Numerical Solution Scheme Research Articles

A reference-component coordinate system approach to model the mass transfer of a droplet with binary volatiles

A theoretical framework based on the reference-component centered coordinates was modified to enable the prediction study of the simultaneous absorption and evaporation of droplets consisting of two volatiles. A new equation of Robin boundary condition was imposed at the droplet-ambience interface, coupling with a new numerical scheme for solution. Experimental validation was performed with the following situations: evaporation of single pure droplet and bicomponent droplet, and simultaneous absorption and evaporation of droplet. The model predicted the mass profiles reasonably well for droplet evaporation while over-prediction was found for the case of simultaneous absorption and evaporation of droplet. Further preliminary evaluation has found the necessity to encounter the phenomenon of mass flux depression when predicting the simultaneous absorption and evaporation of droplet. This will provide a potential predictive tool for the processes which involves droplet absorption, such as antisolvent-vapor precipitation and gas scrubbing.

The exact Riemann solver for the shallow water equations with a discontinuous bottom

A new exact Riemann solver for the shallow water equations with a discontinuous bottom is proposed. The algorithm is based on the approach to overcome the non-uniqueness of the Riemann problem solution by assuming that the ‘true’ solution should have the discharge at the bottom discontinuity, which continuously depends on the initial conditions. The solver ensures the existence and uniqueness of a solution for arbitrary initial conditions. The exact Riemann solver is embedded in the Godunov scheme for numerical solution of the shallow water equations to demonstrate its advantage. It is shown that the proposed solver allows one to significantly reduce the number of computational cells for stationary flows and non-stationary ones if the mesh can resolve non-stationary features of the flow. In addition, the practical problem of rainfall-runoff is considered, and the results obtained using the exact Riemann solver show good agreement with observation data.

A methodology for solving the population balance equation based on an embedded reduced order representation

Polydispersed particles are frequently described in terms of a property distribution whose evolution is governed by the population balance equation (PBE). In the present article, we develop a numerical solution scheme for the univariate, spatially inhomogeneous PBE that allows for the economical resolution of the particle size distribution at a moderate accuracy and is robust with respect to growth-dominated applications. The method combines the notion of characteristic curves in particle size space with a constrained Galerkin projection and is informed by a training step for the identification of representative shape functions. This step is based on the proper orthogonal decomposition (POD) of snapshot matrices containing adaptive grid solutions of the PBE obtained in simplified flow configurations. In order to assess the viability, accuracy and convergence properties of the combined PBE-POD approach, we analyze the pure growth and dispersion in a laminar plane jet of a polysized particle population.

A numerical scheme based on the collocation and optimization methods for accurate solution of sensitive boundary value problems

Despite the significant advances in the numerical solution of nonlinear boundary value problems, most of the existing methods still encounter with a high sensitivity to the initial guess. The aim of this paper is to propose a less sensitive robust numerical scheme for accurate solution of sensitive boundary value problems. For this purpose, an orthogonal collocation approach for discretization of the problem is utilized. Thereby, the problem is converted to the solution of nonlinear algebraic equations. However, due to the difficulties of solving the obtained system of nonlinear equations, particularly in providing the proper initial guess, the obtained system of equations is transferred to an optimization problem in which the values of the solution at collocation points are considered as decision parameters. The method finds good results even using not good initial guess for decision parameters and even using a small number of discretization points. Numerical results of five benchmark examples are presented and the efficiency of the method is reported.

Review of the strain-based formulation for analysis of plane structures Part I: Formulation of basics and the existing elements

Since the introduction of the finite element approach, as a numerical solution scheme for structural and solid mechanics applications, various for mulation methodologies have been proposed. These ways offer different advantages and shortcomings. Among these techniques, the standard displacement-based approach has attracted more interest due to its straightforward scheme and generality. Investigators have proved that the other strategies, such as the force-based, hybrid, assumed stress, and as sumed strain provides special advantages in comparison with the classicfinite elements. For instance, the mentioned techniques are able to solve difficulties, like shear locking, shear parasitic error, mesh sensitivity, poor convergence, and rotational dependency. The main goal of this two-part study is to present a brief yet clear portrait of the basics and advantages of the direct strain-based method for development of high-performance plane finite elements. In this article, which is the first part of this study, assump tions and the basics of this method are introduced. Then, a detailed review of all the existing strain-based membrane elements is presented. Although the strain formulation is applicable for different types of structures, most of the existing elements pertain to the plane structures. The second part of this study deals with the application and performance of the reviewed elements in the analysis of plane stress/strain problems.

LDG approximation of a nonlinear fractional convection-diffusion equation using B-spline basis functions

This paper develops new numerical schemes for solution to nonlinear fractional convection-diffusion equations of order β∈[1,2]. We propose the local discontinuous Galerkin methods by adopting linear, quadratic, and cubic B-spline basis functions and prove stability and optimal order of convergence O(hk+1) for the fractional diffusion problem. This method transforms the equation into a system of first-order equations and approximates the solution of the equation by selecting the appropriate basis functions. The B-Spline functions significantly improve the accuracy and stability of the method. The performed numerical results demonstrate the efficiency and accuracy of the proposed scheme in different conditions and confirm the optimal order of convergence.

Efficient geometric nonlinear elastic analysis for design of steel structures: Benchmark studies

Most structural design codes require the consideration of geometric nonlinearities or second-order effects. Analysis of second-order forces and moments in structural members requires the solution of a nonlinear system of equilibrium equations. There are many ways to solve such a system, with varying levels of accuracy and computational expense. Numerical solution schemes for the geometric nonlinear analysis of structures have been shown to produce extremely accurate results when applying small load increments and/or implementing multiple linear analyses or iterations per increment. This paper proposes the use of an approximate solution scheme that utilizes two linear analyses within a single load increment, thereby simplifying the second-order elastic analysis to a predictor-corrector type algorithm with improved computational speed. Further contributing to its efficiency is the ability to use the analysis results from within the serviceability design process in the predictor step, thereby requiring only one linear corrector analysis per load combination investigated. The predictor-corrector scheme is proposed as a substitute for either an exact geometric nonlinear elastic analysis or the approximation of second-order results by amplifying first-order results. Twenty-two steel benchmark frames are used to assess the method's accuracy in comparison with a more exact solution scheme. Comparisons are also made, where possible, with interstory drift amplifiers applied to first-order results. The findings demonstrate the method's ability to maintain sufficient accuracy while significantly improving computational efficiency. Use of the method is demonstrated with two illustrative examples, and advantages and limitations are discussed, as well as insights regarding frame sensitivity to second-order effects.

Effect of Viscous Damping Models on Displacement Ductility Demands for SDOF Systems

The viscous damping is a highly idealized but mathematically convenient way of representing the energy dissipation mechanisms not related to the hysteretic response. In this study, the tangent-stiffness proportional Rayleigh damping (TSPRD) appropriate for inelastic hysteretic structures outfitted with the supplemental damping devices is utilized in inelastic dynamic analysis of simple single-degree-of-freedom (SDOF) structures. A numerical procedure of dynamic equilibrium equation for SDOF systems with TSPRD model is derived for development of the general numerical solution scheme. By specifying various combinations of tangent-stiffness and mass proportional damping terms in TSPRD model, the influence of viscous damping models on displacement ductility demands is investigated. The results indicate that the difference in ductility demands for various damping models is considerable, depending on the periods of vibration, relative lateral strength, hysteretic models, stiffness deterioration and initial damping ratios. The constant-strength displacement ductility spectra, by considering both tangent-stiffness proportional and mass proportional damping, are developed in terms of site conditions, hysteretic rules and initial damping ratios.

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

Numerical computations and theoretical investigations of a dynamical system with fractional order derivative

This manuscript is devoted to consider population dynamical model of non-integer order to investigate the recent pandemic Covid-19 named as severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) disease. We investigate the proposed model corresponding to different values of largely effected system parameter of immigration for both susceptible and infected populations. The results for qualitative analysis are established with the help of fixed-point theory and non-linear functional analysis. Moreover, semi-analytical results, related to series solution for the considered system are investigated on applying the transform due to Laplace with Adomian polynomial and decomposition techniques. We have also applied the non-standard finite difference scheme (NSFD) for numerical solution. Finally, both the analysis are supported by graphical results at various fractional order. Both the results are comparable with each other and converging quickly at low order. The whole spectrum and the dynamical behavior for each compartment of the proposed model lying between 0 and 1 are simulated via Matlab.

Solution of the Fokker–Planck Equation by Cross Approximation Method in the Tensor Train Format

We propose the novel numerical scheme for solution of the multidimensional Fokker–Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.

On entropy generation due to transient MHD radiative free convection with induced magnetic field in a porous medium channel

AbstractThe communication pertains to transient magnetohydrodynamics radiative free convection flow with induced magnetic field in a porous medium channel of width L. The setup is subjected to a uniform magnetic field normal to the channel walls and stands for the induced magnetic field. A Cartesian coordinate system is chosen where x*‐axis is taken along the left channel wall and y*‐axis is normal to it. At the time t* = 0, both the walls and the fluid bear a fixed temperature . At the time t* > 0, that is, the onset of the convection, the temperatures of the walls at y* = 0 and at y* = L are just changed to and , respectively, where < and < . The radiative flux qr in the energy equation is assumed to follow the Rosseland approximation. The Keller‐box method has been invoked to solve the governing system encompassing the momentum, energy, and magnetic field equations. The numerical solution strategy yields velocity and temperature distributions, skin friction, and Nusselt number. The velocity and temperature fields are used to compute entropy. Distributions for entropy and Bejan number are shown graphically through two‐dimensional (2D) and 3D plots and discussed. The central objective of this study is to analyze inherent thermodynamic irreversibility. The study has direct applications in plasma aerodynamics and nuclear engineering control where the magnetic field is used as a tool to control flow and heat characteristics. The present study is novel in terms of the objective of the study and the implicit finite difference solution strategy called the Keller‐box method. From the 2D and 3D plots, it is found that the entropy and Bejan number experience qualitative and quantitative variations for varying Brinkman number, radiation parameter, Darcy number, magnetic Prandtl number, Grashoff number, and time. The skin friction and Nusselt number at the channel walls witness qualitative variations for varying parameters.

On the application of variational theory to urban networks

The well-known Lighthill–Whitham–Richards (LWR) theory is the fundamental pillar for most macroscopic traffic models. In the past, many methods were developed to numerically derive solutions for LWR problems. Examples for such numerical solution schemes are the cell transmission model, the link transmission model, and the variational theory (VT) of traffic flow. So far, the eulerian formulation of VT found applications in the fields of traffic modelling, macroscopic fundamental diagram estimation, multi-modal traffic analyses, and data fusion. However, these studies apply VT only at the link or corridor level. To the best of our knowledge, there is no methodology yet to apply VT at the network level. We address this gap by developing a VT-based framework applicable to networks. Our model allows us to account for source terms (e.g. inflows and outflows at intersections) and the propagation of spillbacks between adjacent corridors consistent with kinematic wave theory (KWT). We show that the trajectories extracted from a microscopic simulation fit the predicted traffic states from our model for a simple intersection with both source terms and spillbacks. We also use this simple example to illustrate the accuracy of the proposed model, and the ability to model complex bottlenecks. Additionally, we apply our model to the Sioux Falls network and again compare the results to those from a microscopic KWT simulation. Our results indicate a close fit of traffic states, but with substantially lower computational cost. The developed methodology is useful for extending existing VT applications to the network level, for network-wide traffic state estimations in real-time, or other applications within a model-based optimization framework.

Computing Dynamic User Equilibrium on Large-Scale Networks Without Knowing Global Parameters

Dynamic user equilibrium (DUE) is a Nash-like solution concept describing an equilibrium in dynamic traffic systems over a fixed planning period. DUE is a challenging class of equilibrium problems, connecting network loading models and notions of system equilibrium in one concise mathematical framework. Recently, Friesz and Han introduced an integrated framework for DUE computation on large-scale networks, featuring a basic fixed-point algorithm for the effective computation of DUE. In the same work, they present an open-source MATLAB toolbox which allows researchers to test and validate new numerical solvers. This paper builds on this seminal contribution, and extends it in several important ways. At a conceptual level, we provide new strongly convergent algorithms designed to compute a DUE directly in the infinite-dimensional space of path flows. An important feature of our algorithms is that they give provable convergence guarantees without knowledge of global parameters. In fact, the algorithms we propose are adaptive, in the sense that they do not need a priori knowledge of global parameters of the delay operator, and which are provable convergent even for delay operators which are non-monotone. We implement our numerical schemes on standard test instances, and compare them with the numerical solution strategy employed by Friesz and Han.

Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model

A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation

In this paper, we use a numerical method for solving the nonlinear Black–Scholes partial differential equation of the European option under transaction costs, which is based on the nonstandard discretization of the spatial derivatives. The proposed scheme, in addition to the unconditional positivity, is stable, consistent, and monotone. In order to illustrate the efficiency of the new method, numerical results have been performed by four models.

Bernstein polynomial induced two step hybrid numerical scheme for solution of second order initial value problems

This paper presents a two-step hybrid numerical scheme with one off-grid point for the numerical solution of general second-order initial value problems without reducing to two systems of the first order. The scheme is developed using the collocation and interpolation technique invoked on Bernstein polynomial. The proposed scheme is consistent, zero stable, and is of order four($4$). The developed scheme can estimate the approximate solutions at both steps and off-step points simultaneously using variable step size. Numerical results obtained in this paper show the efficiency of the proposed scheme over some existing methods of the same and higher orders.

Numerical solution strategy for natural convection problems in a triangular cavity using a direct meshless local Petrov-Galerkin method combined with an implicit artificial-compressibility model

Thermal analysis of natural convection in a cavity is solved by numerical approach. The direct meshless local Petrov–Galerkin method combined with an implicit artificial-compressibility model is proposed here to simulate laminar natural convective heat transfer in triangular cavities. The spatial terms of the governing equation are discretized using the direct meshless local Petrov–Galerkin method with the Dirac function as the test function. The method uses an artificial-compressibility scheme to couple the pressure with the velocity components directly. The semi-algebraic system is solved using an implicit backward Euler pseudo-time method. Kovasznay flow problem is presented to examine the accuracy of proposed method. The validation of the obtained results by using experimental data is also provided, and four test problems — natural convection in an equilateral triangular cavity, a right-angled triangular cavity, an isosceles triangular cavity, and an irregular triangular cavity — are solved by using the proposed method. The numerical results obtained using the proposed method showed excellent agreement with those obtained using the conventional methods available in the literature.