Nonlinear system

Summary

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear com

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https://en.wikipedia.org/wiki/Nonlinear_system

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The Adomian Decomposition Method for Nonlinear Partial Differential Equations

Natalie Sauerwald

The goal of this report is to study the method introduced by Adomian known as the Adomian Decomposition Method (ADM), which is used to find an approximate solution to nonlinear partial differential equations (PDEs) as a series expansion involving the recursive solution of linear PDEs. We first describe the method, giving two specific examples with different nonlinearities and show exactly how the method works for these problems. Some analytical convergence results are then given, along with numerical solutions to the examples demonstrating these convergence results. A discussion of parameters inside of these nonlinearities follows, both for polynomial nonlinearities and for the more complicated hyperbolic sine nonlinearity problem. Finally, we compare the ADM with the Picard method, pointing out some important differences and demonstrating them by solving the given examples with both methods and comparing the results.

2013

Open flows, such as wakes, jets, separation bubbles, mixing layers, boundary layers, etc., develop in domains where fluid particles are continuously advected downstream. They are encountered in a wide variety of situations, ranging from nature to technology. Such configurations are characterised by the development of strong instabilities resulting in observable unsteady dynamics. They can be categorised as oscillators which present intrinsic dynamics through self-sustained oscillations, or as amplifiers, which exhibit a strong sensitivity to external disturbances through extrinsic dynamics. Over the years, different linear and nonlinear approaches have been adopted to describe the dynamics of oscillators and amplifiers. However, a simplified physical description that accurately accounts for the nonlinear saturation of instabilities in oscillators as well as that of the response to disturbances in stable amplifier flows is still missing. In this thesis, this question is addressed by introducing a self-consistent semi-linear model. The model is formally constructed by a set of equations where the mean flow is coupled to a linear perturbation equation through the Reynolds stress. The full nonlinear fluctuating motion is thus approximated by a linear equation. The nonlinear dynamics of oscillators is studied in the cylinder wake, where the most unstable eigenmode of finite amplitude is coupled to the instantaneous mean flow for different oscillation amplitudes. This family of solutions provides an instantaneous mean flow evolution as a function of an equivalent slow time. A transient physical picture is formalised, wherein a harmonic perturbation grows and changes the amplitude, frequency, growth-rate and structure due to the modification of the instantaneous mean flow by the Reynolds stress forcing. Eventually this perturbation saturates when the flow is marginally stable. In contrast to standard linear stability analysis around the mean flow, the iterative solution of the model provides a priori an accurate prediction of the instantaneous amplitude, frequency and growth rate, as well as the flow fields, without resorting to any input from numerical or experimental data. Regarding noise amplifiers, the nonlinear saturation of the large linear amplification to external disturbances is studied in the framework of the receptivity analysis of the backward facing step flow. The self-consistent model is first introduced for harmonic forcing and later generalised to stochastic forcing by reformulating it conveniently in frequency domain. The results show an accurate prediction of the response energy as well as the flow fields. Hence, a similar picture is revealed, wherein the Reynolds stress dominates the saturation process. Despite the difference in the dynamics of the described flows, they share the same nonlinear saturation mechanism: the mean flow distortion.

EPFL2015

A micromechanical approach to the behaviour of single wood fibers and wood fracture at cellular level

Marjan Sedighi Gilani

Mechanical and fracture behaviors of wood are defined by the morphology and mechanical properties of wood fibers and their bonding medium. Parallel orientation of wood fibers makes them the most influential microstructural elements from the mechanical point of view. On the other hand, in wood fracture, the difference between the properties of fiber and bonding medium (which make weak cleavage plates) plays a more important role. Experiments show that the mechanical behavior of a single wood fiber under axial tension is complex, although the cause of this complexity has still not been clearly understood. In this thesis, in order to explain the mechanism underlying the mechanical behavior of wood fibers and the fracture of wood specimens at fiber level, a micromechanical approach has been used. Confocal laser scanning microscopy was used to investigate the pattern of the distribution of microfibrils in different wood fibers. It was shown that the microfibril angle within a single fiber is non-uniform and this non-uniformity in radial walls of earlywood fibers, which contain the bordered pits, is higher than tangential walls of earlywood fibers and also higher than in latewood fibers. Tensile and cyclic tensile tests on single spruce fibers were carried out and their non-linear and force-history dependent behaviors under axial tension were shown. It was found that the fiber behavior is affected by the range of microfibril angle non-uniformities and other defects. After a certain force limit, wood fiber undergoes irreversible strains and the elastic limit of the fiber increases in the tensile loading. To explain these results, a model based on the assumption of helical and non-uniform distribution of cellulose microfibrils in the fiber and damage of the hemicellulose and lignin matrix after yielding, was proposed. The model indicated that multi-damage and evolution of microfibrils in the damages segments are the main governing mechanisms of the tensile behavior of wood fiber. Difficulties of considering the porous and heterogeneous microstructure of wood in a continuum-based fracture model, led us to develop a mixed lattice-continuum model. The three-dimensional geometry of lattice, composed of different beam elements which represent the bonding medium and alternation of earlywood and latewood fibers, enabled us to detect the propagation of cracks in both cross sections and longitudinal sections at the fiber level. Model showed that in Mode I fracture, parallel to the fibers, the location of the developed crack and the resulting stress-strain curves have a good agreement with the experimental evidence.

EPFL2006

EPFL2015

A micromechanical approach to the behaviour of single wood fibers and wood fracture at cellular level

Marjan Sedighi Gilani

EPFL2006