Finite Difference Method Research Articles

Optical memory and counter using a graphene based hybrid plasmonic temporal integrator

This paper presents design and analysis of an optical memory and counter based on ultra-compact temporal integrators (INTs) using a graphene hybrid plasmonic add-drop ring resonator (GHP-ADRR) and pulley-type ring resonator (GHP-PRR) for optical signal processing. Due to the valuable features of graphene hybrid plasmonic technology, the footprint of these INTs is equal to 4 × 3.5 µm2 for GHP-ADRR and 5.4 × 3.6 µm2 for GHP-PRR. Also, the performance of the INTs has been analyzed by the three-dimensional finite-difference time-domain method in the frequency and time domains, and the accuracy of the results has been compared with those of the math counterparts and also key specifications of the first-order temporal INTs including phase jump, insertion loss, 3 dB bandwidth, rise time, integration time window, and energy efficiency have been investigated. Based on the results, both circuits have better performance than the photonic counterparts. Furthermore, the performance of these INTs has been evaluated in detail as a high-speed optical memory and counter. It has been illustrated that due to the greater quality factor of the GHP-PRR, this circuit has more accuracy for realizing the first-order integration, optical memory, and counter than the GHP-ADRR-based INT.

Hydrodynamic analysis of floating VLFS using multi-domain boundary element method

The present study emphasizes the investigation of rigid and flexible Very Large Floating Structures (VLFS) by analyzing the reflection and transmission coefficients, plate deflections and wave force on the floating structure. The study involves analyzing the hydrodynamic characteristics of a porous and rigid VLFS, and hydroelastic analysis of flexible VLFS. The analysis is performed using the coupled Multi-Domain Boundary Element Method (MDBEM) and Finite Difference Method (FDM) at finite water depth. The flexible VLFS is modelled based on the Euler–Bernoulli thin plate theory and the study is performed using small amplitude wave theory. The study evaluates the hydroelastic behaviour in terms of structural deflection and hydrodynamic parameters by varying the structural porosity. The reflection and transmission coefficients are analyzed to show the extent of wave propagation on the lee side and the seaside of the structure. The wave force coefficients obtained signify the importance of the structure's orientation for the oncoming waves. The numerical results are validated with the results available in the literature. The analysis is performed for different structural and material properties to obtain the minimal hydroelastic and hydrodynamic response to assess the optimum design criteria and suitability of the type of VLFS, thereby maintaining the stability of the structure for safer operations.

Stability and Accuracy of a Meshless Finite Difference Method for the Stokes Equations

Stability and Accuracy of a Meshless Finite Difference Method for the Stokes Equations

A Novel Case of Cooling and Heating in Rectangular Lid-Driven Cavities: Interplay of Richardson Numbers in Streamlines and Isotherms

The thermal and dynamic behavior of SiO2 nanofluid was studied in a rectangular lid-driven cavity using the finite difference method. A non-adiabatic lid and a hot section at the bottom wall were considered in different heating and cooling cases. Three novel study cases were studied: a standard temperature at Th (heat conduction through the left-side walls), a high hot temperature, 2Th (heat conduction through the left-side walls), and a 2Tc high cold temperature (heat conduction through right-side walls). The Richardson number was varied between 10 and 100, and the lid direction. With a Richardson number of 10, the streamlines in the different cases tended to the formation of a central vortex with small vortices on the side walls, and the isotherms tended to a central one near the lower wall’s heated section and the homogenized temperature in the center of the cavity. At a Richardson number of 100, the streamlines produced a division in the cavity through a central vortex due to the heating of the bottom wall; this affected the isotherms, generating a prominent one in the center of the cavity and others near it. The generating decreased in the temperature near the bottom and top walls but increased in the middle of the cavity. The standard temperature case tended to behave similarly to the high cold temperature case but presented different temperatures, while the high hot temperature case generally maintained a slightly different behavior. These effects were more noticeable with the lid direction opposite X.

Nonlinear Coupled Equations Free-Boundary Problem in Atherosclerosis: Symmetry Analysis of Solutions and Numerical Simulation

Atherosclerosis, a chronic inflammatory cardiovascular disease closely related to plaque formation during arteriosclerosis, poses a significant threat to global health. To deepen the understanding of the multifaceted interactions driving atherosclerosis progression and provide theoretical support for designing targeted therapeutic strategies, this study establishes a nonlinear coupled atherosclerotic free-boundary model integrating inflammatory immune cells, cytokines, and oxidized low-density lipoprotein. By applying the compression mapping principle, the local and global existence and uniqueness of solutions are proven, while revealing certain symmetries in the model’s solution structure. Under specific assumptions, a quasi-steady-state approximate model is derived, and the existence of its solution is demonstrated. Through numerical simulations of the quasi-steady-state approximate model using the finite difference method, the temporal and spatial evolution of pro-inflammatory macrophages and oxidized low-density lipoprotein is analyzed. The findings highlight the model’s strength in capturing the intricate dynamics of atherosclerosis, uncovering underlying mechanisms and identifying therapeutic targets. By evaluating inflammatory dynamics across plaque types and stenosis levels, the experimental design further validated the model’s ability to replicate clinical processes and reinforced its predictive accuracy. Notably, in the process of model analysis and solution, symmetries in the equations and boundary conditions play a crucial role in determining the solution properties. However, the current one-dimensional model has limitations. Future research should focus on developing higher-dimensional models and integrating more influencing factors to enhance the model’s clinical applicability and deepen the understanding of this complex disease.

Local and global stability analysis of HIV/AIDS by using a nonstandard finite difference scheme

This study presents a mathematical model incorporating both asymptomatic and symptomatic HIV-infected individuals to analyze the dynamics of HIV/AIDS. This expanded model offers a more comprehensive understanding of the epidemic’s spread. We calculate the basic reproduction number (R0) to quantify the virus’s transmission potential. To achieve accurate and robust simulations, we introduce the Nonstandard Finite Difference Scheme (NSFD). Compared to traditional methods like RK-4, NSFD offers improved dynamical consistency and numerical precision, leading to enhanced stability and efficiency in simulating infectious diseases like HIV/AIDS. Local and global stability analysis are performed using the Routh-Hurwitz method. The NSFD method effectively captures the dynamics of HIV propagation under various scenarios, providing valuable insights into HIV/AIDS progression. We demonstrate the superiority of the NSFD approach compared to existing methods, paving the way for further research in modeling viral infections.

Investigation of temperature jump, first and second-order velocity slip effects on blood-based ternary nanofluid flow in convergent/divergent channels

Purpose High surface area-to-volume ratios make nanoparticles ideal for cancer heat therapy and targeted medication delivery. Moreover, ternary nanofluids (TNFs) may possess superior thermophysical properties compared to mono- and hybrid nanofluids due to their synergistic effects. In light of this information, the objective of this article is to examine the blood-based TNF flow within convergent/divergent channels under velocity slip and temperature jump. Design/methodology/approach Leading partial differential equations corresponding to the problem are transformed into a system of nonlinear ordinary differential equations by using similarity variables. The bvp4c code that uses the finite difference method is used to obtain a numerical solution. Findings The effect of nanoparticles may change depending on the characteristics of flow near the wall. The properties and proportions of the used nanoparticles become important to control the flow. When TNF was used, an increase in the Nusselt number between 4.75% and 6.10% was observed at low Reynolds numbers. At high Reynolds numbers, nanoparticles reduce the Nusselt number and skin friction coefficient values under some special flow conditions. Importantly, the effects of second-order slip on engineering parameters were also investigated. Furthermore, the Nusselt number increases with increasing shape factor. Research limitations/implications Obtained results of the study can be beneficial in both nature and engineering, especially blood flow in veins. Originality/value The main innovations of this study are the usage of blood-based TNF and the examination of the effect of shape factor in convergent/divergent channels with second-order velocity slip.

Impact of electric field on donor energy and polarisability in GaAs variant toroidal quantum ring

ABSTRACT This study investigates the electronic properties of a GaAs Variant Toroidal Quantum Ring (VTQR) containing an off-centre donor impurity, focussing on the effect of an external electric field on donor energy and polarisability. Based on the effective mass approximation, we solved the Schrödinger equation using the finite difference method. The results show a typical effect at a toroidal angle of 180 ∘ , where the strength of the electric field leads to a decrease in the donor energy. Intriguingly, at toroidal angles of 90 ∘ and 60 ∘ , the donor energy increases with increasing field strength, indicating improved electron confinement. The probability density distributions shown reveal the location of the electron as influenced by the applied field. Furthermore, polarisability reaches its maximum around an angle of about 80 ∘ for all electric field strengths. This work provides a better understanding of the behaviour of electron in quantum ring systems and suggests potential for the engineering of tunable quantum devices.

An axial elastic deformation of ceramets on elastohydrodynamic lubrication of wavy-tilt-dam mechanical face seals

Purpose This paper aims to investigate the lubrication characteristics of siliconized graphite with a wavy-tilt-dam (WTD) pattern applied to the hydrodynamic face seals. Design/methodology/approach It focuses on two friction pairs, carbon graphite versus tungsten carbide (CG-TC) and siliconized graphite versus siliconized graphite (SG-SG), through a three-dimensional elastic hydrodynamic lubrication numerical model that integrates finite difference method and finite element method. The consequence of axial elastic deformation of sealing pair materials on film thickness, film pressure, cavitation and sealing performance for a WTD mechanical face seal under full working conditions of ΔP = 0.8, 5.3 and 15.8 MPa are analyzed theoretically. Findings The nuclear hydrodynamic WTD face seal generates a convergent gap and exhibits a dual-characteristic behavior of hydrodynamic and hydrostatic effects under various ΔP. Compared to the CG-TC, the SG-SG shows a lower minimum film thickness, decreasing by 3.9%, 17.3% and 35.1%. The flow leakage rate decreases by 47.8%, 52.1% and 75.4%. In addition, the film stiffness increases by 46.8%, 49.8% and 97.8%. Thus, the SG-SG better deals with the dynamic tracking problem, and the sealing performance is stable. The strength and hardness of siliconized graphite enhance WTD sealing performance and improve cavitation control in high-pressure applications. Originality/value The lubrication characteristics of the siliconized graphite with a WTD pattern could inform the future design of hydrodynamic shallow groove wavy seals in boiler feedwater engineering implements under high-pressure conditions for the nuclear power industry. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-10-2024-0382/

Classical Layer-Resolving Scheme for a System of Two Singularly Perturbed Time-Dependent Problems with Discontinuous Source Terms and Spatial Delay

In this paper, a classical layer-resolving finite difference scheme is formulated to solve a system of two singularly perturbed time-dependent initial value problems with discontinuity occurring at (y,t) in the source terms and Robin initial conditions. The delay term occurs in the spatial variable, and the leading term of the spatial derivative of each equation is multiplied by a distinct small positive perturbation parameter, inducing layer behaviors in the solution domain. Due to the presence of perturbation parameters, discontinuous source terms, and delay terms, initial and interior layers occur in the solution domain. In order to capture the abrupt change that occurs due to the behavior of these layers, the solution is further decomposed into smooth and singular components. Layer functions are also formulated in accordance with layer behavior. Analytical results and bounds of the solution and its components are derived. The formulation of a finite difference scheme involves discretization of temporal and spatial axes by uniform and piecewise uniform meshes, respectively. The formulated scheme achieves first-order convergence in both time and space. At last, to bolster the numerical scheme, example problems are computed to prove the efficacy and accuracy of our scheme.

Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration

Constructing a freeform surface that accurately satisfies both integrable condition and Snell’s law under a given invariant source–target map is challenging for freeform design. Here, we propose a fixed-point iteration method to address this problem. This process involves solving a set of balanced gradient equations in the form of fixed-point iterations that are derived from equivalent integrability conditions and Snell’s law. By using the convergence theorem of fixed-point iteration, a unique solution for the balanced gradient equations exists, which is determined by the natural geometric properties of the freeform surface and is independent of the mapping. The gradient operators on the left-hand side of the equations are converted into a differential matrix form via a finite difference scheme. In one iteration, differential operations are forward-performed on the right-hand side of the equations, and the system of linear equations is solved on the left-hand side of the equation. The constructed freeform surfaces work well in both the paraxial and nonparaxial regions, and convergence in the nonparaxial region is faster than that in the paraxial region. The robustness and high efficiency of the proposed method are demonstrated with several design examples.

Prediction of both diffusive and hydraulic conductance in the pore network model extracted from 3D images using deep learning

Abstract Pore network modeling (PNM) with network extracted from 3D images is widely used for studying mass transport in porous media, with its accuracy heavily dependent on the precise determination of hydraulic and diffusive conductance values of local pore pairs. Deep learning (DL) prediction based on true pore geometry offers a promising approach for conductance estimation. However, due to arbitrarily orientation and the lack of explicit inlet and outlet surfaces in 3D extracted pore pair geometries, existing studies [24,25] have relied only on 2D cross section images for hydraulic conductance prediction. A custom finite difference method was recently developed, capable exclusively of estimating diffusion conductance from 3D pore geometry [26]. In this work, a convolutional neural network (CNN) was trained to estimate both diffusive and hydraulic conductance values, with reference ground truth values obtained by applying the Lattice Boltzmann Method (LBM) on 3D pore pair geometries. The predicted pore pair conductance values from the CNN closely matched the LBM results on unseen segmented image data. This study highlights the importance of using the full 3D pore pair geometry for accurate hydraulic conductance evaluation, rather than relying solely on 2D throat cross sections. To evaluate the generality of our method, we calculated the permeability and diffusivity of 3D porous media across a wide range of porosities using PNM with local conductance values predicted by our DL model. Comparisons with results of LBM indicate that while diffusivity was accurately predicted, permeability was underestimated. This underestimation is attributed to the over-segmentation of pore spaces, which resulted in an underestimated flow rate.

The role of machine learning in analyzing magnetic field localization and vortex generation in tetra-hybrid nanofluid: A numerical study

This research distinguishes itself by integrating machine learning algorithms to assess the impact of confined magnetic fields on vortex dynamics in hybrid nanofluid flow, through the inclusion of both vertical and horizontal magnetic field strips. Specifically, the study focuses on a vertical cavity with an aspect ratio of 1:5, where a bottom lid moves horizontally from left to right to drive the flow. We have applied a limited magnetic field consisting of vertical, and horizontal strips. The authors have developed MATLAB codes to implement an algorithm for solving the governing equations of the nanofluid flow and heat transfer. The algorithm is based on the Stream-Vorticity formulation and uses a finite difference method. The algorithm can examine how several parameters, such as magnetic field strength (0–300), nanoparticle volume fraction (0–20%), and Reynolds number (Re) (1–50), impact the characteristics of nanofluids in terms of flow and thermal properties. The results demonstrate that magnetic fields influence the stress distribution of the flow pattern and the temperature distribution. Further, the presence of a magnetic field also affects stress distribution. Moreover, it has been determined that the Nusselt number (Nu) experiences a 60% increase due to the magnetic field, while there is a remarkable rise attributed to the Re. Similarly, significant changes are observed in skin friction under both parameters. These findings carry implications for designing and operating devices.

Combined Flow and Temperature Nonuniformity in Compact Cross‐Flow Three‐Fluid Heat Exchangers

ABSTRACTCryogenic processes often involve the heat interaction of multiple fluids and are highly sensitive to even minor variations in operating conditions. Hence, flow and temperature nonuniformity adversely impact the performance of thermal systems, and their combined occurrence can be especially detrimental. The transient response and thermal effectivity of a cross‐flow three‐fluid heat exchanger are investigated while considering the flow and temperature nonuniformity at the inlet section. Four modes of combined flow and temperature nonuniformity are opted, which replicates the pragmatic situation existing in the working of heat exchangers, and the effectivity of the thermal system is evaluated by its alignment with the uniform flow mode. Flow nonuniformity is implicated in both the inlet section and within the core all along with the effect of fluid back mixing. The “Dankwert's” boundary condition is specifically used to account for the effects of axial dispersion within the fluids. The model incorporates longitudinal heat conduction in partition walls, with the “finite difference method” employed to solve the partial differential equations. To the best of our knowledge, the present study is the first attempt to analyze both the transient response and effectivity of three‐fluid heat exchangers under combined flow and temperature nonuniformity. It is learned that thermal effectivity is degraded by 31.5% and enhanced by 1% under flow nonuniformity mode QPQ and temperature nonuniformity mode 1 respectively. Under the combined mode of both, the effectivity settles down somewhere amid both and is found to be degraded by 8%. The higher NTU comes up with a sharp peak in temperature distribution, which makes the core more prone to cracks.

Experimental Research and Significance Analysis of Advanced Interpolation Methods for Optimizing System State Items and Processing Parameters

The Finite-Difference Method (FDM) plays a pivotal role in the field of stability prediction, particularly in the modeling and stability analysis of cutting process dynamics. However, traditional approaches to optimizing the FDM often treat system state terms and time-delay terms as a monolithic entity, failing to explicitly distinguish between them, which leads to a lack of specificity in selecting optimization targets. In this study, an innovative approach is introduced by incorporating the third-order Newton interpolation method and the fourth-order Hermite interpolation method. By comparing the computational accuracy and convergence speed, it is found that the 3N-FDM (third-order Newton Finite-Difference Method) exhibits superior overall performance, and it is clearly pointed out that increasing the order does not always result in better outcomes. Additionally, this study selects different discretization numbers, denoted as m, for comparative analysis to thoroughly evaluate their impact on computational accuracy. Experimental validation demonstrates the high accuracy of the 3N-FDM. Through a one-way ANOVA (analysis of variance) of tool wear and workpiece surface roughness, it is revealed that changes in system state terms have the most significant impact on the feed rate f, followed by the cutting depth ap, and finally the spindle speed n. Based on the experimental results and analysis mentioned above, this study concludes that optimizing system state terms can more effectively explore the combined influences of processing parameters on processing quality, production efficiency, and tool wear.

Spatiotemporal patterns in a predator–prey model with anti-predation behavior and fear effect

In this study, a predator–prey model incorporating prey’s fear effect and anti-predation behavior has been developed. The functional response takes the square root of the prey population and adds the predator’s loss term. We prove the existence of the system’s equilibrium point and derive the conditions for its local stability, as well as discuss how diffusion affects the system’s stability. Bifurcation analysis of the parameters reveals that the fear effect leads to a decrease in the predator population in the coexistence equilibrium. In addition, we apply the finite difference method and the finite element method to simulate the diffusion model in rectangles and irregular customized regions, respectively. The results demonstrate that the system can eventually reach a uniform stable state after experiencing oscillations and fluctuations at certain parameters and initial values.

Modeling Hybrid Crossover Dynamics of Immuno‐Chemotherapy and Gene Therapy: A Numerical Approach

ABSTRACTA piecewise fractional differential equation (deterministic–stochastic differential equation or vice versa) has appeared in recent literature. Piecewise operators are used to study crossover real data effectively. By using stochastic–deterministic piecewise hybrid fractional derivatives, with hybrid fractional‐order and variable‐order fractional operators, this paper extends the deterministic model of immuno‐chemotherapy with gene therapy and time delay. The immuno‐chemotherapy with gene therapy combines traditional chemotherapy and immunotherapy with genetic engineering techniques to enhance the immune system's ability to target and destroy cancer cells. This offers a multifaceted approach to cancer treatment, potentially enhancing effectiveness while minimizing side effects. Two approximation techniques are used to solve the proposed model numerically. In the hybrid fractional deterministic models, we use the nonstandard Caputo proportional constant finite difference method, and in the stochastic models, we use the nonstandard Milstein technique. We examine the stability analysis of these methods to ensure their efficiency. Both the theoretical results and the efficiency of the methods are confirmed by numerical tests. New piecewise operators are illustrated by the curves presented. Studying the introduced piecewise fractional differential immuno‐chemotherapy model with gene therapy and time delay along with a stochastic case greatly explained the dynamics of immuno‐chemotherapy interaction.

Computational Approaches to Compressible Micropolar Fluid Flow in Moving Parallel Plate Configurations

In this paper, we consider the unsteady flow of a compressible micropolar fluid between two moving, thermally isolated parallel plates. The fluid is characterized as viscous and thermally conductive, with polytropic thermodynamic properties. Although the mathematical model is inherently three-dimensional, we assume that the variables depend on only a single spatial dimension, reducing the problem to a one-dimensional formulation. The non-homogeneous boundary conditions representing the movement of the plates lead to moving domain boundaries. The model is formulated in mass Lagrangian coordinates, which leads to a time-invariant domain. This work focuses on numerical simulations of the fluid flow for different configurations. Two computational approaches are used and compared. The first is based on the finite difference method and the second is based on the Faedo–Galerkin method. To apply the Faedo–Galerkin method, the boundary conditions must first be homogenized and the model equations reformulated. On the other hand, in the finite difference method, the non-homogeneous boundary conditions are implemented directly, which reduces the computational complexity of the numerical scheme. In the performed numerical experiments, it was observed that, for the same accuracy, the Faedo–Galerkin method was approximately 40 times more computationally expensive compared to the finite difference method. However, on a dense numerical grid, the finite difference method required a very small time step, which could lead to an accumulation of round-off errors. On the other hand, the Faedo–Galerkin method showed the convergence of the solutions as the number of expansion terms increased, despite the higher computational cost. Comparisons of the obtained results show good agreement between the two approaches, which confirms the consistency and validity of the numerical solutions.

Pricing weather derivatives and managing weather risks under regime switching

Abstract A large number of empirical studies have shown that regime switching is an important characteristic of temperature processes. Thus, this work develops an efficient finite difference scheme with one-sided difference to price weather derivatives with the partial differential equations (PDEs) of regime-switching temperature models. For management purposes, an initial management model framework is then proposed to manage portfolios incorporating weather derivatives. This work also performs statistical analysis on the temperature dataset from Toronto, Canada and recalibrates the regime-switching temperature models, as one parameter value is found incorrect in the literature. Numerical experiments indicate that the finite difference scheme is much more competitive than the Monte Carlo simulations and the lattice approaches. The PDEs pricing approach has great potential to be used in practical portfolio management problems involving weather risks.

Nonlinear Transient Heat Transfer and Optimisation of Multidirectional (1D/2D/3D) Functionally Graded Porous Composites

In this paper, the steady and transient heat transfer behaviours of multidirectional (1D/2D/3D) functionally graded composite plates are investigated. Also, two different porosity types, i.e., even and uneven distributions, are considered along with material gradation. Here, the spatial/temperature-dependent effective material properties of these highly heterogeneous composites with porosity are evaluated using the extended Voigt’s homogenisation scheme via multi-variable power-law functions and further verified with the representative volume element (RVE) scheme. In continuation, generalised mathematical models are developed by including material nonlinearity using the cubic-polynomial-based temperature-dependent constitutive model. The Poisson’s and Laplace’s equations are utilised for steady and transient heat transfer problems, respectively, and the weak form is derived using the Galerkin method. Further, coupled finite element method (FEM)–finite difference method (FDM) is adopted via Picard’s successive iteration and Crank–Nicolson technique to compute the nonlinear transient temperature. The proposed model is verified by comparing it with the previously reported results. In addition, the effects of heterogeneity, porosity, power-law indices and boundary conditions on the steady/transient heat transfer responses of multidirectional functionally graded plates are examined and discussed in detail. At last, the optimum material distributions are obtained using the response surface methodology (RSM) by minimising the heat transfer responses.