Finite Difference Method Research Articles

Thermotunable mid-infrared metamaterial absorption material based on combined hollow cylindrical VO2 structure

We have designed a metamaterial absorption material based on the VO2 phase transition effect. The aim is to realize thermally tunable broadband absorption in the mid-infrared band. The metamaterial absorption material is a three-layer structure. From top to bottom, the combined hollow cylindrical VO2 resonant layer, the SiO2 dielectric layer and the Ti substrate. In this work, we perform numerical simulations using the three-dimensional time-domain finite element difference method. The simulation shows that the absorption material's average absorption intensity is adjustable by temperature between 0.09 and 0.95. At a temperature of 342 K, the absorption material has an absorption bandwidth of 17.3 μm (6.43 μm to 23.73 μm) with an absorption rate above 90%. And the average absorption in this band range is 94.95%. And this band covers the wavelength of the atmospheric transparency window (8 μm∼14 μm), which suggests that our absorption material has great application prospects. In this work, we first analyze the electromagnetic field at the absorption material's resonant wavelength and explain the absorption material's absorption mechanism. Then, we investigated the absorption effect of the absorption material under the VO2 structure with different temperatures. And we explain temperature's effect on the absorption material's absorption effect by analyzing the electric field's change on the absorption material's surface at different temperatures for the same wavelength. The effect of the absorption material structural parameters on the absorption material absorption performance to determine the optimal structural parameters was then investigated. Finally, we investigate the absorption curves of the absorption material under different columnar VO2 layers and demonstrate the innovative and unique nature of the designed absorption material. The absorption material we designed has a simple structure and is easy to configure. And the absorption material has high average absorption rate and a wide absorption bandwidth. We therefore believe that the metamaterial absorption material has great application prospects in optoelectronic detection, infrared imaging and infrared stealth.

Numerical Investigation of Bedding Rock Slope Potential Failure Modes and Triggering Factors: A Case Study of a Bridge Anchorage Excavated Foundation Pit Slope

The analysis of slope failure modes is essential for understanding slope stability. This study investigated the failure modes and triggering factors of a rock slope using the limit equilibrium method, finite differences method, and exploratory factor analysis. First, the limit equilibrium method was used to identify potential sliding surfaces. Then, the finite differences method was employed to study deformation and failure features in a slope. Stability factors were calculated considering specific conditions such as rainfall, prestressing loss, and earthquakes using the strength reduction method. Finally, exploratory factor analysis was utilized to identify the triggering factors of each failure mode. The results revealed that failure modes were categorized into two types based on the positions of the sliding surface. The main triggering factors for Failure Mode 1 were rainfall and prestress loss, while for Failure Mode 2 they were earthquake loading and prestress loss. This study offers a comprehensive exploration of potential failure modes and their triggering factors from mechanical and statistical perspectives, enriching our understanding of potential failure modes in rock slopes.

3D modelling of ground magnetic-source airborne TEM data for anisotropic media based on the rational Krylov method

With the rapid development of the transient electromagnetic method (TEM), the numerical simulations of three-dimensional anisotropic media have garnered attention. This study used the rational Krylov method to realise the three-dimensional modelling of the ground magnetic-source airborne transient electromagnetic data for arbitrary anisotropic media. First, based on Maxwell's equations, a time-domain electromagnetic equation was derived for anisotropic media. The finite difference method was employed to discretise the equation, and the electromagnetic field was then expressed as the product of an exponential matrix and a vector related to the source. Subsequently, the rational Arnoldi algorithm was adopted to construct the rational Krylov subspace orthogonal basis vectors, and the electromagnetic fields at different times were obtained based on a model reduction strategy. The accuracy of the proposed algorithm was verified through the comparisons of its results with those of other software packages. We calculated the response for different anisotropic parameters of the typical three-dimensional anisotropic models, and analysed the influence of the anisotropic media on the ground magnetic-source airborne transient electromagnetic data. The proposed three-dimensional forward TEM algorithm provides important technical support for identifying the anisotropic geological bodies and lays the foundation for further inversion research.

Thermo-pressure coupling model for gas hydrate depressurization exploitation in South China Sea

The sediment type in the “Shenhu” sea area of the South China Sea is mud sediment, with poor reservoir physical conditions, and the seabed is loose with a small pressure window. These conditions pose heightened safety challenges for the extraction of hydrates in this region. This study develops a multiphase flow model that accounts for the endothermic decomposition of hydrates and employs the finite difference method for its solution. Utilizing this model, the multiphase flow characteristics during depressurization extraction at a specific well in the “Shenhu” area are investigated. Building on this, the study analyses the impact of varying pump rates and geothermal gradients on multiphase flow properties, hydrate yield, and engineering safety. Based on the analytical findings, recommendations are proposed to balance hydrate production with engineering safety, effectively mitigating potential engineering accidents during depressurization extraction. The outcomes of this research offer technical guidance for the commercial exploitation of hydrates in the “Shenhu” area of the South China Sea, laying a foundation for future related studies.

Multiscale-mechanical analysis on self-healing microcapsules under asphalt pavement

To investigate the stress state of self-healing microcapsules in asphalt pavement during their operational phase, this study utilizes the finite difference method to establish a macroscopic structural model of asphalt pavement and a multi-scale model of representative volume elements (RVE)containing self-healing microcapsules. It systematically analyzes the effects of microcapsule content, capsule wall modulus, and wall thickness on the its mechanical performance under different asphalt pavement. Experimental findings indicate that an increase in microcapsule content initially increases and subsequently decreases the maximum principal stress and shear stress of the capsule wall, suggesting an optimal content range for achieving optimal mechanical response. Moreover, an increase in capsule wall modulus results in increased maximum principal stress, minimum principal stress, and maximum shear stress borne by the capsule wall, underscoring the importance of capsule wall modulus for the mechanical properties of microcapsules. The impact of increasing wall thickness on minimum principal stress shows an initial increase followed by a decrease, emphasizing the critical role of appropriate wall thickness selection in facilitating prompt response of microcapsules to pavement cracks. This research provides essential theoretical foundations for the design and application of microcapsules, thereby enhancing the self-healing capability of asphalt pavement.

Squeezing hydromagnetic nanofluid flow between two parallel plates with joule heating effects and induced magnetic field

Over the years, cooling high energy devices has become a major challenge to most industries. Thus, squeezing hydromagnetic nanofluid flow between two parallel plates has garnered alot of attention. In the present study, an unsteady, incompressible, hydromagnetic gold and water nanofluid flow squeezed between two parallel plates with Joule heating and a magnetic field that is induced is considered. The model considers the impacts of Joule heating,Viscous dissipation and induced magnetic field. In contrast with other studies which may have focussed on only one or two of these aspects this comprehensive approach provides a more accurate depiction of the flow pattern. Employing the finite difference method, the flow's non-linear differential equations are numerically solved and then implemented in MATLAB. The effect of changing various parameters on flow variables such as velocity, magnetic induction and thermal profiles are determined and the results obtained are presented graphically. Increasing magnetic parameter and Prandtl magnetic number caused a decline in temperature,velocity and magnetic induction profiles. Further, increasing the Eckert number and Joule heating parameter lead to an increase in temperature profiles. This findings have a direct impact on high-energy cooling devices which frequently experience heat dissipation. Additionally, engineers can ensure optimal performance and the longevity of plate heat exchangers in industries.

Numerical Analysis of Low-Enthalpy Deep Geothermal Energy Extraction Using a Novel Gravity Heat Pipe Design

Geothermal energy, derived from the Earth’s internal heat, can be harnessed due to the geothermal gradient between the Earth’s interior and its surface. This heat, sustained by radiogenic decay, varies across regions, and is highest near volcanic areas. In 2020, 108 countries utilised geothermal energy, with an installed capacity of 15,950 MWe for electricity and 107,727 MWt for direct use in 2019. Low-enthalpy sources require binary systems for power production. Open-loop systems face issues like scaling, difficult water treatment, and potential seismicity, while closed-loop systems, using abandoned petroleum or gas wells, reduce costs and environmental impacts greatly. The novel geothermal gravity heat pipe (GGHP) design eliminates parasitic power consumption by using hydrostatic pressure for fluid circulation. Implemented in an abandoned well in north-east (NE) Slovenia, the GGHP uses a numerical finite difference method to model heat flow. The system vaporises the working fluid in the borehole, condenses it at the surface, and uses gravitational flow for circulation, maintaining efficient heat extraction. The model predicts that continuous maximum capacity extraction depletes usable heat rapidly. Future work will explore sustainable heat extraction and potential discontinuous operation for improved efficiency.

Triangular finite differences using bivariate Lagrange polynomials with applications to elliptic equations

This paper proposes finite-difference schemes based on triangular stencils to approximate partial derivatives using bivariate Lagrange polynomials. We use first-order partial derivative approximations on triangles to introduce a novel hexagonal scheme for the second-order partial derivative on any rotated parallelogram grid. Numerical analysis of the local truncation errors shows that first-order partial derivative approximations depend strongly on the triangle vertices getting at least a first-order method. On the other hand, we prove that the proposed hexagonal scheme is always second-order accurate. Simulations performed at different triangular configurations reveal that numerical errors agree with our theoretical results. Results demonstrate that the proposed method is second-order accurate for the Poisson and Helmholtz equation. Furthermore, this paper shows that the hexagonal scheme with equilateral triangles results in a fourth-order accurate method to the Laplace equation. Finally, we study two-dimensional elliptic differential equations on different triangular grids and domains.

Numerical simulation of one-dimensional perovskite solar cell model

Perovskite solar cell (PSC) is one of the advanced third-generation solar cells that have rapid efficiencies but instability issues in terms of air, moisture, and UV light sensitivity becomes a barrier to commercialization. The instability issue is due to the charge accumulation at the interface of the PSC reducing its efficiency. This research focuses on the operation of PSCs through a one-dimensional (1D) drift-diffusion model for planar heterojunction PSCs. The model also accounts for the electric potential by Poisson’s equation, the ion generation, and the recombination rate. The method of lines technique is applied to solve the model for the perovskite layer numerically using the finite difference method which is then solved forward in time using the ‘ode15s’ solver in MATLAB. It is highlighted that the comparison with the experimental data from the reference shows good agreement. The effect of parameter thickness variation of the perovskite layer upon the efficiency of PSCs is analyzed. The result shows that the best efficiency obtained is 19.77% obtained at thickness 0.25 𝜇𝑚. The results may prove useful for a guideline of the cell thickness that predicts the perovskites cell performance.

Optical absorption properties in multi-layer InGaN/GaN core-shell quantum dots: The influences of ternary mixed crystal effect and hydrogen-like impurity

Based on the effective mass approximation and the compact density matrix theory, the electronic states and optical absorption properties of intersubband transitions between the conduction-band energy levels in zinc-blende InGaN/GaN/InGaN/GaN core-shell quantum dots (CSQDs) are investigated by using the finite difference method. The effects of hydrogen-like impurity and In component on the wave functions, transition energies, and optical absorption properties are discussed in detail. The results show that hydrogen-like impurity can cause the peak positions of optical absorption coefficients (OACs) and refractive index changes (RICs) to move to high energy area, and the peak values of OACs rise. Increasing In component of InGaN crystal in the core region of a CSQD has the same effect. But an opposite tendency will be found when increasing In component in the shell-well of a CSQD, i.e. optical absorption peaks move to low energy area and their values decrease. Comparing to the shell-well, the core region has more powerful confinement effect on electrons. So the better absorption properties are more likely to be obtained for this type of multi-layer InGaN/GaN CSQDs when In component in the core is larger than that in the shell-well.

Unconditionally optimal high-order weighted compact nonlinear schemes with sharing function for Euler equations

Weighted compact nonlinear schemes (WCNS) are a type of high-order shock-capturing finite difference schemes commonly used in various applications. Their spatial discretizations involve a nonlinear interpolation step and a linear difference step. However, the nonlinear interpolation step requires significantly more computational resources compared to the linear difference step. Therefore, simplifying the interpolation step is an effective way to improve the efficiency of these schemes. In this paper, we propose a new approach to construct WCNS schemes based on the sharing function for Euler equations. This approach uses a set of common nonlinear interpolation weights for different components, resulting in a significant improvement in efficiency compared to the original WCNS schemes that use different sets of weights for each component. To ensure accuracy at critical points of any orders, we employ nonlinear weights that guarantee unconditionally optimal high order. Additionally, this new approach may reduce oscillations caused by non-characteristic interpolation. Furthermore, we also develop a new shock detector using the sharing function, enabling us to employ characteristic interpolation for nonsmooth regions and linear component-wise interpolation for the rest. We validate the proposed schemes based on the sharing function through numerical examples of one- and two-dimensional Euler equations, demonstrating their effectiveness in terms of shock-capturing capability and efficiency.

Numerical simulation for an initial-boundary value problem of time-fractional Klein-Gordon equations

This paper mainly presents numerical solutions to an initial-boundary value problem of the time-fractional Klein-Gordon equations. We developed a numerical scheme with the help of the finite difference methods and the predictor-corrector methods to find numerical solutions of the considered problems. The proposed scheme is based on discretizing the considered problems with respect to spatial and temporal domains. Numerical results are derived for some illustrative problems, and the outputs are compared with the exact solution in the integer order case. The solution behavior and 3D graphics of the discussed problems are demonstrated using the proposed scheme. Finally, the proposed scheme, which does not require solving large systems of linear equations, can be extended and modified to handle other classes of time-fractional PDEs.

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-1σ formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme.

Distance-regularized level set inversion of magnetic data

Inversion is one of the key and difficult issues in potential field data processing and interpretation, and high-precision inversion has always been a popular research topic. We propose and develop the distance-regularized level set inversion of magnetic data (DRLSI-M) method. A double-well function is introduced in the objective function of the level set inversion, we take advantage of its mathematical properties to minimize the gradient of the level set function so that it reaches a minimum value at points 0 and 1. As a result, the distance-regularized term not only replaces the costly reinitialization process so that the level set function maintains a signed distance property but also stabilizes the evolution of the level set function. When solving for the minimum value of the objective function, we transform the optimization problem into an initial-boundary value problem and solve it with the finite difference method. The distance-regularized level set inversion method and the reinitialization level set inversion method are tested using three models: a single-level-set double-inclined plate model, a double-level-set double-inclined plate model, and a multiple-level-set three-cuboid model. Through the model tests, the soundness of the proposed method is verified, and compared with the reinitialization level set inversion method, the proposed method avoids the limitations such as low efficiency and instability caused by the reinitialization of the level set inversion method. Furthermore, we apply the proposed method for the inversion of aeromagnetic data from the Jinchuan copper-nickel mine. The results are consistent with the known geological information, thus validating the practicality and effectiveness of the new method. The distance-regularized level set inversion method provides a theoretical basis for deep mine exploration.

An investigation of heat transfer and optimization of entropy in bio-convective flow of Eyring-Powell nanomaterial with gyrotactic microorganisms

Entropy generation in nanofluid flows is a critical parameter that influences the performance, efficiency, and sustainability of thermal systems. Understanding and optimizing entropy creation can lead to noteworthy advancements in numerous engineering applications, from renewable energy systems to industrial processes and biomedical equipments. The purpose of this article is to examine the entropy produced in the bioconvection flow of Eyring-Powell nanomaterial with gyrotactic microorganisms. The mathematical equations representing the flow are modeled considering the flow towards a porous surface of a cylinder. Together with the effects of the governing parameters, the mass and heat transfer components of the problem are discussed. The thermal effects of different natures are used in a new way. Overall, entropy creation is designed with regard to the second law of thermodynamics. Activation energy, chemical reaction, thermal radiation, and internal friction force effects are accounted for the development of the mathematical model. Coupled non-linear dimensional equations have been altered into ordinary differential equations (ODEs) and then treated using the MATLAB Finite Difference Method (FDM). Consequence of diverse sundry parameters on entropy production, thermal field, mass concentration profile, Bejan quantity, and motile density of microorganisms is deliberated via plots. Through tables, engineering quantities are analyzed. According to the findings, the velocity profile rises as the curvature and Eyring-Powell fluid parameters rise, while it falls when the magnetic parameter is enhanced. Additionally, it is noted that as the curvature variable enhances, the rate of heat transfer and skin friction coefficient decays.

Calibrating resistance factors of pile groups based on individual pile proof load tests

Pile load tests have been utilized to reduce the uncertainty of pile resistance, thus leading to a higher resistance factor used in the Load and Resistance Factor Design (LRFD). Previous studies have primarily focused on calibrating resistance factors for single piles based on load tests. This calibration hinges upon the resistance bias factor of single piles, defined as the ratio of measured resistance to predicted resistance. Due to the redundancy in the pile group system, it is conventionally assumed that if the individual piles within the group achieve a lower reliability index (e.g., 2.0–2.5), the pile group as a whole attains the target reliability index of 3. However, the approach is empirical as it does not consider system redundancy directly. Moreover, this empirical approach disregards the correlation between resistance bias factors of individual piles, which is inherently influenced by the spatial variability of soils. In this study, the random finite difference method (RFDM) is employed to evaluate the correlation between resistance bias factors of individual piles in spatially variable soils. The resultant correlation matrix is subsequentially employed in Bayes’ theorem to update resistance bias factors using individual pile load test results and their corresponding test locations. The updated resistance bias factors are then used for the direct calibration of resistance factors for pile groups within the framework of LRFD. A pile group subject to vertical loading in undrained clays is adopted for illustration. Comparative analyses between the proposed approach and the empirical approach demonstrate that the latter tends to overestimate the resistance factor. Furthermore, the proposed approach enables the determination of optimal locations for conducting subsequent load tests based on previous test results.

An Efficient Convergent Approach for Difference Delayed Reaction-Diffusion Equations

It is usually not possible to solve partial differential equations, especially the delay type, with analytical methods. Therefore, in this article, we present an efficient method for solving differential equations of the difference delayed reaction-diffusion type, which can be generalized to other delayed partial differential equations. In the proposed approach, we first convert the delayed equation into an equivalent non-delayed equation by inserting the corresponding delay function with an effective technique. Then, using a pseudo-spectral method, we discretize the obtained equation in the Legendre-Gauss-Lobatto collocation points and present an algebraic system with an equal number of equations and unknowns which can be solved by quasi-Newton methods such as Levenderg-Marquardt algorithm. The approximate solutions can be obtained with exponential accuracy. The convergence analysis of the method is fully discussed and four examples are presented to evaluate the results and compare with one of the conventional methods used to solve partial differential equations, that is, the compact finite difference method.

Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms

This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reaction-diffusion system incorporating gradient terms, employing numerical finite difference approximations. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed time-stepping. The numerical findings obtained demonstrate that the blow-up time sequence exhibits convergence as the space step decreases. Moreover, thenumerical orders of convergence for the blow-up time goes well with the theoretical orders observed in the numerical solutions.

High order conservative finite difference WENO scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations play an important role in the high-energy-density-physics fields, such as astrophysics and inertial confinement fusion (ICF). It describes the interaction between radiation and high-energy-density plasmas including electron and ion in the assumption that radiation, electron and ion are in their own equilibrium state, which means they can be characterized by their own temperatures. The 3-T RH system consists of the density, momentum and three internal energy (electron, ion and radiation) equations. In this paper, we propose a high order conservative finite difference weighted essentially non-oscillatory (WENO) scheme solving one-dimensional (1D) and two-dimensional (2D) 3-T RH equations respectively. Following our previous paper [7], we introduce the three new energy variables, and then design a finite difference scheme with both the conservative property and arbitrary high order accuracy. Based on the WENO interpolation and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a class of fifth order conservative finite difference schemes in space and third order in time. Compared with the Lagrangian method we proposed in [7], which can only reach second order accuracy for 2D 3-T RH equations if straight-line edged meshes are used, the finite difference scheme can be easily designed to arbitrary high order accuracy for multi-dimensional 3-T RH equations. The finite difference formulation is also much less expensive in multi-dimensions than finite volume schemes used in [7]. Furthermore, our method can handle fluids with large deformation easily. Numerous 1D and 2D numerical examples are presented to verify the desired properties of the high order finite difference WENO schemes such as high order accuracy, non-oscillation, conservation and adaptation to severely distorted single-material radiation hydrodynamics problems.

A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

In this paper, we consider the numerical method for generalized Allen–Cahn equation with nonlinear mobility and convection term. We propose a linear second-order finite difference scheme which preserves the discrete maximum bound principle (MBP). The scheme is discretized by stabilized Crank–Nicolson in time, upwind scheme for convection term and central-difference scheme for diffusion term. We show that the proposed scheme preserves the discrete MBP under some constraints on temporal step size and stabilizing parameter. Optimal L∞-error estimate is obtained for our scheme. Several numerical experiments are performed to validate our theoretical results.