Method For Numerical Solution Research Articles

Spectral Collocation Method for Numerical Solution to the Fully Nonlinear Monge-Ampère Equation

Spectral Collocation Method for Numerical Solution to the Fully Nonlinear Monge-Ampère Equation

Analyzing the Diversion Effect of Debris Flow in Cross Channels Utilizing Two-Phase Flow Theory and the Principle of Energy Conservation

The movement process of debris flow in the complex roads system is important for risk evaluation and emergency rescue. This paper presents an in-depth study of the diversion effect of debris flow in cross channels, a common branching structure in both natural and engineered environments, especially in the field of urban debris flow prevention. A mathematical model is established based on the conservation of mass, momentum, and energy, and a solid–liquid two-phase motion equation for debris flow is derived from two-phase flow theory. A numerical solution method, combining the finite difference method and finite volume method, is employed to discretize and solve the equation. The model’s validity and effectiveness are confirmed through a numerical simulation of a typical engineering case and comparison with existing experimental data or theoretical results. This study reveals that debris flow at cross channels exhibits a diversion phenomenon, with some debris flow continuing downstream along the main channel and some diverting into the branch channel. The diversion rate, defined as the ratio of outlet flow to inlet flow of the branch channel, indicates the magnitude of this effect. This research shows that the solid–liquid ratio, inflow, width ratio, height ratio, and angle of the cross channel significantly impact the diversion effect. A series of numerical simulations are conducted by altering these parameters as well as the physical properties of debris flow and boundary conditions. These simulations analyze changes in flow rate, velocity, pressure, and other parameters of debris flow at cross channels, providing insights into the factors and mechanisms influencing the diversion effect. This research offers a robust instrument for comprehending and forecasting the dynamics of urban debris flows. It contributes significantly to mitigating the effects of debris flows on city infrastructure and enhancing the safety of city dwellers.

A 1+ Mechanism Model for Predicting the Mixed-Oil Concentration in Multiproduct Pipelines

Summary Petroleum products are frequently transported successively through the same multiproduct pipeline. Due to turbulent and convective diffusion mass transfer, two adjacent oils will mix with each other, forming a mixed-oil segment. Accurate and rapid prediction of mixed-oil concentration is crucial for the precise management of mixed-oil segments. Conventional 1D modeling methods exhibit shortcomings in accurately representing the asymmetric distribution characteristics of mixed-oil concentration curves, and high-dimensional models are not practically applied due to their prohibitive computational time costs. Building on the 1D model framework, this paper proposes a “1+” mechanism model by considering the convective mass transfer behavior between the turbulent core region and the laminar boundary layer, and new governing equations and corresponding numerical solution methods are also introduced. Simulation experiments affirm the ability of the new model to characterize the asymmetric distribution features of mixed-oil concentration curves, along with its high computational efficiency in engineering applications. This is demonstrated by the computational time of approximately 30 seconds for simulating a pipeline of 300 km in length (Δx = 10 m, Δt = 1 second, CPU: i5-12500H, RAM: 16 GB). When applied to pipelines in industrial scenarios, the new model is shown to accurately predict the distribution of mixed-oil concentration curves. The research findings are significantly beneficial in assisting field personnel to gain advanced insights into the mixed-oil concentration distribution at the station, enabling timely and well-informed strategies for handling mixed-oil segment, thereby enhancing the operational efficiency of multiproduct pipelines.

Development of an optimal adaptive finite element stabiliser for the simulation of complex flows

An optimal adaptive multiscale finite element method(AMsFEM) for numerical solutions of flow problems modelled by the Oldroyd B model is developed. Complex flows experience instabilities due to a phenomena known as the high Weissenberg number problem. The stabilisers are terms in-cooperated into the variational formulation when applying the finite element method. For the selected few stabilisers, numerical experiments are performed to study the convergence of the solutions. These demonstrate that adaptive strategies reduce the computational load of flow simulation. A best performing combination of choice of stabiliser and adaptive strategy is suggested.

Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂t−i△Γ where △Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

Pell Collocation Approach for the Nonlinear Pantograph Differential Equations

Pantograph equations, which we encounter in the branches of pure and applied mathematics such as electrodynamics, control systems and quantum mechanics, are essentially a particular form of the functional differential equations characterized with proportional delays. This study focuses on exploring the approximate solution to the Pantograph differential equation. Since there is no analytic solutions for this equation class, only the approximate solutions can be obtain. For this purpose, Pell Collocation Method which is one of the numerical solution methods is chosen. As the result of applying the method to the equation, an algebraic equation system has been gained and then the approximate solution has been found by using MATHEMATICA via the given initial conditions. The method is applied to the some test examples and then the results are presented by both graphically and by table. The error estimations show that the method works accurately and efficiently.

Free Vibration of Ogival Circular Arch

This study aims to analyze the free vibration of an ogival arch in which two symmetrical arched shapes are discontinuously met at the mid-arc. Differential equations governing the free vibration of an ogival circular arch were derived, where the rotatory inertia and shear deformation effects are included. The governing equations were numerically solved to calculate natural frequencies and mode shapes. In numerical solution methods, the symmetric and anti-symmetric boundary conditions at the mid-arc were focused rather than the boundary conditions of supported end due to the discontinuity of the mid-arc. For the first time, the free vibration problem for discontinuous arches, such as ogival arches, is solved in this study. Calculation results of this study for natural frequencies are compared well with those of the finite element method. The effects of various arch parameters on natural frequencies were highlighted and discussed in detail.

Fourier–Gegenbauer Pseudospectral Method for Solving Periodic Higher-Order Fractional Optimal Control Problems

Optimal control (OC) problems involve determining the best control strategy for a system to achieve the desired outcome. Fractional-order OC models have emerged in recent decades to represent complex systems that exhibit memory effects and nonlocal interactions. These models offer advantages over the traditional integer-order models. However, a key challenge with existing fractional calculus methods to define fractional derivatives, such as Riemann–Liouville, Caputo, and Grünwald–Letnikov derivatives, is that they often fail to maintain the periodic nature of dynamical systems. This limitation hinders their application to real-world scenarios, in which periodicity is crucial. This study proposes a novel model for periodic higher-order fractional OC problems (PHFOCPs). We achieve this by employing recently developed periodic fractional derivatives that specifically preserve the periodicity within the model. This study outlines also a robust numerical solution method for this new class of problems. The numerical method is applicable for any positive fractional order [Formula: see text] and leverages several key techniques, including a useful change of variables, Fourier collocation, and Fourier and Gegenbauer quadratures, to retain numerical stability and high accuracy. We theoretically prove the exponential convergence of the proposed numerical method for sufficiently smooth periodic functions. The effectiveness of the proposed approach is further validated through various numerical simulations. In essence, this work presents a novel framework for the OC of periodic systems using fractional calculus, overcoming the limitations of classical methods and offering a powerful tool for various scientific and engineering applications.

Towards accelerating particle‐resolved direct numerical simulation with neural operators

AbstractWe present our ongoing work aimed at accelerating a particle‐resolved direct numerical simulation model designed to study aerosol–cloud–turbulence interactions. The dynamical model consists of two main components—a set of fluid dynamics equations for air velocity, temperature, and humidity, coupled with a set of equations for particle (i.e., cloud droplet) tracing. Rather than attempting to replace the original numerical solution method in its entirety with a machine learning (ML) method, we consider developing a hybrid approach. We exploit the potential of neural operator learning to yield fast and accurate surrogate models and, in this study, develop such surrogates for the velocity and vorticity fields. We discuss results from numerical experiments designed to assess the performance of ML architectures under consideration as well as their suitability for capturing the behavior of relevant dynamical systems.

Large Deflection Analysis of Functionally Graded Beam by Using Combining Method

In this study, the large deflection behavior of a circular cross-section beam is examined using the Combining Method (CM). The CM is a numerical solution method that uses block diagrams in the Matlab-Simulink program and weighting coefficients in the Differential Quadrature Method (DQM). The beam material considered is Functionally Graded Material (FGM). Boundary conditions of the beam are taken as clamped-free (C-F) and a singular load is assumed to be applied from the free end of the beam. Geometric nonlinear analysis is performed while calculating the large deflection equations of the beam and performing numerical analysis. The effects of increasing the force applied to the beam, changing the beam cross-section in the longitudinal direction, and changing the material index of the FGM on the extreme deflection behavior of the beam were examined. For comparison purposes, the results obtained from CM are compared with the results obtained from both SolidWorks-Simulation and Ansys-Workbench programs. As a result of the analysis, increasing the applied force causes the x and y coordinates of the end point of the beam to decrease. The change in geometry and material index greatly affects the large deflection occurring in the beam.

Research on the prediction model of energy separation effect of vortex tube based on trajectory deflection characteristics and its numerical solution method

In this paper, in order to establish the energy separation mechanism of the vortex tube, the hydrodynamic behavior of the compressible fluid in the asymmetric cavity space is investigated, and a numerical model of the trajectory deflection behavior is deduced and established; in order to form the optimal design method of the structural parameters of the vortex tube, the force situation of the fluid microelements entering different regions of the vortex chamber of the vortex tube is analyzed, and the trajectory deflection equations are corrected by combining with the expansion behavior of the fluid and the characterizing equations of vortex strength, transportability, and vortex initiation characteristics are given. The characterization equations of vortex strength, transportability and vortex initiation characteristics are given, and the numerical simulation of their influence parameters is carried out; in order to realize the prediction of the vortex tube performance of a given structure, the multifactor Pearson thermodynamic map is used to correlate and analyze the experimental data of vortex tubes reported publicly in the past years, and the polynomial regression equations are designed and established for the prediction of the vortex tube's energy separation effect and the confidence level and the degree of coincidence of the prediction results are examined. The confidence level and degree of agreement of the prediction results were examined. It is found that: the trajectory deflection motion of the compressible fluid in the asymmetric cavity space is the result of the combined effect of structural air pressure bias and the expansion behavior of the incident fluid; in order to improve the vortex strength in the vortex tube, the vortex initiation chamber space should be as small as possible; the increase of the diameters of the hot-end pipe and the cold-end pipe is conducive to the enhancement of vortex strength, but at the same time, it weakens the vortex transport in the heat pipe; the vortex initiation chamber size has a negative correlation with the hot-end temperature rise, and the inlet fluid pressure has a The negative correlation between the size of the vortex chamber and the temperature rise at the hot end, the positive correlation between the increase of inlet fluid pressure and the resulting temperature rise, and the strong correlation between the inlet fluid pressure and the friction coefficient on the effect of energy separation; the predictive equations for the effect of energy separation obtained by the fitting are in good agreement with the real situation.

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

Разработка и исследование подхода к обработке сигналов цифровых измерительных трансформаторов тока и напряжения

During the development of secondary systems of stations and substations, including relay protection, automation, control, electricity metering, etc., qualitative changes have recently occurred due to the introduction of digital technologies. The introduction of digital technologies is supported by innovative development programs such as “Digital Transformation 2030”. The introduction of digital technologies is associated with the use of digital data transmission channels according to uniform standards. Projects of “digital substations” are being actively implemented with the transmission of digital information at all levels, including the primary equipment, the transmission of information both from current and voltage measuring transformers, and from switching devices. Currently, digital instrument transformers are actively being developed and implemented, in which information about measured currents and voltages is presented in digital code. New approaches to measure primary currents and voltages, and transmit information pose new challenges, and open prospects for improving signal processing methods used for relay protection purposes. Thus, the relevant aim of the study is the development and research of approaches to digital processing of signals of digital instrument current and voltage transformers. To solve the problems posed within the framework of this study, analytical and numerical solution methods have been used. Numerical methods are applied using software products such as MathCAD and the Python programming language to generate signals and display them on graphs. To substantiate the reliability, verification of the results obtained by different methods is carried out. The authors have formulated an approach to digital processing of signals obtained using current sensors of a digital instrument transformer. The authors have studied the approach to determine synchronized current vector values by processing information received from primary current converters. The characteristics of the algorithm for obtaining a vector have been analyzed and the dependences of errors on the influence of distorting factors have been plotted. The proposed approach is a development of the existing approach to obtain a vector by getting information from current sensors operating on different physical principles. The results show the advantages of the proposed method. The proposed digital signal processing algorithm is more technically expensive since it contains at least two metrologically calibrated measuring channels instead of one. However, all other things being equal, it allows to increase the accuracy and speed of the digital processing algorithms taken as a prototype. The results obtained can be used to formulate algorithms for measuring relay protection devices.

Fractional-order brucellosis transmission model between interspecies with a saturated incidence rate

In this study, brucellosis dynamics between interspecies are discussed with the Atangana-Baleanu fractional derivative to examine the transmission of brucellosis by its behavior. The recovered compartment, recruitment, and natural death rate for humans are considered for the fractional order model to analyze the transmission dynamics in more detail from an epidemiological point of view. Additionally, the saturated incidence rate is suggested for brucellosis as indirectly transmitted to individuals from the environment. By fixed point theory, it is verified that developed fractional transmission dynamics have a unique solution. The model under consideration employs the Adams-type predictor-corrector method for numerical solution. All comparative results are plotted by MATLAB.

Review of Adams-Bashforth method for numerical solution of first order ordinary differential equations

This project work presents a comprehensive review of the Adams-Bashforth method for the numerical solution of explicit first-order ordinary differential equations (ODEs). The study begins with a historical overview of the development of ordinary differential equations (ODEs), tracing back to the seminal works of Newton, Leibniz, and their contemporaries. The evolution of differential equations as a distinct mathematical discipline and their wide-ranging applications across various fields are discussed. Furthermore, the paper provides an in-depth analysis of the Adams-Bashforth method, a prominent numerical technique for solving ODEs. The method is derived and analyzed using Mathematica Software, demonstrating its efficacy and accuracy in approximating solutions to ODEs. A comparison with existing methods such as the Euler method and the Runge-Kutta method highlights the advantages of the Adams-Bashforth method, particularly in terms of computational efficiency and accuracy. Moreover, fundamental concepts related to ODEs, including the initial value problem, existence, and uniqueness of solutions, are explored in the context of numerical solution methods. The properties of linear multistep methods and their relevance to solving ODEs efficiently are also discussed. In conclusion, this project work provides valuable insights into the Adams-Bashforth method as a powerful tool for the numerical solution of first-order ODEs. By leveraging modern computational tools such as Mathematica Software, the method offers a practical and efficient approach to solving complex differential equations encountered in various scientific and engineering applications.

A computational- experimental model of reactor kinetic for investigating the linearity response of in-core neutron detectors of a low power research reactor

In this study, the linear response of fission chamber (FC) neutron detectors in Isfahan miniature neutron source reactor (MNSR) is investigated through experiments and calculations at different powers of the reactor. The linearity of the response of the main neutron detector in the MNSR is important due to its safety functions in reactor power control. Additionally, determining the linear response region of this detector presents a linear neutron dosimetry field even outside of the reactor core, i.e. at the outlet of external beam tubes. Moreover, it is very difficult to remove and calibrate this neutron detector that has been installed in the reactor core for a long time. For this purpose, an alternative method for measuring the linearity of the response of a neutron detector is extended by investigating the kinetic behavior of reactor. The aim of this method is to calculate the delayed neutrons population in the reactor core and compare with the FC detector response. In this method, the numerical solution method is used to solve the point kinetic equation with 15 delayed neutron groups (6 delayed neutron groups and 9 photo-neutron groups) after inserting a large negative reactivity into the MNSR core by considering temperature and xenon reactivity feedbacks during a coupling procedure. To obtain the xenon reactivity feedback, xenon concentration is calculated by coupling the 135Xe and 135I production equations and the εpPFnlPTnl parameter is obtained by an experimental method. Furthermore, the value of fission cross section (Σf) is calculated using a validated model in the Dragon code. The large reactivities are inserted at two initial power of 150 W and 30 kW to investigate the response of the main FC neutron detector in a cold xenon-free condition and by considering temperature and xenon reactivity feedbacks, respectively. The results demonstrate that the response of the MNSR FC detector is linear in the neutron flux range between 5 × 107 to 1 × 1012 n.cm−2.s−1, which is equivalent to reactor power of 1.5 W to 30 kW. This innovative method can be applied to calibrate the neutron detectors in other research and power reactors.

Multiple-arc optimization of low-thrust Earth–Moon orbit transfers leveraging implicit costate transformation

This work focuses on minimum-time low-thrust orbit transfers from a prescribed low Earth orbit to a specified low lunar orbit. The well-established indirect formulation of minimum-time orbit transfers is extended to a multibody dynamical framework, with initial and final orbits around two distinct primaries. To do this, different representations, useful for describing orbit dynamics, are introduced, i.e., modified equinoctial elements (MEE) and Cartesian coordinates (CC). Use of two sets of MEE, relative to either Earth or Moon, allows simple writing of the boundary conditions about the two celestial bodies, but requires the formulation of a multiple-arc trajectory optimization problem, including two legs: (a) geocentric leg and (b) selenocentric leg. In the numerical solution process, the transition between the two MEE representations uses CC, which play the role of convenient intermediate, matching variables. The multiple-arc formulation at hand leads to identifying a set of intermediate necessary conditions for optimality, at the transition between the two legs. This research proves that a closed-form solution to these intermediate conditions exists, leveraging implicit costate transformation. As a result, the parameter set for an indirect algorithm retains the reduced size of the typical set associated with a single-arc optimization problem. The indirect heuristic technique, based on the joint use of the necessary conditions and a heuristic algorithm (i.e., differential evolution in this study) is proposed as the numerical solution method, together with the definition of a layered fitness function, aimed at facilitating convergence. The minimum-time trajectory of interest is sought in a high-fidelity dynamical framework, with the use of planetary ephemeris and the inclusion of the simultaneous gravitational action of Sun, Earth, and Moon, along the entire transfer path. The numerical results unequivocally prove that the approach developed in this research is effective for determining minimum-time low-thrust Earth–Moon orbit transfers.

Comparative study and validation of a new analytical method for hydraulic modelling of bidirectional low temperature networks

Bidirectional Low Temperature Networks (BLTNs) are an innovative technology of district heating and cooling systems. Because of their intrinsic feature of supplying simultaneous heating and cooling either directly from an energy source/sink or through distributed heat pumps and chillers, circulation pumps installed at each building extract water from the network and a distinct pressure cone is subsequently created. Variations in the amount (and direction) of the extracted water call for accurate hydraulic design and analysis to ensure safe and adequate network operation. This paper proposes a new analytical solution method for fast hydraulic modelling of BLTNs. The analytical method was compared against three numerical solution methods and validated using different current and future operating scenarios of a real-world bidirectional network. In all modelled cases, the calculated mass flows and pressure drops using the new analytical method yielded very small Normalised Mean Bias Error (NMBE) and Coefficient of Variation of the Root Mean Square Error (CV(RMSE)), suggesting excellent agreement between the analytical and numerical solution methods. The new analytical method offers better performance over its numerical counterparts due to its explicit nature, ease of implementation, and quick modelling time.

Analysis of Underlapped Symmetrically Ported Valve-Controlled Asymmetric Cylinder Drive

The valve-controlled cylinder drive system is the most common type among hydraulic applications. Nonlinear behaviour in such systems is inevitable when the valve spool is around its null position. We utilised the component linking method to investigate the nonlinearities in a Moog valve-controlled asymmetric cylinder drive system by simulation in Fortran, in which a generalised concept is introduced and validated by comparing to the experimental results. An X factor is proposed in the generalised concept to describe the asymmetric cylinder state, which is a constant when the cylinder is extending or retracting, but numerically calculated when the valve spool is in the underlap region. This analytical solution is approximately 200 times more computationally efficient than the numerical solution method. This paper utilises the component linking method to simulate the Moog valve-controlled asymmetric cylinder drive system in Matlab Simulink, and proposes an analytical solution for the X factor when the valve spool is in the underlap region. This analytical solution is approximately 200 times more computationally efficient than the numerical solution method.

A Stable Finite Volume Method for Numerical Solution of Time-Tempered Fractional Sine–Gordon Equations

A Stable Finite Volume Method for Numerical Solution of Time-Tempered Fractional Sine–Gordon Equations