Finite Difference Method Research Articles

High-efficiency polarization-selective two-dimensional diffraction gratings based on silver cuboid arrays

A two-dimensional (2D) diffraction grating based on silver cuboid arrays is demonstrated. With the structural optimization by three-dimensional (3D) finite-difference time-domain method, the proposed grating exhibits favorable polarization selectivity. At wavelength 1550 nm, high diffraction efficiency and polarization correlation can be realized in the case of two-port output ((0, ±1) or (±1, 0)) under TE polarization, as well as five-port output ((0, 0), (0, ±1) and (±1, 0)) under TM polarization. Under the two-port output state, the diffraction efficiency is nearly 48% with the extinction ratio of 16.44 dB and insertion loss of 0.20 dB. Under the five-port output status, the efficiency is larger than 19% for each diffraction order, meanwhile, the total efficiency and uniformity are 97% and 0.58%, respectively. The proposed grating with positive implications has good potentials in polarization multiplexing and polarization-correlated grating interferometry.

Enhancing interpolation and approximation error estimates using a novel Taylor-like formula

In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the P1-interpolation error estimate. Following the same principles, we have also developed a novel numerical scheme for the heat equation that provides a better error estimate compared to the classical implicit finite differences scheme. Numerical illustrations confirm this behavior.

Isogeometric collocation method to simulate phase-field crystal model

PurposeThe purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.Design/methodology/approachFirst, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.FindingsIn the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.Originality/valueThis research work introduces the IGC method as a simulation technique for the phase-field crystal model.

Influence of launch light beam conditions on the bandwidth in multimode graded-index microstructured POFs

The time-dependent power flow equation (TD PFE) is used to theoretically examine the wavelength dependency of the bandwidth in a multimode graded-index microstructured polymer optical fiber (GI mPOF) with a solid core. The TD PFE is numerically solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINNs). Our numerical results show that the bandwidth decreases with an increasing wavelength from 476 to 522 and finally to 568 nm. With further increasing of the wavelength from 568 to 633 nm, the bandwidth increases. In this way, we have demonstrated that the maximum bandwidth of the analyzed GI mPOF in the analyzed wavelength region is obtained at 633 nm. Such bandwidth behavior is a consequence of the different influences of the analyzed wavelengths on the parameters that characterize the refractive index distribution and the maximum propagating principal mode number of the analyzed GI mPOF. We also showed that, when the radial offset of the incident light beam increases, the bandwidth decreases. This results from a greater modal dispersion when higher guided modes are excited. As a result, based on the observations reported in this work, it is simpler to modify the GI mPOF for a particular application at various wavelengths.

On Numerical Integration and Conservation of Cell-Centered Finite Difference Method

On Numerical Integration and Conservation of Cell-Centered Finite Difference Method

A reaction–diffusion model of major emerging infectious diseases in a spatially heterogeneous environment and case study

During the outbreak of major emerging infectious diseases, virus droplets can survive in the environment as aerosols for hours to days, and their impact on human infectious diseases is often overlooked. In addition, the speed of transmission of infectious diseases is often closely related to transportation. Therefore, studying the impact of environmental viruses and transportation on disease development is significant for effective infectious disease prevention and control. We proposed a degenerate reaction–diffusion infectious disease model ([Formula: see text]) considering environmental virus interference and established a well-posedness and threshold system for this model. We have obtained the system solution approaches the disease-free equilibrium ([Formula: see text]) when the basic reproduction number [Formula: see text]. The system has at least one positive steady state solution (PSS) when [Formula: see text]. This paper used a non-standard finite difference method discretization model while data fitting and parameter estimation were performed based on data provided by the Health Commission. Further sensitivity analysis was conducted on [Formula: see text]. At the same time, we also discussed the impact of various parameters in the early stages of the outbreak of major emerging infectious diseases on the development of the disease. Research found that even if the contact rate between people is controlled at a shallow level, the disease may persist. In the early stages of major emerging outbreak of infectious diseases, immediately reducing the use of transportation can effectively reduce the speed of disease spread.

Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

Numerical solution of a hydrodynamic model with cavitation using finite difference method at arbitrary meshes

In this paper, we investigate the implementation of the finite difference method on arbitrary meshes in conjunction with a fixed-point algorithm for the lubrication problem of a journal bearing with cavitation, considering the Elrod-Adams model. We establish numerical properties of the generalized finite difference scheme and provide several illustrative examples.

A finite difference scheme for (2+1)D cubic-quintic nonlinear Schrödinger equations with nonlinear damping

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of n≥2 suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete L2-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete L2-norm and H1-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.

In-Plane Vibrations of Elastic Lattice Plates and Their Continuous Approximations

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson’s ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately.

Implicit finite difference simulations for unsteady oscillating flow of Walters-B nanofluid with microbes using the Cattaneo–Christov model

The thermal importance of nanoparticles in different fields of engineering is still a developing issue that necessitates additional focused consideration. The suspension of non-Newtonian fluids with nonmaterials has various applications in areas such as enhanced thermal systems, energy processes, aerospace engineering, and chemical industries. Following such motivations, this work aims to analyze the heat and mass transfer flow of Walters-B nanofluid in the presence of microorganisms. The fluid motion is presumed to be nonsteady over a surface that undergoes periodic acceleration. The motivations for studying nanofluid’s oscillating flow are its use in effective mixing processes, combustion stability, and process efficiency. The text provides a detailed explanation of the saturation of porous media. A modified theory for heat and mass fluxes, known as the Cattaneo–Christov approach, suggests an extension in heat equations. Moreover, it supports the interaction of radiation phenomena for enhancing heat transfer. It is noteworthy that the entire problem is represented using extremely nonlinear partial differential equations (PDEs). The CFD study utilizing the renowned implicit finite difference method (FDM) has been effectively employed to propose a numerical solution for the problem. The numerical results are deemed valid and confirmed within certain limiting cases, based on the available data. The evaluation of velocity, skin friction coefficient, Nusselt number, and Sherwood number as a function of time has been demonstrated for various parameters. It is asserted that the wall shear force exhibited an oscillating pattern with a greater magnitude as a result of the viscoelastic parameter. The Nusselt number and Sherwood number exhibit slight acceleration as a result of changes in the Prandtl number and Schmidt number, respectively. The proposed model is specifically designed for use in heat exchangers, vibrational systems, compact designs, and vibration control.

Hydroelastic modelling of a deformable wave energy converter including power take-off

Given the advantages of flexible wave energy converters (FlexWECs), such as deformation-led energy harnessing and structural loading compliance, there has been a significant interest in FlexWECs in both academia and industries. To simulate the FlexWEC interaction with ocean surface waves, a 3D computational fluid-structure interaction approach is developed in this study. The fluid and solid governing equations are discretized using finite difference and finite element methods, respectively. An immersed boundary method is used to couple the two independent grid systems. A novel numerical technique is introduced to model the dielectric elastomer generator (DEG) as the power take-off (PTO). The wave energy capture performance is analysed for different PTO configurations and at various wave conditions. Based on the obtained results, the PTO damping coefficient and the relative wavelength range that maximizes the capture width ratio (CWR) are determined. The wavefield results also reveal the presence of wave-height enhancement and attenuation points around a single FlexWEC, providing potential site selection references when deploying multiple FlexWECs in an array.

Exploration of linear and exponential heat source/sink with the significance of thermophoretic particle deposition on ZnO-SAE50 nano lubricant flow past a curved surface

The present research focuses on exploring the impact of linear and exponential heat source/sink on double-diffusive MHD convective flow heat and mass transport of ZnO – SAE50 nano lubricants in an exponentially stretching curved sheet along with the effect of thermophoretic particle deposition. The heat transmission is considered to be caused by both radiation and convection. The mathematical modelling of the considered problem results in a coupled system of partial differential equations, using suitable similarity transformation variables these equations are reduced to a coupled non-linear system of ordinary differential equations and are solved numerically by adopting the finite difference method to obtain the solution for the field variables. The quantity and spatial distribution of linear heat sources/sinks affect the physical dynamics and optimum status of the entire system and the application of exponential heat sources/sinks can improve heat dispersion and energy efficiency in industrial operations such as nuclear reactors and steel production. The thermophoretic particle deposition parameter controls particle concentration by allowing particles to accumulate on surfaces or in specific areas of the fluid. A rise in Hartmann number causes a mechanical damping force to act in the opposite direction of the fluid motion, lowering the speed of the fluid in the curved sheet. The heat and mass transmission properties are analyzed through the Levenberg – Marquardt backpropagating algorithm and the correlation coefficient has ideal values of unity for training, validation, testing, and overall. In light of this, the anticipated behaviour of the LMNN-BPA indicated that the heat transfer profile numerical data and the network output are in good consensus.

Physics-Informed Fully Convolutional Networks for Forward Prediction of Temperature Field and Inverse Estimation of Thermal Diffusivity

Abstract Physics-informed neural networks (PINNs) are a novel approach to solving partial differential equations (PDEs) through deep learning. They offer a unified manner for solving forward and inverse problems, which is beneficial for various engineering problems, including heat transfer analysis. However, traditional PINNs suffer from low accuracy and efficiency due to the fully-connected neural network framework and the method to incorporate physical laws. In this paper, a novel physics-informed learning architecture, named physics-informed fully convolutional networks (PIFCNs), is developed to simultaneously solve forward and inverse problems in thermal conduction. The use of fully convolutional networks (FCNs) significantly reduces the density of connections. Thus, the computational cost is reduced. With the advantage of the nodal-level match between inputs and outputs in FCNs, the output solution can be used directly to formulate discretized PDEs via a finite difference method, which is more accurate and efficient than the traditional approach in PINNs. The results demonstrate that PIFCNs can flexibly implement Dirichlet and Neumann boundary conditions to predict temperature distribution. Remarkably, PIFCNs can also estimate unknown thermal diffusivity with an accuracy exceeding 99%, even with incomplete boundaries and limited sampling data. The results obtained from PIFCNs outperform those obtained from PINNs.

Transient study on heat transfer and flow performance of alkali metal heat pipe in space nuclear powered heat pipe radiator

The non-direct contact radiator has a simple structure and high reliability, making it more suitable for long-life nuclear powered spacecraft. This article is based on Fortran language and innovatively combines molecular dynamics models to establish a two-dimensional transient analysis program for thermo-hydraulic performance of rubidium heat pipes. The unsteady Navier Stokes control equation, heat conduction equation, and continuity equation is calculated by the finite difference method. This study adopt molecular dynamics theory for the first time to deal with condensation/ evaporation at the interface of alkali metal heat pipes under the background of space nuclear power. The delay method and multi grid method are used to ensure convergence and accelerate calculation speed. The transient thermo-hydraulic performance of different positions of heat pipes in non-direct contact radiators during start-up are analyzed. Current research can provide scientific reference for the upgradation of heat pipe radiators for nuclear powered spacecraft.

2-D acoustic equation prestack reverse-time migration based on optimized combined compact difference scheme

Abstract Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective reverse time migration method suitable for coarse grid conditions.

Ultrabroadband invisibility cloaking design of a concentric multilayered cylindrical metamaterial

An ultrabroadband invisibility cloak is designed using a genetic algorithm (GA) for a cylindrical target with an infinite length. The cloaking band is 0.4 to 0.7µm, covering almost the entire visible light range. The invisibility cloaking medium consists of concentric multilayered cylindrical metamaterials (CM). When the cloaking target is covered with a well-designed 26+ layer CM, the average scattering cross-section is 6% or less of the uncloaked target’s cross-section. The cloaking performance of the structure designed by the GA was superior to that designed by analytical calculations based on a combination of the transformation optics and the effective medium approximation. The cloaking is confirmed through the time variation of the magnetic fields calculated using the finite-difference time-domain method. In addition, the mechanism of the cloaking is discussed.

Machine learning based analysis of entropy production and wall shear stress in blood flow through stenosed artery

The building up of cholesterol and various substances on the inner boundary of arterial walls forms a plaque (stenosis), which results in the narrowing of walls of arteries, and this exerts additional stress on the walls. Shear stress at the wall of arteries may cause the formation of plaque, plaque growth, and expansive arterial remodeling, and it may also result in an increased risk of plaque rupture. The purpose of this research work is to use machine learning approaches to analyze the wall shear stress and entropy generation through irreversible processes of blood flow in a porous stenosed artery within the context of magnetohydrodynamic consideration. As a way of illustrating blood’s non-Newtonian behavior, the Casson fluid model is used. A blood flow model in the context of mild stenosis has been developed, and it has subsequently been computationally addressed using an explicit finite difference method. The velocity, temperature, entropy generation, Bejan number, and wall shear stress are visualized for different values of pertinent variables. The result reflects the increased velocity in the stenosed region. A hybrid machine learning method using artificial neural network and particle swarm optimization is implemented to optimize the entropy generation and wall shear stress. The network is trained using Stochastic gradient–descent training procedure as well as Levenberg–Marquardt training procedure, and the outcomes of both training procedures are compared. Using this machine learning process, the minimum entropy production has been found for specific values of the Hartmann number, dimensionless temperature difference, and pulsatile component of pressure gradient. This machine learning-based optimization approach enhances the ability to forecast quantitative changes in affecting parameters, which will contribute to minimized damage to the arterial wall, improved blood flow efficiency and the delivery of necessary components to tissues.

Multiaxial ratcheting-fatigue evaluation of pressurized elbow pipe under strong cyclic loading using damage-coupled cyclic plasticity model

Continuous damage mechanics (CDM) is based on the theories of continuous medium mechanics and continuous medium thermodynamics, which considers damage is an irreversible dissipative process within the materials, and employs the field-theoretic approach of image-only science to study the internal macroscopic damage evolution law of the materials and its influence on the deterioration of the macroscopic mechanical properties of the materials. Hence, within the framework of CDM, a damage-coupled cyclic plasticity constitutive model is proposed based on combined isotropic and Chen-Jiao-Kim (CJK) kinematic hardening rule for evaluating the ratcheting-fatigue behavior of 90° elbow pipes under strong cyclic loading in this paper. Stress return mapping and numerical solution procedures of the proposed model are formulated based on the backward finite difference method. Formulation of the consistent tangent operator is presented for the Finite Element implementation of the plasticity model in ABAQUS UMAT. The FE analysis of non-pressurized and pressurized elbow pipes under cyclic displacement-controlled loading is respectively performed using the implemented constitutive model. The results reveal that the predicted results of the damage-coupled cyclic plasticity constitutive model are better than combined isotropic-kinematic hardening model, and well consistent with the experimental data.

A Novel Geothermal Wellbore Model Based on the Drift-Flux Approach

Geothermal energy, being a clean energy source, has immense potential, and accurate wellbore modeling is crucial for optimizing the drilling process and ensuring safety. This paper presents a novel geothermal wellbore model based on the drift-flux approach, tested under three different temperature and pressure well conditions. The proposed model integrates the conservation equations of mass, momentum, and energy, incorporating the gas–liquid two-phase flow drift-flux model and heat transfer model. The key features include handling the heat transfer between the formation and the wellbore, addressing the slip relationship between the gas and liquid phases, and accounting for wellbore friction. The nonlinear equations are discretized using the finite difference method, and the highly nonlinear system is solved using the Newton–Raphson method. The numerical simulation, validation, and comparison with existing models demonstrate the enhanced accuracy of this model. In our tests, the model achieved a high accuracy in calculating the bottom-hole pressure and temperature, with mean relative errors (MREs) significantly lower than those of other models. For example, the MREs for the bottom-hole pressure and temperature of the Rongxi area well in Xiongan, calculated by this model, are 1.491% and 1.323%, respectively. These results offer valuable insights for optimizing drilling parameters and ensuring drilling safety. Comparisons indicate that this approach significantly outperforms others in capturing the complex dynamics of geothermal wellbores, making it a superior tool for geothermal energy development.