Numerically Research Articles

Electromagnetic scattering by curved surfaces and calculation of radiation force: Lattice Boltzmann simulations

This study aims to investigate the effectiveness of the lattice Boltzmann method (LBM) in studying the scattering of electromagnetic waves by curved and complex surfaces. The computation of Maxwell’s equations is done by solving for a pair of distribution functions, which evolve based on a two-step process of collision and streaming. LBM bypasses the need for expansion via vector spherical harmonics and thus is amenable to scatterers with complex geometries. We have employed LBM to compute the scattering width and radiation force for perfect electrically conducting and dielectric cylinders of circular and elliptical cross sections. Both smooth and corrugated surfaces are studied, and the results are compared against known analytical and numerical solutions from other methods. To ensure the broad applicability of the method, we have explored a wide range of parameter space—the dielectric constant and particle size to the wavelength ratio spanning Rayleigh, Mie, and geometrical optics regimes. Our simulations have successfully reproduced well-known analytical and numerical solutions, confirming the accuracy and reliability of the LBM for scattering calculations by complex-shaped objects.

Dynamics of massive and massless particles in the spacetime of a wiggly cosmic dislocation

AbstractIn this paper, we investigate the spacetime containing both small-scale structures (wiggles) and spatial dislocation, forming a wiggly cosmic dislocation. We study the combined effects of these features on the dynamics of massive and massless particles. Our results show that while wiggles alone lead to bound states and dislocation introduces angular momentum corrections, their coupling produces more complex effects, influencing both particle motion and wave propagation. Notably, this coupling significantly modifies radial solutions and eigenvalues, with the direction of motion or propagation becoming a critical factor in determining the outcomes. Numerical solutions reveal detailed aspects of particle dynamics as functions of dislocation and string parameters, including plots of trajectories, probability densities, and energy levels. These findings deepen our understanding of how a wiggly cosmic dislocation shapes particle dynamics, suggesting new directions for theoretical exploration in cosmological models.

Adaptive Factored Incomplete Inverse Matrices

ABSTRACTThe solution of large sparse linear systems is an essential part in many scientific fields. Numerical solution of such systems is usually performed using iterative methods in conjunction with effective preconditioning schemes. A new class of factored approximate inverses is proposed, namely adaptive factored incomplete inverse matrices, which are computed using a recursive Schur complement‐based approach. This class of approximate inverses does not require the knowledge of a sparsity pattern, which is formed adaptively during computation. For this to be possible, a flexible sparse storage scheme was designed. Several numerical dropping strategies are showcased and discussed, along with the effects of reordering to the preconditioner density and the corresponding convergence behavior. Dropping based on monitoring the growth of elements in the incomplete LU factorization is also presented and discussed along with improvements during computation of the successive Schur complements. A static sparsity pattern variant is also provided and discussed. Implementation details and analysis, for computing the proposed scheme, are given along with the computational complexity and memory requirements. Numerical results depicting the effectiveness and applicability of the proposed scheme are also presented.

Approximations of random periodic solutions for path‐dependent stochastic differential equations with finite/infinite delay

The aim of this work is to show the numerical approximation of random periodic solutions for path‐dependent stochastic differential equations with a finite and an infinite delay, in which the drifts are only dissipative on average with respect to the time parameter. The key findings are as follows: (i) by using a combination of synchronous coupling approaches and continuous‐time Euler–Maruyama schemes, we study the existence and uniqueness of numerical random periodic solutions to path‐dependent stochastic differential equations on an infinite time horizon; (ii) by introducing a new technical method, we investigate the strong convergence in the mean‐square sense of a numerical approximation to path‐dependent stochastic differential equations in the case of an infinite time horizon; (iii) to the best of our knowledge, our result is the first one upon numerical approximations of random periodic solutions for path‐dependent stochastic differential equations (SDEs) with infinite delay. We expect that our approximated method can be extended to a more general structure.

Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering

Abstract This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u ″ ″ + q u = f u^{\prime\prime} ^{\prime\prime} +qu=f . The equation entails two distinct boundary conditions: pinned–pinned conditions on u u and u ′ u^{\prime} , and clamped–clamped conditions on u u and u ″ {u}^{^{\prime\prime} } . To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

Numerical solution of time-dependent two-parameter singularly perturbed problems via trigonometric quintic B-spline collocation technique

Abstract This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection-diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

Statistical Inference of Uncertain Autoregressive Model via the Principle of Least Squares

In the study of uncertain autoregressive models, how to estimate the unknown parameters and uncertain disturbance term in the models is always a key problem. In view of this, this paper proposes a statistical inference method based on the principle of least squares to determine the unknown parameters and uncertain disturbance term in an uncertain autoregressive model, and designs a numerical algorithm to calculate the numerical solutions of the corresponding estimators. Then, the uncertain hypothesis test is used to verify the applicability of the estimated uncertain autoregressive model, and point forecast and interval forecast are also made for the time series of future moments. Finally, a case study of the Consumer Price Index for all items in U.S. cities is provided to illustrate the effectiveness of the approach proposed in this paper.

Nanocząstki krzemu w zrównoważonym rolnictwie

The increasing population, shrinking arable land, and climate change prompt the search for new solutions in agriculture. In a sustainable approach, agriculture should be based on improving the quality and quantity of yields while maintaining biodiversity and protecting the natural environment. Nanotechnology, present in many areas of our lives, offers opportunities to support the development of sustainable agriculture on many levels. Among the numerous solutions and nanomaterials, silicon, which is a natural component of the ecosystem, deserves special attention. In its nanoparticle form, it acquires new, unique properties. This article focuses on the significant role of silicon nanoparticles in organic farming, with particular emphasis on their function as nanofertilizers. The authors analyze the impact of silicon nanoparticles on plant growth and development and their potential in mitigating the negative effects of abiotic stress factors caused by drought, salinity, and exposure to metals. Additionally, the beneficial effects of silicon nanoparticles on plants growing under biotic stress conditions induced by microorganisms such as bacteria and fungi are presented. The paper includes a review of original research results conducted in recent years in this area, as well as possible mechanisms and strategies of silicon nanoparticle action at the physiological, cellular, and molecular levels. The issue of the safety of using nanoparticles in agriculture and the prospects for their further use as a factor enhancing the resistance and productivity of crops are also discussed.

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

Wave Propagation in Prestressed Structures with Geometric Nonlinearities Through Carrera Unified Formulation

This paper deals with the analysis of wave propagation characteristics in various prestressed structures with geometric nonlinearities using the Carrera Unified Formulation (CUF). CUF provides a versatile platform to model a wide range of structures and nonlinearities that can take care of all wave propagation aspects. In this work, different geometric nonlinearities for which representative governing equations have been derived and numerical solutions have been obtained through a unified approach are considered. The study investigates in detail the effect of prestress and geometric nonlinearity on wave propagation behavior. The results indicate that prestress has a very influential effect on modal frequency and dispersion characteristics for wave propagation. Specifically, three CUF-modeled beams are considered herein, having a sandwich, metallic portal, and metallic box cross section, respectively. Initially, the principal cross-sectional modal shapes of the unstressed, linear, and full nonlinear (i.e., full three-dimensional Green–Lagrange strain matrix) beam with a prestress are investigated, among which torsional and flexural modes can be recognized. Afterward, the equilibrium curves of such structures for various geometrical nonlinear approximations are traced, highlighting that most types of nonlinearity induce a hardening behavior in the system, which increases with the preload, directly leading to a variation in modal frequencies. The dispersion relations of the full nonlinear structure examined as a function of the applied preload are further compared, enriching the investigation by exploiting wave finite element method capabilities. This knowledge paves the way toward the design and optimization of prestressed systems with enhanced acoustic performance, and that fosters the development of sound absorption, noise insulation, and structural isolation.

The Donnan equilibrium is still valid in high-volume HDF.

Clinical studies have shown that hemodiafiltration reduces morbidity and mortality of dialysis patients compared to hemodialysis alone. This is attributed to its superior middle molecule clearance compared to standard hemodialysis. However, doubts arose as to whether a high convective flux through the dialyzer membrane has an influence on the equilibrium concentration of small ions, especially that of sodium. Due to the presence of negatively charged impermeable proteins on the blood side, the Gibbs-Donnan effect leads to an asymmetric distribution of membrane permeable ions on both sides of the membrane. In thermodynamic equilibrium, the concentrations of those ions can easily be calculated. However, the convective fluid flow leads to deviations from thermodynamic equilibrium. In this article, the effect of a convective flow on the ion distribution across a semipermeable membrane is analyzed in a theoretical model. Starting from the extended Nernst-Planck equation, including diffusive, convective, and electrostatic effects, a set of differential equations is derived. An approximate solution for flow speeds up to 0.1 ms-1 as well as a numerical solution are given. The results show that in any practical dialysis setting the convective flow has negligible influence on the electrolyte concentrations.

Compressional rheology model for vacuum filtration-consolidation of sludge in geotextile tubes

Recently, the disposal of high-water-content dredged sludge through geotextile tubes with vacuum-assisted prefabricated horizontal drains (PHD) has gained growing popularity for its convenience and efficiency. However, existing simulations for this typical coupled filtration-consolidation process using conventional consolidation models exhibit deficiencies due to the absence of effective stress and invalidity of Darcy’s law in the particle-wandering filtration phase. In this study, based on the compressional rheology theory, a two-dimensional coupled filtration-consolidation model constituted by the compressive yield stress Py(ϕ) and hindered setting factor r(ϕ) is developed to elucidate the solid-solid and solid-fluid interactions during slurry dewatering. A novel approach for measuring consistent Py(ϕ) and r(ϕ) relationships is proposed. The numerical solution is derived utilizing the alternating direction implicit (ADI) difference method and verified against a classical one-dimensional compressional rheology model and a field trial. Further analysis suggests that constitutive parameters, e.g., gel point, affect the dewatering efficiency by determining the relative degree of soil disorder, obstruction to particle movement, and vacuum transmission effect, while design parameters, e.g., PHD spacing and tube height, impact the magnitude and radiation range of vacuum pressure to influence the overall efficiency.

A spatial stochastic frontier model with fixed effects and endogenous environmental variables

ABSTRACT Stochastic frontier analysis (SFA) models often address fixed effects and endogeneity as separate issues. This study bridges this gap by proposing a unified approach within an ML (maximum likelihood) framework to manage both challenges simultaneously. The ML function is derived in a closed form, but its maximisation requires a numerical solution. To assess the properties of the ML estimator, a Monte Carlo simulation was performed. The simulation shows that the estimator performs reasonably well and gains efficiency asymptotically. The applicability of the estimator was further assessed through an empirical exercise that examined technical efficiency of the production function in a panel of European countries, managing spatiality, endogeneity, and fixed effects.

Studying a cylindrical moving boundary problem in ductal carcinoma in situ

In this paper, a mathematical model of ductal carcinoma in situ (DCIS) is considered as a moving boundary problem in cylindrical coordinates. Using the front-fixing technique, first, we transform the problem into a nonlinear parabolic problem on a fixed domain, and then some essential properties of the solution, such as the uniqueness of the solution, are shown. A numerical approach based on the finite difference method is established for solving the fixed domain problem. The solvability of the proposed finite difference method is proved, and the stability and convergence of the numerical solution are shown. For a test case problem, analytical and numerical results are compared to demonstrate the ability and effectiveness of the numerical method.

Wave turbulence in a rotating channel: numerical implementation and results

The analysis of Scott (J. Fluid Mech., vol. 741, 2014, pp. 316–349) is implemented numerically. Decaying turbulence is confined to a channel between two infinite, parallel, rotating walls. The Rossby and Ekman numbers are supposed small, the former condition making nonlinearity small, while the latter allows the turbulence to persist for the many rotational periods needed for the small nonlinearity to be effective. The flow is expressed as a combination of inertial waveguide modes, indexed by a two-dimensional wave vector $\boldsymbol{k}$ and an integer n. The $n = 0$ modes form a two-dimensional component of the flow, whereas the remainder is the wave component, on which attention is focused in this article. Assuming statistical axisymmetry and homogeneity in directions parallel to the walls, the second-order moments of the mode amplitudes yield a spectral matrix ${A_{nm}}(k,t)$ (where $k = |\boldsymbol{k} |$ ), of which the diagonal elements describe the distribution of energy over different modes. Wave-turbulence analysis provides an equation governing the time evolution of ${A_{nn}}$ , $n \ne 0$ , the wave spectra, which forms the basis for the present work. The initial distribution of energy is Gaussian and depends on a parameter $\varXi$ , the initial spectral width. The problem has two other parameters, ${\beta _w}$ and ${\beta _v}$ , which correspond to two distinct viscous dissipative mechanisms: wall damping due to boundary layers and volumetric damping by viscous effects throughout the flow. Results obtained by numerical solution include the time evolution of the total wave energy, E, and the detailed description of its distribution over k and n provided by ${A_{nn}}(k)$ .

On semi-orthogonal matrices with row vectors of equal lengths

When does a rectangular matrix with an orthonormal set of column vectors have row vectors of equal lengths? The column spaces of such matrices are multidimensional generalizations of the projection plane used in isometric perspective. We show that in the absence of unexpected linear relations, any rectangular matrix can be row-scaled so that if we were to orthonormalize the column vectors, the row vectors would attain equal lengths in the process. We use Grassmann coordinates to reduce the question into an instance of the famous matrix scaling problem, and with the help of existing theory introduce simple numerical solutions.

Static and Transient Response Analysis of Arbitrary Laminated Composite and Sandwich Shells by the FSDT Meshless Method

This paper, for the first time, presents a novel meshless method based on the first-order shear deformation theory (FSDT) and moving-least squares (MLS) approximation to carry out static and transient response analysis on arbitrary laminated and sandwich shell structures. The novelty of the method lies in the implementation of MLS in the curvilinear shell formulation, and the derivation of meshless governing equations on the static and transient response of arbitrary laminated and sandwich shells according to the principle of minimum potential energy and Hamilton’s principle, respectively. The concept of a Convected coordinate system is adopted to deal with arbitrary curvilinear surfaces. Both the position and displacement vectors of shells are approximated by MLS, which is conceptually the same procedure as that in the isoparametric finite element method (FEM). The numerical integration of the stiffness matrices is conducted by the simple Gaussian integral in the Convected coordinate system. Because the moving-least squares (MLS) approximation does not satisfy the Kronecker delta condition, the full transformation method is used to modify the meshless governing equations. Numerical examples are calculated to investigate the static and transient response analysis of laminated and sandwich structures with different geometrical shell shapes, including square plates, spherical shells, doubly-curved shells, and cylindrical shells. The numerical tests show that the proposed method is reliable, and robust and produces accurate results compared to the analytical and numerical solutions taken from the previous literature.

A Rapid Numerical Method for Nonlinear Generalized Time-Fractional Kawahara Equations via Domination Polynomials of Complete Graph.

Abstract The key objective of this study is to examine the nonlinear Time-fractional generalized Kawahara equation (N-TF-GKE), a significant nonlinear equation that appears in a variety of physical domains, including ion-acoustic waves in plasmas, shallow water waves, and nonlinear acoustics. This Equation exhibits complex dynamic properties, most notably the appearance of shock waves, solitary waves, and chaotic solutions. We use a graph theoretic polynomial approach to obtain numerical solutions of the N-TF-GKE. We present the dominance polynomials of complete graphs (DPC), a new class of graph polynomials, to develop an efficient operational integration matrix to approximate the N-TF-GKE. The N-TF-GKE transformed into a collection of nonlinear algebraic equations with the standard collocation points. We acquire the numerical solutions for the Domination polynomial using Newton's approach. We validate the efficiency and robustness of our domination polynomial collocation method (DCM) with a set of numerical experiments and comparative analyses.

A comparative study of numerical methods for approximating the solutions of a macroscopic automated-vehicle traffic flow model

In this paper, a particle method is used to approximate the solutions of a “fluid-like” macroscopic traffic flow model for automated vehicles. It is shown that this method preserves certain differential inequalities that hold for the macroscopic traffic model: mass is preserved, the mechanical energy is decaying and an energy functional is also decaying. To demonstrate the advantages of the particle method under consideration, a comparison with other numerical methods for viscous compressible fluid models is provided. Since the solutions of the macroscopic traffic model can be approximated by the solutions of a reduced model consisting of a single nonlinear heat-type partial differential equation, the numerical solutions produced by the particle method are also compared with the numerical solutions of the reduced model. Finally, a traffic simulation scenario and a comparison with the Aw-Rascle-Zhang (ARZ) model are provided, illustrating the advantages of the use of automated vehicles.

Assessment of the air pollutants on automobiles emission dispersion: the effects of barrier height and the road surface temperature.

In the paper, a mathematical model was constructed that describes the specifications of the wind flow and the dispersion of pollutants, taking into account the variable temperature on the roadway surface, which varies depending on the time for some quarter of the city of Almaty. The impact of the traffic tidal flow was studied based on the data of measuring passing vehicles as a source of pollution by the CFD and on the spatial distribution of pollutants for various types of pollution. A test problem was performed to validate the numerical algorithm and the mathematical model. From the obtained numerical solutions, it was determined that the existing barriers along the road have a positive effect on pedestrian zones regardless of the type of pollution. It was also found that, taking into account the variable temperature on the carriageway, the presence of barriers with a height of 4 m shows favorable behavior on the adjacent areas, in which the average concentration value drops by 6.4 times for the pedestrian zone, which is located on the left side and 2.9 times for the pedestrian zone, which is on the right side compared to the option without a barrier.