Finite Difference Method Research Articles

X-SLAM: Scalable Dense SLAM for Task-aware Optimization using CSFD

We present X-SLAM, a real-time dense differentiable SLAM system that leverages the complex-step finite difference (CSFD) method for efficient calculation of numerical derivatives, bypassing the need for a large-scale computational graph. The key to our approach is treating the SLAM process as a differentiable function, enabling the calculation of the derivatives of important SLAM parameters through Taylor series expansion within the complex domain. Our system allows for the real-time calculation of not just the gradient, but also higher-order differentiation. This facilitates the use of high-order optimizers to achieve better accuracy and faster convergence. Building on X-SLAM, we implemented end-to-end optimization frameworks for two important tasks: camera relocalization in wide outdoor scenes and active robotic scanning in complex indoor environments. Comprehensive evaluations on public benchmarks and intricate real scenes underscore the improvements in the accuracy of camera relocalization and the efficiency of robotic navigation achieved through our task-aware optimization. The code and data are available at https://gapszju.github.io/X-SLAM.

Multi-layer non-hydrostatic free-surface flow model with kinematic seafloor for seismic tsunami generation

Earthquake induced tsunamis pose devastating threat to coastal communities worldwide. Accurate description of tsunami generation by kinematic fault rupture is of importance to investigate earthquake events and evaluate tsunami impacts. The paper develops a multi-layer non-hydrostatic model with a kinematic bottom boundary condition and conducts comprehensive validation to examine the model’s capability in resolving seismic tsunami generation. The two-dimensional governing equations of the multi-layer non-hydrostatic free-surface flow system equipped with kinematic seafloor displacement is first introduced in Cartesian coordinates. A combined finite difference and finite volume scheme is utilized to discretize the governing equations with flow variables arranged on a staggered Arakawa C-grid. Application of pressure correction technique to the discretized formulations yields Poisson-type equations from which the non-hydrostatic pressure is solved for the next time step to complete the temporal integration. Based on existing analytical solutions, four groups of tsunami generation cases considering broad ranges of source parameters, including horizontal scale, rise time, and rupture velocity, are designed to demonstrate performance of the proposed non-hydrostatic model with one-, two-, and three-layers as well as the hydrostatic one. Comparison of 59 generation cases including extreme scenarios indicates the non-hydrostatic model performs better than the hydrostatic model in reproducing the entire waveform and predicting the maximum wave amplitude. High modeling accuracy can be achieved through incorporation of more layers. The proposed multi-layer non-hydrostatic model is a powerful tool for investigating earthquake source mechanisms and evaluating coastal tsunami hazards.

Investigation of an absorption plasmonic electro-optical modulator based on the free carrier dispersion effect in the aluminum doped zinc oxide layer

A plasmonic electro-optical modulator, which is based on the free carrier dispersion effect, has been introduced here. The structure of the proposed modulator is a substrate of fused silica/gold/aluminum doped zinc oxide/hafnium dioxide/gold/air. The free carrier dispersion effect occurs in the aluminum doped zinc oxide (AZO) layer. As the electrical permittivity of the AZO layer is near zero (the epsilon near zero effect) at a wavelength of 1.55 μm, the amplitude of an electrical field is high in this layer, therefore, the modulator is highly sensitive to the varying refractive index of this layer. By applying a voltage to two gold layers, the electrical charge density changes in the AZO layer. Therefore, the real and imaginary parts of the refractive index have been changed, which led to a change in the absorption of the modulator. In order to obtain the charge density distribution in the AZO layer, the Poisson equation is solved by using the finite difference method. To investigate modulator absorption, the Nelder-Mead method is implemented in order to solve the dispersion equation numerically. Finally, the magnetic field, the electric field, and the time average of the Poynting vector have been given by using the least squares approximation method.

Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.

Theoretical Evaluation of Lubrication Performance of Thrust-Type Foil Bearings in Liquid Nitrogen

The development of reusable liquid rocket turbopumps has gradually highlighted the disadvantages of rolling bearings, particularly the contradiction between long service life and high rotational speed. It is critical to explore a feasible bearing scheme offering a long wear life and high stability to replace the existing rolling bearings. In this study, liquid nitrogen is adopted to simulate the ultra-low temperature environment of liquid rocket turbopumps, and theoretical evaluations of the lubrication performance of thrust-type foil bearings in liquid nitrogen are conducted. A link-spring model for the bump foil structure and a thin-plate finite element model for the top foil structure are established. The static and dynamic characteristics of the bearings are analyzed using methods including the finite difference method, the Newton–Raphson iteration method, and the finite element method. Detailed analysis includes the effects of factors such as rotational speed, fluid film thickness, thrust disk tilt angle, and the friction coefficient of the bump foil interface on the static and dynamic characteristics of thrust-type foil bearings. The research results indicate that thrust-type foil bearings have a good load-carrying capacity and low frictional power consumption. The adaptive deformation of the foil structure increases the fluid film thickness, preventing dry friction due to direct contact between the rotor journal and the bearing surface. When faced with thrust disk tilt, the direct translational stiffness and damping coefficient of the bearing do not undergo significant changes, ensuring system stability. Based on the results of this study, the exceptional performance characteristics of thrust-type foil bearings make them a promising alternative to rolling bearings for the development of reusable liquid rocket turbopumps.

Modeling time-dependent deformation in concrete: A fractional calculus method with finite element implementation

This study presents a novel numerical method to simulate the time-dependent behavior of concrete materials. The deformation response of both traditional and fractional calculus (FC) viscoelastic models is analyzed by the proposed method, in which models’ one-dimensional constitutive relationships are extended to the three-dimensional form. By applying numerical and finite difference methods based on Caputo-type fractional integration, the stress-strain behavior of the FC viscoelastic model is discretized over the time scale. This method exhibits broad application prospects, and can be employed to represent various forms of viscoelastic models. For the concrete material, suitable FC viscoelastic models are developed. To facilitate practical engineering applications of the proposed model, a numerical solution algorithm is implemented in the finite element (FE) analysis through the User-defined Material Mechanical Behavior (UMAT) interface of the commercial FE software ABAQUS. Finally, FE results are compared with the experimental results from several concrete creep tests. The consistency between the FE results and experimental data confirms the effectiveness of the proposed model in describing concrete behavior. The proposed method and model provide theoretical and numerical support for a more profound understanding and simulation of the time-dependent deformation of RC specimens in practical engineering applications.

Fractional non-Fourier modeling of laser drilling process

In this paper, a novel fractional non-Fourier model is employed to simulate the laser drilling process, addressing limitations inherent in classical heat conduction equations, including the well-known heat equation paradox associated with infinite heat propagation velocity. This model approach combines spatial approximation via the Meshless Local Petrov-Galerkin method with temporal approximation using the Grünwald-Letnikov finite difference scheme. The study assesses the impact of employing fractional orders, both constant and variable over time, on numerical results, and validates the model using experimental data.

A priori test of scale-similarity parameterizations for subgrid-scale fluxes in convective storms

Abstract This study evaluates a parameterization scheme for subgrid-scale (SGS) fluxes based on the scale-similarity assumption and employing a large-eddy simulation of an idealized backbuilding convective system. In this parameterization, the SGS fluxes are decomposed into the “Leonard term” which depends only on the resolved scale components, the “Reynolds term” which depends only on the SGS components, and the “cross term” which corresponds to the interaction between the resolved scale and SGS components. Assuming a linear relationship between the Leonard term and the Reynolds and cross terms, SGS fluxes are expressed as the product of an empirical coefficient and the Leonard term. The Leonard term reasonably represents the SGS flux derived by a smooth filter operation, including the counter-gradient vertical SGS transport of potential temperature, which cannot be represented by a traditional eddy-diffusivity model. The dependence of the empirical coefficient on filter width is also evaluated. This dependence is related mainly to the Reynolds term, the magnitude of which varies widely with filter width. The estimation based on the spectral decomposition of the Reynolds term explains the obtained dependence of the empirical coefficient for the vertical flux on filter width. In contrast, the variation of the empirical coefficient with filter width is not required to obtain the horizontal flux. For the parameterization of SGS fluxes in kilometer-scale models that use finite difference or volume methods, the Leonard term is expressed by the horizontal gradient of variables on a discrete grid. The Leonard term on a discrete grid also accurately represents the amplitude and spatial pattern of the SGS flux.

Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

Prediction of waves in tanks excited by ship motion and moving walls with a three-dimensional fully linear finite difference approach

Mitigation of waves in shipboard swimming pools excited by ship motions is a topic of current interest that has not yet been extensively investigated. Various damping mechanisms have been applied to mitigate waves excited in tanks, such as porous baffles or different types of internal baffles. However, most of these devices are installed in LNG tanks to prevent serious sloshing effects. Furthermore, these are passive devices and, hence, unable to control wave-induced excitations. Our focus was on the validation and verification of the three-dimensional (3D) fully linear Finite Difference Method (FDM) based on the previously developed method of Qi et al. (2024) for predicting waves in a transversely placed swimming pool excited by ship motions and piston-type actuators. We validated our FDM approach against comparative Computational Fluid Dynamics (CFD) simulations as well as experimental model test measurements. The CFD tools solved the Reynolds-averaged Navier-Stokes (RANS) equations, relied on an appropriate turbulence model and the Volume of Fluid (VOF) method to capture the free surface of the liquid in the model tank, and employed the overset technique to specify the motions of the active actuators. Our FDM considered ship-induced roll excitations as well as simultaneous roll and sway excitations, albeit only at frequencies outside the range of the pool's resonance frequency. For these periods, our approach accurately predicted not only the mitigated waves in the pool, but also the optimum amplitude of the actuators needed to mitigate such waves. Our FDM can be applied to obtain a preview of excited waves under various conditions, leveraging its principal advantage of extreme computational efficiency.© 2017 Elsevier Inc. All rights reserved.

A unified model for mechanical deformations of single stranded DNA-microbeam biosensors considering surface charge effects

A unified energy model for mechanical deformation of the microbeam included by single stranded DNA (ssDNA) adsorption considering the surface charge density on substrate is investigated. First, a free energy model is established to quantify the deflection and axial displacement of the microbeam, the ion concentration in the solution, as well as the electric potential distribution inside and outside the ssDNA film with a uniform surface charge density. Then, the governing equations and corresponding boundary conditions are obtained by minimizing the free energy with three types of variational variables. And, the relationship of the electric potential difference between the two sides of the ssDNA film and the deformation of microcantilever and simply support beam are determined in different solutions (1: 1, 1: 2, 1: 3). Moreover, the numerical solutions of the electric potential distribution inside and outside the ssDNA film are investigated by using the finite difference method and Newton's iteration method. The results suggest that the nonlinear model for electric potential distribution should be used under higher surface charge density. By fitting and comparing the present predictions with Arroyo-Hernandez's experimental data, the effectiveness of the model is verified. The study shows that the surface charge density can modulate the magnitude and direction of ssDNA-microbeam deformation, which also indicates that there exists a critical surface charge density, causing the failure of ssDNA-microbeam detection. These conclusions are helpful for designing the efficient biosensors based on DNA-microbeam.

A refined dynamic liquid dehumidification model for the packed dehumidifier considering time-varying thermal mass of the desiccant

Liquid desiccant dehumidification technology is a promising, energy-efficient dehumidification approach. It can flexibly adapt to variations in renewable energy supply and users' demands, making it an inherently dynamic process. However, optimizing this process from a dynamic perspective is often neglected. This is mainly due to the challenges in establishing a dynamic dehumidification model, where the desiccant's thermal mass and heat and mass transfer coefficients should be time-dependent. In this paper, a refined dynamic dehumidification framework using the finite difference method is developed, where governing equations for the mass and species conservations are rebuilt taking the thermal mass and desiccant concentration as time-dependent variables. Auxiliary equations determining evolutions of the desiccant's thermal mass and heat and mass transfer coefficients are derived based on the continuity equation and analogy method, respectively. Subsequently, the experiments are carried out and serve as a benchmark to check the simulations. It suggests the simulated results using the refined model align well with the experimental data and demonstrate around 50 % better predictive accuracy compared to the existing method, indicating the feasibility of the refined model. The refined model also contributes to rapid adaptation of the dehumidification process to meet specific needs. In a specific case, this could result in energy savings of 60 % for the regeneration of the desiccant solution. This study provides a reference for quantifying evolutions of the liquid desiccant's thermal mass based on an understanding of the physical process. It lays a solid foundation for modeling the dynamic dehumidification process at the design stage and assists in operating an energy-saving dehumidification system.

Analytical and numerical study of thermo-solutal convection in pulsating flows between two coaxial cylinders

This research reports an analytical and numerical study of thermo-solutal convection in pulsating flows between two coaxial cylinders under a pressure gradient and subject to Dirichlet-type boundary conditions for temperature and concentration along the vertical walls of the annulus. The numerical procedure used in this work is mainly based on the solution of the momentum equations and coupled with the energy and concentration equations. The finite difference method is adopted to solve governing equations using the implicit Crank-Nicolson time marching scheme in the fluid domain and the alternating direction implicit method inside the inner cylinder. The effect of the various parameters such as the fluid-solid conductivity ratio, thermal diffusivity ratio, buoyancy ratio, Lewis number, Richardson number, aspect ratio, and radius ratio on the heat and mass transfer characteristics and the flow structure is presented and discussed. The governing equations are solved analytically after separating into a steady case and an oscillatory case, where an analytical solution for velocity profile, temperature, and concentration distributions is obtained in terms of Bessel’s functions, while the analytical solutions of temperature and concentration profiles are keeping symmetry with the increase of the radius ratio. The results show that the analytical profile can reproduce the three fields obtained numerically and thus for the whole range of parameters.

Numerical treatment of singular ODEs using finite difference and collocation methods

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class.In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes.

A new approach to problems of thermoelasticity in stresses

It is known that the plane problems of the theory of elasticity and thermoelasticity are usually reduced to solving the biharmonic equation with respect to the Airy stress function. Solutions to spatial problems can also be defined using the Maxwell and Morera stress functions, etc. In this work, for solving a spatial thermoelastic problem, in the framework of the Beltrami-Michell equations, Poisson-type differential equations are found for each component of the stress tensor. In this case, the right-hand sides of these equations depend on temperature and elastic constants. The boundary conditions consist of the usual surface forces and “additional” boundary conditions derived from the equilibrium equations. The differential equations and boundary conditions of the plane problem of thermoelasticity in stresses are discussed in more detail. Grid equations for plane and spatial problems of thermoelasticity were compiled by the finite-difference method and solved by the iterative method. The problems of a thermoelastic rectangle and a parallelepiped located in a given temperature field are solved numerically. The validity of the formulated boundary value problems and the reliability of the results obtained are substantiated by comparing the numerical results with the solutions of the problems of a thermoelastic plate and a parallelepiped known in the literature.

Highly coherent supercontinuum generation in a photonic crystal fiber based on GeSe2–As2Se3–PbSe chalcogenide in the mid-infrared region

We numerically demonstrated highly coherent mid-infrared (IR) super-continuum (SC) generation using all-normal dispersion (AND) GeSe2−As2Se3−PbSe chalcogenide photonic crystal fiber (PCF). The structural parameters of this PCF were optimized to obtain an ultra-flat-AND region and high nonlinearity characteristics using the full-vectorial finite difference method with the perfectly matched layer boundary conditions to calculate the linear properties of the fiber. We show that, pumping a light source at 9.8 µm pump wavelength with an input pulse energy of 3.5 nJ and pulse width of 200 fs, corresponding to the peak power of 17.5 kW, into a 10-cm-long fiber, an ultra-flat-coherent mid-IR SC extended from 2851.8 nm to 13538.8 nm (bandwidth of 10687 nm) is obtained. This ultra-wide and flattened spectrum has excellent stability and coherence properties that can be used for mid-IR applications such as biological imaging, optical coherence tomography, and frequency metrology.

Controlling numerical diffusion in solving advection-dominated transport problems

Numerical schemes for advection-dominated transport problems are are evaluated in a comparative study. Explicit and implicit finite difference methods are analyzed together with a global random walk algorithm in the frame of a splitting procedure. The efficiency of the methods with respect to the control of the numerical diffusion is investigated numerically on one-dimensional problems with constant coefficients and two-dimensional problems with variable coefficients consisting of realizations of space-random functions.

Effects of axial electric and transverse magnetic fields on a rotating electro-osmotic flow in micro-parallel plates

This paper explores the combined influence of an axial electric field and a perpendicular magnetic field imposed on rotating micro-parallel plates immersed in an electrolyte solution. A specialized computer program was developed to solve the velocity as well as the EDL potential fields using the finite difference method, employing the Debye-Hückel (DH) approximation to linearization the EDL potential. The study examines the influence of various non-dimensional parameters, including rotational speed (ω), Hartmann number (Ha), Debye-Hückel parameter (κ), and the non-dimensional parameter ‘S’, on axial, and transverse velocities, wall shear stress, and net flow rate. Results demonstrate that, both velocity components decrease with increased rotational speed and Hartmann number, while the net flow rate increases with the Debye-Hückel parameter for both rotating and non-rotating systems. The impact of these parameters on shear stress was also analyzed. Analysis of Ekmann spirals in the velocity plane revealed closed spirals at a higher rotational speed and open spirals at lower speeds, with spiral size reducing as rotational speed increases.

Fractional dynamics of entropy generation in unsteady mixed convection of a reacting nanofluid over a slippery permeable plate in Darcy–Forchheimer porous medium

AbstractThis paper presents a theoretical investigation of the inherent irreversibility in unsteady fractional time derivative mixed convection of a reacting nanofluid with heat and mass transfer mechanism over a slippery permeable plate embedded in a Darcy–Forchheimer porous medium. The model fractional partial differential equations are obtained based on conservation laws and numerically solved using the implicit finite difference scheme. The study displays and discusses the effects of various emerging parameters on the overall flow structure, such as velocity profiles, temperature distribution, nanoparticles concentration profiles, skin friction, Nusselt number, Sherwood number, entropy generation rate, and Bejan number. It was found that an increase in dimensionless time and fractional parameters leads to a decrease in both the entropy generation rate and the Bejan number. The study revealed that fractional order derivatives can capture intrinsic memory effects, non‐local behaviour, and anomalous diffusion in the nanofluid flow process. This can ultimately lead to better engineering system design and control.