Finite Difference Equations Research Articles

Groundwater Assessment using Numerical Flow Model: A Case Study in Gash Sub-Basin-Kassala State, Sudan

The main objective of groundwater modelling in Gash River Sub-basin, is to investigate the effect of hydrologic, hydrogeological parameters and stresses on hydrodynamic behaviour through the implementation of a realistic three-dimensional groundwater flow model. Severe decline of water level due to uncontrollable heavy abstraction, exposes a water scarcity problem especially in summer seasons. The model was developed for four geological layers encompassing two aquifer zones. The improved three-dimensional visual MODFLOW Code was selected, implemented and run using WHS method to solve the finite difference equation using trail-and-error calibration procedure at Kassala Area. The transient model was successfully calibrated with acceptable results of model calibration criteria. The contour maps of the simulated heads were performed as potentiometric surface. The general flow direction of the groundwater is from southeast towards northwest part of the area and from Gash River course towards the east and west directions as detected from gradual decreasing of potential line’s values in those directions, confirming the aquifer recharge from Gash River. The similarity of potentiometric surface contour maps of the two aquifers confirm the aquifers hydraulic interactions. It is found that the increasing pumping rate caused considerable increase in drawdown as detected from pumping rate incremental scenarios. Moreover, incremental pumping rate scenarios also reflected increasing river leakage into the aquifer system due to disturbance of water balance due to water level decline. The components of water budgets were calculated and its percentage was performed for the hydrologic balance. The difference between inflow and outflow of the water balance shows a deficit in most stress periods of the model simulations. Calibration fitness was accomplished at most of the observation wells suggesting that the groundwater model is an accurate representation of the actual historic groundwater system and confirm the validity of the model to forecasting purposes. It was found that the model is more sensitive to hydraulic conductivity and least sensitive to specific yield (Sy). Hence, precaution should be revealed for hydraulic conductivity in forecasting model usage.

Unsteady Hybrid (Ag–CuO/Water) Nanofluid Flow and Heat Transfer due to a Stretching Sheet with Variable Temperature

In this paper, the focal aims are (i) to explore the transient boundary-layer flow and heat transfer of an electrically conducting hybrid (Ag–CuO water) nanofluid along a vertical stretching surface (sheet) having non-zero slot velocity at variable temperature, and (ii) to discuss the influences of momentous parameters involved on the heat transfer and skin friction coefficient graphically. The “Tiwari-Das nanofluid model” is used. The central equations (PDEs) are converted into finite difference equations by the powerful Crank Nicolson technique and numerically solved using the Thomas algorithm. The achieved outcomes for a specific case of the challenge are compared with an analytical solution computed using the Laplace transform technique and discovered to be in excellent accord.

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation.
 Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written.
 The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

Quantum Curves, Resurgence and Exact WKB

We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of q-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.

Novel Real-Time Compensation Method for Machine Tool’s Ball Screw Thermal Error

The real-time compensation of thermal error in ball screws is an effective means to improve the accuracy of machining tools. However, the trade-off between robustness and computational efficiency of existing ball screw thermal error models is complicated and not conducive to practical, high-precision, real-time error compensation. Focusing on this problem, we propose an iterative prediction model of screw thermal error based on a finite difference equation. By assuming an approximately linear relationship between heat generation and the ball screw’s convection power and feed speed, a simplified and more efficient identification of physical parameters needed for the iterative model is achieved. The proposed method is integrated with a three-axis drilling and tapping machine powered by an HNC–848D controller. A test piece machine using the proposed real-time thermal error compensation method exhibited a maximum machining error of 13 µm, compared to the 71 µm of an uncompensated specimen. The proposed method is demonstrated to improve machining accuracy, especially in the X- and Y- axes, and overcome the limitations of traditional thermal error prediction models.

Investigating the stability of an explicit ADE-FDTD scheme for modeling graphene: Avoiding erroneous conclusions

In recent years, an explicit auxiliary differential equation finite difference time domain (ADE-FDTD) scheme has been frequently used in graphene simulations. In this respect, a differencing scheme in which the magnetic field and the current density are collocated in time was developed and it has been recently stated that this explicit ADE-FDTD implementation is “conditionally stable with the maximum time-step size bounded by the common Courant–Friedrichs–Lewy (CFL) limit”. In this communication, the stability of this explicit ADE-FDTD scheme is revisited and it is shown that the conventional CFL constraint is not retained, and a time-step less than the CFL limit must be used in all domain grids to insure the stability in the whole system. Although this time-step stringent can be of little effect for some graphene applications, and in order to avoid drawing erroneous conclusions, it is shown that when the given ADE-FDTD scheme is used for simulating graphene structures that include noble and other plasmonic materials the derived time-step stringent can be of significant effect. To retain the CFL stability constraint, alternative formulation based on the Runge–Kutta (RK) time-differencing scheme is also presented in this communication. Contrary to the other existing CFL-stable ADE-FDTD approaches, the presented RK-FDTD method is equally applicable to both magnetized and unmagnetized graphene structures without incurring additional computational cost. Furthermore, the RK-FDTD scheme is extended for modeling the complex-frequency-shifted perfectly matched layer (CFS-PML) mesh truncating technique. Finally, the stability and accuracy of the explicit ADE/RK-FDTD schemes are demonstrated through numerical examples.

Load transfer analysis of vertically loaded bored piles in sea reclamation areas considering the effect of the pile-gravel interface roughness

A load transfer model for vertically loaded bored piles in sea reclamation areas with gravel is proposed in this paper. The focus is on quantifying the mobilization of pile-side friction, which is closely related to the shear behaviors of the pile-gravel interface. The surface profile of the gravel particles and the concrete piles are assumed to be circle and wave curves, respectively. The shear displacement and dilatancy calculation are based on the macro and micro geometry relationship between gravel particles and pile surface. The shear stress calculation of the pile-gravel interface is dealt with by empirical elastoplastic equations. A simple slip-line field is also built to identify the critical shear displacement. The finite difference equations are used to derive the load-displacement curves for vertically loaded bored piles in sea reclamation areas with gravel. The model predictions are compared with field observations and particle flow numerical simulation to validate the performance of the proposed method, and the results prove to be reasonable. Moreover, parameter studies are conducted and the results prove that the pile can bear more load when the particle diameter is similar to the pile's rough surface, and both the increases of wave angle and the relative friction coefficient can promote the bearing capacity.

A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields

In many instances, conditions in the subsurface are highly variable, while site investigations only provide sparse measurements. Consequently, subsurface models are usually inaccurate. These characteristics reflect uncertainty and mean significant engineering and environmental hazards. Such uncertainty should be quantified, in order, ultimately, to be reduced. To this end, in this paper, a new two-dimensional approach to quantify stratigraphic uncertainty is proposed and described. The approach is based on non-homogeneous random fields and considers categorical quantities to represent geological structures. Unlike other reported approaches in related literature, we provide evidence on the usefulness of the approach to: (i) comprehensively explore the many diverse and potential geological structures' configurations to exhaust uncertainty quantification; (ii) facilitate the representation of non-homogeneous fields in a computational inexpensive fashion; and (iii) ameliorate the complication of producing spatial Markov chains or Bayesian models to quantify uncertainty using finite difference equations. The proposed approach is demonstrated using a case analysis.

Poisson stable motions of monotone and strongly sub-linear non-autonomous dynamical systems

This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo periodic, pseudo recurrent and Poisson stable) motions of monotone sub-linear non-autonomous dynamical systems. The main results we establish in the framework of general non-autonomous (cocycle) dynamical systems.We apply our general results to the study of the problem of existence of different classes Poisson stable solutions of some types of non-autonomous evolutionary equations (Ordinary Differential Equations, Functional-Differential Equations with finite delay and Difference Equations).

Infinite elliptic hypergeometric series: convergence and difference equations

We derive finite difference equations of infinite order for theta-hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, and we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion of the convergence of $q$-hypergeometric series for ${|q|=1}$, $q^n\neq 1$, to the elliptic level and prove the convergence of infinite very-well poised elliptic hypergeometric $ _{r+1}V_r$-series for restricted values of $q$. Bibliography: 13 titles.

Fluid Micelle Network for Image Super-Resolution Reconstruction.

Most existing convolutional neural-network-based super-resolution (SR) methods focus on designing effective neural blocks but rarely describe the image SR mechanism from the perspective of image evolution in the SR process. In this study, we explore a new research routine by abstracting the movement of pixels in the reconstruction process as the flow of fluid in the field of fluid dynamics (FD), where explicit motion laws of particles have been discovered. Specifically, a novel fluid micelle network is devised for image SR based on the theory of FD that follows the residual learning scheme but learns the residual structure by solving the finite difference equation in FD. The pixel motion equation in the SR process is derived from the Navier-Stokes (N-S) FD equation, establishing a guided branch that is aware of edge information. Thus, the second-order residual drives the network for feature extraction, and the guided branch corrects the direction of the pixel stream to supplement the details. Experiments on popular benchmarks and a real-world microscope chip image dataset demonstrate that the proposed method outperforms other modern methods in terms of both objective metrics and visual quality. The proposed method can also reconstruct clear geometric structures, offering the potential for real-world applications.

Numerical solution of problem of equilibrium of parallelepiped in stresses

Usually, planar and spatial problems of the theory of elasticity in stresses are solved using the Airy or Maxwell stress function, respectively. Solving a plane problem usually comes down to solving a biharmonic equation. In the spatial case, it can be solved by the Castigliano variational method. This work is devoted to the mathematical and numerical solution of the spatial problem of elasticity theory directly concerning stresses. In this case, the boundary value problem consists of three equilibrium equations and three Beltrami-Michell equations with the corresponding boundary conditions. At the same time, equilibrium equations are considered as missing boundary conditions. Symmetric finite-difference equations are constructed, and the problems of equilibrium of a parallelepiped under the action of domed and uniformly distributed loads are solved numerically. The reliability of the results obtained is substantiated by comparing the numerical results with the wellknown Filonenko-Borodich solution and with the results of solving the boundary value problem of the equilibrium of the parallelepiped with respect to displacements.

Course stability models of a wide-width tractor unit

The purpose of the research is to provide for solving dynamic problems of course stability of a wide-width sowing and row-crop-tractor unit. For constructing a computational scheme for the generated problem applied the central finite-difference method with a second-order accuracy. When developing a machine algorithm, the fractional grid steps are converted by formulas and a vector equation is obtained. The results of scientific and practical research and testing to determine and evaluate performance of machine-tractor units (MTU) consisting of inter row spacing sowing and cultivators with four-wheel tractors were obtained. Studies on appropriateness of increasing coverage area of cotton units based on field size have shown that the shift productivity of eight-row units at planting was lower than that of four-row units when the field area was less than 3 hectares. When the field area was 3 hectares or more, the MTU’s productivity increased by 32–34%. A similar trend has been observed in the inter-row processing units. Field experiments have shown that it is advisable to maintain the existing four-row machine system on the basis of existing energy vehicles for 27-30% of cotton fields. Cauchy problem for finite-difference equations of dynamic problems of course stability of a wide-cut sowing and tilled-tractor unit was formed, in the form of algebraic equations. As a result, it was possible to build 6-, 8- and 10-row cultivator units based on four-wheel tractors.

Calculating a beam of variable section lying on an elastic foundation

In this article, there has been studied the bending state of a reinforced concrete beam with a variable cross-sectional height along its length that rests along its entire length on a brick wall. The beam is under the action of arbitrarily located concentrated forces. The study has been performed on the basis of the original inhomogeneous differential equation of the 4th order, taking into account the external load and the bedding coefficients of the elastic foundation. Using the finite difference method, typical resolving finite difference equations have been obtained. A study of the influence of the degree of elasticity of the base, with a change in the value of the bed coefficients and elasticity parameters, was conducted. The results confirm the reliability of theoretical and practical calculations. Given theoretical provisions and applied results can be used in scientific research in the field of mechanics of a deformable solid body, as well as in practical design.

A finite difference-based approach for strain demand prediction of inelastic pipes subjected to permanent ground displacements

Permanent ground displacement triggered by geohazards constitutes a major threat to the integrity of long-distance pipelines. In this study, a novel and simple method is proposed to evaluate inelastic pipes’ behavior under ground movements considering soil-pipe interaction based on the finite difference method. The existing finite difference-based method previously proposed by the authors for strain analysis of buried pipelines subjected to ground movements excludes material nonlinearity in pipes, which limits its applicability in engineering practice. To remedy this situation, the method presented herein maintains the well-established concepts of the existing finite-difference-based method but also introduces the scheme to consider the inelastic material behavior of steel pipes. More specifically, the expressions of internal axial force and bending moment, required in the finite difference equations, are explicitly derived based on the actual stress distribution on the pipe cross-section. The proposed method is validated against the finite element method (one-dimensional beam model) in terms of the strain and deformation demands using two indicative case studies, referred to as symmetric and non-symmetric soil resistance conditions, respectively. The comparison between finite element analysis and the proposed method indicates good agreement. Additionally, the proposed method is compared with four existing analytical methods for pipes subjected to strike-slip fault displacement. The proposed method is a simple but general technique to analyze pipes’ response under a wide range of geohazards, and thus can be potentially used as an alternative method for preliminary design, safety prescreening, and reliability-based assessment of pipes against geohazards.

Iterative solver with folded preconditioner for finite element simulation of magnetotelluric fields

We propose an iterative solver with a folded preconditioner for large sparse systems of linear algebraic equations (SLAEs), which are obtained as a result of a vector finite element approximation of electromagnetic fields in magnetotelluric problems. The solver is based on the computational scheme of the Conjugate A-Orthogonal Conjugate Residual method (COCR), diagonal preconditioner, and residual smoothing. To build a finite element approximation, the primary-secondary field approach is used. To construct a preconditioner, we use a matrix, the elements of which are the weights of the representation of gradients of nodal basis functions as a linear combination of edge basis functions. For Dublin Test Model 1 in a wide frequency range from 0.0001 Hz to 10 Hz, we compare the accuracy and computational efficiency of our code with the results of other authors, as well as compare the computational efficiency of the proposed diagonal folded preconditioned (DFP) COCR solver with other solvers. It is shown that the apparent resistivities obtained on the coarsest mesh with our code differ by no more than a few percent from the solution on a finer mesh and from some solutions obtained by other authors using the finite element, finite difference, and integral equation methods. The computational time required for our code is 8 times less than the minimum time presented by other authors (with a corresponding recalculation of the performance of the computers used). It is also shown that the DFP COCR solver practically does not increase the number of iterations at low frequencies and its computational time is about 20 times less than the time required for the COCR solver with a standard diagonal preconditioner. A comparison of the DFP COCR solver with the Parallel Direct Sparse solver (PARDISO) shows that even at low and medium frequencies, the computational time of the DFP COCR solver is about 20 times less than the computational time of the PARDISO, and at high frequencies this difference reaches already 50 times.

Existence of solutions of discrete fractional problem coupled to mixed fractional boundary conditions

In this paper, we introduce a two-point nonlinear boundary value problem for a finite fractional difference equation. An associated Green’s function is constructed as a series of functions and some of its properties are obtained. Some existence results are deduced from fixed point theory and lower and upper solutions.

Вплив шунтування високотемпературних тензорозісторів на точність виміру статичних деформацій елементів ГТД

The development of gas turbine engines (GTE), for various purposes, is inextricably linked with an increase in their main characteristics. Simultaneously, the parameters of the working fluid increase, in particular, the temperature of the gas flow and the intensity of loads on structural elements with an increasing frequency of rotation of the rotors. The strength reliability of highly heated GTE elements is a factor that determines the life of the engine as a whole. The most common cases of damage to GTE elements are caused by static stresses, and mainly relate to the blade apparatus, compressor and gas turbine housings, combustion chambers and rotor elements operating in the temperature range of 200–750 °C. Errors in measuring static deformations of parts are usually associated with insufficient compensation for the temperature increment of resistance by the sensitive element (SE) of the strain gauge and with the occurrence of shunt currents between the SE and the body of the part through a binder insulator. The change in the electrical resistance of the strain gauge SE is perceived by the measuring system as an imaginary deformation. The measurement error due to shunting increases significantly with an increase in the temperature of the part under study, since this significantly reduces the specific electrical resistance of the binder insulator. A strain gauge sensor with two sensitive elements is considered in this work. The lower CE of the strain gauge sensor is located in the insulator-connector directly in the vicinity of the body of the part and perceives its main deformation. The upper Euro is located above the lower. The main axes of the elements are rotated relative to each other by 90 ° C and plays the role of temperature-compensating element and at the same time registers the transverse deformation of the part. An electrical model of the potential distribution in a strain gauge is presented. To determine the magnitude of shunt currents, Kirchhoff's rule was applied to a linear electric circuit, and finite-difference differential equations for the sum of currents in all nodes of the electric model were recorded. Finite-difference differential equations are transformed into a matrix one, the solution of which allows to obtain leakage currents in all nodes of the electric model of the strain gauge. The total leakage currents in the lower and upper SE strain gauge for different cases, as well as the relative errors of deformation measurement due to shunting, are obtained.

Modeling the multifractal dynamics of COVID-19 pandemic

To describe the COVID-19 pandemic, we propose to use a mathematical model of multifractal dynamics, which is alternative to other models and free of their shortcomings. It is based on the fractal properties of pandemics only and allows describing their time behavior using no hypotheses and assumptions about the structure of the disease process. The model is applied to describe the dynamics of the COVID-19 pandemic from day 1 to day 699 from the beginning of the pandemic. The calculated parameters of the model accurately determine the parameters of the trend and the large jump in daily diseases in this time interval. Within the framework of this model and finite-difference parametric nonlinear equations of the reduced SIR (Susceptible-Infected-Removed) model, the fractal dimensions of various segments of daily incidence in the world and variations in the main reproduction number of COVID-19 were calculated based on the data of COVID-19 world statistics.

A numerical model for geometrically nonlinear analysis of long free spanning offshore pipelines enhanced with buoyancy modules

Despite a detailed oil and gas pipeline route assessment, the emergence of highly long free spans induced by seafloor roughness in deep waters, accompanied by significant gaps between the pipeline and the seafloor, can sometimes be unavoidable. In those cases, cost-efficient span correction can be achieved by attaching buoyancy modules along certain sections of the free span.This paper provides a simple numerical technique for static analysis of submarine pipeline free spans enhanced with buoyancy modules and resting on span shoulders of nonequal elevations.The entire pipeline is modeled as a single continuous segment considering geometrical nonlinearity induced by large deformations. The solution technique is based on consistent minimization of the total potential energy of the system discretized as a Riemann sum and the solution of the resulting algebraic system of nonlinear finite difference equations with a Newton-Raphson technique.The feasibility of the proposed technique is demonstrated through several pipeline free span scenarios. The effects of buoyancy module section length, its distribution, effective axial tension, span shoulders soil stiffness, and span shoulders elevation gap are considered.The proposed technique provides a fast and accurate alternative to special-purpose finite element commercial software, particularly throughout the intermediate stages of the design process.