System Of Discrete Equations Research Articles

Stability estimates for radial basis function methods applied to linear scalar conservation laws

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ2-norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be ℓ2-stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations.

A Three-Grid High-Order Immersed Finite Element Method for the Analysis of CAD Models

The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an efficient, robust, high-order immersed finite element method for complex CAD models. Our approach relies on three adaptive structured grids: a geometry grid for representing the implicit geometry, a finite element grid for discretising physical fields and a quadrature grid for evaluating the finite element integrals. The geometry grid is a sparse VDB (Volumetric Dynamic B+ tree) grid that is highly refined close to physical domain boundaries. The finite element grid consists of a forest of octree grids distributed over several processors, and the quadrature grid in each finite element cell is an octree grid constructed in a bottom-up fashion. The resolution of the quadrature grid ensures that finite element integrals are evaluated with sufficient accuracy and that any sub-grid geometric features, like small holes or corners, are resolved up to a desired resolution. The conceptual simplicity and modularity of our approach make it possible to reuse open-source libraries, i.e. openVDB and p4est for implementing the geometry and finite element grids, respectively, and BDDCML for iteratively solving the discrete systems of equations in parallel using domain decomposition. We demonstrate the efficiency and robustness of the proposed approach by solving the Poisson equation on domains described by complex CAD models and discretised with tens of millions of degrees of freedom. The solution field is discretised using linear and quadratic Lagrange basis functions.

Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme

Hepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0$$\\end{document}, stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when R0=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0=1$$\\end{document}. It is concluded that the endemic equilibrium is globally asymptotically stable when R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0>1$$\\end{document}, by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

Effects of Initial Small-Scale Material Nonlinearity on the Pre-Yield and Pre-Buckling Response of an Externally Pressurized Ring

The effects of initial small-scale material nonlinearity on the pre-yield and pre-buckling response of externally pressurized metallic (plane strain) perfect rings (very long cylindrical shells) is investigated. The cylindrically curved 16-node element, based on an assumed quadratic displacement field (in surface-parallel coordinates) and the assumption of linear distribution of displacements through thickness (LDT), is employed to obtain the discretized system equations. The effect of initial small-scale material nonlinearity (assumed hypo-elastic) on the deformation and stress in the pre-yield and pre-buckling regime of a very long relatively thin metallic cylindrical shell (plane strain ring) is numerically investigated. These numerical results demonstrate that the enhanced responses for metallic rings due to initial small-scale nonlinearity are significant enough to not miss attentions from designers and operators of submersibles alike.

Three-dimensional elastoplastic post-buckling analysis of porous FG plates resting on Winkler/Pasternak foundation using meshless RRKPM

This paper presents a three-dimensional (3D) analysis of the elastoplastic post-buckling behavior of porous functionally graded (FG) plates resting on Winkler/Pasternak foundations under uniaxial and biaxial in-plane loadings using an enhanced meshless approach. The principle of virtual work is used to derive the governing equations, and 3D nonlinear Green–Lagrange strains are considered. Plastic deformation is modeled employing the Prandtl–Reuss flow law and the isotropic hardening von Mises criterion. A novel meshless radial basis reproducing kernel particle approach is used to obtain the discretized system of equations. The Newton–Raphson method, coupled with the arc-length technique, is used to compute post-buckling paths of porous FG plates. Numerical assessments show that post-buckling paths are significantly influenced by porosity parameter, porosity distribution, foundation parameters, material gradient, plate thickness, loading ratio, and boundary conditions (BCs).

Connection between the symmetric discrete AKP system and bilinear ABS lattice equations

In this paper, we show that all the bilinear Adler–Bobenko–Suris (ABS) equations (except Q2 and Q4) can be obtained from symmetric discrete AKP system by taking proper reductions and continuum limits. Among the bilinear ABS equations, a simpler bilinear form of the ABS H2 equation is given. In addition, an 8-point 3-dimensional lattice equation and an 8-point 4-dimensional lattice equation are obtained as by-products. Both of them can be considered as extensions of the symmetric discrete AKP equation.

ДИНАМІЧНА МОДЕЛЬ РЕЗОНАНСНОГО ПЕРЕТВОРЮВАЧА ДЛЯ ВПЛИВУ ЗІ СТОРОНИ ЖИВЛЕННЯ

In the paper, a discrete dynamic model of a full-bridge resonant converter with a symmetrical operating mode has been obtained, which describes the resonant converter as a transfer link with an supply voltage input side and a load current output side. The dynamic model is based on a linear mathematical model of the resonant converter built according to the superposition principle. The structure of the resonant converter with the processes outline function is given and analyzed. The structure of the discrete dynamic model of the resonant converter of the nth order is presented. It is proved that the transfer function of the discrete dynamic model for the outline function can be determined by the transfer function of the continuous system. The resulting dependencies describing the discrete transfer functions of the resonant converter are used to obtain the discrete dynamic model of the double-circuit transformer resonant converter in a synchronous rectifier. For this, the sequence of actions is defined: definition of the transfer function of the continuous dynamic model; obtaining the system of equations in vector-matrix form describing electromagnetic processes in the converter; definition of the system of discrete equations; definition of the transfer function of the circuit. The obtained theoretical results were verified by means of simulation modeling of the considered double-circuit resonant converter. The proposed dynamic model makes it possible to analyze the influence of instability and low-frequency fluctuations of the supply voltage of the resonant converter on the output values. The presented results can be used in the development and improvement of single- or multi-circuit bridge high-frequency resonant converters as the secondary power sources and converters used in ship demagnetization systems. References 25, figures 8, tables 2.

Finite element simulations of double diffusion in a staggered cavity filled with a power-law fluid

Double diffusion refers to a phenomenon where two different components of a fluid (such as heat and mass) exhibit distinct diffusive behaviors. In this study, we employ finite element-based numerical simulations to investigate the phenomenon of double diffusion in a non-Newtonian fluid within a staggered cavity. Mathematically, this system can be understood by coupling the two-dimensional continuity, momentum, energy, and concentration equations. Since the governing equations have been written in a dimensionless form, Galerkin's finite element method is used to find a solution. The velocity profile and temperature are calculated in a function space of quadratic polynomials (P2), while the pressure is calculated in a linear (P1) finite element function space. Discrete systems of nonlinear algebraic equations are resolved through the implementation of Newton's method with appropriate damping and PARDISO solver in the inner loops for solving the sparse linear systems. In this work, the data are presented graphically in the form of streamlines, isotherms, iso-concentration, average Nusselt numbers, average Sherwood numbers, and kinetic energy distribution. Code validation and grid independence study are also provided. Moreover, convective mass transfer is significantly correlated with the Lewis number, as demonstrated by the results. As the power-law index increases, convection also exhibits enhanced as a means of transmitting heat and mass.

Investigation of MHD oxytactic microorganisms with NEPCMs in rectotrapezoidal enclosure with FEM: Applications to energy storage technologies

Nanoparticle-enhanced phase change materials (NEPCMs) have gained significant attention in recent years due to their unique ability to enhance the thermal properties and overall performance of phase change materials (PCMs). The present work investigates the bioconvection flow of nano-encapsulated phase change material within a rectotrapezoidal cavity. The left wall of the is fixed at the constant high temperature whereas the right vertical wall is kept at cold temperature. The higher order Galerkin finite element method (GFEM) has been employed to simulate the proposed problem. The discrete systems of nonlinear equations are linearized by implementing the adaptive Newtons method. Influence of various pertinent parameters such as inclined magnetic field, oxytactic bacteria, radiation and Lewis number has been analyzed. It is observed that the average Sherwood and Nusselt number exhibit a positive correlation with the Hartmann and radiation parameter. Moreover, the average heat transfer exhibits a decreasing trend as the Hartmann, bioconvection Rayleigh and Peclet increase.

Power-series solutions of fractional-order compartmental models

Compartmental models based on coupled differential equations of fractional order have been widely employed in the literature for modeling. An abstraction of these models is given by a system with polynomial vector field. We investigate the use of power series for solving generic polynomial differential equations in any dimension, with Caputo fractional derivative. As is well known, power series convert a continuous formulation into a discrete system of difference equations, which are easily solved by recursion. The novelty of this paper is that we rigorously prove that the series converge on a neighborhood of the initial instant, which is an analogue of the Cauchy–Kovalevskaya theorem. Besides, these series are proved to be continuous with respect to the fractional index. For applications, a general-purpose symbolic implementation of truncated power series is developed, and its execution is illustrated for the fractional SIR epidemiological model.

An accurate coupled method for analysis of transcranial magneto-acoustic-electrical stimulation

In this paper, a numerical method for simulating transcranial magneto-acoustic-electrical stimulation (TMAES) is presented. The smoothed finite element method (SFEM) is applied to calculate the magnetic field and the Rayleigh integral (RI) is used for solving the ultrasonic field. Three-node triangular and four-node tetrahedral elements are set as the background cells for constructing the smoothing domains (SDs). A gradient smoothing technique (GST) is applied over each SD to obtain the magnetic flux density. The discretized system equations are obtained by using the generalized smoothed Galerkin weakform. To solve the sound pressure, the ultrasonic transducer is discretized into a series of micro-elements. The acoustic pressure in the problem domain is then equal to the superposition of that generated by the individual micro-element. Finally, the current density can be calculated by the magneto-acoustic coupling effect. Numerical examples demonstrate that SFEM can improve accuracy, convergence rate, and efficiency to 10, 1.8, and 4 times those of FEM, respectively. In the calculation of coupled fields, SFEM-RI also improves the accuracy by 2.5 times. Additionally, the presented method possesses the property of insensitivity to mesh distortion.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

Power Dissipation and Wear Modeling in Wheel–Rail Contact

This paper is concerned with the modeling of power dissipation due to friction and its relation with wear estimation in wheel–rail contact. Wear is a complex multi-scale and multi-physical phenomenon appearing in rolling contact. Wear is generated by high contact stress and the work of friction forces. This phenomenon leads to the appearing of the worn material in the form of wear debris between contacting surfaces. In contact models, wear is usually described in terms of the wear depth function. This function modifies the gap between the contacting bodies as well as the shape of the surfaces of the wheel and rail in contact. In this paper, besides the wear depth function, the dissipated energy, rather than the contact stress, is taken into account to evaluate the wear impact on rail or wheel surfaces. The dissipated energy allows us to more precisely evaluate the wear debris amount as well as the depth of wear and its distribution along the contact interface. A two-dimensional rolling contact problem with frictional heat flow is considered. The elasto-plastic deformation of the rail is considered. This contact problem is governed by a coupled system of mechanical and thermal equations in terms of generalized stresses, displacement and temperature. The finite element method is used to discretize this problem. A discretized system of equations with nonpenetration and friction conditions is transformed and formulated as a nonlinear complementarity problem. The generalized Newton method is applied to numerically solve this mechanical subproblem. The Cholesky method is used to find the solution of the heat-conductive problem. The dissipated power is evaluated based on the resultant force and slip at a reference point. Numerical results including the distribution of slip velocity, power factor and wear rate are provided and discussed.

Arbitrary mesh‐moving velocity‐based space‐time finite element method for large deformation analysis of solids

SummaryThis paper presents a new velocity‐based space‐time finite element method for computing the large deformation of solids over arbitrary Lagrange/Eulerian moving meshes. The proposed method is oriented toward quasi‐static deformation problems with hypoelastic and neo‐Hookean hyperelastic constitutive models. The Cauchy stress in the weak form of the linear momentum equation is explicitly described in terms of deformation velocity through consistent time integration of objective stress rates transferred in the arbitrary moving mesh. Consequently, the resultant system equation only contains the space‐time nodal velocity as the primary unknown. An iterative algorithm for solving the discretized system equation is implemented, whereby the nonlinearities induced by geometrical change in the computational domain as well as nonlinear constitutive equations are computed in an iterative manner updating the stress and the strain over the moving mesh of the deforming domain. Three benchmark problems, such as uniform beam compression, thick‐cylinder compression with superimposed rigid body rotation and extrusion of a column involving material loss, have been solved in order to examine the accuracy of the proposed numerical scheme. The numerical results have revealed the proper computation of the stress and the displacement of the solids undergoing the large deformation and rotation over the arbitrary Lagrange/Eulerian moving meshes, and have shown the validity and applicability of the proposed method with linear convergence of the iterative algorithm.

A Fast Reduced-Order Model for Radial Integration Boundary Element Method Based on Proper Orthogonal Decomposition in the Non-Uniform Coupled Thermoelastic Problems

To efficiently address the challenge of thermoelastic coupling in functionally graded materials, we propose an approach that combines the radial integral boundary element method (RIBEM) with proper orthogonal decomposition (POD). This integration establishes a swift reduced-order model to transform the high-dimensional system of equations into a more manageable, low-dimensional counterpart. The implementation of this reduced-order model offers the potential for rapid numerical simulations of functionally graded materials (FGMs) under thermal shock loading. Initially, the RIBEM is utilized to resolve the thermal coupling issue within the FGMs. From these solutions, a snapshot matrix is constructed, capturing the solved temperature and displacement fields. Subsequently, the POD modes are established and a POD reduced-order model is constructed for the boundary element format of the thermally coupled problem. Finally, a system of low-order discrete differential equations is solved. Numerical experiments demonstrate that the results obtained from the reduced-order model closely align with those of the full-order model, even when considering variations in structural parameters or impact loads. Thus, the introduction of the reduced-order model not only guarantees solution accuracy but also significantly enhances computational efficiency.

An implicit time integration approach for simulation of corona discharges

Corona discharge is a type of electric discharge that occurs when a small conductor carries a voltage strong enough to ionize the surrounding air. The resultant ions are drifted by the electric field and transfer their momentum to neutral particles via elastic collisions, creating a flow effect known as ionic wind that has promising application in flow control. Over the last two decades, a large amount of numerical works have been realized to study this effect. However, simulation of corona discharges with explicit time integration usually requires an excessive amount of time to finish, ranging from days to months in two dimensions, depending on mesh size and physical end time. The reason lies in the profound disparity between physical time scales in a discharge: for example, electron density evolves on the order of picoseconds while the time scale of airflow dynamics is on the order of milliseconds. The objective of this work is therefore to reduce the CPU time of corona discharges. We propose an implicit time strategy in the scope of the Local Field Approximation plasma model. We investigate some algorithms, namely the Lie operator splitting, the Douglas-Rachford method and a nonlinear variant of the Gauss-Seidel algorithm, to solve the system of discrete equations, which are adapted to take into consideration a constraint on the electron density. This constraint is introduced to compensate for sources of electrons from physical processes that are not accounted for in the plasma model; it is expressed by a minimum density, or floor density, imposed on the electrons. Therefore, simplified plasma models can be employed for discharge simulations, for instance a four-specie, four-reaction model is used in this article. The investigation is done on a one-dimensional wire-to-wire corona discharge. After having identified the most suitable algorithm, we carry on to two-dimensional simulations and compare the numerical results with experiment data for different levels of floor density.

A fast preconditioning iterative method for solving the discretized second-order space-fractional advection–diffusion equations

Space-fractional advection–diffusion equations (SFADEs) arise in many application areas. In this paper, we propose fast second-order numerical methods for solving the one- and two-dimensional SFADEs defined on a finite domain. The Crank–Nicolson difference scheme is utilized to discretize the temporal derivatives, the weighted and shifted Grünwald difference operators are employed to discretize the spatial fractional derivatives in SFADEs. We analyze the stability and convergence of the difference schemes by using the matrix analysis method. The coefficient matrix of the discretized system of linear equations has the structure of the sum of an identity matrix and non-symmetric Toeplitz matrices. New τ-matrix approximate preconditioners are proposed for the discretized system of linear equations for both one- and two-dimensional SFADEs, respectively. The generalized minimal residual (GMRES) methods combined with the proposed preconditioners are applied to solve the linear systems. We analyze the convergence rate of the GMRES method for solving the preconditioned linear systems. Numerical results demonstrate the effectiveness of the proposed methods.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

Quasi-Random Discrete Ordinates Method to Radiative Transfer Equation with Linear Anisotropic Scattering

The modeling of energy transport via radiative transfer is important to many practical high temperature engineering applications. Furthermore, the computation of solutions to the radiative transfer equation (RTE) plays a fundamental part in it. The quasi-random discrete ordinates method was developed as an alternative to mitigate the ray effect found in the classical discrete ordinates method solutions. The former method was originally developed for transport problems with isotropic scattering and it is here extended and tested to problems with linear anisotropic scattering. Its main idea is to approximate the integral term of the RTE by a quasi-Monte Carlo integration. The discrete system of differential equations arising from it can be solved by a variety of classical discretization methods, here computed with a SUPG finite element scheme. The novel developments are tested for selected manufactured solutions. The achieved good results indicate the potential of the novel method to be applied to the solution of radiative transfer problems with anisotropic scattering.

Efficient numerical strategies for an implicit volume fraction transfer scheme for single crystal plasticity including twinning and secondary plasticity on the example of magnesium

AbstractIn this work an efficient algorithm for a fully implicit single crystal plasticity routine including twinning and secondary plasticity is presented and implemented for the example of magnesium. The material model uses the volume fraction transfer scheme for the evolution of twinning, that is, plastic slip in newly formed twins (“secondary plasticity”) is resolved. This is considered particularly important, if the volume fractions of some twins reach the order of magnitude of the parent phase. However the resolution of secondary plasticity also implies a large number of unknowns, which is significantly reduced by a newly proposed algorithm. For magnesium a hardening model based on basal, prismatic, pyramidal a and pyramidal c+a slip modes as well as a tension and a compression twinning mode and yields a total of 18 slip systems and 12 twinning systems. As a special feature of the proposed algorithm, the total amount of 246 unknowns, due to simultaneous slip in the parent phase and the twinned phases, is reduced to only 31 unknowns. Additionally, thermodynamic consistency is ensured by including the second law with the Clausius–Duhem equation, which considers the change of free energy upon twinning as an additional driving force for twinning. Further, the setup of the time discrete nonlinear equation system using midpoint rule, as well as the analytical solution of the algorithmic tangent are given in detail. Finally, the implemented model is tested in finite element simulations and compared to single‐ and polycrystal compression and tension experiments.