Finite Difference Method Research Articles

Numerical Analysis of Temperature Distribution in AA1100 Friction-Stir Welded Joints by Finite Difference Approximation

Numerical Analysis of Temperature Distribution in AA1100 Friction-Stir Welded Joints by Finite Difference Approximation

Analytical and numerical solutions to the Klein–Gordon model with cubic nonlinearity

In this paper, the nonlinear Klein–Gordon equation’s exact solutions are obtained through the application of an appropriate transformation based on He’s semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method’s accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters.

Numerical simulation of 3D seismic wave based on alternative flux finite-difference WENO scheme

Summary High-frequency non-physical oscillations may occur due to shock waves in seismic wavefield and dynamic rupture simulation. In this study, we introduced the alternative flux finite-difference weighted essentially non-oscillatory (AWENO) scheme to address potential shock wave issues in computational seismology effectively. The wavefield of the body-fitted curvilinear domain was accurately computed through conservative grid mapping, ensuring accurate implementation of free surface boundary conditions on irregular surfaces using characteristic boundary conditions and minimizing artificial boundary reflections with exponential decay absorbing layers. Finally, we compared our scheme with the GRTM for flat surfaces and the CGFDM3D-EQR for irregular surfaces to demonstrate its correctness and accuracy, and validated its non-oscillatory characteristics. The aforementioned scheme is anticipated to assume a significant function in simulating more intricate seismic wavefields or dynamic ruptures.

A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

Data fusion based on short-term memory Kalman filtering using intermittent-displacement and acceleration signal with a time-varying bias

Accurate displacement information plays a vital role in various applications from structural health monitoring to damage detection in physical systems. Due to practical reasons, displacement sensing is often performed using sensors that provide less accuracy or a low sampling rate. This study proposes an algorithm for displacement signal estimation that fuses high sampling rate acceleration data (which consists of unknown time-varying bias) with low sampling rate displacement data. In contrast to the present literature, the state vector in the proposed state-space formulation consists of short-term memory of the past realizations of displacement and velocity, and bias terms which allow (i) online estimation of the displacements by avoiding multi-rate Kalman filtering, and (ii) estimate time-varying bias in the measured acceleration signal. The state equations comprise the finite difference approximations of the time derivative of velocity and displacement which act as a moving average filter to provide smooth estimates, thus preventing the need for offline smoothing procedures. By inclusion of bias terms in the state vector, the drifts in displacement and velocity caused due to time-varying bias in measured acceleration are easily eliminated. The proposed method was found to provide accurate estimates of displacement, velocity, and time-varying bias in measured acceleration signals when tested on different types of bias, including but not limited to constant, linear, and parabolic variations. Additional tests on three oscillators subjected to three earthquake ground motions showed significantly better state estimates for stiff oscillators and improved estimates for flexible oscillators. Experimental validation on a base-excited two-story frame further affirms the capability of the method to estimate the hidden bias in the measured acceleration and provide an accurate estimate of displacement.

On a stabilization of the Ingard-Myers impedance boundary condition and its time domain implementation

It has been well-known that the Ingard-Myers impedance condition, while simple to apply, is subject to the hydrodynamic Kelvin-Helmholtz-type instability due to its use of a vortex sheet in modeling the flow at the liner boundary. Recently, in the development of a time domain boundary element method for acoustic scattering by treated surfaces, it was found that by neglecting a certain second-order spatial derivative term in the Ingard-Myers formulation, the hydrodynamic instability can be avoided. The present paper aims to provide further analysis of this modified condition, hereby referred to as the Truncated Ingard-Myers Impedance Boundary Condition (TIMIBC). It will be shown, based on the dispersion relations of linear waves, that the instability intrinsic to the Ingard-Myers condition is eliminated in the proposed new formulation. Quantitative assessments on the accuracy of the TIMIBC for scattering of acoustic waves by lined surfaces are carried out, and its effectiveness is demonstrated by a numerical example. It is found that the TIMIBC provides a good approximation to the original Ingard-Myers condition for flows of low to mid subsonic Mach numbers. As such, the proposed TIMIBC can offer a practical solution for overcoming the intrinsic instability associated with the Ingard-Myers condition. Moreover, time domain implementation of the TIMIBC is also discussed and illustrated with a numerical example using a finite difference scheme. In particular, a minimization procedure for finding the poles and coefficients of a broadband multipole expansion for the impedance function is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured.

Efficient reliability assessment of slope stability with insightful sensitivity analysis concerning the safety factor

The limit state functions (LSFs) of slope stability are nonlinear, implicit, or derived from numerical simulations, and additional surrogate models are usually used for its reliability problems. Alternatively, an efficient numerical procedure using a variant of the HLRF-BFGS algorithm is proposed to address this issue without additional surrogate models. This proposed procedure simultaneously conducts reliability assessment and sensitivity analysis concerning the safety factor by suggested calculations of partial derivatives using the finite difference method (FDM). Two highly nonlinear numerical examples are presented to validate its convergence and to investigate the step length of the FDM. Results show that the proposed procedure can get considerable accuracy while the HLRF algorithm cannot converge, and its required iterations are about 30% and 81% less than those of the limited step-size iteration algorithm. In addition, a step length not greater than 0.1 is suggested after investigations. Application to a classical two-layered slope is illustrated. Reliability results show that the proposed procedure can get the same accuracy as those of surrogate models after only 5 iterative steps. Sensitivity results show that the cohesion of the upper soil is the only sensitivity factor, and the most possible potential slide surface lies in the upper soil.

Analysis of the flow and thermal-fluid–solid coupling of crude oil in circular pipe caused by variable pressure gradient

Abstract Globally, enhanced oil recovery (EOR) has become a pressing issue as the demand for crude oil continues to increase. This study investigates the flow and thermal-fluid–solid coupling of crude oil in a rod pump during hot water recovery and obtains the maximum recovery of crude oil in a vertical pipeline through numerical analysis. The pressure gradient in the pump barrel was first developed and deduced based on the ideal gas state equation and Bernoulli’s equation. According to the rheological experiment results, it was proven that the light crude oil conforms to the Newtonian constitutive equation. Subsequently, the momentum equation of crude oil flowing in the pipeline and fluid–solid coupling heat transfer equations were established and solved using the finite difference method. The effects of the thermal recovery temperature T w , wall thickness c, and stroke time n of the rod pump on flow Q are discussed. In particular, the flow Q within 1 min first increases and then slows down with the increase in stroke time n and reaches its maximum value at n = 7 r/min. Furthermore, flow Q decreases with an increase in c but increases as T w increases; c = 1.2 cm, T w = 363 K is the best oil recovery scheme.

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

Selection of Euler deconvolution solutions using the enhanced horizontal gradient and stable vertical differentiation

Abstract Euler deconvolution is widely used for interpreting magnetic anomalies as it estimates the edges and depths of magnetic sources. Since this method was proposed, there has been an intensive effort to mitigate its primary deficiencies, namely, the generation of many spurious solutions and the high noise sensitivity. To select the most significant solutions, we adopt the strategy of constraining the moving window to the source edges, whose locations are estimated using the enhanced horizontal gradient amplitude method. On the other hand, we reduce noise propagation by performing a stable calculation of the vertical derivatives. For this purpose, we use the β-VDR method, a finite-difference method that yields a robust approximation of the vertical derivatives of magnetic data. The accuracy of the proposed technique is demonstrated on synthetic magnetic anomalies, providing the depths more precisely and being insensitive to noise. Application of this technique is also demonstrated on aeromagnetic anomalies from the Olympic Peninsula (USA), where the obtained result is in good agreement with known information of the study region.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

A space marching method for sonic boom near field predictions

The conventional sonic boom propagation prediction method widely adopted in supersonic aircraft design involves a two-step procedure. In the first step, a compressible viscous or inviscid Computational Fluid Dynamics analysis is applied to the aircraft geometry at flight conditions to produce a flow field solution near the aircraft. Then in the second step, a one dimensional nonlinear Burgers’ equation model is used to propagate sonic boom signature traces from the near field solution at flight altitude to the ground along a ray path. For an accurate ground signature prediction the near field signature must be accurately modeled at a sufficient distance from the aircraft flight path in order to minimize errors in the one dimensional propagation model. This is a very challenging task for general purpose CFD tools in a design environment because the cost of maintaining highly accurate off-body solutions increases dramatically as the radial distance is enlarged in the computational domain. It is also particularly difficult to apply these tools for wave propagation because the algorithms are normally lower order and numerically dissipative and dispersive. In this work a space marching procedure based on an optimized higher order finite difference method is developed and applied in conjunction with a CFD solution concentrated in the close vicinity of the aircraft. This new approach is much more efficient, compared to previous methods, in providing highly accurate near field signatures for full carpet ground predictions. Results indicate that near field signatures retain more waveform shape information farther from the aircraft geometry while reducing the CFD cost significantly. The predicted ground signature is also shown to converge in shape as the radial distance of the near field signature grows, which is indicative of a more ideal initial condition being supplied to the one dimensional wave propagation to ground. This feature is very difficult to replicate in a tractable manner with common CFD approaches used in design.

Numerical simulation of natural convection in a rectangular enclosure filled with porous medium saturated with magnetic nanofluid using Buongiorno's two‐component model

AbstractMotivated by studying emerging nanofluid‐based magnetic fuel cells and hybrid direct absorber solar collectors, a numerical study is presented for buoyancy‐driven flow in a vertical enclosure containing a porous medium saturated with magnetized nanofluid flow under a transverse static magnetic field. The enclosure features adiabatic side walls and vertical heat slits, ensuring consistent cold temperatures on the lower and upper bounded walls. The side walls of the regime are insulated, and the hot slits are centrally located on these walls. The finite difference method (FDM) is applied to employ the transformed dimensionless vorticity–stream function (VSF) formulation of the transport equations. The impact of pertinent parameters on isotherm, streamline, iso‐concentration, and average Nusselt and Sherwood numbers are visualized with contour plots and graphs. Increasing Darcy number values tend to amplify the isotherm magnitudes. Higher Hartmann (magnetic) number values, on the other hand, lead to a reduction in local Nusselt number profiles but do not significantly modify the local Sherwood number. The porous medium permeability, as simulated via the Darcy number, has a strong impact on streamlines, thermal contours, and iso‐concentrations. A reduction in Darcy's number suppresses local Nusselt and Sherwood numbers, whereas an elevation in Rayleigh's number enhances them. Increasing the Buongiorno nanoscale Brownian motion parameter enhances local Nusselt and Sherwood numbers at both cold walls of the enclosure. The simulations provide a deeper insight into enclosure flows involving electrically conducting nanofluids in advanced processing systems for nanomaterials and hybrid fuel cells utilizing electromagnetic and liquid fuel technologies.

Comparative analysis of a fractional co-infection model using nonstandard finite difference and two-step Lagrange polynomial methods

In this work, Caputo fractional model for co-dynamics of two-strains (Alpha and Delta variants) of SARS-CoV-2 and tuberculosis (TB) is formulated. We investigates the basic mathematical analysis, existence of unique solution, stability analysis, sensitivity analysis and simulations of the proposed model. In the basic mathematical analysis, it is proved that the model’s solutions are non-negative, bounded and locally asymptotically stable at disease free equilibrium (DFE) point. The existence of unique solution is proved by using Banach fixed point theorem (BFPT). Stability analysis is carried out by employing Hyers–Ulam stability criteria. Sensitivity analysis is performed to assess the impact of the model parameters on the general dynamics of the model by implementing partial rank correlation coefficient (PRCC) technique. Simulations of the proposed model obtained from non-standard finite difference scheme (NSFDs) are compared with the simulations obtained from two-step Lagrange polynomial method (TLPM). It is concluded that NSFDs gives more accurate and realistic results than TLPM. To assess some control measures necessary for reducing the co-spread of both diseases, comparative simulations are carried out for the infected classes. Also, numerical assessments showed that preventive efforts against SARS-CoV-2 variants could results in the reduction of TB prevalence and the co-infection of both diseases.

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.

Quadratic American Strangle Options in Light of Two-Sided Optimal Stopping Problems

The aim of this paper is to examine some American-style financial instruments that lead to two-sided optimal hitting problems. We pay particular attention to derivatives that are similar to strangle options but have a quadratic payoff function. We consider these derivatives in light of much more general payoff structures under certain conditions which guarantee that the optimal strategy is an exit from a strip. Closed-form formulas for the optimal boundaries and the fair price are derived when the contract has no maturity constraints. We obtain the form of the optimal boundaries under the finite maturity horizon and approximate them by maximizing the financial utility of the derivative holder. The Crank–Nicolson finite difference method is applied to the pricing problem. The importance of these novel financial instruments is supported by several features that are very useful for financial practice. They combine the characteristics of the power options and the ordinary American straddles. Quadratic strangles are suitable for investors who need to hedge strongly, far from the strike positions. In contrast, the near-the-money positions offer a relatively lower payoff than the ordinary straddles. Note that the usual options pay exactly the overprice; no more, no less. In addition, the quadratic strangles allow investors to hedge the positions below and above the strike together. This is very useful in periods of high volatility when large market movements are expected but their direction is unknown.

Sensitivity of a Process for Heating Thin Metal Film Described by the Dual-Phase Lag Equation with Temperature-Dependent Thermophysical Parameters to Perturbations of Lag Times

In the paper, an equation with two delay times (dual-phase lag Equation (DPLE)) in a version that takes into account the dependence of thermophysical parameters (volumetric specific heat and thermal conductivity) on temperature is considered. In particular, an analysis of the sensitivity of transient temperature field in relation to disturbances in delay times (the relaxation and thermalization times) is performed. The sensitivity model concerns the process of heating an ultrathin metal layer with a laser beam. First, the equation with two delay times in the case of temperature-dependent thermophysical parameters is presented. Next, the sensitivity equations with respect to delay times are derived using the direct method. The algorithms for solving the basic and sensitivity tasks are also briefly presented. At the stage of computations, an authorial program based on the implicit scheme of a finite-difference method is developed. In the final part of the paper, examples of numerical solutions (for layers made from gold and nickel) are presented. The research conducted here shows that disturbances in the temperature field are clearly visible and depend, on the one hand, on the thermophysical parameters of the material, and on the other hand, on the intensity of heating with an external heat source.

A novel investigation of Casson fluid flow over an infinite plate with constant temperature by constructing the absorbing boundary conditions

Casson fluid has been widely used in blood, honey, chocolate, and various polymer fluids, for which the investigation of its underlying flow and heat transfer mechanisms in complex environments is of vital significance. In this paper, the Casson fluid flow over an infinite moving plate with constant temperature is studied, which leads to a coupled system of the velocity and temperature fields. To treat the infinite coupled system more accurately, the artificial boundary conditions in terms of fractional derivatives are developed using the Laplace transform, which transfers the problem in the infinite domain to a model in the bounded domain. The finite difference method is applied to solve the new bounded coupled system. Furthermore, three numerical experiments are presented to demonstrate the validity and accuracy of the transferred model and the numerical scheme. The influence of some important physical parameters on the velocity and temperature fields is also explored. The main findings are that the viscosity of the Casson fluid decreases with an increasing Grashof number leading to an accelerated flow, the thermal conductivity diminishes as the Prandtl number increases resulting in a weakened thermal effect within the flow, and the velocity at the fixed position decreases as the Casson parameter increases.

A boundary element analysis of quasi-potential inviscid incompressible flow in multiply connected airfoil wing

AbstractThe quasi-potential flow in a doubly linked airfoil wing with a sharp trailing edge can be solved using a high-order boundary-element method (BEM). A bicubic Hermite spline interpolation is used to express the unknown, with node derivatives given by finite difference method. The revolutionary traits apply to bleeding-edge settings; specifically, challenges relating to the three-dimensional problem of multiply connected domains are investigated. Numerical validation examples were utilized to assess the implemented BEM approach.

A novel hybrid finite element – Finite difference approach for 3D Magnetotelluric modeling

A new method is developed to solve the Magnetotelluric (MT) forward problem in three dimensions (3D). The finite-differences (FD) and the finite-elements (FE) methods are considered to be popular numerical techniques to solve the partial differential equations in electromagnetics. The FD approach is known to be easily applicable to a given problem while the FE method is much more flexible when a topography is present, however, it is also computationally more expensive to solve the linear matrix equations for the FE due to the number of non-zero entries it populates when compared to the FD’s. For that reason, a new algorithm is offered to mix the FE and FD in a single mesh using hexahedra and rectangular prisms in 3D. By doing that, it is possible to take advantage of the beneficial aspects of these two methods. It was found out that the newly developed hybrid approach (HYB) is almost as fast as the FD method while being very memory efficient which is an important aspect due to the limited memory space of a GPU which is employed to solve the systems iteratively. It was found that the HYB method is as accurate as the FE method while it is a more reasonable choice for larger models due to the solution speed of the FE method as it does not scale well with the increased model size. On average, a speed-up of 1.6 is found between the FE and HYB methods with the three models tested and it increases with the increasing model size. Using a GPU during the iterative solution made it possible to solve the linear systems in seconds and with the help of the HYB method, the memory consumption was found to be less than half of what the FE method needed. This ultimately means that much larger models can be simulated on the limited memory that a GPU has. Finally, the technique shown to implement the HYB approach makes the deployment of it to existing FE codes so much easier.