Finite Difference Method Research Articles

Theoretical model and parameter sensitivity analysis of the filter cake formation during slurry shield tunnelling

The filter cake formation during slurry shield tunneling in high permeability soil layers is complex and difficult to monitor. There is no reliable method to evaluate the mechanical characteristics during the filter cake formation. A theoretical model is proposed to describe the filter cake formation and development in this study. The finite difference method is used to simulate the whole filter cake formation process. In addition, the influence of the key bentonite slurry parameters and of the deep infiltration on the filter cake formation is investigated based on the proposed model. The results show that the excess pore water pressure within the filter cake is not uniformly distributed. The slurry support effect is manifested by a pressure drop generated on both sides of the filter cake. The development process of the filter cake can be summarized into three stages: prior infiltration filling of the filter cake, accumulation and thickening of the new filter cake, and compression and consolidation of the prior filter cake. The slurry deep infiltration results reflected by the membrane specific resistance have the most significant effect on the filter cake development. Compared with the slurry concentration, the viscosity has a greater influence on the formation and final thickness of the filter cake.

A hybrid discrete exterior calculus and finite difference method for anelastic convection in spherical shells

The present work develops, verifies, and benchmarks a hybrid discrete exterior calculus and finite difference (DEC-FD) method for density-stratified thermal convection in spherical shells. Discrete exterior calculus (DEC) is notable for its coordinate independence and structure preservation properties. The hybrid DEC-FD method for Boussinesq convection has been developed by Mantravadi et al. (2023). Motivated by astrophysics problems, we extend this method assuming anelastic convection, which retains density stratification; this has been widely used for decades to understand thermal convection in stars and giant planets. In the present work, the governing equations are splitted into surface and radial components and discrete anelastic equations are derived by replacing spherical surface operators with DEC and radial operators with FD operators. The novel feature of this work is the discretization of anelastic equations with the DEC-FD method and the assessment of a hybrid solver for density-stratified thermal convection in spherical shells. The discretized anelastic equations are verified using the method of manufactured solution (MMS). We performed a series of three-dimensional convection simulations in a spherical shell geometry and examined the effect of density ratio on convective flow structures and energy dynamics. The present observations are in agreement with the benchmark models.

A Finite Difference Scheme for the Large-Deflection Analysis of Non-Prismatic Cantilever Beams

A Finite Difference Scheme for the Large-Deflection Analysis of Non-Prismatic Cantilever Beams

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

Re-constructing the River Bed Using the Streamline-Generation Method

River bed reconstruction plays an essential part in supporting the hydrodynamic simulation and understanding the morphological processes of a river. The streamlines can be solved using Laplace equations. The equation is first numerically solved in a computational environment and then adapted to the whole considered physical field to solve the resulting streamlines in a physical domain. One of the goals of this research is to determine the bottom line of the riverbed, and doing this through a field survey is very expensive, the methods presented in this research help a lot in reducing costs. By determining the concave line, the path of the flood in the river bed is determined, so one of the practical achievements and efficiencies of this research is flood trending at a low cost. In the present study, the bottom line is determined for the meandering Qinhe River, a distributary of the Yellow River, China, and Gaz River, located in Khuzestan Province, Iran. The method is based on the following steps:• Reconstruction of 2D river based on the Laplace equation.• Use the Finite Element Method to solve streamlines in the physical domain.• Use the Finite Difference Method to solve streamlines in the physical domain.

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

Two Decoupled and Linearized Block-Centered Finite Difference Methods for the Nonlinear Symmetric Regularized Long Wave Equation

Two Decoupled and Linearized Block-Centered Finite Difference Methods for the Nonlinear Symmetric Regularized Long Wave Equation

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

Finite element method for fractional order parabolic obstacle problem with nonlinear source term

This paper considers the finite element method for a fractional order parabolic obstacle problem in time with a nonlinear source term. An error estimation of the spatially semi-discrete problem is discussed. A fully discrete problem is proposed by introducing the finite difference approximation for the time variable. The L2-unconditional stability and convergence of the approximate solution have been proven.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

A future scenario earthquake and ground motion hazards for Kathmandu, Nepal

Since Nepal is an earthquake-prone country due to the collision of the Indian and Eurasian plates, it is crucial for its capital of Kathmandu to evaluate ground motion hazards from a future large earthquake. For this purpose, we constructed a realistic scenario earthquake with realistic rupture parameters in a likely location. To obtain the location, a new distribution of the rupture zones of large historical earthquakes along the Main Himalayan Thrust (MHT) was obtained by integrating the distribution from a previous study and the results of trench surveys. Global Navigation Satellite System observations indicated that the plate boundary is strongly coupled from the southern boundary of the MHT to a depth of approximately 10 km and there is almost no lateral change in the coupling. This implies that all regions along the MHT have similar rates of strain increase. Therefore, it is most probable that the rupture zone of the oldest previous event will rupture as a scenario earthquake. In the new distribution, the 1255 earthquake is the oldest. However, large earthquakes occurred in 1934 and 2015 within its rupture zone. Thus, we adopted the area obtained by removing the 1934 and 2015 rupture zones from the western part of the 1255 rupture zone. The relationship between the rupture area size and seismic moment of the 2015 earthquake lies between the scaling formulas for crustal earthquakes and plate-boundary earthquakes, but is closer to the former. Therefore, using this and the scheme for characterized source models, we determined realistic rupture parameters. We then simulated broadband ground motions in Kathmandu using these rupture parameters, our 3-D velocity structure models, and a hybrid method combining the finite-difference method and the stochastic Green’s function method. We obtained the peak ground accelerations (PGAs) of simulated ground motions, and calculated the seismic intensities in the Modified Mercalli intensity scale from the PGAs as indexes of hazards for Kathmandu. Intensities IX coincide with the center of the Kathmandu Valley, and intensities VIII and VII are found in the area surrounded by the sedimentary boundary and the southernmost part of the valley.Graphical abstract

A quantitative method of calculating transient nonlinear heat partition coefficient between clutch friction discs with deep learning

In the current clutch temperature field study, the generally used constant heat partition coefficient tends to overestimate the separator disc temperature and underestimate the friction disc temperature. Although some researchers have found the characteristics of the time-varying heat partition coefficient, a suitable method is still needed to apply it to temperature calculations. This study provides a quantitative method for the application of the transient nonlinear heat partition coefficient to temperature calculations. The finite difference method is adopted to figure out the time-varying curve of the heat partition coefficient by coupling the contact temperature of the friction components. The numerical results show that the heat partition coefficient is independent of rotation speed with three stages: initial value, rapid time-varying, and steady-state. Different from the analytical method, we apply a deep learning method to train the quantisation function to characterise these three stages, avoiding complex formula derivation. As a result, the quantitative function can characterise the time-varying heat partition coefficient accurately, with an average error of 0.19%, 3.05% and 0.62% for the inert, time-varying, and steady-state stages, respectively. In addition, the accuracy of applying the quantisation function in temperature simulation is verified by friction experiments, and the error is less than 8%. This is superior to the results of solving the temperature field by a constant heat partition coefficient.

Topological optimization in 3D-magnetostatics: development of adjoint methods using the equations of magnetic moments

Purpose The purpose of this paper is mainly to develop the adjoint method within the method of magnetic moment (MMM) and thus, to provide an efficient new way to solve topology optimization problems in magnetostatic to design 3D-magnetic circuits. Design/methodology/approach First, the MMM is recalled and the optimization design problem is reformulated as a partial derivative equation-constrained optimization problem where the constraint is the Maxwell equation in magnetostatic. From the Karush–Khun–Tucker optimality conditions, a new problem is derived which depends on a Lagrangian parameter. This problem is called the adjoint problem and the Lagrangian parameter is called the adjoint parameter. Thus, solving the direct and the adjoint problems, the values of the objective function as well as its gradient can be efficiently obtained. To obtain a topology optimization code, a semi isotropic material with penalization (SIMP) relaxed-penalization approach associated with an optimization based on gradient descent steps has been developed and used. Findings In this paper, the authors provide theoretical results which make it possible to compute the gradient via the continuous adjoint of the MMMs. A code was developed and it was validated by comparing it with a finite difference method. Thus, a topology optimization code associating this adjoint based gradient computations and SIMP penalization technique was developed and its efficiency was shown by solving a 3D design problem in magnetostatic. Research limitations/implications This research is limited to the design of systems in magnetostatic using the linearity of the materials. The simple examples, the authors provided, are just done to validate our theoretical results and some extensions of our topology optimization code have to be done to solve more interesting design cases. Originality/value The problem of design is a 3D magnetic circuit. The 2D optimization problems are well known and several methods of resolution have been introduced, but rare are the problems using the adjoint method in 3D. Moreover, the association with the MMMs has never been treated yet. The authors show in this paper that this association could provide gains in CPU time.

Compact Low Loss Ribbed Asymmetric Multimode Interference Power Splitter

Optical power splitters (OPSs) are utilized extensively in integrated photonic circuits, drawing significant interest in research on power splitters with adjustable splitting ratios. This paper introduces a compact, low-loss 1 × 2 asymmetric multimode interferometric (MMI) optical power splitter on a silicon-on-insulator (SOI) platform. The device is simulated using the finite difference method (FDM) and eigenmode expansion solver (EME). It is possible to attain various output power splitting ratios by making the geometry of the MMI central section asymmetric relative to the propagation axis. Six distinct optical power splitters are designed with unconventional splitting ratios in this paper, which substantiates that the device can achieve any power splitter ratios (PSRs) in the range of 95:5 to 50:50. The dimensions of the multimode section were established at 2.9 × (9.5–10.9) μm. Simulation results show a range of unique advantages of the device, including a low extra loss of less than 0.4 dB, good fabrication tolerance, and power splitting ratio fluctuation below 3% across the 1500 nm to 1600 nm wavelength span.

High-Q Multiband Narrowband Absorbers Based on Two-Dimensional Graphene Metamaterials

In this paper, an absorber with multi-band, tunable, high Q, and high sensitivity, based on terahertz periodic two-dimensional patterned graphene surface plasmon resonance (SPR), is proposed. The absorber consists of a bottom metal film separated by a periodically patterned graphene metamaterial structure and a SiO2 dielectric layer, where the patterned graphene layer is etched by “+” and “L” shapes and circles. It has simple structural features that can greatly simplify the fabrication process. We have analyzed the optical properties of a graphene surface plasmon perfect metamaterial absorber based on graphene in the terahertz region using the finite-difference method in time domain (FDTD). The results show that the absorber device exhibits three perfect absorption peaks in the terahertz bands of f1 = 1.55 THz, f2 = 4.19 THz, and f3 = 6.92 THz, with absorption rates as high as 98.70%, 99.63%, and 99.42%, respectively. By discussing the effects of parameters such as the geometrical dimensions of patterned graphene metamaterial structure “+” width W1, “L” width W2, circular width R, and the thickness of the dielectric layer on the absorption performance of absorber, as well as investigating the chemical potential and relaxation time of patterned-layer graphene material, it was found that the amplitude of the absorption peaks and the frequency of resonance of absorber devices can be dynamically adjusted. Finally, we simulated the spectra as the surrounding refractive index n varied to better evaluate the sensing performance of the structure, yielding structural sensitivities up to 382 GHz/RIU. Based on this study, we find that the results of our research will open new doors for the use of multi-band, tunable, polarization-independent metamaterial absorbers that are insensitive to large-angle oblique incidence.

Ultra-high Q-factor quasi-BIC BaTiO3 metasurface for electro-optic modulation

Metasurfaces play a crucial role in trapping electromagnetic waves with specific wavelengths, serving as a significant platform for enhancing light–matter interactions. In all kinds of dynamic modulation metasurfaces, electro-optic modulation metasurfaces have attracted much attention due to its advantages of fast, stable and high efficiency. In order to respond to the extremely weak refractive index change of the electro-optical effect of the materials, the metasurfaces are required to support optical signals with high Q values. The quasi-bound state in the continuum (Q-BIC) is often used to enhance the light-field modulation capability of metasurfaces and to improve the modulation sensitivity of electro-optic modulators due to its ability to generate high Q-factor resonances. However, the design of an electro-optic modulation metasurface that facilitates the application of voltage and achieves modulation efficiency of nearly 100% is still in urgent need of development. In this study, single-crystal BTO metasurfaces are modeled using finite-difference time-domain method, and the structural symmetry is broken to introduce a Q-BIC resonance to generate a high Q-factor optical signal of 2.45 × 104 for high-depth electro-optic modulation. By simulating an applied electric field of 143 V/mm on the metasurface, a slight refractive index change of BTO of 8 × 10−4 was produced, leading to an electro-optical intensity modulation depth of 100%. Furthermore, the nanostructure of the metasurface was carefully designed to facilitate nano-fabrication and voltage application, and it is ideal for the development of low-power, CMOS-compatible, and miniaturized electro-optic modulation devices. Although the results of this study are based on simulations, they provide a crucial theoretical basis and guidance for the realization of efficient and realistic design of dynamic metasurfaces.

Finite Difference Method for Infection Model of HPV with Cervical Cancer under Caputo Operator

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. The model has two equilibrium points: the disease-free point and the endemic point. The examination of the system’s local and global stability is provided in terms of the basic reproductive number Rp°. The global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. The results demonstrate that in the infection model, if Rp°<1, then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst Rp°>1, the endemic equilibrium is considered to exist. Simulations are implemented via a finite difference method with Grünwald-Letnikov discretization approach for Caputo derivative operator to define how changes in parameters impact the dynamic behavior of the system using Matlab.

Nonlinear Dynamic Properties of Rigid–Elastic–Liquid Coupled Ball Bearings Considering External Load Excitation

In this study, a novel dynamic model of a ball bearing based on rigid–elastic–liquid (REL) coupling structure considering external load excitation is established, and the dynamic response of its outer ring under external load excitation is examined. First, the differential equation of the journal motion (i.e. the outer ring) is derived based on Newton’s law, and the oil film force induced by the REL coupling structure is determined by using the central finite difference method, planar bending theory of thin ring, and Simpson’s rule. Further, the fourth and fifth-order variable step length Runge–Kutta method is used to solve the kinematic differential equation. Second, the dynamic characteristics of the outer ring of the bearing under different speeds and elastic support structural parameters are examined. Finally, the feasibility of the theoretical model is verified through experiments. The results show that compared with the traditional squeeze film damper (SFD) system, the ball bearing with REL coupling structure is more stable, and the vibration damping effect is better under higher bearing speed, lower squirrel cage stiffness, and smaller elastic ring flexibility coefficient. The above findings provide useful insights into the dynamic characteristics of REL coupled ball bearings under unbalanced excitation, which can serve as a reference for the optimization design of such ball bearings.

Threshold current density for diffusion-controlled stability of electrolytic surface nanobubbles

Understanding the stability mechanism of surface micro/nanobubbles adhered to gas-evolving electrodes is essential for improving the efficiency of water electrolysis, which is known to be hindered by the bubble coverage on electrodes. Using molecular simulations, the diffusion-controlled evolution of single electrolytic nanobubbles on wettability-patterned nanoelectrodes is investigated. These nanoelectrodes feature hydrophobic islands as preferential nucleation sites and allow the growth of nanobubbles in the pinning mode. In these simulations, a threshold current density distinguishing stable nanobubbles from unstable nanobubbles is found. When the current density remains below the threshold value, nucleated nanobubbles grow to their equilibrium states, maintaining their nanoscopic size. However, for the current density above the threshold value, nanobubbles undergo unlimited growth and can eventually detach due to buoyancy. Increasing the pinning length of nanobubbles increases the degree of nanobubble instability. By connecting the current density with the local gas oversaturation, an extension of the stability theory for surface nanobubbles [Lohse and Zhang, Phys. Rev. E 91, 031003(R) (2015)] accurately predicts the nanobubble behavior found in molecular simulations, including equilibrium contact angles and the threshold current density. For larger systems that are not accessible to molecular simulations, continuum numerical simulations with the finite difference method combined with the immersed boundary method are performed, again demonstrating good agreement between numerics and theories.