Various Types Of Nonlinearities Research Articles

Simulation of nonlinear benchmarks and sheet metal forming processes using linear and quadratic solid–shell elements combined with advanced anisotropic behavior models

A family of prismatic and hexahedral solid‒shell (SHB) elements with their linear and quadratic versions is presented in this paper to model thin 3D structures. Based on reduced integration and special treatments to eliminate locking effects and to control spurious zero-energy modes, the SHB solid‒shell elements are capable of modeling most thin 3D structural problems with only a single element layer, while describing accurately the various through-thickness phenomena. In this paper, the SHB elements are combined with fully 3D behavior models, including orthotropic elastic behavior for composite materials and anisotropic plastic behavior for metallic materials, which allows describing the strain/stress state in the thickness direction, in contrast to traditional shell elements. All SHB elements are implemented into ABAQUS using both standard/quasi-static and explicit/dynamic solvers. Several benchmark tests have been conducted, in order to first assess the performance of the SHB elements in quasi-static and dynamic analyses. Then, deep drawing of a hemispherical cup is performed to demonstrate the capabilities of the SHB elements in handling various types of nonlinearities (large displacements and rotations, anisotropic plasticity, and contact). Compared to classical ABAQUS solid and shell elements, the results given by the SHB elements show good agreement with the reference solutions.

A Monte Carlo Investigation of Unit Root Tests and Long Memory in Detecting Mean Reversion in I(0) Regime Switching, Structural Break, and Nonlinear Data

The potential observational equivalence between various types of nonlinearity and long memory has been recognized by the econometrics community since at least the contribution of Diebold and Inoue (2001). A large literature has developed in an attempt to ascertain whether or not the long memory finding in many economic series is spurious. Yet to date, no study has analyzed the consequences of using long memory methods to test for unit roots when the “truth” derives from regime switching, structural breaks, or other types of mean reverting nonlinearity. In this article, I conduct a comprehensive Monte Carlo analysis to investigate the consequences of using tests designed to have power against fractional integration when the actual data generating process is unknown. I additionally consider the use of tests designed to have power against breaks and threshold nonlinearity. The findings are compelling and demonstrate that the use of long memory as an approximation to nonlinearity yields tests with relatively high power. In contrast, misspecification has severe consequences for tests designed to have power against threshold nonlinearity, and especially for tests designed to have power against breaks.

Pseudospectral solutions to some singular nonlinear BVPs

We solve by Laguerre collocation method two second order genuinely nonlinear singularly perturbed boundary value problems and a fourth order nonlinear one formulated on the half-line. First, we are concerned with solutions to a model equation for low Reynolds number flow, the so-called Cohen, Fokas, Lagerstrom model. Then we compute monotonously increasing solutions, the so-called "bubble type solutions", to density profile equation for the description of the formation of microscopical bubbles in a nonhomogeneous fluid. We compare the results obtained in the latter case with those carried out by shooting or polynomial collocation both coupled with domain truncation. A fourth order nonlinear boundary value problem supplied with mixed boundary conditions in origin as well as at infinity is also solved. The solution and its first three derivatives are obtained and compared with their counterparts obtained by Keller's box scheme. The collocation based on Laguerre functions avoids the domain truncation, exactly impose any kind of boundary conditions at infinity and resolve with reliable accuracy the sharp interior or exterior layer of the solutions. The method is fairly easy implementable and efficient in treating various types of nonlinearities. The accuracy of the method is also tested on the classical Blasius problem (a third order one). We solve with the same method the eigenvalue problem obtained by linearization around the constant solution, which corresponds to the case of a homogeneous fluid (without bubbles), and observe that this solution is stable.

Flatness of Semilinear Parabolic PDEs—A Generalized Cauchy–Kowalevski Approach

A generalized Cauchy-Kowalevski approach is proposed for flatness-based trajectory planning for boundary controlled semilinear systems of partial differential equations (PDEs) in a one-dimensional spatial domain. For this, the ansatz presented in “Trajectory planning for boundary controlled parabolic PDEs with varying parameters on higher-dimensional spatial domains” (T. Meurer and A. Kugi, IEEE Trans. Autom. Control, vol. 54, no, 8, pp. 1854-1868, Aug. 2009) using formal integration is generalized towards a unified design framework, which covers linear and semilinear PDEs including rather broad classes of nonlinearities arising in applications. In addition, an efficient semi-numerical solution of the implicit state and input parametrizations is developed and evaluated in different scenarios. Simulation results for various types of nonlinearities and a tubular reactor model described by a system of semilinear reaction-diffusion-convection equations illustrate the applicability of the proposed method.

Finite-element simulation of contact interaction with the use of concepts of contact pseudomedium mechanics

In the present paper, we analyze an algorithm for solving problems of elastic and elastoplastic contact between rigid deformable bodies taking into account the presence of additional media with various properties in the contact region. These may be nonlinear properties of surface rough layers and films. To take into account the influence of such layers, the finite-element scheme uses contact finite elements with nonlinear properties. We describe an algorithm developed for solving problems with various types of nonlinearities. Using the example of experimental studies described in detail in [1, 2], we identify the nonlinear properties of contact finite elements. Several examples are used to demonstrate the algorithm operating capability.

Generalized solutions to a semilinear wave equation

Semilinear wave equations in space dimension n ⩽ 9 with singular data and various types of nonlinearities are considered. We employ the framework of the algebra G L 2 of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter ε such that it becomes globally Lipschitz for each such ε . This produces a family of Cauchy problems, to which we construct a family of solutions which in turn determines an element of G L 2 which serves as a generalized solution to the original equation. For cubic nonlinear terms the equation is directly solved in G L 2 without regularizations. Finally, in certain cases, it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one.

Damage Diagnosis of Rotors: Application of Hilbert Transform and Multihypothesis Testing

In this paper, a combined approach to damage diagnosis of rotors is proposed. The intention is to employ signal-based as well as model-based procedures for an improved detection of size and location of the damage. In the first step, Hilbert transform signal-processing techniques allow for a computation of the signal envelope and the instantaneous frequency, so that various types of nonlinearities due to damage may be identified and classified based on measured response data. In the second step, a multihypothesis bank of Kalman filters is employed for the detection of the size and location of the damage based on the information of the type of damage provided by the results of the Hilbert transform.

Relativistic mean-field theory and the high-density nuclear equation of state

The properties of high-density nuclear and neutron matter are studied using a relativistic mean-field approximation to the nuclear matter energy functional. Based on ideas of effective field theory, nonlinear interactions between the fields are introduced to parametrize the density dependence of the energy functional. Various types of nonlinearities involving scalar-isoscalar ($\sigma$), vector-isoscalar ($\omega$), and vector-isovector ($\rho$) fields are studied. After calibrating the model parameters at equilibrium nuclear matter density, the model and parameter dependence of the resulting equation of state is examined in the neutron-rich and high-density regime. It is possible to build different models that reproduce the same observed properties at normal nuclear densities, but which yield maximum neutron star masses that differ by more than one solar mass. Implications for the existence of kaon condensates or quark cores in neutron stars are discussed.

Nonlinear time series models for multivariable dynamic processes

Several paradigms are available for developing nonlinear dynamic input-output models of processes. Polynomial models, threshold models, models based on spline functions, and polynomial models with exponential and trigonometric functions can describe various types of nonlinearities and pathological behavior observed in many physical processes. A unified nonlinear model development framework is not available, and the search of the appropriate nonlinear structure is part of the model development effort. Various artificial neural network structures and nonlinear time series model structures are presented and illustrated by developing a model from data sets generated by a series of example systems. The use of a nonlinear model development paradigm which is not compatible with the types of nonlinearities that exist in the data can have a significant effect on model development effort and model accuracy.

Stress Distribution in Granular Media and Nonlinear Wave Equation

We propose phenomenological equations to describe how forces propagate within a granular nledium. Trie linear part of these equations is a wave equation, where the vertical coordinate plays trie rote of time, and trie horizontal coordinates trie raie of space. This means that (in two dimensions) trie stress propagates along light-cones; trie angle of these canes is related (but not equal to) trie angle of repose. Dispersive corrections to trie picture, and various types of nonlinearity are discussed. Inclusion of nonlinear terms may be able to describe trie arching phenomenon, which has been proposed to explain trie nonintuitive hor- izontal distribution of vertical pressure (with a local minimum or dip under the apex of trie pile) observed experimentally. However, for physically motivated parameter choices, a hump, rather than a dip, is predicted. This is aise true of a perturbative solution of the continuum stress equations for nearly-flot piles. The nature of trie force fluctuations is aise briefly discussed.

Asymptotic behavior of positive solution branches of elliptic problems with linear part at resonance

We consider Dirichlet problems for semilinear elliptic equations whose linear part is at resonance. For various types of nonlinearities we provide sufficient conditions for the solvability of these problems for certain types of forcing terms. Depending upon the dimension of the independent variable and the shape of the domain, we obtain multiple existence results and show that in some cases infinitely many solutions may exist.

Forced Vibration Analysis of a Nonlinear Structure Connected in Series. 1st Report. Suggesting the Incremental Transfer Influence Coefficient Method.

The incremental transfer influence coefficient method is developed in order to analyze the periodic steady-state vibrations of a nonlinear system with a multi-degree-of-freedom system on a personal computer, combining the concepts of both the method of harmonic balance and the transfer influence coefficient method through the incremental method. The present method has some merits, that is, high computation speed, high numerical computational accuracy, applicability to the various types of nonlinearity and so on. For the stability problem, general and simplified methods are suggested in order to determine the stability of the periodic solutions from the behavior of the solutions of the corresponding variational equation, which is described as a multi-degree-of-freedom system with parametric excitations. The former is applicable to the variational equation in general and with high accuracy. The latter gives the sufficient condition for certain main unstable regions of the variational equation.

Image enhancement by nonlinear signal processing

We present a nonlinear joint transform processor that can perform image enhancement. Image enhancement results are obtained for different degrees of nonlinearity applied to the joint power spectrum. The first-order harmonic term at the output plane produces the enhanced image which has the exact Fourier phase of the input image and a Fourier magnitude of the input modified by the nonlinearity. For compression types of nonlinearity, the thresholding will redistribute the energy in the Fourier magnitude of the image by increasing the magnitude of the higher spatial frequencies. Thus, the fine details of the image that are contained in the high spatial frequencies of the joint power spectrum are enhanced. We investigate the effects of various types of nonlinearity on the enhanced images. Analytical expressions for the enhanced images obtained by the nonlinear technique will be provided. Computer simulations of the nonlinear processor for image enhancement are presented to study the performance of the system. We show that the nonlinear technique produces reasonably good results and that it provides much better light efficiency when compared with the block spatial filtering technique.

Image enhancement by nonlinear joint transform processing

A nonlinear joint transform processor is presented that can perform image enhancement. Image enhancement results are obtained for different degrees of nonlinearity applied to the joint power spectrum. The first order harmonic term at the output plane produces the enhanced image which has the exact Fourier phase information of the input image and a Fourier magnitude of the input modified by the nonlinearity. For compression types of nonlinearities, the thresholding will redistribute the energy in the Fourier magnitude of the image by increasing the magnitude of the higher spatial frequencies. Thus, the fine details of the image that are contained in the high spatial frequencies of the joint power spectrum are enhanced. The effects of various types of nonlinearities on the enhanced images are investigated. Analytical expressions for the enhanced images obtained by the nonlinear techniques will be provided. Computer simulations of the nonlinear processor for image enhancement are presented to study the performance of the system. It is shown that the nonlinear technique produces reasonably good results.

Stochastic Stability of Nonlinear Oscillators

The stochastic stability of a nonlinear oscillator parametrically excited by a stationary Markov process is considered. The stochastic stability problem in terms of a mean first passage time is formulated. Specifically, if the mean first passage time of the energy E of the oscillator to a given energy level is finite,then the oscillator is unstable. A method of averaging is used to derive a Fokker–Planck equation in the energy variable. The stability criterion depends on the nature of the boundary points $E = 0$ and $E = \infty $ and is expressed in terms of a Feller-type criterion. The stability condition is derived for various types of nonlinearities, including Coulomb friction. In contrast, it is observed that the standard stability criterion, in terms of Lyapunov exponents, is inconclusive for this type of problem.

A first passage time approach to stochastic stability of nonlinear oscillators

We consider the stochastic stability of parametrically excited nonlinear noisy oscillators. We formulate the stochastic stability problem in terms of first passage times. Specifically we calculate the probability that the energy remains bounded by a preassigned level E c, for all time. The stability criterion is then expressed in terms of a Feller-type condition. We show that if the criterion is satisfied, the probability that the first passage time ≈τ from E to E c is finite, approaches zero as E approaches zero, so that the oscillator is stochastically stable. If the criterion is not satisfied, ≈τ is finite with probability one, so that the oscillator is stochastically unstable. If ≈τ is finite, we also calculate the mean first passage time to E c. Our stability condition is derived for various types of nonlinearities, including Coulomb friction. In contrast, we observe that the standard stability criterion, in terms of Lyapunov exponents, is inconclusive for this type of problem.

Strain-softening and large-displacement analysis in structural plasticity

In the past few years, several computer systems for the nonlinear structural analysis by finite elements have been conceived and implemented. Any type of nonlinearity was considered in a unified computational scheme consisting of a combination of the step by step (incremental) approach and the Newton-Raphson procedure for the solution of the resulting nonlinear system at each loading step. Such a unified approach may be successful from the point of view of system organization and implementation, but too expensive, or even incorrect, for particular types of nonlinearities. This paper considers some elastic-plastic problems in which two major aspects have been disregarded: (1) plasticity confined to a few elements of the structure, and (2) possibility of unloading phenomena. The latter imposes a very severe limitation off the size of the loading step, with obvious computational disadvantages. With these considerations in mind, the development of a computer system for Structural Plasticity Analysis problems (or STRUPL-ANALYSIS) is currently in progress. The purpose of the paper is to present the concepts involved in this system and its differences with regard to other existing codes. Particular interest is paid to limit load and stability analysis for various types of material and geometrical nonlinear behaviour of the structure. The calculation scheme is first developed for elastic-perfectly plastic structures and then is extended to structures for which strain hardening, strain softening and large displacement effects need be considered. A numerical example illustrates the application of the system to various types of nonlinearities.

Nonlinear optical resonators (excited by external radiation) (review)

A review is given of theoretical and experimental investigations of optical resonators excited by an external light beam and filled with a nonlinear medium. An analysis is made of various types of nonlinearities (Kerr, Raman, Brillouin, and quadratic nonlinearities, saturation in a medium of two-level particles) and equations which describe the dynamics of optical oscillations in a resonator, are derived. Descriptions are given of the main properties such as limitation of the transmitted beam intensity, bistability and hysteresis, intensity pulsations, and amplification of amplitude-frequency deviations in the optical beam. Transient processes are also discussed. In the description of the experimental results, not only the resonator parameters and the observed properties are indicated but attention is also paid to identification of the type of nonlinearity. An analysis is also made of hybrid systems in which the nonlinearity is simulated (and deliberately increased) by radioelectronics.

Some new solutions to ferroresonance problem in power system

A parametric equation of the envelope on the G?1(jw) plane for the general polynomial-type nonlinearity is derived. An analytical method has been suggested for the evaluation of threshold frequency for the occurrence of ferroresonance. Mathematical equations have been deduced for computing the jump-to points for quintic nonlinearity. A generalised equation for asymptotic values of percentage unstable zone for various types of nonlinearities has been derived. New definitions for jump-up severity and jump-down severity are introduced. A system due to Swift has been analaysed for the purposes of illustration.