Finite Difference Method Research Articles

Numerical Study on the Effect of Trapezoidal‐Wave Shaped Partition on Natural Convection Flow Within a Porous Enclosure

ABSTRACTThe use of a trapezoidal‐wave shaped diathermal partition to reduce natural convection flow and heat transfer within a fluid‐saturated, differentially heated porous enclosure is investigated in this study. This work is motivated by the need to control and reduce convective heat transfer in differentially heated porous enclosures, impacting applications like energy‐efficient building materials, thermal insulation, and improved heat exchangers. The study aims to disrupt convection currents and minimize thermal transfer. The Darcy flow model, representing fluid flow in porous media, is applied here and solved using the successive accelerated replacement (SAR) scheme with a finite difference method. Key parameters are varied to explore their effects on thermal and flow patterns. These parameters include the partition's length (with values between ), height (spanning from ), and distance from the left wall of the enclosure (), along with the modified Rayleigh number, which ranges from . Through computational visualization of streamlines and isotherms, this study examines how changes in partition geometry influence flow deviations. Results indicate that the trapezoidal partition allows flexibility in adjusting its geometrical parameters, effectively reducing convection flow strength without significant compromise. A drop of 41.13% in flow strength and about 56% reducted in heat transfer is achieved for trapezoidal partition with smaller edge length These findings suggest that such a partition setup can significantly improve thermal management in systems where fluid‐saturated porous enclosures are subject to differential heating.

Dynamic characteristics of hybrid water-lubricated herringbone groove journal bearing

Purpose The water-lubricated hydrodynamic herringbone groove journal bearing (HGJB) is capable of running at high speed. However, when running at a low speed, it suffers from a low load-carrying capacity due to the weak hydrodynamic effect. To overcome this problem, this study proposes a hybrid water-lubricated HGJB and aims to investigate its dynamic characteristics. Design/methodology/approach A hybrid lubrication model applicable to the hybrid water-lubricated HGJB is established based on the boundary fitted coordinate system, which considers the turbulent, thermal and tilting effects, and the finite difference method is used to calculate the dynamic characteristics of the hybrid water-lubricated HGJB. Findings The result shows that the hybrid HGJB has larger dynamic coefficients and better system stability compared with the hydrodynamic HGJB when running at low speed. Furthermore, the stiffness of hybrid HGJB are mainly governed by the hydrodynamic effect rather than the hydrostatic effect when running at high speed. Originality/value The proposed hybrid water-lubricated HGJB shows excellent dynamic characteristics at either low speed or high speed; and the hybrid water-lubricated HGJB has a large load-carrying capacity when running at low speed and has a good dynamic stability when running at high speed. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-06-2024-0233/

CALCULATION OF STATIC DEFORMATION OF AXISYMMETRIC SHELLS OF ROTATION WITH DIFFERENTIAL MODEL

In the paper differential equations of static geometrically nonlinear deformation of axisymmetric shell of rotation are obtained. The resolving functions are projections of vectors in the global coordinate system. The equations allow describing any geometry of meridian (breaks, curvature jumps), large deformations, changing of shell thicknesses during deformation, also cross shears characteristic for thick shells. For the numerical solution, the approach based on the finite difference method is applied, which is realized in the own software package for the calculation of the mechanics of spatial rod systems – DARSYS. The calculations of test problems of the internal pressure inflation of cylindrical, spherical, elliptical, conical shells, as well as a combined conicalcylindrical shell with a meridian break are presented. Graphs of convergence of displacements at the reference points as a function of mesh density and under load variation are given, and deformed meridian configurations are plotted. The solutions obtained in ANSYS by different finite elements of Shell type were used as a reference for comparison. APDL scripts for parametric calculations of the test problems are given in the text of the paper. The proposed approach to the calculation of static deformation of shells of rotation has shown good agreement with finite element modeling in ANSYS (including thick shells) and in the future will be extended to the modeling of dynamic deformation and the possibility of solving coupled problems of interaction with liquid or gas. The given equations of the axisymmetric shell are a special case of the general equations, the development and application of which are beyond the scope of this paper, and the obtained solution results are the first stage of testing the developed complex approach to the calculation of static and dynamic deformation of shells, alternative to finite-element modeling.

VARIATIONAL-DIFFERENCE SOLUTION OF DEFORMATION AND BUCKLING PROBLEMS OF ELASTOPLASTIC SHELLS OF REVOLUTION WITH ELASTIC FILLER UNDER COMBINED QUASI-STATIC AND DYNAMIC AXISYMMETRIC LOADINGS

The paper suggests a formulation and method for a numerical solution of deformation and buckling of elastoplastic shells of revolution with elastic filler under quasi-static and dynamic loadings. The problem is solved in a two-dimensional plane or generalized axisymmetric formulation with torsion. The governing system of equations is written in a Cartesian or cylindrical coordinate system. Modeling of deformation of an elastic-plastic shell is carried out based on the hypotheses of the theory of shells of the Timoshenko type, taking into account geometric nonlinearities. Kinematic relations are written in velocities and formulated in the metric of the current state. The elastoplastic properties of the shell are described by the flow theory with nonlinear isotropic hardening. Filler modeling is based on continuum mechanics hypotheses. The filler material is assumed to be linearly elastic. The variational equations of motion of structural elements (both shells and filler) are reduced from the three-dimensional equation of the balance of virtual powers of the work of continuum mechanics taking into account the accepted hypotheses of the theory of shells or a flat deformed state or generalized axisymmetric deformation with torsion. The modeling of the contact interaction between the shell and the filler is based on the condition of nonpenetration along the normal and slippage along the tangential. The finite-difference method and an explicit time integration scheme of the cross type are used to solve the defining system of equations. Approbation of the technique was carried out on the problem of buckling of a steel cylindrical shell with an elastic filler under quasi-static and dynamic compression by an external pressure that linearly increases with time. The results of the numerical study are compared with calculations performed using two other approaches developed earlier by the authors. The first approach is based on full-scale modeling of the process of deformation of the shell and filler within the framework of continuum mechanics. In the second approach, a simplified formulation is used, in which the deformation of the shell is modeled according to the hypotheses of the theory of non-sloping shells of the Timoshenko type taking into account geometric nonlinearities, and the filler is modeled according to the Winkler foundation hypothesis. The developed approaches make it possible to model the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates taking into account geometric shape imperfections, to study buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static combined loadings in plane and axisymmetric deformations.

APPLICATION OF CABARET AND WENO SCHEMES FOR SOLVING THE NONLINEAR TRANSPORT EQUATION IN THE MODELING OF SOUND WAVE PROPAGATION IN THE ATMOSPHERE

The most convenient model for describing the phenomenon of shock wave propagation in the atmosphere is the extended Burgers equation. This work investigates the influence of the numerical scheme on the results of solving the equation, which accounts for the nonlinear nature of shock wave propagation in the atmosphere. This equation is a key component of the extended Burgers equation and defines the transformation of the perturbed pressure profile during its propagation. Two numerical schemes were applied for the solution: CABARET and WENO, which are quasi-monotonic finite difference schemes that allow for solutions without significant numerical oscillations. An analysis of the applicability of these schemes for solving the considered problem was conducted.

Inverse design of narrowband thermal emitter with tandem films structures using simulated annealing algorithm

An inverse design method of narrow-band thermal emitter with a tandem films structures is proposed here, which is accomplished by the simulated annealing algorithm. Using this method, two kinds of structures are designed, which peak absorption rate is 0.8261 at 5.34 μm with a Q-factor of 175, and 0.8931 at 3.56 μm with a Q-factor of 132. The electric field distribution within the structure is analyzed using the finite-difference time-domain method (FDTD), and behavior similar to a Tamm plasmon state. The simulation and experiment of the inverse design method shows good agreement. Such inverse design method is very promising in mid-infrared simulator and detection.

Heat transfer in two-zone thick-walled pipes with axial varying periodic temperature boundary condition

The combined heat transfer in thick-walled pipes is investigated by also considering axial heat conduction. The problem considers a two-zone infinite pipe. There is a constant outer wall temperature boundary condition in the upstream region of the pipe. In the downstream region, the outer wall temperature is assumed to vary periodically in the axial direction spatially. This problem, where the flow is assumed to be laminar, is solved numerically by the finite difference method. The effects of the basic parameters of wall thickness ratio, Peclet number, wall-fluid thermal conductivity coefficient ratio and wall-fluid thermal diffusivity coefficient ratio on the interface heat flux are investigated. The effects of the axial dimensionless frequency on the results are also taken into account in the studies. The results obtained are highly dependent on the parametric values. It is observed that the wall thickness ratio is more effective than the other parametric values. In addition, it can be said that the pipe wall and fluid axial conduction cause heat transfer to the upper flow region at a non-negligible scale.

Hierarchical Bayesian Data Selection

There are many issues that can cause problems when attempting to infer model parameters from data. Data and models are both imperfect, and as such there are multiple scenarios in which standard methods of inference will lead to misleading conclusions; corrupted data, models which are only representative of subsets of the data, or multiple regions in which the model is best fit using different parameters. Methods exist for the exclusion of some anomalous types of data, but in practice, data cleaning is often undertaken by hand before attempting to fit models to data. In this work, we will employ hierarchical Bayesian data selection; the simultaneous inference of both model parameters, and parameters which represent our belief that each observation within the data should be included in the inference. The aim, within a Bayesian setting, is to find the regions of observation space for which the model can well-represent the data, and to find the corresponding model parameters for those regions. A number of approaches will be explored, and applied to test problems in linear regression, and to the problem of fitting an ODE model, approximated by a finite difference method. The approaches are simple to implement, can aid mixing of Markov chains, and are broadly applicable to many inferential problems.

Irreversible effects for SWCNT‐MWCNT/water based flow with Coriolis force over rotating frame: A numerical study

AbstractIn this study, two different types of nanofluids are used: One is the simple nanofluid consist of single‐wall carbon nanotubes (SWCNT), and the other is the hybrid nanofluid composed of single and multi‐wall carbon nanotubes (SWCNT‐MWCNT). The famous Xue model is incorporated for the thermal energy analysis. An important property related to the nanoparticles is the Brownian motion and thermal‐migration, which is described by using Buongiorno's model. The appropriate similarity transformations are used to convert the governing nonlinear partial differential equations into the nonlinear coupled ordinary differential equations for the both velocity and temperature profiles. The numerical solution is obtained for the present problem by using bvp4c Matlab routine. This code is based on the finite difference method that implements the three‐stage Lobatto IIIa formula. The comparative analysis for the entropy generation in rotating frame due to heat transfer, viscous heating, and mass diffusion is the main focus in this study. The impact of different parameters such as, rotational motion , thermomigration and Brownian motion , volume fractions of SWCNT & SWCNT‐MWCNT ,, Eckert number , Lewis number , chemical reaction parameter , Brinkman number the temperature ratio parameter and concentration difference parameter on axial and transverse velocity, temperature profile, entropy generation, and Bejan number are obtained. The obtained results for these different variables are presented through graphs and tabular form. When the solid volume fraction is enhanced, the flow and heat distribution are observed to increase. The Nusselt and Sherwood numbers for both nanofluids are investigated. The decay in Nusselt number is observed as the values of the heat source/sink parameter, thermophoresis, Brownian motion, and Eckert number are increased. However, in the case of Sherwood enhancement an increment in the value of the chemical reaction parameter, thermophoresis, Brownian motion, and Lewis number is observed. The entropy generation is enhanced as a result of increment in Brinkman number and temperature ratio. However, in the case of Bejan number, opposite trend was seen in the entropy generation. For the hybrid nanofluid, the enhancement in both heat transmission and irreversibility effect is observed as compared to simple nanofluid.

Bifurcation and sensitivity analysis along with the numerical simulations of susceptible breaking-out quarantine treatment recovered computer model

Internet worms have received a lot of attention because of their massive hazards to the internet. As the internet worms propagate quickly, therefore, the automatic control on the internet is required. In this paper, we present a new epidemic model called susceptible-breaking out-quarantine-treatment-recovered (SBQTR) model which integrates dynamic quarantine and treatment approaches to control the spread of internet worms. By using the SBQTR model, we can calculate the basic reproduction number [Formula: see text], which determines the existence of worms. The model has two equilibrium states: virus-free equilibrium (VFE) and viral equilibrium (VE). The stability analysis of both the equilibrium points is carried out and impact of treatment is examined under various parametric values. The sensitivity analysis of model is carried out to determine the impact of model parameters on virus transmission. Furthermore, we established nonstandard finite difference (NSFD) schemes and Runge–Kutta algorithm of order 4 (RK-4) for SBQTR model. The stated NSFD technique retains all of the significant properties of system (1), whereas RK-4 mechanism gives negative solutions and doesn’t converge to the equilibrium points of system (1). Numerical results together with comparison for various values of step size ([Formula: see text]) are available. Bifurcation value of infection coefficient is also investigated. It is shown that an appropriate treatment is essential to control the spread of worms in computer network.

A robust and computationally efficient guided wave tomography of pipes

ABSTRACT Guided wave ultrasonic tomography of pipe usually uses the helical modes propagated between parallel transducer arrays to compensate for the missing incident angle . Nevertheless, separating mixed signals from various helical modes during inversion is challenging, and this method necessitates multiple replications of the two-dimensional unfolding area , leading to high computational costs. It also faces practical problems such as environmental noise, transducer differences, and poor coupling . To address these problems, this paper proposes a finite difference method that utilizes helical mode information without multiple replications or the need to separate signals. Additionally, an objective function based on zero-delay cross-correlation is employed to mitigate the effects of noise, transducer inconsistencies, and coupling issues that distort signal amplitudes. The proposed method reduces the inversion space of high-order helical modes of the pipeline to the area of a two-dimensional unfolding, and improves the reliability and accuracy of inversion through objective function optimisation. In numerical simulations, the algorithm achieves thickness loss imaging with errors less than 0.1 mm and demonstrates good performance in experimental tests .

WELL-POSEDNESS AND CONVERGENCE ANALYSIS OF THE MULTI-LEVEL MONTE CARLO METHOD FOR A SYSTEM OF PDEs WITH RANDOM COEFFICIENTS MODELING DRUG TRANSPORT IN TUMORS

In this paper, we consider a mathematical model of drug transport in tumors given by a system of PDEs with random coefficients and initial data. The existence of a strong solution to this model in any dimension is proved under the assumption that the random coefficients are uniformly bounded. Along with the finite difference method (FD), we applied a multilevel Monte Carlo method to simulate these random PDEs. We also derived the overall convergence rate and estimated the total computation cost. Finally, some numerical results are presented to confirm our theoretical results. Additionally, we provide a comparison between the stochastic and deterministic approaches for solving the drug transport equation, demonstrating the efficiency of the method we adopted.

Design and FDTD simulation of a remote-controllable visible-light plasmonic waveguide and refractive-index-sensitive switch

In this paper, a plasmonic switch based on metal-insulator-metal (MIM) waveguide structure is proposed, whose transmission characteristics can be remotely controlled by a rake switch design. The distance from the remote control unit to the bus waveguide is more than 1 um and still maintains a very high efficiency. The refractive-index-dependent transmission spectrum of this filter was simulated using the finite-difference time-domain method. The results show that the on/off switching of wave propagation in the bus waveguide can be achieved by changing the refractive index of the control unit 1 µm away from the bus waveguide. A change in refractive index of only 0.2 is required to simultaneously control the switching off and on of four waves in the waveguide in the visible band, with corresponding switching ratios of 40.5, 74.3, 48.6, and 15.1, showing great potential for applications in refractive index sensors and remotely controllable band stop filters.

The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number

The mean dissipation rate of turbulent energy reaches a constant value at high Taylor–Reynolds numbers (Rλ). This value is associated with the well-scaling dissipation spectrum in Kolmogorov units, where the maximum corresponds to the bottleneck peak. Even the scalar dissipation rate at the high Rλ considered in the present direct numerical simulations attains a constant value as Sc increases. In this scenario, the maximum of the scalar dissipation spectra reaches its peak within the bottleneck, starting at Sc>0.5. A qualitative explanation for the formation of the two bottlenecks is related to the blockage of energy transfer from large to small scales in the inertial ranges. Within the bottleneck, the self-similar, ribbon-like structures transition into the rod-like structures characteristic of the exponential decay range. Investigating the viscous dependence of the bottleneck’s amplitude may be aided by examining the evolution of a passive scalar. As Sc decreases, the scalar spectra undergo changes across the wave number k range. The bottleneck is dismantled, and at very low Sc values, the spectrum tends towards Batchelor’s theoretical prediction, diminishing proportionally to k−17/3. To comprehend the flow structures responsible for the bottleneck, visualizations of θ∇2θ and probability density functions at various Sc values are presented and compared with those of ui∇2ui. The numerical method employed for generating three-dimensional spectra and quantities such as energy and scalar variance dissipation in physical space must be accurate, particularly in resolving small scales. This paper additionally demonstrates that the second-order finite difference scheme conserving kinetic energy and scalar variance in the inviscid limit in viscous simulations accurately predicts the exponential decay range in one-dimensional and three-dimensional turbulent kinetic energy and scalar variance spectra.

Using RBF-FD for calculation of hydroelastic vibrations of axisymmetric orthotropic shells of rotation

Geometrically nonlinear differential equations describing the dynamic deformation of axisymmetric shells of rotation are derived on the basis of general equations for solving functions in the global coordinate system. The equations take into account thinning/thickening at large longitudinal strains as well as transverse shear for thick shells. The motion and pressure of an ideal incompressible fluid is described by a displacement potential. To obtain the numerical solution, the finite difference method based on spline interpolation by polyharmonic radial basis functions is applied. The calculation method is implemented in software package. Good agreement of the calculated displacements with the results of modeling by different finite elements in ANSYS is obtained. The frequencies of the hydroelastic vibrations of the tanks are compared with those obtained by the finite element and boundary element method, as well as with results from published articles by other researchers.

A Comprehensive Model and Numerical Study of Shear Flow in Compressible Viscous Micropolar Real Gases

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases under shear stress. We formulate the governing equations by incorporating viscosity and micropolar effects and transform the obtained system into the mass Lagrangian coordinates. Two numerical methods, Faedo–Galerkin approximation and finite-difference methods, are used to solve it. These methods are compared using several benchmark examples to assess their accuracy and computational efficiency. Both methods demonstrate good performance, achieving equally precise results in capturing essential flow characteristics. However, the finite-difference method offers advantages in speed, stability, and lower computational demands. This research bridges gaps in existing models and establishes a foundation for further investigations into complex flow phenomena in micropolar real gases.

Convergence properties of the radial basis function-finite difference method on specific stencils with applications in solving partial differential equations

We consider the problem of approximating a linear differential operator on several specific stencils using the radial basis function method in the finite difference scheme. We prove a linear convergence order on a non-equispaced five-point stencil. Then, we discuss how the convergence rate can be boosted up to the second-order on an equispaced stencil. Moreover, we show that including additional nearby nodes (six to twelve) in the stencil does not improve the convergence rate, thus increasing the computational load without enhancing convergence. To overcome this limitation, we propose a stencil that accelerates the convergence up to four using a nine-point stencil, unlike existing approaches which are based on thirteen-point equispaced stencils to achieve such an order of convergence. To support our findings, we conduct numerical experiments by solving Poisson equations and a parabolic problem.

A robust seismic wavefield modeling method based on minimizing spatial simulation error using L2-norm cost function

To reduce the spatial simulation error generated by the finite difference method, previous researchers compute the optimal finite-difference weights always by minimizing the error of spatial dispersion relation. However, we prove that the spatial simulation error of the finite difference method is associated with the dot product of the spatial dispersion relation of the finite-difference weights and the spectrum of the seismic wavefield. Based on the dot product relation, we construct a norm cost function to minimize spatial simulation error. For solving this optimization problem, the seismic wavefield information in wavenumber region is necessary. Nevertheless, the seismic wavefield is generally obtained by costly forward modeling techniques. To reduce the computational cost, we substitute the spectrum of the seismic wavelet for the spectrum of the seismic wavefield, as the seismic wavelet plays a key role in determining the seismic wavefield. In solving the optimization problem, we design an exhaustive search method to obtain the solution of the norm optimization problem. After solving the optimization problem, we are able to achieve the finite-difference weights that minimize spatial simulation error. In theoretical error analyses, the finite-difference weights from the proposed method can output more accurate simulation results compared to those from previous optimization algorithms. Furthermore, we validate our method through numerical tests with synthetic models, which encompass homogenous/inhomogeneous media as well as isotropic and anisotropic media.

Digital twin dynamic force-thermal physics sub-cell for CFRP drilling process

The digital transformation of cutting processes is anticipated to advance digital twin (DT) automated machining, particularly for difficult-to-cut materials. This paper develops a dynamic physical sub-cell for the CFRP drilling process, facilitating real-time force-thermal analysis during tool degradation and enhancing DT machining technology. A comprehensive dynamic model was developed to account for force-thermal effects as the tool wears. This model included five correction functions that consider the wear characteristics of the drill bit, allowing the improved cutting force model to track the dynamic force trajectory accurately. The finite difference method was employed to analyze drilling temperature and heat accumulation, with the heat source generated by the cutting force. The heat source intensity was determined by the heat generation efficiency function. Furthermore, the cutting force at the exit was calculated, as the exit temperature alters the material's thermal properties at the entrance. Taking experimental data from 100 holes at each feed rate as a baseline, optimization algorithms determined the values of each correction coefficient within the comprehensive model. The model's validity was verified by comparing experimental results, showing average errors of 5.14N for thrust, 0.0165N·m for torque, and 7.04°C for temperature. This model was packaged as a dynamic physical sub-cell demo, which can be integrated into a DT machining system for rapid response to multifaceted force-thermal attributes.

A local radial basis function-compact finite difference method for Sobolev equation arising from fluid dynamics

In this article, a new high-order, local meshless technique is presented for numerically solving multi-dimensional Sobolev equation arising from fluid dynamics. In the proposed method, Hermite radial basis function (RBF) interpolation technique is applied to approximate the operators of the model over local stencils. This leads to compact RBF generated finite difference (RBF-FD) formula, which provides a significant improvement in the accuracy and computational efficiency. In the first stage of the proposed method, the time discretization is performed by Crank–Nicolson finite difference scheme along with temporal Richardson extrapolation technique. In the second stage, the space dimension is discretized by applying the local radial basis function-compact finite difference (RBF-CFD) method. By performing some numerical simulations and comparing the results with existing methods, the high accuracy and computational efficiency of the proposed method are clearly demonstrated. The numerical results show that the presented method has fourth-order accuracy in both space and time dimensions. Finally, it can be concluded that the proposed method is a suitable alternative to the existing numerical techniques for the Sobolev model.