system are solved with the use of the alternate direct implicit method (ADI). The impact of the dimensionless parameters on the flow and thermal effect including entropy generation, with the fluid structure interaction cases are discussed. The following ranges have been used for simulations for the relevant dimensionless parameters: Grashof Number (103≤Gr≤107), Rayleigh number (103≤Ra≤106), Eckert number (0≤Ec≤5×10-5), the Forschheimer resistance (0≤Γ,γ≤10), radiation parameter (0≤Rd≤5) and Prandtl number (0≤Pr≤10). The present numerical work, compared with previously published work are made, and excellent agreement is discovered. According to the simulation results, the top right wall and base of the heated wall are where entropy is being generated most. Furthermore, as the Forschheimer resistance increases the flow rate, the heat transfer, and entropy production reduce. As Pr decreases at Ra=105 the buoyancy force takes control of the flow fields when the porous medium’s resistance weakens. As a result, the fluid flow becomes stronger.

Efficient Numerical Simulation of Biochemotaxis Phenomena in Fluid Environments.

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis-fluid equations and incompressible Navier-Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier-Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied.

Large-Strain Consolidation Analysis for Clayey Sludge Improved by Horizontal Drains

The use of prefabricated horizontal drains (PHDs) with combined surcharge and vacuum preloading is an effective improvement approach for dredged clayey slurries. However, there is no large-strain analysis method for PHD-induced consolidation of high-water-content sludge, even though it is a typical large-strain problem. This study develops a two-dimensional plane strain model based on Gibson’s large-strain theory, considering horizontal and vertical flows, nonlinear hydraulic conductivity, and compressibility during the consolidation process. The alternative direction implicit (ADI) difference method is used to solve the governing equation. The proposed model is verified by the data from an analytical one-dimensional (1D) large-strain model and from field measurements. Compared with the improved small-strain models, the proposed model produces a slower consolidation of sludge. Furthermore, the analyses incorporating geometrical and mechanical nonlinearities show that, in comparison with the external load and the horizontal permeability, the spacing of the PHDs (horizontal and vertical) and the vertical permeability are more crucial factors significantly impacting consolidation efficiency. Specifically, smaller PHD spacing and greater vertical permeability lead to a more efficient consolidation.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

Production of entropy due to combined buoyancy and Marangoni convection in a cylindrical annular enclosure

The main purpose of the current analysis is to numerically describe the entropy generation due to collective thermocapillary and buoyancy determined convection in a perpendicular cylindrical porous cavity. The annular enclosure is occupied with MgO–Ag/water hybrid nanofluid. Two separate warmers of equal portions are located in the interior wall. The exterior wall of the cylindrical enclosure is set to be cold. The unheated section of the internal wall as well as the upper and lower borders are presumed to be adiabatic. The finite difference technique is applied to crack the non-dimensional leading equations. An Alternating Direction Implicit (ADI) procedure is utilized to simulate the vorticity and energy equations. The stream function equation is resolved by using the Successive Over Relaxation (SOR) scheme and the central differencing scheme is used to evaluate the velocity terms. The nature of fluid flow, thermal behaviour, entropy generation and thermal transportation rate are examined graphically for several physical and geometrical constraints. The attained outcomes disclose that there is a dominance of fluid friction in the generation of entropy inside the annulus and the average entropy production is lowest in the case of annular enclosure rather than the rectangular enclosure.

Analysis of the Load-Bearing Characteristics of Gas-Extruded Membrane Bearings Based on the Alternating Direction Implicit Method

In this study, a novel gas-extruded membrane bearing was developed, and an optimization algorithm was applied to solve a Reynolds equation that describes the load-bearing characteristics of this bearing. This was effective in improving the solution rate of the Reynolds equation, significantly reducing the difficulty of obtaining a solution, avoiding high programming difficulty, and achieving a high solution accuracy. Through a comparative analysis, the error in the accuracy of the alternating implicit difference method was addressed, and the traditional finite element method for solving the same model was verified, with an average error of 2% reached to verify its applicability. The algorithm was also used to analyze the load-bearing capacity of the gas-extruded membrane bearing. This revealed not only a positive correlation of the average load-bearing capacity of the gas-extruded membrane bearing with the frequency and amplitude of vibration but also a negative correlation with radial clearance, with the cut-off frequency reaching 19 Khz. The load-bearing capacity of the gas-extruded membrane bearing proposed in this paper reached 1.28 N, which indicated an error of 3.28% with the theoretical approach. To sum up, this research provides an important reference for the design and manufacture of novel gas-extruded membrane bearings.