Finite Difference Method Research Articles

Significance of finite difference approach for application of Cattaneo-Christov theory conveying radiative ternary-hybrid nanofluids flow

This paper investigates the characteristics of ternary-hybrid nanoliquid for time-dependent Darcy-Forchheimer flow. Formulation is based for mixture of water subject to ternary nanoparticles with different shapes like (platelet, spherical and cylindrical). Heat transfer consists of convective condition, radiation and heat generation. Appropriate variables are used to generate dimensionless nonlinear system. Analysis is carried out for Cattaneo-Christov heat flux and variable thermal conductivity. Finite difference method (FDM) is utilized to construct solutions. Salient characteristics of sundry variables are discussed. Analysis is carried out for Nusselt number and skin friction. Present results may have relevance in renewable energy process, polymer extrusion and metallurgy.

A new fourth-order compact finite difference method for solving Lane-Emden-Fowler type singular boundary value problems

We develop a novel fourth-order compact finite difference scheme to solve nonlinear singular ordinary differential equations. Such problems occur in many fields of science and engineering, such as studying the equilibrium of an isothermal gas sphere, reaction–diffusion in a spherical permeable catalyst, etc. These problems are challenging to solve because of their singularity or nonlinearity. By our proposed method, we can easily solve these complex problems without removing or modifying the singularity. To construct the new fourth-order compact difference method, Initially, we created a uniform mesh within the solution domain and developed a compact finite difference scheme. This scheme approximates the derivatives at the boundary nodal points to handle the problem’s singularity effectively. Employing a matrix analysis approach, we discussed the convergence analysis of the methods. To demonstrate its efficacy, we apply our approach to solve various real-life problems from the literature. The new method offers high-order accuracy with minimal grid points and provides better numerical results than the nonstandard finite difference method and exponential compact finite difference method.

A novel mapping method for equivalent stiffness and equivalent damping of [formula omitted]-DOF series hydraulic robot

Impedance control is a commonly used approach for robots to interact with environments. At present, the impedance control is mostly implemented at the foot end, while the impedance control at the joint space offers faster response speed and is a more favorable option. However, due to the influence of installation position and moment arm of hydraulic drive unit (HDU), it is more complicated to establish the mapping relationship of equivalent stiffness and equivalent damping between the foot end and joint HDU. To address this challenge, this paper provides a general calculation expression of mapping matrices between the foot end and joint HDU, which also considers the influence of above nonlinear factors. Additionally, the accuracy of mapping matrices is verified through the impedance control, and the “finite difference method” (FDM) is used to enhance the accuracy of validation. Finally, the accuracy of mapping matrices is validated through the single leg of a 2-DOF series hydraulic robot.

Injected phase change seismic source construction and monitoring experiment

Scientific assessment of the transport channels and distribution of oil, gas, water, and other fluids is the key to enhancing oil and gas production capacity and ensuring the safe and efficient development of mineral resources. Traditional microseismic monitoring has been suffering from limitations such as ambiguous attribute characteristics of the seismic source and low signal-to-noise ratios of the data, leading to poor location accuracy and large uncertainties. We develop a phase change seismic source monitoring method that allows for a detailed characterization of the spatial distribution of subsurface fluid transport by accurately locating the equivalent seismic source within supportive fractures. A phase change seismic source is developed and constructed based on the phase change material, enabling control over phase change temperature, source energy, and source scale through the principle of phase change energy storage. The source introduces an active approach to seismic signal generation, diverging from the conventional passive method reliant on rock rupture. Then, 3D homogeneous models are constructed for ground monitoring and in-well monitoring using impulse source propagation theory. The numerical simulation of the wavefield uses the finite-difference method, and the equivalence of discrete sources is confirmed using the Pearson correlation function. The reliability of the monitoring method is verified through a seismic monitoring experiment and comprehensive feasibility analysis.

A novel temporal finite element method to solve static viscoelastic problems

A novel temporal finite element method is presented to solve static viscoelastic problems, which is more flexible and appropriate to describe temporal variation of displacement than conventional finite difference or numerical integral methods. By using virtual work principle, a spatial finite element based governing equation is derived in terms of displacement and its derivatives. Then two temporal finite element models are developed using Gurtin variational principle and weighted residual technique. A kind of hybrid shape functions with polynomial and trigonometric basis is stressed to give more flexible descriptions of time varying variables. A criterion of stability analysis is derived, which can numerically be conducted when the constitution of shape functions and step size are prescribed. A recursive algorithm toward end is developed by which the temporal FE analysis can be conducted via a matrix power product with the initial condition, instead of step marching. The proposed approach is available for the viscoelastic model described by linear differential equations with order h ≤ 2, and can conveniently be combined with well-developed numerical algorithms to tackle with boundary value problems, such as FEM, SBFEM etc. Various numerical examples, including those with static/harmonic loads, creep, stress singularity and heterogeneous structures, etc. are provided to illustrate the efficiency of the proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with analytical or ABAQUS-based solutions.

Gas-Kinetic Unified Algorithm for Aerodynamics Covering Various Flow Regimes by Computable Modeling of Boltzmann Equation

The gas-kinetic unified algorithm (GKUA), to solve the modeling of the Boltzmann equation, has been developed to study the aerothermodynamics problems with the effects of wall activation energy covering various flow regimes. The unified velocity distribution function equation could be accordingly presented on the basis of the Boltzmann-Shakhov model. To remove the dependence of the distribution function on velocity space, the conservational discrete velocity ordinate method has been developed for hypervelocity flows. The gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by the operator splitting technique. The discrete velocity numerical integration method with the Gauss-type weight function has been developed to evaluate the macroscopic flow variables. Specially, to model the real physical process between gas molecules and the surface, the Maxwell-type gas-surface interaction model has been presented by the MD (molecular dynamic) simulation to obtain the energy adaptability coefficient. The multi-processing domain decomposition strategy and parallel implementation of high parallel efficiency and expansibility designed for the gas-kinetic numerical method is presented with good load balance and data communication efficiency. To validate the accuracy and feasibility of the present algorithm, the supersonic flows past two-dimensional circular cylinder are simulated covering various flow regimes. The results are in good agreement with the related theoretical, DSMC (Direct Simulation Monte Carlo), N-S (Navier-Stokes), and experimental data. The hypersonic reentry flows with the effects of wall activation energy around the Tianzhou-5 cargo spacecraft are simulated by the present GKUA and the massive parallel strategy. It has been confirmed that the present algorithm from the gas-kinetic point of view probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes during the re-entry and disintegration of the large-scale spacecraft.

Optimal Point-Wise Error Estimate of Two Second-Order Accurate Finite Difference Schemes for the Heat Equation with Concentrated Capacity

Optimal Point-Wise Error Estimate of Two Second-Order Accurate Finite Difference Schemes for the Heat Equation with Concentrated Capacity

Calculation of Single and Multiple Low Reynolds Number Free Jets with a Lattice-Boltzmann Method

Numerical calculations of low-Reynolds-number freejets with a Lattice Boltzmann Method are presented. The calculated-time-averaged axial velocity of a round jet with Re=1030 matches experimental data, including the length of transition from laminar to turbulent flow. Special care was needed for the inlet conditions in order to reproduce the vena contracta phenomenon. The results for round jets with Re=1000/1500/2000 show good agreement with Finite Difference Method calculations from the literature. In principle, there is a strong sensitivity to the inlet conditions, suggesting a need in future experimental work to measure in detail the velocity profiles and turbulence quantities at the nozzle outlet. The application of turbulence at the inflow boundary of the calculation domain is often used to emulate sources of disturbances in experiments. The present study demonstrates the need to investigate the impact of turbulence level and length scale at inlet independent of each other. Finally, the calculation for a bundle of nine jets with a square inlet led to the finding that the velocity decay of the central jet is maximal when the spacing between the jets is ca. one jet diameter.

Hybrid Modelling Using Lattice Boltzmann and Finite Difference Methods of Dikes Effect on Sediment Transport and Morphological Processes with a Quasi-Tridimensional Approach

Hybrid Modelling Using Lattice Boltzmann and Finite Difference Methods of Dikes Effect on Sediment Transport and Morphological Processes with a Quasi-Tridimensional Approach

Novel method to estimate horizontal variability of shear wave velocity through multichannel analysis of surface waves

Scale of fluctuations (SOFs) of spatially variable soil properties have been regarded as one of the important parameters for performing reliability-based design in geotechnical engineering. However, the information required to estimate the SOFs in practice is limited, especially in the horizontal direction. In this study, the potential use of Multichannel Analysis of Surface Waves (MASW) to estimate the SOFs of shear wave velocity (VS) in horizontal direction is investigated through a series of numerical investigations. 2D random field models with a vertically increasing trend of VS were first simulated with different levels of variability controlled by the coefficient of variation (COV) and SOF. Afterwards, a large number of numerical MASW tests were performed using a 2D finite difference method, where the survey lines were progressively shifted. Results show that the COV of VS can be determined from the scatter of the dispersion curves, whereas the horizontal SOF of VS can be appropriately estimated from the phase velocity profile presented in the wavelength form. Additionally, in-situ MASW tests were conducted to estimate the horizontal SOF, and the obtained results align with those estimated by different methods. It is highlighted that the accuracy of the estimation depends on survey length. The interpretation using a long array reflects an averaged site condition of the survey area, thus losing the variability information. Favorable predictions are produced when survey length is shorter than 1.0 SOF. However, it should be noted that use of short survey length may limit the depth of investigation.

Continuous [C] Content Prediction Model in Molten Steel of the All-Scrap Electric Arc Furnace

The continuous [C] content prediction model in molten steel of the all-scrap electric arc furnace was established through the explicit finite difference method combined with numerical simulation based on the thermodynamics and kinetics of the decarburization reaction. Initially, a method to figure out the average [C] content of mixed input scrap was proposed, which provided a basis for subsequent calculations. Furthermore, scrap melting rate, mixing side-blowing / bottom stirring, and oxygen blowing intensity adjustment of oxygen lance had been considered as the three key elements. Decreasing scrap melting time was beneficial to improving the EAF efficiency because it accounts for 80% of the total operation time. The research found that the scrap melting rate was mainly affected by the solid-liquid phase heat transfer and [C] content transfer, and the heat transfer played a dominant role. Applying mixing side blowing / bottom stirring could lead to an improvement in the mass transfer coefficient of the decarburization reaction, and the value was increased from the range of 0.0059 s-1-0.021 s-1 by using single side-blowing to the range of 0.0079 s-1-0.023 s-1. The mass transfer coefficient could be enhanced by the oxygen intensity adjustment of oxygen lances when mixing side blowing / bottom stirring was used. This was a benefit of deeper jet impact zones on the molten steel surface caused by high oxygen flow. After the model was applied to an industrial site, the prediction accuracy of the endpoint [C] content within the error range of ±0.02% and ±0.01% reached 91.2% and 85.7%, respectively.

Thermal performance analysis of a domestic-size absorption cooling system incorporating nanofluid with thermal insulation in hot climate of Algeria

The increased demand for energy consumption for air conditioning, due to rising temperatures, has created challenges related to thermal comfort in residential environments .In the present study, a simulation of an air conditioning system for a small building was conducted, considering various absorption cooling scenarios to assess system performance through dynamic and economic analysis. The proposed system employs a parabolic trough collector in conjunction with a storage tank, designed to cool a 12 m² room in Adrar, Algeria. The study aims to identify the optimal scenario based on temperature, heat, and system coefficient of performance indicators. A computer program was developed using characteristic equations and finite difference methods to analyze the system's behavior with different nanofluid concentrations and thermal insulation. Simulation results show that scenario with nanofluid and insulation the best performance is achieved with a 4% nanofluid concentration, a 9 m² solar collector, a 0.3 m³ storage tank, and 0.05 m thick polystyrene insulation. The system can operate for 9 h, maintaining an indoor temperature of 27°C with a peak cooling load of 3 kW. The economic analysis estimates an investment cost of 4535.4$ and Net Present Value 3,370.40$ with a payback period of 12 years and 3 months.

Treatment and delay control strategy for a non-linear rift valley fever epidemic model

Rift Valley Fever (RVF) is a viral disease affecting animals and humans, causing symptoms such as fever, liver damage, and bleeding, particularly prevalent in Africa. This study focuses on numerical solutions for a non-linear delayed dynamic epidemiological model of RVF. It extends a control problem incorporating the susceptible, infected, treated, recovered vector to analyze the impact of measures such as mosquito repellent and treatment. The goal is to examine how time delays in implementing control measures affect the dynamics of an epidemic. The model considers delay factors such as mosquito replication, hospitalization, travel restrictions, and isolation due to the lack of proper vaccination. The study explores the model’s aspects, including the reproduction number, equilibrium points, and stability. Local and global implications are examined using techniques such as the Lyapunov function and the Brauer-F lemma. Numerical analysis employs the non-standard finite difference method, establishing the local stability of the equilibrium through the effective reproduction number Rrvf and sensitivity analysis. The research highlights the importance of treatment and delay strategies in reducing RVF transmission, emphasizing the critical need for immunization and preventive measures.

Machine learning analysis for heat transfer enhancement in nano-encapsulated phase change materials within L-shaped enclosure with heated blocks

Phase change materials(PCMs) are crucial to energy storage systems due to their enhanced thermal properties. They significantly boost energy efficiency and promote sustainability. Nevertheless, the low thermal conductivity of PCMs presents a significant challenge, which is addressed by utilizing nano-encapsulation to enhance energy efficiency in energy storage systems. Motivated by this, the current study conducts a theoretical investigation to explore the heat transfer characteristics in a buoyancy-driven Nano-Encapsulated Phase Change Materials(NEPCM) nanofluid within an L-shaped porous enclosure with the impacts of a heated block and magnetic field. Furthermore, the fusion temperature plays a crucial role in initiating phase change in NEPCM, thereby impacting the heat transfer process. Hence, identifying the optimal fusion temperature is essential. To accomplish this, a machine learning approach was employed to identify the ideal fusion temperature. A dataset of 160 data points across four different fusion temperature values was used in this analysis. Additionally, the machine learning model analyzed how variations in fusion temperatures impact physical parameters. An in-house Matlab code is utilized to solve the dimensionless fluid transport equations employing the finite difference method. The results indicate that increasing the nanoparticle volume fraction significantly enhances the heat transfer rate across all physical parameters. Specifically, under higher thermal buoyancy force, increasing the volume fraction from 1% to 5% results in a 90.04% increase in the heat transfer rate. The numerical analysis demonstrates that heat transfer rates improve significantly when the fusion temperature is adjusted to 0.5, a result further validated by machine learning techniques. At this temperature, thermal buoyancy force increases by 0.98% and 2.68% compared to values of 0.1 and 0.9, respectively, while the Stefan number shows increases of 159.42% and 87.48% under these conditions; thereby, the heat transfer rate increases at this value. This computational study provides important insights into the significance of fusion temperature, emphasizing the need to determine its optimal value for improving heat transfer. Identifying this optimal value can enhance the efficiency of thermal energy storage systems and improve cooling performance in electronic devices.

Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

Optimizing heat and mass transfer in Carreau nanofluid with mixed nanoparticles in porous media using explicit finite difference method

Optimizing heat and mass transfer in Carreau nanofluid with mixed nanoparticles in porous media using explicit finite difference method

Global tsunami modelling on a spherical multiple-cell grid

A model of shallow water equations (SWEs) on a spherical multiple-cell (SMC) grid of 4-level (2.5-5-10-20 km) spatial resolutions is used to simulate tsunami propagation on global ocean surfaces. The unstructured SMC grid retains rectangular cells of the longitude-latitude grid so that efficient finite-difference schemes could be used. It also supports multi-resolutions like mesh refinement to resolve small islands and coastline details while keeping models compact enough to fit into available computers. Two earthquake-induced tsunami cases are simulated and compared with available observations. Results indicate that the modelled tsunami arrival times agree well with observations while tsunami wave heights are underestimated, particularly the observed runups on remote coastal lands. Possible reasons for this underestimation include the smoothing scheme used to suppress numerical oscillations and the missing of initial kinetic energy input from the earthquakes. Another reason is the limitation of the SWEs to describe coastal bores and breaking waves in coastal waters. A possible tsunami scenario induced by a landslide in the Canary Islands is also simulated to assess its potential impact on Atlantic coastal regions. Model results indicate that this kind of tsunami may cause severe damage to local areas but its effects on far fields, like the UK and American coastal regions are small. As the initial landslide disturbance is overly simplified, this study does not give a true representation of a real landslide tsunami but rather a qualitative assessment of its impact on the Atlantic Ocean. More realistic initial condition and improved model representation of coastal processes are needed for further studies of this possible landslide hazard.

Deep Learning for Solving Partial Differential Equations: A Review of Literature

Partial Differential Equations (PDEs) are fundamental in modeling various phenomena in physics, engineering, and finance. Traditional numerical methods for solving PDEs, such as finite element and finite difference methods, often face limitations when applied to high-dimensional and complex systems. In recent years, deep learning has emerged as a promising alternative for approximating solutions to PDEs, offering potential improvements in both efficiency and scalability. This paper provides a comprehensive review of the literature on deep learning-based methods for solving PDEs, focusing on key approaches such as Physics-Informed Neural Networks (PINNs), deep Galerkin methods, and neural operators. These methods leverage the expressiveness of neural networks to capture underlying physics while avoiding the curse of dimensionality associated with classical techniques. We explore the theoretical foundations, advantages, and limitations of these deep learning models, along with their applications in diverse fields like fluid dynamics, quantum mechanics, and financial modeling. Additionally, this review examines recent advancements in hybrid models that combine traditional numerical methods with deep learning approaches to enhance accuracy and stability. Through this review, we highlight key trends and open challenges in the field, paving the way for future research at the intersection of deep learning and computational mathematics.

Numerical Solution of Burgers Equation Using Finite Difference Methods: Analysis of Shock Waves in Aircraft Dynamics

In this research, the Lax, the Upwind, and the MacCormack finite difference methods are applied to the experimental solving of the one-dimensional (1D) unsteady Burger's Equation, a Hyperbolic Partial Differential Equation. These three numerical analysis-solving methods are implemented for accurate modeling of shock wave behavior high-speed flows that are necessary for aerospace engineering design. This research analysis proves that the MacCormack technique is the one that treats the differential equations with second-order accuracy. This method is quite preferred when it comes to numerical simulations because of its advanced level of accuracy. Although the Upwind and Lax methods are slightly less accurate, they show the development of shock waves that give visualizations to better understand the flow dynamics. Also, in this study, the impact of varying viscosity coefficients on fluid flow characteristics by using the lax (a numerical method for solving the viscous Burgers equation) is investigated. This identification of the phenomenon sheds light on the behavior of boundary layers, which, in turn, can be used to improve the design of high-speed vehicles and lead to a greater understanding of the area of ​​fluid dynamics.

Theoretical Analysis of a Novel Rock Wall to Limit Heating Demands in Historical Buildings

In the near future, the building sector will continue to absorb the greatest share of primary energy worldwide. It is necessary to find innovative solutions that promote energy efficiency through renovation measures, especially in historical buildings, for which refurbishment is constrained by several issues. In this study, we propose a novel Trombe Wall configuration that is easily integrable and based on a rock wall made of caged stone to use as a thermal accumulator. The system was investigated preliminarily using a transient Finite Difference Method (FDM) code to analyse the temperature field inside the rock wall. Successively, FDM results were employed as input data in TRNSYS simulations to determine the savings achievable in thermal heating requirements. The results demonstrated that the proposed solution, in the considered climate and on a reference historic building, can produce monthly heating savings varying between 26% and 85%. So, the rock wall results in a reliable solution for buildings in which refurbishment is difficult, allowing for preserving aesthetic features and improving energy efficiency by rationally using solar radiation.