Finite element method resolution of non-linear Helmholtz equation

A non-paraxial beam propagation method for non-linear media is presented. It directly implements the non-linear Helmholtz equation without introducing the slowing varying envelope approximation. The finite element method has been used to describe the field and the medium characteristics on the transverse cross-section as well as along the longitudinal direction. The finite element capabilities as, for example, the non-uniform mesh distribution, the use of adaptive mesh techniques and the high sparsity of the system matrices, allow one to obtain a fast, versatile and accurate tool for beam propagation analysis. Examples of spatial soliton evolution describe phenomena not predicted in the frame of the slowing varying envelope approximation.     

Propagation of finite amplitude sound waves radiated from a pulsating sphere

The propagation of weakly non-linear acoustic waves is studied, which are radiated from a harmonically pulsating sphere in an inviscid perfect gas. A representation of the solution is presented for a far field equation of the first order, which is closely related to the solution obtained by the method of renormalization. The applicability of the method to the present problem is proved within the first order approximation. The asymptotic solution in the far field is obtained up to the second order by using the method of renormalization. A uniformly valid solution is, to the second approximation, constructed by matching the near field solution with the far field solution. Also investigated is the effect of dimensionless parameters and the second order correction on the acoustic shock formation distances and the non-linear distortion of waveforms.     

Limit cycle oscillation of missile control fin with structural non-linearity

Non-linear aeroelastic characteristics of a deployable missile control fin with structural non-linearity are investigated. A deployable missile control fin is modelled as a two-dimensional typical section model. Doublet-point method is used for the calculation of supersonic unsteady aerodynamic forces, and aerodynamic forces are approximated by using the minimum-state approximation. For non-linear flutter analysis structural non-linearity is represented by an asymmetric bilinear spring and is linearized by using the describing function method. The linear and non-linear flutter analyses indicate that the flutter characteristics are significantly dependent on the frequency ratio. From the non-linear flutter analysis, various types of limit cycle oscillations are observed in a wide range of air speeds below or above the linear divergent flutter boundary. The non-linear flutter characteristics and the non-linear aeroelastic responses are investigated.     

A note on the ultraspherical polynomial approximation method of averaging

The method of averaging based on ultraspherical polynomial approximation has been applied to several non-linear oscillation problems. In this paper it is shown that the above method can be interpreted as a Galerkin technique with an appropriate weight function. Even though the analysis is not rigorous enough mathematically, it does provide insight into this generalized method of averaging.     

Non-linear effect of wall linings on sound attenuation in ducts

The equivalent surface source method is extended to the analysis of high intensity sound propagation in a duct whose wall is partially treated with a sound absorbing material. The propagation of sound in the gas is assumed to be linear, but the acoustic resistance of the sound absorbing material is assumed to be a function of the normal acoustic velocity. The problem is reduced to a non-linear integro-differential equation for the fluid particle displacement at the lined wall surface, which can be solved by a successive approximation method. Numerical examples show that the non-linear effect decreases or increases the peak sound attenuation rate of the lowest mode depending upon the linear component of the resistance. The dependence of the attenuation spectrum on modal phase difference of multi-mode incident waves is heavily affected by the non-linear effect. In the case of incident waves of multi-circumferential modes, different circumferential modes are generated by the non-linear effect.     

A non-linear observer for unsteady three-dimensional flows

A method is proposed to estimate the velocity field of an unsteady flow using a limited number of flow measurements. The method is based on a non-linear low-dimensional model of the flow and on an expansion of the velocity field in terms of empirical basis functions. The main idea is to impose that the coefficients of the modal expansion of the velocity field gives the best approximation of the available measurements, while at the same time satisfying the non-linear low-order model as closely as possible. Practical applications may range from feedback flow control to the monitoring of the flow in non-accessible regions. The proposed technique is applied to the flow around a confined square cylinder, both in two- and three-dimensional flow regimes. Comparisons are provided with existing linear and non-linear estimation techniques.     

The effect of spin on the non-linear resonant motions of a dynamical system

The motions of a two degree of freedom mechanical oscillator in a state of internal resonance due to the non-linear coupling between its modes are analyzed by the method of multiple scales. The system is connected by a motor to a vertical shaft driven at a constant spin rate relative to inertial space. It is shown that the non-linear resonance phenomenon can effectively be controlled by properly changing the spin rate of the motor. In addition, the transition curves that separate the non-linear resonant and the non-resonant motions of the system are also determined analytically by a straightforward perturbation method. The analytical expression for the transition curves is used in connection with the multiple scale analysis to yield a refined approximation for the main characteristics of the non-linear resonant motion.     

On the berger approximation: A critical re-examination

Since the first paper by Berger some two decades ago, the simplification known as the Berger approximation has been invoked by the authors of several score papers in spite of the fact that no rational mechanical basis for the approximation could be found. Many recent papers have raised doubts on its applicability. In this paper, using certain well known results from the two dimensional theory of elasticity, a plausible explanation for the origin of the Berger method is suggested. The arguments can be developed further to show that other specious Berger-like approximations can be developed, all of them leading to uncoupled non-linear equations yielding different overall results. Further, it is shown that such methods fail to predict the non-linear behaviour with respect to important parameters and that whatever accuracy is obtained in the solution of a particular problem can at best be attributed to fortuity.     

The dynamic analysis of circular plates and shallow spherical shells

In the investigation reported here an attempt has been made to study the influence of Berger's approximation on the non-linear transient response of circular plates and shallow spherical shells. The governing equations of motion obtained from Berger's approximation are solved by using the rapidly converging Chebyshev series spacewise and the Houbolt scheme for integration in the time domain. Results calculated when using Berger's approximation are compared with exact results. It is shown that Berger's method yields very accurate values for plates and shells under transient loading, in the case of immovable edge conditions.     

Shear and rotatory inertia effects on large amplitude vibration of skew plates

The large amplitude free flexural vibration of elastic, isotropic skew plates is investigated, the effects of transverse shear and rotatory inertia being included. By use of Galerkin's method and the extended Berger approximation, solutions are obtained on the basis of an assumed vibration mode. The non-linear period vs. amplitude behavior is of the hardening type and the non-linear period is found to increase when the effects of transverse shear and rotatory inertia are considered in the analysis. The influence of these effects on aspect ratios and skew angles of thin and moderately thick skew plates is investigated both at small and large amplitudes.     

A Field Method for Integrating Equations of Motion of Nonlinear Mechanico-Electrical Coupling Dynamical Systems

We deal with the generalization of the field method to weakly non-linear mechanico-electrical coupling systems. The field co-ordinates and field momenta approaches are combined with the method of multiple time scales in order to obtain the amplitudes and phase of oscillations in the first approximation. An example in mechanico-electrical coupling systems is given to illustrate this method.     

On approximate theories of single-mode rectangular waveguides

We present a new scheme of computation for the CEVAR (cosine-exponential field based variational method) for modelling wave propagation in single mode rectangular waveguides. The method avoids non-linear optimization. We show that the fully optimized weighted index approach is identical to the CEVAR method in which the Rayleigh quotient is optimized to obtain the best separable approximation to the field propagating in single mode guides with rectangular dimensions.     

Analytic-numerical approach to non-linear problems in dielectric waveguides

A combined analytic-numerical approach is presented for investigations of electromagnetic signal propagation in dielectric waveguides containing a non-linear medium. The approach is based on the resolvent method for the Volterra integral equation describing an electromagnetic process. The proposed algorithm uses the exact expression for the solution to the integral equation for the electromagnetic field in the planar waveguide with time jump changing of the medium properties. In the case of a non-linear dielectric as the waveguide core the continuous change of the medium parameters is replaced by jump changes of the latter. This is the only approximation made in the algorithm. The resolvents that are constructed give the exact solution on each jump step. The results from some example numerical calculations are presented.     

An intrinsic method of harmonic analysis for non-linear oscillations (a perturbation technique)

An intrinsic method of harmonic analysis for application to problems concerning non-linear periodic oscillations is presented. The method is designed to overcome the observed shortcomings of the conventional method of harmonic balance, and it is described with the aid of an example which was used earlier to demonstrate the inconsistency of the results obtained through the method of harmonic balance. The new technique is a perturbation method and it eliminates the danger of omitting any possible contributions of various harmonics to a particular approximation by engaging all the necessary harmonics and excluding the others automatically. Thefore the new approach yields ordered forms of consistent approximations for the non-linear problem without requiring knowledge about the number of harmonics a priori, in contrast to the conventional technique.     

A finite slow-motion approximation method for self-gravitating systems

We present a finite approximation procedure for weakly self-gravitating quasi-stationary systems applied to a Lorentz-invariant, non-linear scalar model theory of gravity. We derive lowest order radiation-reaction terms in the equations of motion and compare them with those terms derived using the usual slow-motion method.     

Optical non-linearities associated to hydrogenic impurities in InAs/GaAs self-assembled quantum dots under applied electric fields

Abstract

The effects of the hydrogenic impurity on the electron-related non-linear optical processes in a InAs/GaAs dome-shaped quantum dot with a wetting layer under applied electric fields are studied within the density-matrix formalism. The one-electron energy levels and wave functions are calculated using the effective mass approximation and the finite element method. The non-linear optical absorption, relative refractive index change and non-linear optical rectification associated with interlevel transitions are calculated under a strong probe field excitation for both in-plane and z-polarisation of the incident light. According to our results as the electric field increases the absorption and dispersion peaks decrease and exhibit red shift. Hydrogenic impurity located at the origin induces a blue shift in the optical responses. For the optical absorption coefficient the peaks magnitude is enhanced by the impurity presence independent of the electric field strengths, whereas the non-linear optical rectification is larger in the case with impurity only for zero applied electric field.     

Weighted linearization technique for period approximation in large amplitude non-linear oscillations

Period approximations for conservative, one-dimensional, non-linear oscillating systems are considered. The approximation technique used is linearization of the non-linear restoring function governing the system. While in previous papers the best period approximations in the small amplitude limit have been discussed, the best large amplitude results are investigated herein, with use of linearization based on weight functions of the power type. The non-linear functions examined here are odd polynomials, sine, hyperbolic sine, tangent, and hyperbolic tangent.     

FDTD modelling of spatial soliton propagation

The finite difference time domain method is used as a tool to model non-linear optical devices. Following this approach, Maxwell's equations are directly integrated in the time domain, avoiding any mathematical approximation. As an example, spatial soliton propagation and interaction will be modelled by the FDTD technique. Some new results will be discussed, to highlight the envisaged breakthroughs allowed by the method. This revised version was published online in November 2006 with corrections to the Cover Date.