Effective Numerical Algorithm Research Articles

Implementation, Realization and an Effective Solver of Two-Equation Turbulence Models

Currently, when the Reynolds-Averaged Navier–Stokes (RANS) equations are solved using turbulence modelling, most often the one-equation model of Spalart and Allmaras is used. Then, it is only necessary to solve the RANS equations in conjunction with a single transport equation for modeling turbulence. For this model, considerable assessment and analysis has been performed, allowing the possibility of a reliable solution method for an eddy viscosity required to compute the Reynolds stresses in the RANS equations. Such evaluation along with analysis has not been performed for similar performance with two-equation models of the k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} type. The primary objective of this paper is to present and discuss the components of an effective numerical algorithm for solving the RANS equations and the two transport equations of k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document}-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} type turbulence models. All the important details of the turbulence model as actually implemented are given, which is sometimes not done in various papers considering such modeling. The viability and effectiveness of this solution algorithm are demonstrated by solving both two-dimensional and three-dimensional aerodynamic flows. In all applications, a linear rate of convergence without oscillations or other evidence of unstable behavior is observed. This behavior is also particularly true when the proposed algorithm is applied to systematically refined mesh sequences, which is generally not observed with algorithms solving more than one transport equation. Thus numerical integration errors are systematically reduced, allowing for a significantly more reliable assessment of the effectiveness of the model itself. Additionally, in this paper analysis of the solution algorithm, including linear stability, is also performed for a particular flow problem.

Taking into account a priori information in the iterative reconstruction of images of foundry products

Methods of restoring images and properties of non-destructive testing objects based on solving inverse problems (problems of restoring distribution functions of unknown characteristics of an object based on the results of indirect measurements) are considered. Management methods are based on solving inverse problems and allow you to get the most complete information about the distributed properties of an object. The need to attract additional information imposes serious restrictions on the development of universal applied algorithms for solving incorrectly set tasks. As a rule, individual additional information is available for each specific non-destructive testing task. An effective numerical algorithm for solving an incorrectly posed problem should be focused on taking this information into account at each stage of the solution search. When solving an applied problem, it is also necessary that the algorithm corresponds to both the measuring capabilities and the capabilities of available computing tools. The problem of low-projection X-ray tomography is always associated with a lack of initial data and can only be solved using a priori information. To introduce the necessary additional information into the numerical algorithm, the methods of iterative reconstruction of tomographic images are identified as the most suitable. One of the approaches to the presentation of this kind of information is described. A practical solution to this problem will expand the scope of the X-ray tomography method.

Analysis and achievement for fractional optimal control of Hepatitis B with Caputo operator

Deciphering the effect of long memory on dynamic behavior is important for controlling disease transmission. Employing fractional order calculus, here, we propose a model with Caputo operator to emphasize the regulation effect of the long memory on the optimal control of Hepatitis B. Considering four different health classes of populations, susceptible, acute infections, chronic infections, and recovered, we try to elucidate the existence of the optimal solution of the fractional optimal control problems (FOCP) by applying a strategy of isolation, treatment, and vaccination. Initially, we discuss the locally asymptotically stability of the equilibrium solutions of the fractional differential model. Sensitivity analysis and numerical simulations are also carried out to verify how parameter changes affect the system dynamics, that is, the threshold effect depending on the basic reproduction number, which provides inspiration for constructing the controlled system. After that, the necessary conditions for the optimality of the Hepatitis B system are derived to focus on the Caputo fractional operator with a singular kernel. An effective numerical simulation algorithm based on trapezoidal approximation is used to validate the control effect in the sense of the transient response by comparing the difference between the fractional-order and integer-order derivatives, and the numeric results can vividly show the attenuation effect of memory on control. This study may provide a powerful rationale for uncovering the importance of the long-memory effects in disease control.

Mathematical modeling of salt concentration change process in two-layer aqueous media

A mathematical model was developed to monitor and predict changes in salt concentration in groundwater in a two-layer environment, representing the process of salinization and swamping. A qualitative analysis of scientific research on the problem is provided. Since the process is represented by a differential equation, effective numerical computational algorithms were developed to solve the problem of salt concentration change in groundwater based on a stable disclosure scheme with high-order accuracy relative to time and spatial variables. Mathematical models of salt transport during geo-filtration processes were improved, and computational algorithms were developed, taking into account such important parameters as the physical and mechanical properties: water loss coefficient, diffusion coefficients, and filtration rates depending on the water level. The results obtained based on these developed numerical algorithms were analyzed. Before introducing new technologies, recommendations were developed to predict changes in salt content in water using an effective numerical algorithm. Based on the laws of motion of wastewater flows, it was noted that it is possible to conduct experiments using algorithms to determine the changes in distance and velocity of wastewater propagation with water-soluble chemicals and active properties in soil layers.

Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods

In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the velocity satisfy the tangential condition. The first-order backward Euler scheme and second-order BDF scheme are used to discretize the time separately, and the stability of the two schemes is analyzed. The mixed finite element pair (P2,P1) is applied to discretization of space. Finally, numerical examples are given to verify the accuracy and effectiveness of the proposed method.

Shifted Bernstein–Legendre polynomial collocation algorithm for numerical analysis of viscoelastic Euler–Bernoulli beam with variable order fractional model

In this paper, a kinetic equation of Euler–Bernoulli beam is established with variable order fractional viscoelastic model. An effective numerical algorithm is proposed. This method uses a combination of shifted Bernstein polynomial and Legendre polynomial to approximate the numerical solution. The effectiveness of the algorithm is tested and verified by mathematical examples. The dynamic behavior of viscoelastic beams made of two materials under various loading conditions is studied.

Solving Surface Deformation of Radio Telescope Antenna by Artificial Neural Network

Recent investigations have derived the relation between the near-field plane amplitude and the surface deformation of reflector antenna, namely deformation-amplitude equation (DAE), which could be used as a mathematical foundation of antenna surface measurement if an effective numerical algorithm is employed. Traditional algorithms are hard to work directly due to the complex mathematical model. This paper presents a local approximation algorithm based on artificial neural network to solve DAE. The length factor method is used to construct a trial solution for the deformation, which ensures the final solution always satisfies the boundary conditions. To improve the algorithm efficiency, Adam optimizer is employed to train the network parameters. Combining the application of the data normalization method proposed in this paper and a step-based learning rate, a further optimized loss function could be converged quickly. The algorithm proposed in this paper could effectively solve partial differential equations without boundary conditions such as DAE, which at the same time contains the first-order and the second-order partial derivatives, and constant terms. Simulation results show that compared with the original algorithm by Fast Fourier transform, this algorithm is more stable and accurate, which is significant for the antenna measurement method based on DAE.

A Meshless Method for Retrieving Nonlinear Large External Forces on Euler-Bernoulli Beams

We retrieve unknown nonlinear large space-time dependent forces burdened with the vibrating nonlinear Euler-Bernoulli beams under varied boundary data, comprising two-end fixed, cantilevered, clamped-hinged, and simply supported conditions in this study. Even though some researchers used several schemes to overcome these forward problems of Euler-Bernoulli beams; however, an effective numerical algorithm to solve these inverse problems is still not available. We cope with the homogeneous boundary conditions, initial data, and final time datum for each type of nonlinear beam by employing a variety of boundary shape functions. The unknown nonlinear large external force can be recuperated via back-substitution of the solution into the nonlinear Euler-Bernoulli beam equation when we acquire the solution by utilizing the boundary shape function scheme and deal with a small-scale linear system to gratify an additional right-side boundary data. For the robustness and accuracy, we reveal that the current schemes are substantiated by comparing the recuperated numerical results of four instances to the exact forces, even though a large level of noise up to 50% is burdened with the overspecified conditions. The current method can be employed in the online real-time computation of unknown force functions in space-time for varied boundary supports of the vibrating nonlinear beam.

Change of the Most Probable Escape Path in a Fast-Slow Insect Outbreak System Under Different Noise Amplitude Ratios

Abstract Noise-induced escape from the domain of attraction of a stable state in a fast-slow insect outbreak system is investigated. According to Dannenberg's theory (et al., 2014, “Steering Most Probable Escape Paths by Varying Relative Noise Intensities,” Phys. Rev. Lett., 113(2), p. 020601), only noise amplitude ratio μ will lead to the change of the most probable escape path (MPEP). Therefore, the research emphasis of this paper is to extend their study and discuss the variation of the MPEP in more detail. First, for the case of μ = 1, the MPEP almost moves along the critical manifold. Via projecting the full system onto the critical manifold, a reduced system is obtained, and the action of the MPEP in the full system can be partly evaluated by that in the reduced system. In order to test the accuracy of the computed MPEP, based on the iterative action minimizing method (IAMM), a new relaxation method, which can reduce the central processing unit time, is then presented. Then, as μ converges to zero, an improved analytical method is given, through which a better approximation for the MPEP at the turning point is obtained. And then, when the value of μ is moderate, the MPEP will peel off the critical manifold. To determine the changing point on the critical manifold, an effective numerical algorithm is presented. In brief, a complete investigation on the structural change of the MPEP in a fast-slow insect outbreak system under different noise ratios is given, and the results of the numerical simulation match well with the analytical ones.

Shifted Legendre polynomials algorithm used for the numerical analysis of viscoelastic plate with a fractional order model

An effective numerical algorithm is presented to analyze the fractional viscoelastic plate in the time domain for the first time in this paper. The viscoelastic behavior of the plate is described with fractional Kelvin–Voigt (FKV) constitutive model in three-dimensional space. A governing equation with three independent variables is established. Ternary unknown function in the governing equation is solved by deriving integer and fractional order differential operational matrices of the shifted Legendre polynomials. Error analysis and mathematical example are presented to verify the effectiveness and accuracy of proposed algorithm. Finally, numerical analysis of the plate under different loading conditions is carried out. Effects of the damping coefficient on vibration amplitude of the viscoelastic plate are studied. The results obtained are consistent with the current reference and actual situation. It shows that shifted Legendre polynomials algorithm is suitable for numerical analysis of fractional viscoelastic plates.

Augmented saddle-point formulation of the steady-state Stefan–Maxwell diffusion problem

Abstract We investigate structure-preserving finite element discretizations of the steady-state Stefan–Maxwell diffusion problem, which governs mass transport within a phase consisting of multiple species. An approach inspired by augmented Lagrangian methods allows us to construct a symmetric positive definite augmented Onsager transport matrix, which in turn leads to an effective numerical algorithm. We prove inf-sup conditions for the continuous and discrete linearized systems and obtain error estimates for a phase consisting of an arbitrary number of species. The discretization preserves the thermodynamically fundamental Gibbs–Duhem equation to machine precision independent of mesh size. The results are illustrated with numerical examples, including an application to modelling the diffusion of oxygen, carbon dioxide, water vapour and nitrogen in the lungs.

Computational Study on Skin Tissue Freezing Using Three-Phase Lag Bioheat Model

Abstract This paper considers the three-phase lag (TPL) bioheat model to study the phase change phenomena in skin tissue during cryosurgery. The considered TPL model is based on the model of thermo-elasticity, i.e., the combination of the rate of thermal conductivity and new phase lag (τv) due to thermal displacement. An effective heat capacity-based numerical algorithm is established to solve the nonlinear governing equation for biological tissue freezing. Space and time derivatives appearing in the mathematical model are approximated using the radial basis function (RBF) and finite difference method (FDM), respectively. The impact of three nonclassical models, single-phase lag (SPL), dual-phase lag (DPL), and TPL, on the freezing process is studied. The effects of phase lags involved in the models on freezing are also part of this study.

LOCALIZATION ANALYSIS OF DAMAGE FOR ONE-DIMENSIONAL PERIDYNAMIC MODEL

Peridynamics is a recently developed extension of continuum mechanics, which replaces the traditional concept of stress by force interactions between material points at a finite distance. The peridynamic continuum is thus intrinsically nonlocal. In this contribution, a bond-based peridynamic model with elastic-brittle interactions is considered and the critical strain is defined for each bond as a function of its length. Various forms of length functions are employed to achieve a variety of macroscopic responses. A detailed study of three different localization mechanisms is performed for a one-dimensional periodic unit cell. Furthermore, a convergence study of the adopted finite element discretization of the peridynamic model is provided and an effective event-driven numerical algorithm is described.

Blind motion deconvolution for binary images

Binary images are prevalent in digital systems and have a wide range of applications including texts, fingerprint recognition, handwritten signatures, stellar astronomy, barcodes, and vehicle license plates. The recorded binary images are often degraded by blur and additive noise due to environmental effects and imperfections in the imaging system. In this paper, we study the problem of recovering the sharp binary image and the blur kernel from the motion degraded observation. We propose a new minimization model by using the binary prior of image pixel and the l0 norm of image gradient to enforce the estimated image to be binary and the image gradient to be sparse respectively. An effective numerical optimization algorithm is applied for solving the proposed model. Extensive experiments for blind binary image deconvolution demonstrate that the proposed method outperforms some existing state-of-the-art methods in terms of visual quality and peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM).

Analysis of a fractional model for HIV CD$ 4^+ $ T-cells with treatment under generalized Caputo fractional derivative

In this paper, a mathematical model of generalized fractional-order is constructed to study the problem of human immunodeficiency virus (HIV) infection of CD$ 4^+ $ T-cells with treatment. The model consists of a system of four nonlinear differential equations under the generalized Caputo fractional derivative sense. The existence results for the fractional-order HIV model are investigated via Banach's and Leray-Schauder nonlinear alternative fixed point theorems. Further, we also established different types of Ulam's stability results for the proposed model. The effective numerical scheme so-called predictor-corrector algorithm has been employed to analyze the approximated solution and dynamical behavior of the model under consideration. It is worth noting that, unlike many discusses recently conducted, dimensional consistency has been taken into account during the fractionalization process of the classical model.

Nonlinear mathematical model and numerical algorithm for monitoring and predicting the concentration of harmful substances in the atmosphere

The article developed a nonlinear mathematical model and an effective numerical algorithm for monitoring and forecasting the process of spreading industrial emissions in the atmosphere, taking into account the orography of the area and the speed of movement of fine substances. Multidimensional partial differential equations describe the model with corresponding initial, internal, and boundary conditions of the third kind to consider mass transfer across the interface. In the work, to prove the adequacy of the developed mathematical apparatus, the calculated data are compared with field data taken from a specific object, in which the computational algorithm provides sufficient accuracy of the solution, based on which it can be concluded that the developed mathematical model and the numerical algorithm have a certain advantage over others numerical methods.

Unsteady flow of gas in a semi-infinite porous medium: a numerical investigation by using RBF-DQM

The study aims to investigate how an effective numerical algorithm can be used to solve the unsteady isothermal flow of gas through a semi-infinite micro-nano porous medium. The unsteady gas equation used here is a nonlinear, second order differential equation with two points of boundary value on the semi-infinite domain. The study uses RBFs-DQ method in which the derivative value of function with respect to the point is directly approximated by a linear combination of all functional values in the entire domain. The main purpose of using this method is to determine the weight of coefficients. The study also used Gaussian (GS) function to approximate the solution of the mentioned equation. The efficiency and accuracy of this method are verified by the comparison made between our results and other numerical methods including shooting method, RBF.G and Wavelet Legendre collocation method. As a results, by comparison made to other numerical methods this showed a Meshfree (RBF-DQ) method which its validity is equal or even more valid than numerical methods.

MATHEMATICAL MODELING OF CANCELLOUS BONE TISSUE ADAPTATION APPLIED TO THE HUMAN MAXILLODENTAL SYSTEM

The bone tissue in different parts of the skeleton conforms to Wolff’s law: it aims to become optimal for the loading which acts on the corresponding bone; the bone is remodelling by means of osteosynthesis and resorption mechanisms. The modern problems of biomechanics demand research on the history of formation of bone structures in the course of time at both physiological and pathological loadings. Ever changing loadings of different nature have influence on development and functioning of the trabecular bone tissue. The mandible is one of the most liable to external and internal changes bones. Very often one has to deal with pathological changes caused by incorrect loading of different regions of bone tissue due to dysfunction of a dentition, a temporomandibular joint and so on. For example, the Popov-Godon’s syndrome which connects with tooth loss is accompanied by pathological remodelling of the surrounding bone tissue. Thus, the mathematical modeling of the cancellous bone tissue behavior in the human maxillodental system is one of the most topical problems of biomechanics and medicine. Trabecular bone tissue is a heterogeneous, porous, anisotropic material. Heterogeneity of spongy structure can be described by methods of quantitative stereology. At the same time, structural features of the trabecular bone can be described by means of the fabric tensor. This is possible to implement if there is both a constitutive relation which connects the stress tensor, the fabric tensor, and the strain tensor, and kinetic equations which describe the evolution of the fabric tensor and bone density. An initial boundary value problem on the trabecular bone tissue remodelling is stated. The effective numerical algorithm allowing to solve the problem is developed. This algorithm is implemented as a complex of problem-oriented programs. Verification of the model and identification of its parameters are carried out. All numerical calculations are performed using the ANSYS software. Trabecular bone tissue evolution is demonstrated on the set of model examples when the stress–strain state is changed. The results demonstrate different character of influence of changes of loading conditions on process of structure formation which follows from Wolff’s law.

Prospects in elongation of railway transition curves

The paper presents an analysis of the elongation of transition curves in relation to railway track alignment correction and modernisation. The analysis is based on numerical computations for a wide range of parameters describing a typical railway geometrical layout with transition curves. The differences between the horizontal ordinates of the existing layout and the layout with the elongated transition curves are evaluated and assessed. The analytical model comprises formulated appropriate theoretical dependencies for each characteristic zone of the layout. The impacts of a circular arc radius and a route intersection angle on the elongation process are considered. Based on the theoretical assumptions, an effective numerical algorithm for comparative analysis of the transition curves’ elongation variants is developed. The value of a new circular arc radius is determined in an optimisation process. For this purpose, the indicators for process evaluation and the constraints determining the feasibility of the design variants are defined.

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

AbstractThe aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.