两级敏度分析求解非线性稳态多宗量热传导反问题

提出基于两级敏度分析的稳态非线性热传导反问题的求解策略,建立了便于非线性正演和反演的有限元模型,由此可直接求导进行敏度分析,同时还考虑了非均质和分布参数的影响。对非线性热物性参数和边界条件的反演,进行了数值验证,取得了令人满意的结果,并对信息误差作了初步探讨。     

Non-linear phase-mismatch compensated by focusing for efficient THG in gases

A possibility of removing the limitation imposed by non-linear refraction on the attainable conversion efficiency is considered for third-harmonic generation (THG) in gaseous media. A numerical analysis has shown that by a proper focusing, the THG efficiency can approach values close to unity in spite of a substantially large value of the inhomogeneous non-linear refraction. The dependence of the conversion efficiency and the optimum linear phase-mismatch on the focusing tightness is presented for the steady-state and pulsed regimes.     

A semi-classical approach to two-frequency solitons in a three-level cascade atomic system

A semi-classical theory of two intense optical fields interacting with a third-order non-linear medium composed of a three-level cascade atomic system is presented. It is predicted that non-linear atom-field interactions allow the formation of two-frequency bright, dark and grey spatial solitons. We demonstrate through numerical simulations and analytic stability analysis that the bright and grey solitons are stable.     

Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies

A method for analyzing multidegree-of-freedom systems having a repeated natural frequency subjected to a parametric excitation is presented. Attention is given to the ordering of the various terms (linear and non-linear) in the governing equations. The analysis is based on the method of multiple scales. As a numerical example involving a parametric resonance, panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis. Some of the analytical results are verified by a numerical integration of the governing equations.     

Non-linear behaviour of electron acoustic wave dynamics in a magnetized plasma with non-thermal hot electrons

Non-linear dynamical behaviour of electron acoustic waves (EAWs) is studied in a magnetized non-thermal plasma (containing inertial cold electrons, inertialess hot electrons following non-thermal distribution function, and static ions) via a fluid dynamical approach. A linear dispersion relation is derived and the propagation of two possible modes and their evolution are studied through the different plasma configuration parameters, such as non-thermality and external magnetic field strength. In a non-linear perturbation regime, a reductive perturbation technique is employed to derive the non-linear evolution equation and the analysis is executed for travelling plane waves in terms of a non-linear dynamical system to enlighten the numerous aspects of the phase space dynamics. The results of numerical simulation predict the existence of a wide class of non-linear structures, namely solitonic, periodic, quasiperiodic, and chaotic depending upon different controlling plasma parameters. Also, Poincaré return map analysis confirms these non-linear structures of the EAWs.     

Influence of non-linear forces on beam behaviour in flutter conditions

Appropriate researches on non-linear panel flutter behaviour have been already performed by many authors. In most cases the intent of them focuses on the limit cycle determination, with particular interest towards its amplitude versus the flow dynamic pressure. This paper deals first with a study of all the solutions without damping of beam flutter versus the vibration frequency in non-linear post-critical conditions. A numerical model, which takes into account the influence of the non-linear contribution of the structural forces, due to the axial stretching of the beam, has been implemented. A complete analysis of all the possible non-linear solutions without damping leads to the possibility of characterizing the most appropriate conditions for the presence of the post-critical panel flutter limit cycles. Then the complete model, which also takes into account aerodynamic damping, has been utilized, according to the “Piston Theory”, to verify the state evolution of the fluttering damped beam towards the limit cycle, which is very near to the undamped vibrating beam state with minimum amplitude. This convergence test is an interesting aspect of the numerical results.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

ANALYSIS OF WAVELETS IN CHAOTIC STATES FOR A ONE-DEGREE-OF-FREEDOM SYSTEM WITH VISCOELASTIC RESTRAINTS

A qualitative numerical technique is presented for identifying chaotic states of a non-linear one-sided constrained one-degree of freedom (1-d.o.f.) deterministic system under a dynamic non-conservative load. Discrete wavelet analysis in its classic and packet versions was used to search for the boundaries of chaotic and non-chaotic solutions. The results obtained were verified by an analysis of the Lyapunov exponents of the investigated system. On the basis of numerical tests, one can state that wavelet analysis may be a fast and reliable tool suitable for searching for the boundaries of chaotic and non-chaotic solutions.     

Reconstruction of non-linear time delay models from data by the use of optimal transformations

We propose a new technique for the numerical reconstruction of non-linear delay differential equations from time series by applying the method of optimal transformations, a concept of multiple non-linear regression analysis. By constructing a generalized correlation function, this method allows for testing time series for delay-induced dynamics. Also, the delay times and the governing differential equations are estimated with good numerical accuracy. We present several examples which show that this method is useful also for the analysis of short and noisy time series.     

EXPLICIT PREDICTOR-MULTICORRECTOR TIME DISCONTINUOUS GALERKIN METHODS FOR NON-LINEAR DYNAMICS

Explicit predictor-multicorrector time discontinuous Galerkin (TDG) methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non-linear analysis of structural dynamics problems. The formulation is intended to inherit the accuracy properties of the exact parent implicit TDG methods. To this end, suitable predictors and correctors are designed to achieve third order accuracy, large stability limits and controllable numerical dissipation by means of an algorithmic parameter. As the study of a general non-linear case is rather complex, the analysis of the convergence properties of the resulting algorithms are restricted to conservative Duffing oscillators, for which closed-form solutions are available. It is shown that the main properties of the underlying parent scheme can be retained. Finally, results of representative numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates.     

Analysis of non-linear multilayer waveguides with Kerr-like permittivities

This paper presents a semi-analytical method for calculating the dispersion relationships for stationary non-linear TE waves guided by general multilayer waveguides with Kerr-like permittivities. The non-linear wave equations are solved rigorously for each layer, and the mode index and field distribution are systematically determined so as to satisfy the boundary conditions at all interfaces. To verify the validity of the present method, a non-linear graded-index waveguide is approximated by a number of step layers and the resultant multilayer waveguide is analysed numerically. The convergence properties of mode index and total power flow are presented, and these are compared with direct solutions of the non-linear wave equation by the numerical integration method. The dispersion relationships for a non-linear multiple-quantum-well waveguide have also been investigated under two simple permittivity models. The present method is useful for the analysis of non-linear graded-index waveguides in addition to non-linear multilayer waveguides such as the multiple-quantum-well structure.     

Active control of geometrically nonlinear transient vibration of composite plates with piezoelectric actuators

In this paper, the incremental finite element equations for geometric non-linear analysis of piezoelectric smart structures are developed using a total Lagrange approach by using virtual velocity incremental variational principles. A four-node first order shear plate element model with reduced and selective integration is also developed. Geometrically non-linear transient vibration response and control of plates with piezoelectric patches subjected to pulse loads are investigated. Active damping is introduced on the plates by coupling a self-sensing and negative velocity feedback algorithm in a closed control loop. The numerical results show that piezoelectric actuators can introduce significant damping and suppress transient vibration effectively. The effects of the number and locations of the piezoelectric actuators on the control system are also discussed.     

Non-linear coupled slosh dynamics of liquid-filled laminated composite containers: a two dimensional finite element approach

This paper brings into focus some of the interesting effects arising from the non-linear motion of the liquid free surface, due to sloshing, in a partially filled laminated composite container along with the associated coupling due to fluid-structure interaction effects. The finite element method based on two-dimensional fluid and structural elements is used for the numerical simulation of the problem. A numerical scheme is developed on the basis of a mixed Eulerian-Lagrangian approach, with velocity potential as the unknown nodal variable in the fluid domain and displacements as the unknowns in the structure domain. The FE formulation based on Galerkin weighted residual method along with an iterative solution procedure are explained in detail followed by a few numerical examples. Numerical results obtained by the present investigation for the rigid containers are first compared with the existing solutions to validate the code for non-linear sloshing without fluid-structure coupling. Thereafter the computational procedures are advanced to obtain the coupled interaction effect of non-linear sloshing in laminated composite containers.     

EFFECTS OF BASE POINTS AND NORMALIZATION SCHEMES ON THE NON-LINEAR NORMAL MODES OF CONSERVATIVE SYSTEMS

In this paper, two factors that affect the behaviors of the non-linear normal modes (NNMs) of conservative vibratory systems are investigated. The first factor is the base points (which are equivalent to Taylor series expanding points) of the non-linear normal modes and the second one is the normalization schemes of the corresponding linear modes. For non-linear systems, in general only the approximated NNM manifolds are obtainable in practice, so different base points may lead to different forms of NNM oscillators and different normalization schemes lead to different forward and backward transformations which in turn affect the numerical computation errors. Three different kinds of base points and two different normalization schemes are adopted for comparison respectively. Two examples of non-linear systems with two and three degrees of freedom, respectively, are given as illustration. Simulations for various cases are made. The analysis and the simulation results indicated that, the best base points are the abstract base points determined via the linear normal mode, which would eliminate the third order terms containing velocity (for cubic systems) or quadratic terms (for quadratic systems) in equations of the NNM oscillators. A better invariance of the NNMs would also be maintained with such base points. The best scheme of normalization is the norm-one scheme that would minimize the numerical errors.     

A numerical study of three-dimensional liquid sloshing in tanks

A numerical model NEWTANK (Numerical Wave TANK) has been developed to study three-dimensional (3-D) non-linear liquid sloshing with broken free surfaces. The numerical model solves the spatially averaged Navier–Stokes equations, which are constructed on a non-inertial reference frame having arbitrary six degree-of-freedom (DOF) of motions, for two-phase flows. The large-eddy-simulation (LES) approach is adopted to model the turbulence effect by using the Smagorinsky sub-grid scale (SGS) closure model. The two-step projection method is employed in the numerical solutions, aided by the Bi-CGSTAB technique to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate volume-of-fluid (VOF) method is used to track the distorted and broken free surface. Laboratory experiments are conducted for both 2-D and 3-D non-linear liquid sloshing in a rectangular tank. A linear analytical solution of 3-D liquid sloshing under the coupled surge and sway excitation is also developed in this study. The numerical model is first validated against the available analytical solution and experimental data for 2-D liquid sloshing of both inviscid and viscous fluids. The validation is further extended to 3-D liquid sloshing. The numerical results match with the analytical solution when the excitation amplitude is small. When the excitation amplitude is large where sloshing becomes highly non-linear, large discrepancies are developed between the numerical results and the analytical solutions, the former of which, however, agree well with the experimental data. Finally, as a demonstration, a violent liquid sloshing with broken free surfaces under six DOF excitations is simulated and discussed.     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.     

PHYSICAL AND NUMERICAL MODELLING OF THE DYNAMIC BEHAVIOR OF A FLY LINE

The planar equations of motion for a tapered fly line subjected to tension, bending, aerodynamic drag, and weight are derived. The resulting theory describes the large non-linear deformation of the line as it forms a propagating loop during fly casting. A cast is initiated by the motion of the tip of the fly rod that represents the boundary condition at one end of the fly line. At the opposite end, the boundary condition describes the equations of motion of a small attached fly (point mass with air drag). An efficient numerical algorithm is reviewed that captures the initiation and propagation of a non-linear wave that describes the loop. The algorithm is composed of three major steps. First, the non-linear initial-boundary-value problem is transformed into a two-point boundary-value problem, using finite differencing in time. The resulting non-linear boundary-value problem is linearized and then transformed into an initial-value problem in space. Example results are provided that illustrate how an overhead cast develops from initial conditions describing a perfectly laid out back cast. The numerical solutions are used to explore the influence of two sample effects in fly casting, namely, the drag created by the attached fly and the shape of the rod tip path.     

Analytical method for steady state vibration of system with localized non-linearities using convolution integral and Galerkin method

The analytical method using transfer function or impulse response is very effective for analyzing non-linear systems with localized non-linearities. This is because the number of non-linear equations can be reduced to that of the equations with respect to points connected with the non-linear element. In the present paper, analytical method for the steady state vibration of non-linear system including subharmonic vibration is proposed by utilizing convolution integral and the impulse response. The Galerkin method is introduced to solve the non-linear equations formulated by the convolution integral, and then the steady state vibration is obtained. An advantage of the present method is that stability or instability of the steady state vibration can be discriminated by the transient analysis from convolution integral. The three-degree-of-freedom mass-spring system is shown as a numerical example and the proposed method is verified by comparing with the result by Runge-Kutta-Gill method.     

Dynamical behaviour of non-linear quantum ion acoustic wave in weakly magnetized electron–positron–ion plasma

In order to investigate the propagation characteristics of linear and non-linear ion acoustic waves (IAWs) in electron–positron–ion quantum plasma in the presence of external weak magnetic field, we have used a quantum hydrodynamic model, and degenerate statistics for the electrons and positrons are taken into account. It is found that the linear dispersion relation of the IAW was modified by the externally applied magnetic field. By using the reductive perturbation technique, a gyration-modified Korteweg-de Vries equation is derived for finite amplitude non-linear IAWs. Time-dependent numerical simulation shows the formation of an oscillating tail in front of the ion acoustic solitons in the presence of a weak magnetic field. It is also seen that the amplitude and width of solitons and oscillating tails are affected by the relevant plasma parameters such as quantum diffraction, positron concentration, and magnetic field. We have performed our analysis by extending it to account for approximate soliton solution by asymptotic perturbation technique and non-linear analysis via a dynamical system approach. The analytical results show the distortion of the shape of the localized soliton with time, and the non-linear analysis confirms the generation of oscillating tails.