One-dimensional Nonlinear Model Research Articles

Use of fractional calculus to avoid divergence in newton-like solver for solving one-dimensional nonlinear polynomial-based models

Use of fractional calculus to avoid divergence in newton-like solver for solving one-dimensional nonlinear polynomial-based models

Memory Based Approaches to One-Dimensional Nonlinear Models

Memory Based Approaches to One-Dimensional Nonlinear Models

Nonlinear Dynamics of Atomic Force Microscopy with Viscoelastic Term and Casimir force interactions: An Overview

Abstract Atomic force microscopy (AFM) is an important tool for analyzing the atomic surface of synthetic or biological samples. The atomic force microscopy device consists of a built-in microcantilever that contains a nano tip at its free end. This nano tip finds the surface of the sample which is scanned for data collection. Vibrations caused by irregularities on the sample surface are collected using a laser that reflects from the microcantilever to a photodiode. In this work, we explore the nonlinear dynamics of the mathematical model, which describes the one-dimensional torsional behavior of the microcantilever motion. The term viscoelasticity and the Casimir force were considered, and we established the set of parameters that present chaotic behavior and thus, proposed control techniques to suppress chaos. However, we explored the nonlinear dynamics behavior of a one-dimensional nonlinear AFM model with the twist of the tip of the microbeam containing the analysis tip. And so, we analyze the contribution of the Casimir Force and the viscoelasticity term in the model considered. The aim of this work was to analyze the regions in which the system is in a chaotic and periodic regime, choosing the initial conditions, using classical tools (Lyapunov Exponent, Bifurcation Diagram, Poincare Map, etc.). Determining such regions supports improvements and proposals for future control projects to suppress chaos. Such suppression allows irregularities in the movement of the microcantilever to be periodic, which can reduce noise in sample analyses.

Nonlinear seismic response analysis of liquefiable sites based on effective stress method

In this study, the one-dimensional nonlinear Matasovic model was extended to the two- and three- dimensional stress space by replacing the shear strain with the generalized shear strain. The extended Matasovic model was subsequently combined with Dobry’s excess pore water pressure generation model, and the reliability of the proposed loosely coupled effective stress analysis model was verified through undrained cyclic triaxial tests and a one-dimensional site response analysis. Finally, this method was applied to an engineering site in China to evaluate the nonlinear seismic response of liquefiable sites. The results show that the proposed effective stress analysis method can capture the dynamic behavior of the soil and the generation of pore water pressure during strong shaking. Moreover, there are differences between the proposed effective stress analysis method and the total stress analysis method for liquefiable sites, which indicates the need for careful selection in practical applications.

Nonlinear mathematical model of pressure measurement systems in gas-liquid media

The primary element of the instrumentation for measuring the pressure of a gas-liquid medium is a sensor that supplies data on the pressure of the working medium. It determines the proper functioning of machines, mechanisms, and systems. Increasing the service life, reducing development time, and reducing the cost of sensors is one of the important tasks. Mathematical modeling of pressure measurement systems’ functioning plays an important role at the design stage of such systems. This article examines a nonlinear one-dimensional model of a mechanical system “pipeline – pressure sensor” designed to measure and control the pressure of the working gas-liquid medium in the combustion chambers of engines. In such a system, the sensor is connected to the engine via a pipeline and is located at some distance from it to reduce the impact of vibration accelerations and high temperatures. The purpose of the work is to study the dynamics and stability of joint oscillations of the elastic sensitive element in the pressure sensor and of the working medium in the pipeline for a given law of pressure change in the combustion chamber. The study is provided under the assumption that the working medium is ideal and compressible. To describe the movement of the working medium (gas or liquid), a nonlinear model of fluid and gas mechanics is used. Mathematical description of the process of interest includes an initial boundary value problem, whose formulation contains a nonlinear partial differential equation. To solve it, numerical-analytical method of solution based on the Galerkin method is proposed, which makes it possible to reduce the study of the problem to solving a system of ordinary differential equations. A numerical experiment is carried out and examples of calculating the dynamics of the sensor's sensitive element are presented. The proposed mathematical model makes it possible to determine the law of change in the deviation of the sensor's sensitive element depending on the law of change in pressure in the combustion chamber. The research results are intended for use at the design stage of pressure measurement systems.

Analysis of two spectral Galerkin approximation schemes for solving the perturbed FitzHugh-Nagumo neuron model

In this paper, we mainly study two spectral Galerkin approximation schemes of the FitzHugh-Nagumo (FHN) neuron model for the nerve impulse propagation, which contains a small parameter perturbation and strong nonlinearity. To this end, we consider two kinds of temporal discretization by a semi-implicit Euler method and a Crank-Nicolson method, and the spatial discretization by the spectral Galerkin method for the perturbed FHN neuron model. In particular, in order to obtain the second-order approximation in time for the numerical scheme, we modify the nonlinear term f(un) in the Crank-Nicolson spectral Galerkin (CNSG) scheme. We derive the error estimations with optimal convergence rates of the numerical solutions by the first-order semi-implicit spectral Galerkin (SISG) approximation scheme and the second-order modified CNSG (MCNSG) approximation scheme for the perturbed FHN neuron model. Finally, some numerical examples of one-dimensional and two-dimensional nonlinear FHN models with periodic boundary conditions are carried out to verify that the approximate solutions of the proposed SISG and MCNSG schemes satisfy the proved theoretical results.

Weak form quadrature element analysis of spatial geometrically exact composite beams with torsional warping

Spatial thin-walled laminated composite beams under large deformations with torsional warping effects are studied. A novel two-dimensional cross-section model for composite beams with shear and torsion-warping deformations is proposed wherein the three-dimensional warping and coupling effects caused by inhomogeneous anisotropic materials are considered. A weak form quadrature element formulation is established for nonlinear analysis of geometrically exact composite beams with torsional warping and shear deformations. By combining the two-dimensional cross-section model and the nonlinear one-dimensional beam model, the three-dimensional stress and strain fields can be obtained. Numerical examples are provided to investigate the torsional warping effects and verify the effectiveness of the present formulation. Results of the present beam formulation which are achieved at a significantly lower computational cost are in excellent agreement with those of shell and solid models of commercial codes.

One-dimensional nonlinear model of generalized thermo-electroelasticity

We investigate a one-dimensional restriction of a nonlinear model of thermo-electroelasticity in extended thermodynamics and in the quasi-electrostatic regime (see Ghaleb et al. in Int J Eng Sci 119:29–39, 2017. https://doi.org/10.1016/j.ijengsci.2017.06.010). An additional dependence of the thermal conductivity and the thermal relaxation time on temperature and heat flux is introduced. The aim of the present work is to assess the effect of some quadratic nonlinear couplings between the mechanical, thermal and electric fields. Such couplings are known to have a crucial effect on the stability of the solutions. It is confirmed that there are two speeds of wave propagation of disturbances, the coupled thermoelastic wave and the heat wave. Formulae are provided for both speeds, showing their explicit dependence on temperature, heat flux and electric field. The purely thermal case is briefly considered. The present results may be useful for the description of a broad range of interactions in large polarizable slabs of electro-thermoelastic materials and for the design of such materials.

One-dimensional large strain nonlinear thaw consolidation model for saturated frozen soil

Based on the piece-linear finite difference approach, a one-dimensional large strain thaw consolidation model, TCS1, for saturated frozen soil was established. The model couples the heat transfer considering phase change of pore ice with consolidation deformation, and accounts for large strain, soil self-weight, unfrozen water, variable hydraulic conductivity and compressibility, flexible boundary conditions and time-dependent loading during the process of thaw consolidation. The FORTRAN is used to implement computation of the model. TCS1 was verified by the existing numerical solutions and laboratory tests. The results were in good agreement with the large strain numerical solutions and tests results. Computational examples were conducted for further study of the influence of strain value, loading types and periodic temperature boundaries on the thaw consolidation of frozen soil.

A fast-convolution based space–time Chebyshev spectral method for peridynamic models

AbstractPeridynamics is a nonlocal generalization of continuum mechanics theory which addresses discontinuous problems without using partial derivatives and replacing them by an integral operator. As a consequence, it finds applications in the framework of the development and evolution of fractures and damages in elastic materials.In this paper we consider a one-dimensional nonlinear model of peridynamics and propose a suitable two-dimensional fast-convolution spectral method based on Chebyshev polynomials to solve the model. This choice allows us to gain the same accuracy both in space and time. We show the convergence of the method and perform several simulations to study the performance of the spectral scheme.

Fluid and hybrid simulations of the ionization instabilities in Hall thruster

Low-frequency axial oscillations in the range of 5–50 kHz stand out as a pervasive feature observed in many types of Hall thrusters. While it is widely recognized that the ionization effects play the central role in this mode, as manifested via the large-scale oscillations of neutral and plasma density, the exact mechanism(s) of the instabilities remain unclear. To gain further insight into the physics of the breathing mode and evaluate the role of kinetic effects, a one-dimensional time-dependent full nonlinear low-frequency model describing neutral atoms, ions, and electrons is developed in full fluid formulation and compared to the hybrid model in which the ions and neutrals are kinetic. Both models are quasi-neutral and share the same electron fluid equations that include the electron diffusion, mobility across the magnetic field, and the electron energy evolution. The ionization models are also similar in both approaches. The predictions of fluid and hybrid simulations are compared for different test cases. Two main regimes are identified in both models: one with pure low-frequency behavior and the other one, where the low-frequency oscillations coexist with high-frequency oscillations in the range of 100–200 kHz, with the characteristic time scale of the ion channel fly-by time, 100–200 kHz. The other test case demonstrates the effect of a finite temperature of injected neutral atoms, which has a substantial suppression effect on the oscillation amplitude.

The interplay of regularizing factors in the model of upper hybrid oscillations of cold plasma

A one-dimensional nonlinear model of the so-called upper hybrid oscillations in a magnetoactive plasma is investigated taking into account electron-ion collisions. It is known that both the presence of an external magnetic field of strength B0 and a sufficiently large collisional factor ν help suppress the formation of a finite-dimensional singularity in a solution (breaking of oscillations). Nevertheless, the suppression mechanism is different: an external magnetic field increases the oscillation frequency, and collisions tend to stabilize the medium and suppress oscillations. In terms of the initial data and the coefficients B0 and ν, we establish a criterion for maintaining the global smoothness of the solution. Namely, for fixed B0 and ν>0 one can precisely divide the initial data into two classes: one leads to stabilization to the equilibrium, and the other leads to the destruction of the solution in a finite time. Next, we examine the nature of the stabilization. We show that for small |B0| an increase in the intensity factor first leads to a change in the oscillatory behavior of the solution to monotonic damping, which is then again replaced by oscillatory damping. At large values of |B0|, the solution is characterized by oscillatory damping regardless of the value of the intensity factor ν.

Simulation of High-Power Klystrons with Heterogeneous Focusing Magnetic Field

The work shows a significant effect of the focusing magnetic field on the output characteristics of the klystron. When the calculations are done using nonlinear one-dimensional models, optimization of the parameters makes it possible to obtain versions of devices with the efficiency of 0.8 – 0.9 and higher. However, when testing these options using nonlinear two-dimensional models that take into account the radial motion of electrons, there is a significant discrepancy in the output characteristics obtained from the one-dimensional and two-dimensional models. This is due to the fact that during the motion of the electron beam, the radii of the leading centers of the large particles change the coefficients of interaction of the particle fields with the electromagnetic fields of the resonators change, which leads to a change in the output characteristics of the klystrons: efficiency, output power, and gain. On the other hand, it seemed that setting a large focusing magnetic field to exclude the radial motion of particles could eliminate this drawback, however, another problem arises here - the magnetic system for focusing becomes unacceptably large and it is technically difficult to obtain magnetic induction values of more than 2 T (the weight of the magnetic system can be several hundred kilograms). Therefore, one should choose the magnetic field induction for focusing the electron beam no more than 1T. In this paper, a twodimensional nonlinear mathematical model (2.5D) is proposed that takes into account the azimuthal component in the equations of motion. In the model, the induction of the focusing magnetic field is set in the form of tables. This makes it possible to set the inhomogeneity of the magnetic field at any place in the interaction space of the klystron. The calculation of a powerful relativistic klystron with an accelerating voltage of 1000 kV and the beam current of 250 A was carried out. The use of an inhomogeneous magnetic field makes it possible to reduce the deposition of electrons in the region of the gaps. Therefore, a decrease in the electron deposition leads to an increase in the durability of klystrons.

Nonlinear one-dimensional constitutive model for magnetostrictive materials

This paper presents an analytic model of one dimensional magnetostriction. We show how specific assumptions regarding the symmetry of key micromagnetic energies (magnetocrystalline, magnetoelastic, and Zeeman) reduce a general three-dimensional statistical mechanics model to a one-dimensional form with an exact solution. We additionally provide a useful form of the analytic equations to help ensure numerical accuracy. Numerical results show that the model maintains accuracy over a large range of applied magnetic fields and stress conditions extending well outside those produced in standard laboratory conditions. A comparison to experimental data is performed for several magnetostrictive materials. The model is shown to accurately predict the behavior of Terfenol-D, while two compositions of Galfenol are modeled with varying accuracy. To conclude we discuss what conditions facilitate the description of materials with cubic crystalline anisotropy as transversely isotropic, to achieve peak model performance.

On the Reconstruction of Convection Coefficient in a Semilinear Anomalous Diffusion System

In the paper, we study an inverse problem of recovering a time-dependent convection coefficient from the measured data at an interior/boundary point in a one-dimensional nonlinear subdiffusion model with non-homogeneous boundary conditions. Due to the nonlinearity and non-homogeneous boundary conditions of the system, such an inverse problem is novel and important. We first investigate the unique existence and some regularities of the solution to forward problem by using the transposition method and the fixed point theorem. Then a conditional stability of the inverse problem is obtained based on the regularity of solution for the direct problem and some generalized Gronwall's inequalities. Finally, we transform the inverse problem into a variational problem. The existence and convergence of the regularization solution for the variational problem are proved and we use a modified Levenberg–Marquardt method to find an approximate convection coefficient function. The efficiency and accuracy of the algorithm are illustrated with two numerical examples.

Fazna stabilnost nanosekundnih Ganovih oscilatora

Introduction/purpose: Detailed theoretical and experimental studies have been carried out in order to investigate the problem of a phase stability in electrodynamically uncoupled Gunn oscillators. Methods: The influence of modulating pulse instabilities has been investigated by means of computer simulation in the framework of a nonlinear one-dimensional theoretical model of the GaAs Gunn diode semiconductor active region. Experimental observations were also conducted including microwave measurements and antenna far-field estimation. They confirm the main theoretical results and extend the key work conclusions. Results: It was shown that the initial phase of the microwave oscillation out of the Gunn oscillator is independent of internal noises of the semiconductor structure and can be fixed only by the modulating voltage pulse. Conclusion: Gunn-diodes based microwave oscillators were stabilized using the leading edge of the voltage pulse from the modulating power supply. These results open up serious prospects for designing antenna phased arrays based on Gunn oscillators without mutual feedback (electrodynamically independent).

MATHEMATICAL AND STATISTICAL MODEL OF CLEANING IN A HIGH-GRADIENT MAGNETIC FIELD OF HYDROGENATED FAT FROM SUSPENDED CATALYST

The introduction specifies the object of the study - hydrogenated fat subjected to purification from suspended nickel-containing catalyst in a high-gradient magnetic field. The nickel-containing catalyst, due to the presence of nickel salts in it, is paramagnetic and can be effectively extracted from hydrated fat in the magnetic field by a method of highly gradient magnetic separation. In the experiment technique the magnetic field strength was not less than 600 kA/m, the gradient not less than (20-30)∙104 kA/m2. Based on the results and their discussion, the analysis of empirical dependencies on each factor basis is carried out separately, as well as the construction of classical types of one-dimensional non-linear regression models for each of them. It is shown that the most statistically reliable are the exponential models. It was found that after purification the content of suspended nickel-containing catalyst in the hydrogenated fat has an inverse proportional dependence on the magnetic field strength, height (length) of the filtration zone and filtration rate. A two-dimensional exponential regression model of the content of the nickel-containing catalyst in the hydrogenated fat in the complex dependence on the rate and height of the filtration zone is built. Based on the experimental data at a certain height of the filtration zone and the selected filtration rate, a mathematical-statistical model of change in the concentration of nickel-containing catalyst depending on the magnetic field strength is made. Using the resulting two-dimensional exponential regression model, it is possible to predict the degree of purification of the salomas from the suspended catalyst depending on the specific tension of the magnetic field and to determine the tension of the magnetic field, at which the concentration of nickel as a result of cleaning will reach an advance value.

A novel non-linear one-dimensional unsteady model for thermoacoustic engine and its application on a looped traveling-wave thermoacoustic engine

A novel non-linear one-dimensional unsteady model (1DUM) for thermoacoustic engine is proposed to study on the working mechanism of thermoacoustic engines, with its focus on the unsteady onset process. Numerical simulation of a looped 3-stage traveling-wave thermoacoustic engine (TWTAE) is carried out based on this model. The complete self-excited onset process from static state to saturation is firstly realized in simulation without introducing any external disturbance. The simulation results indicate that the thermal conduction is a crucial factor to excite the oscillation in transient onset process. The onset mechanism and energy balance of thermoacoustic engine is then explained. Experimental work and calculation based on the linear thermoacoustic simulation platform, DeltaEC, are carried out to verify the model. It is shown that the calculated frequency and the proportion of the high-order harmonics by 1DUM agree well with the experiment, and 1DUM presents a reliable acoustic field distribution for the looped multi-stage TWTAE without empirical inputs like DeltaEC, which greatly simplifies the modeling process. Compared with computational fluid dynamics (CFD), 1DUM is more efficient at simulating the non-linear and unsteady onset process.

A one-dimensional model for elastic ribbons: A little stretching makes a big difference

Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness t, width a and length ℓ, assuming t≪a≪ℓ. It takes the form of a rod model with a specific non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically correct and can handle finite rotations. Two popular theories can be recovered as limiting cases, namely Kirchhoff’s rod model for small bending and twisting strains, |κi|≪t∕a2, and Sadowsky’s inextensible ribbon model for |κi|≫t∕a2; we point out that Sadowsky’s inextensible model may be a poor approximation even for ribbons having a very thin cross-section (say, with t∕a as small as 0.02). By way of illustration, the one-dimensional model is applied (i) to the lateral-torsional instability of a ribbon, showing good agreement with both experiments and finite-element shell simulations, and (ii) to the stability of a twisted ribbon subjected to a tensile force. The non-convexity of the one-dimensional model is discussed; it is addressed by a convexification argument.

A Viscoelastic-Plastic Constitutive Model of Creep in Clay Based on Improved Nonlinear Elements

A Triaxial creep experiment was performed with Chengdu clay. The results demonstrate that the deformation of clay includes instantaneous elastic strain, attenuated creep deformation, steady creep deformation, and accelerated creep deformation. In view of the nonlinear properties of creep, an improved nonlinear viscous element describing attenuating creep and an improved fractional order viscous element describing accelerating creep are established, and a one-dimensional nonlinear fractional order five-element creep model is developed with the two improved elements. The model can theoretically describe the entire process of creep. The nonlinear fractional order five-element model and the traditional five-element model were employed to verify the creep test. Compared with the traditional five-element creep model, the nonlinear fractional order creep model more faithfully reproduces the experimental data and more accurately describes the deformation trend in each stage, especially the accelerated creep stage.