Numerically Research Articles

Regularized kernel function methods for the backward heat conduction problem

In this letter, we are interested in the regularized numerical techniques for backward heat conduction problems, which aim at the detection of the previous status of physical field from its present information. It is well known that backward heat conduction problems are ill-posed. The small perturbation in the final status may result in large change in the previous status of physical field. It is difficult to get the stable numerical solutions of backward heat conduction problems. Based on the reproducing kernel theory, the letter aims at finding a new regularized kernel function methods for backward heat conduction problems.

Evaluation of LM 43.8P blade performance at different hub height wind speeds using blade element momentum theory

Abstract An investigation was conducted on a three-bladed rotor similar to AVANTIS AV908. The rotor blade is appropriate for a Class I wind turbine generator and consists of three LM 43.8P blades. The 2.5 MW gearless wind turbine generator has a rated rotational speed of 16 rpm. An analysis of the LM 43.8P blade’s aerodynamics was carried out using the Blade Element Momentum (BEM) Theory to assess the turbine’s performance at different hub height wind speeds. The study employed three BEM models, including the original BEM concept without correction factors, the BEM Theory by DNV/Ris0, and the BEM Theory obtained from GH Bladed. The rotor radius is 45.1 m, and the blade comprises five different airfoils with a design tip speed ratio of 7.557. The axial and tangential induction factors, lift and drag coefficients, aerodynamic forces, and torque profiles as a function of the nondimensional rotor blade were calculated and compared with the numerical solutions obtained from the BEM models at varying upstream wind speeds. The calculation results revealed that with the original BEM concept, the axial induction factor for this particular blade became larger than 0.5 when the upstream wind speed was less than 7.45 m/s. This suggests that the Momentum Theory becomes invalid for lower wind speeds. However, the thrust force, driving force, and torque may be considerably over-predicted with hub height wind speeds larger than 9.50 m/s. It was found that out of the three models considered, the GH Bladed BEM model bears a striking resemblance with the actual power curve of AV908 with a confidence level above 95%, as indicated by a correlation coefficient of 0.99915 and a t-value of 0.51606 using Student’s t-test, implying that the two power curves have no significant difference.

A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

Transfer learning‐based physics‐informed neural networks for magnetostatic field simulation with domain variations

AbstractPhysics‐informed neural networks (PINNs) provide a new class of mesh‐free methods for solving differential equations. However, due to their long training times, PINNs are currently not as competitive as established numerical methods. A promising approach to bridge this gap is transfer learning (TL), that is, reusing the weights and biases of readily trained neural network models to accelerate model training for new learning tasks. This work applies TL to improve the performance of PINNs in the context of magnetostatic field simulation, in particular to resolve boundary value problems with geometrical variations of the computational domain. The suggested TL workflow consists of three steps. (a) A numerical solution based on the finite element method (FEM). (b) A neural network that approximates the FEM solution using standard supervised learning. (c) A PINN initialized with the weights and biases of the pre‐trained neural network and further trained using the deep Ritz method. The FEM solution and its neural network‐based approximation refer to an computational domain of fixed geometry, while the PINN is trained for a geometrical variation of the domain. The TL workflow is first applied to Poisson's equation on different 2D domains and then to a 2D quadrupole magnet model. Comparisons against randomly initialized PINNs reveal that the performance of TL is ultimately dependent on the type of geometry variation considered, leading to significantly improved convergence rates and training times for some variations, but also to no improvement or even to performance deterioration in other cases.

Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

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.

A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent

Weakly singular Volterra integral equations with variable exponent have integral kernels of the form (t−s)−α(t) with 0≤α(t)<1. A Nyström method based on product integration and interpolation by piecewise polynomials is constructed and analysed for the numerical solution of this class of problems. To deal with the weak singularity of typical solutions of such problems, suitably graded meshes are used. A rigorous error analysis proves convergence of the computed solution; moreover, superconvergence is obtained if the quadrature points are well chosen. These results also imply error bounds for the solution of a related iterated collocation method. Numerical experiments demonstrate the sharpness of our theoretical results for the Nyström method.

Investigation on gas–liquid–solid three-phase flow model and flow characteristics in mining riser for deep-sea gas hydrate exploitation

In response to the problem of gas–liquid–solid three-phase flow in deep-sea hydrate extraction pipelines, a gas–liquid–solid three-phase flow model considering the dynamic decomposition of hydrates is established using continuity equations, momentum equations, and energy equations. The numerical solution of the theoretical model is achieved using the finite difference method. Comparing the theoretical model with the experimental results, the results showed that the average error of gas holdup, liquid holdup, solid phase content, gas phase velocity, liquid phase velocity, and solid phase velocity obtained from the theory and experiment are 8.24%, 0.41%, 1.88%, 5.80%, 2.81%, and 2.22%, respectively, which verified the correctness of the theoretical model. On this basis, the influences of hydrate abundance, liquid phase displacement, and wellhead backpressure on the gas–liquid–solid three-phase flow characteristics in the pipeline were investigated, and it was found that the gas holdup rate will increase with the increase in hydrate abundance, liquid phase displacement, and wellhead backpressure, with the influence of hydrate abundance being more sensitive. The liquid holdup rate increases with the increase in hydrate abundance and liquid phase displacement, but decreases first and then increases toward the wellhead position with the increase in wellhead backpressure. The solid phase content decreases with the increase in hydrate abundance, and first increases and then decreases toward the wellhead position as the liquid phase displacement and wellhead backpressure increase. The influence of gas phase velocity on the abundance of hydrates is relatively small, but it increases with the increase in liquid phase displacement. When the wellhead backpressure increases, the instantaneous increase then tends to flatten out. The influence of hydrate abundance on the liquid phase velocity is also relatively small, but it increases with the increase in liquid phase displacement and decreases with the increase in wellhead backpressure. The solid phase velocity will increase with the increase in hydrate abundance and liquid phase displacement, but it will not show significant changes with the change of wellhead backpressure. The research results can provide a theoretical basis for the safety of hydrate mining.

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.

The thermal structure of the Colombian lithosphere: A regional and basin-scale analysis

It is well-established that the thermal state of the lithosphere strongly influences various regional and local geological processes, including crustal deformation, hydrocarbon maturation, hydrogen generation, and geothermal phenomena. Moreover, the thermal structure exhibits high sensitivity to tectonic features, a property of particular significance in Colombia, where three main tectonic plates converge, and lithospheric tearing has been documented. In this contribution, we focus on elucidating the impact of plate architecture on the thermal field in central-eastern Colombia at both shallow and deep levels. To accomplish this, we constructed a series of two-dimensional profiles and derived a numerical solution of the heat equation using the conservative finite difference method. As constraints, we incorporate an inferred distribution of rocks in the deep crust and upper mantle based on global and local lithospheric thickness models. Material parameters for the various rocks, both exposed and inferred, were obtained from the literature. Additionally, we used superficial heat flow estimates and apparent geothermal gradients compiled by the Colombian Geological Survey.Our results suggest a significant influence of the lithospheric Caldas tear on the thermal state of Colombia, with the breaking off occurring in the Nazca plate under the Eastern Cordillera range around 5°N. The modeled asthenospheric heat flow remains approximately 25 mWm−2, except in the northern Eastern Cordillera range, where the background heat flow increases rapidly to 40 mWm−2. Consequently, our model predicts partial melting in the lower crust to the north and a thermally unstable lower crust to the south of the Caldas tear. The material parameters that best fit the surface data suggest the presence of a basement moderately enriched in radioactive elements in the Eastern Llanos basin. After accounting for compaction, we also confirm a strong influence of the tectonic setting on the thermal state of sedimentary basins.

Numerical Solution of Volterra's Integral Equations by Collocation and Taylor Method

In this article, two numerical methods (collocation method and Taylor method) are introduced to solve Volterra integral equation system. In the collocation method, using Bessel polynomials and collocation points, we convert the device into a matrix form and solve the device using the matrix form and obtain an approximate solution for the device. This answer is such that the larger the N becomes, the approximate answer is closer to the exact answer of the device. In Taylor method, With the use of the Taylor series, the system of integral equations is transformed into a matrix equation, and by integrating the output of this new system, a system of algebraic equations is created. By working through this device, we can obtain a rough solution for the Integral Equations device. Using these two methods when the answers are polynomial, we can get real answers.

High order numerical treatment of the Riesz space fractional advection-dispersion equation via matrix transform method

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method.

Analysis of SEIQV Epidemic Model on the Spread of Covid-19 in Jember Regency

Covid-19 is a disease that attacks the respiratory system caused by infection with SARSCoV-2. Efforts to prevent the spread of Covid-19 by vaccination. The spread of disease can be modeled into a mathematical equation. The model used in this study is the SEIQV Model. The disease spread model is analyzed by finding the equilibrium point and the stability of the equilibrium point as well as the basic reproduction number. In the Covid-19 distribution model, a bifurcation analysis is carried out which is needed to determine changes in stability and changes in the number of equilibrium points. Then perform a numerical solution using the fourth-order Runge-Kutta method which is simulated using MATLAB.

Modeling the Dynamic Loads Affecting a Bridge Crane during Start-Up

Introduction. The dynamic loads during the start-up of a bridge crane can cause excessive stress in the structure and components, leading to potential safety hazards and increased wear and tear. To reduce the influence of the dynamic loads, various strategies can be implemented including optimization of the acceleration and deceleration profiles, using the soft start controls, implementing the vibration damping systems. It is vital to ensure that the proper crane maintenance and inspection protocols are in place. By reducing the impact of dynamic loads during the start-up, the overall performance and longevity of a bridge crane can be improved, ultimately enhancing safety and efficiency of the industrial operations. The present research offers a new approach to improving the efficiency and safety of industrial operations by providing a more precise account of the dynamic loads during the start-up of a bridge crane. The objective of this study is to develop a mathematical model for investigating the mechanical properties of the bridge cranes by analyzing the dynamic loads that occur during lifting operations.Materials and Methods. The development of the mathematical model was based on the kinetic model of the system, which included three connecting blocks and two flexible connections for a more accurate description of the bridge crane structure. Lagrange’s equations incorporating the information about the geometry and structure of a bridge crane were used. They made it possible to describe the motion of a system with the multiple elements and degrees of freedom. Processing and analysis of the results of the mathematical model were carried out in the MATLAB program using the Runge-Kutta method.Results. As a result of the research, a mathematical model was developed to study the dynamic loads affecting a bridge crane during lifting operations. Graphs describing the dependences of speed, acceleration, load, and rope angle over time, and their influence on the crane beam were plotted. The changes in these parameters over time, including their maximum values, were analyzed. The reasons for load changes and factors influencing the extension of lifting machines’ service life as well as reducing metal consumption during production thereof were identified.Discussion and Conclusion. The developed mathematical model and its numerical solution using the specialized software (MATLAB) allow for conducting the dynamic analysis of the bridge crane structures and determining the optimal design solutions. The analysis of the factors influencing the load changes leads to the conclusion that the use of this model can significantly reduce the load magnitudes and metal consumption, as well as increase the service life of lifting machines. The results obtained with the developed mathematical model and its numerical solution are useful for optimizing the crane structures, providing compliance with the operational requirements, and extending the service life of lifting machines.

Validity, Verifiability, and Confirmability: A Critique of Multiphase Packed Bed Modeling

The pseudocontinuum models of reactions in packed beds are complicated, and an assessment of the reliability of the predictability of their numerical solution is difficult. The predictability reliability depends on validity and verifiability, whereas the numerical solutions of models of reactions in packed beds cannot be validated or verified. Scientific acceptability cannot commence by metaphysics alone, and the truth of the speculative justifications of the results of the numerical models without robust empirical confirmation is a matter of chance occurrence. Adherence to the principles of noncontradiction and mathematical consistency seems to be the minimal criterion if a pseudocontinuum model is to demonstrate a degree of reliability in prediction, simulation, and design. This article is an exposition of the verifiability, validity, and confirmability characteristics of multiphase multidimensional models of reactions in packed beds. It addresses the difficulties of validation and the complexities of construction of models of reactions in packed beds by modeling kinetic data directly to show that often the claims of validity, verifiability, or confirmability of the results of multidimensional or even one-dimensional models of chemical reactions in packed beds, in spite of robust statistical tools, should be viewed with some degree of skepticism.

Implementation of a Digital Model of Thermal Characteristics Based on the Temperature Field

Introduction. Computer modeling allows engineers to make valid design decisions by accurately assessing the thermal characteristics of design objects. The implementation of digital twin technology in the process of designing technical facilities is the current direction of scientific research and development. To do this, it is necessary to develop computer models whose accuracy meets the requirements for digital twins. However, the scientific literature does not widely present the results of research aimed at implementing digital twin technology in the design process. The general issues related to the use of digital twins in various industries are mainly considered. Therefore, the objective of this study was the development of a digital model and a comparative analysis of the accuracy of calculations of thermal characteristics of the design object.Materials and Methods. The main tool for conducting the research was the methodology proposed by the authors for developing a computer model of thermal characteristics for the implementation of digital twin technology. The numerical solution was implemented through constructing a thermal model for calculating the temperature field based on the finite element method in the ANSYS engineering analysis system from ANSYS, Inc. (USA). For the analytical solution, a computer model of thermal characteristics developed on the basis of the state-space method, implemented in the ANSYS Twin Builder module, was used. The state-space model was matched to the behavior of the original thermal model through approximating the transfer function to the stepwise response of the thermal load using the time domain vector approximation method. Verification of the constructed analytical model was carried out in the engineering calculation system MATLAB from the MathWorks company (USA). The research was carried out for a 400V machine model manufactured by NPO “Stankostroenie” LLC, Sterlitamak (Russia).Results. The developed digital model makes it possible to calculate the thermal characteristics of the design object with high accuracy. The results of the comparative analysis showed a high degree of correspondence between the values of thermal characteristics obtained using the proposed digital model and the results of numerical simulation. The maximum error in calculating thermal characteristics did not exceed 0.1ºC.Discussion and Conclusion. Computer modeling that combines numerical calculation methods and a scientific approach based on digital twin technology, provides obtaining the result as close as possible to the results of experiments. The digital model proposed in the study is an effective solution, since it provides performing calculations to evaluate thermal characteristics in real time, which is one of the most important requirements for the implementation of digital twin technology.

Numerical heat transfer analysis of bio convective flow of non-Newtonian nanofluid with Darcy-Forchheimer effect over a curved stretching surface

The main aim of this study is to investigate the behavior of bio-convective flow of 2nd grade micropolar nanofluid with Darcy-Forchheimer law over a curved porous stretching surface. The fluid flow analysis includes novel aspects such as the nonuniform heat source/sink, Joule heating, magnetic field, chemical reaction, and thermal radiation. Furthermore, the boundary of the stretching surface is acquired the stratification conditions. A suitable conversion method is employed to transform the flow model into a collection of ordinary differential equations, characterized by their nonlinearity. The bvp4c technique on the MATLAB is utilized to acquire numerical solutions for this set of nonlinear equations. The characteristics of various influential parameters on the velocity, temperature, concentration, and microorganism density profiles are observed and discussed. In the present problem, it is observed that by the growth of micropolar parameter enhances the angular fluid velocity. Moreover, larger values of the magnetic parameter, Darcy number, and porosity parameter result in a decrease in the fluid’s linear velocity. The temperature profile exhibits a decreasing pattern with the elevation thermal stratification parameter, while the opposite trend shows by the increment of the radiation parameter and Eckert number. Additionally, the concentration of nanoparticles decreases as the Schmidt number and Brownian motion parameter increases.

On high-order schemes for the space-fractional conservative Allen–Cahn equations with local and local–nonlocal operators

In this study, we focus on two fractional conservative Allen–Cahn equations with a nonlocal space-independent operator (called the RSLM operator) and a local–nonlocal space–time dependent operator (called the BBLM operator), respectively. Recently, scholars have found that the fractional Allen–Cahn equation is better than the classical equation for describing the interface thickness. Subsequently, the fractional conservative Allen–Cahn equation was used for conservative fluid dynamics. We find that two fractional conservative Allen–Cahn equations both hold the mass conservation, energy stability, and maximum bound principle. In this study, we construct second- and third-order numerical algorithms for two fractional conservative Allen–Cahn equations, and prove the mass conservation and energy decrease laws for them. To our knowledge, for the conservative Allen–Cahn equation, there are only a few works for high-order schemes with proof of their energy stability. We then present several numerical examples whose results show that the convergence orders of the numerical solutions reach the second- and third-order in time as expected. We also simulate some phase transition behaviors to be mass conservative and energy stable, as described by the fractional conservative Allen–Cahn equations, and have some interesting discoveries. It is found that the BBLM formulation can better maintain small features, but the RSLM formulation is better at capturing the profile of the interface. We also compare two fractional conservation equations with the fractional Cahn–Hilliard equation, whose changes are similar to those of the RSLM formulation. In particular, we propose the variable-step BDF2 and BDF3 methods, and prove the mass conservation and energy stability under a time-ratio constraint.

Thermo-magnetic convection of MHD Casson fluid in a wavy triangular porous cavity with embedded a cold inverted triangle obstacle

The design of enclosures is crucial in thermal engineering applications and technologies, encompassing areas such as electronics, heat transfer devices, power reactors, heating systems, solar panels, and nuclear power plants. Thermal efficiency in microchannels is optimized and improved by using triangular enclosures with varying aspect ratios. Specifically, the purpose of these triangular enclosures with cool cylinders is to reduce energy loss in nanoscale thermal sinks and thermal exchange devices. The current study aims to explore the thermal radiation effects on magnetohydrodynamic MHD Casson fluid flow within a wavy triangular cavity that includes an embedded cold inverted triangle. The inner inverted triangle is kept at a lower temperature, while the wavy bottom wall of the outer triangular cavity acts as an isothermal heat source at a higher temperature. The numerical solution for this mathematical model is obtained using the open-source finite element software COMSOL. In this study, the wavy triangular enclosure is analyzed for key parameters such as Casson parameter β , Hartmann number Ha , Rayleigh number Ra , numbers of undulation N , radiation Rd and inclination γ . The study presents the local distribution of streamlines, velocity profiles, isothermals, and entropy production, along with the Nu avg . The results reveal the rate of heat transport and total entropy production raise with the increasing number of undulations N and the Casson parameter β , while they decrease with increasing Hartmann number Ha . The number Nu avg rises with the increasing radiation parameter Rd and Rayleigh number Ra . The maximum stream function is observed at an inclination angle of 60 ° .

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.