Euler Method Research Articles

Analysis of the periodicity of movements of mechanisms with elastic links of variable length

Abstract Elastic dynamic systems with open kinematic chains are effectively described by a recursive model of the generalized Newton–Euler method with the provision of computational algorithms with a degree of complexity proportional to the dimension of these systems, i.e. O(n). Such systems, in particular, are elastic manipulators, some of which are discussed in this article. In the case when elastic dynamical systems have closed kinematic circuits, the application of a numerical analysis strategy without reversing the mass matrix is extremely difficult. The analysis of the simplest type of elastic mechanisms with closed circuits, such as a crank–slide mechanism and a four–pin mechanism with elastic connecting rods, revealed a peculiar picture of the periodicity of the functions of elastic movements of connecting rods, which differs significantly from the periods of operation of the mechanisms themselves. In this article, the properties of the periodicity (or pseudo-periodicity) of the elastic displacement function of mechanisms with elastic links of variable length are investigated using examples of numerical dynamic analysis of crank–rocker mechanisms.

An explicit Euler scheme for generalized [formula omitted]-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises

The current paper is devoted to the numerical integration of generalized n-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises. We submitted the Euler integrator for the integral form of this type of differential equation. Convergence analysis shows that the proposed scheme order of strong superconvergence is 1.0 when diffusion terms, in general form gl2(t,ρ), l=1,2, satisfies gl2(t,t)=0. While, the strong convergence order of the scheme is 1/2 if gl2(t,t)≠0. Especially, for convolution diffusion terms gl2(t,ρ)=gl2(t−ρ) our scheme has strong superconvergence order 1.0 when gl2(0)=0. And scheme strongly convergent with order 1/2 if gl2(0)≠0. Finally, four problems are considered to exhibit the efficiency and compatibility of the numerical integrator.

Mathematical analysis of spin transport in topological insulators to optimize memory devices performance using numerical simulations

Understanding spin transport in topological insulators (TIs) is crucial for the development of spin-based technologies, such as magnetic memory, sensors, and quantum bits. To optimize memory device performance, we developed a comprehensive mathematical model that describes the evolution of spin density, as a function of space and time, taking into account spin-momentum locking, spin-orbit coupling, spin dynamics, and diffusive transport processes (including phonon and impurity scattering). Using numerical simulations (finite element method and Backward Euler Method) implemented in Python, we analyzed the spin transport in TIs. Our study demonstrated a critical dependence of spin transport efficiency on device length, with a maximum efficiency of 75% at a length of 8 micrometers. Beyond a critical length of 10 micrometers, efficiency recovered, reaching 80% at 14 micrometers. We observed oscillatory behavior in spin polarization, with amplitude modulation indicating constructive and destructive interference patterns. The inclusion of spin-orbit coupling and Dirac terms in our model revealed a non-uniform spin polarization distribution, with a 30% increase in polarization near the center. Defects and boundaries significantly impacted spin transport, reducing polarization by 40% near the defect region. Through optimization, we achieved a 25% increase in spin transport efficiency and a 20% enhancement in memory device performance. Our results demonstrate the crucial role of optimizing device length, material quality, and interface engineering in achieving efficient spin transport and improved memory device performance, paving the way for the development of high-performance spin-based technologies.

Research of the motion parameters of the entry of a spaceplane into the atmosphere

After the end of operation of the International Space Station in 2028, the Russian Federation plans to develop a national orbital station project. The Russian Space Station will differ from its predecessor in a greater practical aspect. One of the tasks assigned to the station will be the launch and management of a group of small satellites for remote sensing of the Earth, as well as the interaction and maintenance of prospective satellite groups. Due to the limited maneuverability of the orbital station and the potential for a malfunctioning device to be at a significant distance from it, the use of an autonomous spaceplane is proposed to increase the transportation and technical capabilities of the station. In the research, two aerodynamic designs of the spaceplane are presented, and one of them is chosen based on the results of the aerodynamic and weight analysis. The spaceplane configuration and algorithms for its operation on the orbit and descent to the atmosphere are also presented. The goal of the research is to compare the trajectory parameters during the descent of the spacecraft from different descent orbits. For this purpose, a task was formulated to determine the dependence of the area of the descent corridor on the initial parameters. The area of the descent corridor is determined by the boundary conditions, which depend on the operational parameters of the spaceplane. A computational program is written to solve differential equations of flight dynamics of a spaceplane by Euler's method in general and by Runge-Kutta method in a computational case. The results of the research are presented as the dependence of the area of the descent corridor on the altitude of descent. Graphical representations of the primary parameters of the spaceplane descent for the computational case are also presented.

A Nonstandard Finite Difference Scheme for a Mathematical Model Presenting the Climate Change on the Oxygen-plankton System

This paper presents a mathematical model describing climate change in the oxygen-plankton system. The model consists of a system of non-linear ordinary differential equations. The Nonstandard Finite Difference (NSFD) method is applied to discretize the non-linear system. The stability of the continuous and discrete model is presented for the given parameters in the literature. The compatibility of the results has been seen. Moreover, the model is solved by both the NSFD method and the Runge–Kutta–Fehlberg (RKF45) method. The numerical results are compared. Furthermore, the efficiency of the NSFD method compared to classical methods such as the Euler method and the fourth order Runge-Kutta (RK4) method for the bigger step size is shown in tabular form.

A discontinuous Galerkin method for the Brinkman–Darcy-transport problem

In this paper, a weighted discontinuous Galerkin finite element method and an upwind format are presented to solve the coupled Brinkman–Darcy flow and transport model. The flow in a highly permeable region of the model is governed by the Brinkman equations, and the percolation in the porous media is controlled by the Darcy equations, which are coupled by three interface conditions. The permeability coefficients in this model are strongly nonhomogeneous, anisotropic, and discontinuous; the transport equation is a convection-dominated problem; and the velocity field in the porous media domain does not satisfy the divergence-free condition. A weighted discontinuous Galerkin finite element method is used to solve the complex permeability coefficient problem and an upwind scheme is used to solve the convection dominated problem. The interface conditions can be naturally incorporated into the discrete formulation without introducing additional variables. Optimal error estimates are obtained for the semi-discretization and the full discretization with the backward Euler scheme in suitable energy norm. A series of numerical experiments are provided to illustrate the proposed method, including testing the convergence and accuracy of various types of meshes; simulating fluid flow in complex porous media such as obstacles, layered media, and curved interfaces; and studying the coupled flow behavior of surface-subsurface flows with contaminant transport.

Comparing the Numerical Solution of Fractional Glucose–Insulin Systems Using Generalized Euler Method in Sense of Caputo, Caputo–Fabrizio and Atangana–Baleanu

Comparing the Numerical Solution of Fractional Glucose–Insulin Systems Using Generalized Euler Method in Sense of Caputo, Caputo–Fabrizio and Atangana–Baleanu

Approximation‐based implicit integration algorithm for the Simo‐Miehe model of finite‐strain inelasticity

AbstractWe propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non‐iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w‐invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non‐proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial‐boundary value problem is solved.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Analytical and numerical study of diffusion propelled surface growth phenomena

To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

Finite-horizon and infinite-horizon linear quadratic optimal control problems: A data-driven Euler scheme

In this paper, we devote our focus to studying finite-horizon and infinite-horizon linear quadratic optimal control (LQOC) problems in continuous time. We focus on algorithms for approximating optimal controls of these two problems without the knowledge of all system coefficients. Firstly, we transform the backward differential Riccati equation (DRE) corresponding to the finite-horizon LQOC problem into a forward ordinary differential equation. Then, we discretize the forward equation by the Euler method. Subsequently, we use the collected state and input data to calculate all discretized points of the forward equation. At the same time, we obtain the optimal control of the finite-horizon LQOC problem. Moreover, we approximate the optimal control of the infinite-horizon LQOC problem by virtue of an asymptotic behavior between the DRE and an algebraic Riccati equation (ARE) arising in the infinite-horizon LQOC problem. Finally, we validate the obtained results via two practically motivated examples.

Numerical solutions to Hyperbolic Maxwell quasi-variational inequalities in Bean–Kim model for type-II superconductivity

This paper is devoted to the finite element analysis for the Bean–Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with L1-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D H(curl)-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.

Effect of variable conditions on transient flow in a solid–liquid two-phase centrifugal pump

To investigate the influence of altering operational parameters on the transient flow characteristics within the flow channel of a solid–liquid two-phase centrifugal pump, this study employs a particle model based on the Euler–Euler method. Utilizing the standard k–e model, flow field simulations are conducted using the ANSYS-CFX software. Specifically, the study focuses on throttle regulation scenarios, monitoring and comparing the external and internal flow parameters of the solid–liquid two-phase pump with those of a clear water medium centrifugal pump. The results indicate notable modifications in head, efficiency, and shaft power due to the presence of solid particles in the two-phase flow. Decreasing flow rates during throttle regulation lead to fluctuations in pressure and turbulence energy distribution. Furthermore, under identical operational conditions, the variable working conditions of the solid–liquid pump result in increased flow rates of solid-phase particles near the impeller's outer edge, with particles shifting toward the middle and tail of the vane suction surface. This phenomenon exacerbates wear on the vane tail of the suction surface. Moreover, the study identifies that changing operational conditions in the solid–liquid dual-flow centrifugal pump contribute to increased axial forces, consequently leading to pump vibrations. Overall, this research elucidates the transient flow characteristics within the flow channel of solid–liquid two-phase centrifugal pumps under varied operational conditions, serving as a foundational reference for assessing the stability of such pumps.

Numerical Simulation Model of the Infectious Diseases by Comparing Backward Euler Method and Adams-Bash forth 2-Step Method

In this work, the Backward Euler technique and the Adams-Bashforth 2-step method—two numerical approaches for solving the SIR model of epidemiology are compared for performance. An essential resource for comprehending the transmission of infectious illnesses like COVID-19 in the SIR model. While the explicit Adams-Bash forth 2-step approach is well known for its computing efficiency, the implicit Backward Euler method is noted for its stability. The study evaluates the accuracy, strength, and computing cost of the two approaches to determine which approach is best for simulating the spread of infectious illnesses. The SIR Model was easily solved using the Adams Bashforth 2-step analysis and the Backward Euler method. The approaches' solutions are close to the exact requirements. There are important distinctions between the two-step Adams Bashforth and backward Euler procedures. The running time of the Adams Bashforth 2-step backward Euler method is shorter than that of the backward Euler method.

MODELING INFLATION DYNAMICS USING THE LOGISTIC MODEL: INSIGHTS AND FINDINGS

This paper examines applying the logistic model, frequently used in biology, to analyze inflation patterns in dynamic economic systems. The primary objective is to simulate and analyze the complex dynamics of inflation, thus providing new insights into the stability of financial institutions. Numerical methods such as Euler's Method, Runge-Kutta Method (RK4), and Adams-Bashforth-Moulton's method were used to simulate inflation patterns by discretizing the logistic equation. The data utilized in this research were obtained from INSTAT, BoA, MoF, and Eurostat, with quarterly results from 1995 to 2023. The simulation results indicated that the RK4 and Adams-Bashforth-Moulton methods yielded more precise and reliable inflation forecasts than Euler's. The logistic model represented the non-linear aspects of inflation dynamics well, emphasizing the necessity of using suitable numerical approaches. The study's findings highlight the effectiveness of the logistic model in economic analysis, specifically in forecasting inflation trends. Enhanced closure approaches have proven their effectiveness in analyzing intricate economic data, providing crucial insights into the stability of inflation, and informing policy formulation. This study utilizes the logistic model to analyze inflation dynamics, offering a unique methodology for comprehending and forecasting inflation in economic systems. An analysis of several closure techniques reveals a novel aspect of financial modeling tools. The findings indicate that incorporating advanced numerical methods can significantly improve the precision of economic models. These findings significantly impact economic research and policy formulation, especially in devising measures to manage inflation and ensure financial stability.

Discrete energy balance equation via a symplectic second-order method for two-phase flow in porous media

We propose and analyze a second-order partitioned time-stepping method for a two-phase flow problem in porous media. The algorithm is a refactorization of Cauchy's one-leg θ-method: the implicit backward Euler method on [tn,tn+θ], and a linear extrapolation on [tn+θ,tn+1]. In the backward Euler step, the decoupled equations are solved iteratively, with the iterations converging linearly. In the absence of the chain rule for time-discrete setting, we approximate the change in the free energy by the product of a second-order accurate discrete gradient (chemical potential) and the one-step increment of the state variables. Similar to the continuous case, we also prove a discrete Helmholtz free energy balance equation, without numerical dissipation. In the numerical tests we compare this symplectic implicit midpoint method (θ=1/2) with the classic backward Euler method, and two implicit-explicit time-lagging schemes. The midpoint method outperforms the other schemes in terms of rates of convergence, long-time behavior and energy approximation, for both small and large values of the time step.

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Application of Dimension Extending Technique to Unified Hardening Model

This paper provides the process of incremental constitutive integration for the unified hardening model combined with the transformation stress method. The dimension-extending technique takes the hardening function of the hardening/softening model as the same position as the stress components, so that the constitutive integration of the plasticity can be reduced to an initial value problem of differential–complementarity equations, which is solved using the Gauss–Seidel algorithm-based Projection–Correction for the mixed complementarity problem. The Gauss–Seidel based Projection–Correction algorithm does not require the calculation of the Jacobean matrix of the potential function, making it relatively easy to implement in programming. The unified hardening model is proposed based on the modified Cam–Clay model and the sub-loading surface model, and the elastic properties are pressure-dependent. Two processing methods, backward Euler integration and exact elastic property, are used for the variable elasticity properties. The constitutive integration of the increased dimensional unified hardening model is reduced to a special mixed complementarity problem and solved by the proposed algorithm, which does not need to calculate the Jacobean matrix of the potential function, and greatly simplifies the derivation process. Several numerical examples are given to verify the feasibility of the incremental constitutive integration in the unified hardening model, including the single integral point and the boundary value problems. The research results have expanded the scope of use of the Gauss–Seidel based Projection–Correction algorithm.

Numerical study of diffusive fish farm system under time noise

In the current study, the fish farm model perturbed with time white noise is numerically examined. This model contains fish and mussel populations with external food supplied. The main aim of this work is to develop time-efficient numerical schemes for such models that preserve the dynamical properties. The stochastic backward Euler (SBE) and stochastic Implicit finite difference (SIFD) schemes are designed for the computational results. In the mean square sense, both schemes are consistent with the underlying model and schemes are von Neumann stable. The underlying model has various equilibria points and all these points are successfully gained by the SIFD scheme. The SIFD scheme showed positive and convergent behavior for the given values of the parameter. As the underlying model is a population model and its solution can attain minimum value zero, so a solution that can attain value less than zero is not biologically possible. So, the numerical solution obtained by the stochastic backward Euler is negative and divergent solution and it is not a biological phenomenon that is useless in such dynamical systems. The graphical behaviors of the system show that external nutrient supply is the important factor that controls the dynamics of the given model. The three-dimensional results are drawn for the various choices of the parameters.