Uncertainty-based experimental validation of nonlinear reduced order models

This paper presents the first results of a combined experimental–computational investigation focused on the validation of reduced order models of geometrically nonlinear structures in the presence of uncertainty. The validation approach considered here is based on the premise that the model is valid if the experimental results can be considered as random sample responses of the stochastic system of which the reduced order model is the mean. For the situation considered here, the power spectra of the experiments should lie within the 2nd and 98th percentiles of the response (forming the uncertainty band) of the stochastic model. Nominally clamped–clamped beams are considered to demonstrate the entire process. The construction of two mean reduced order models and their stochastic counterparts are first performed. Then, the validation effort is carried out by comparing experimentally obtained power spectra and their corresponding computational uncertainty bands. This process leads, for both reduced order models, to a very good representation of the important upper envelope (98th percentile) of the experimental data but a less good fit of the lower envelope (2nd percentile).     

窄带随机噪声作用下单自由度非线性干摩擦系统的响应

研究了单自由度非线性干摩擦系统在窄带随机噪声参数激励下的主共振响应问题.用Krylov-Bogoliubov平均法得到了关于慢变量的随机微分方程.在没有随机扰动情形,得到了系统响应幅值满足的代数方程.在有随机扰动情形,用线性化方法和矩方法给出了系统响应稳态矩计算的近似计算公式.讨论了系统阻尼项、非线性项、随机扰动项和干摩擦项等参数对于系统响应的影响.理论计算和数值模拟表明,当非线性强度增大时系统的响应显著变小,系统分岔点滞后;随着激励频率的增大系统响应变大,而当激励频率小于一定的值时,系统响应为零;增加干 关键词： 单自由度非线性干摩擦系统 主共振响应 Krylov-Bogoliubov平均法     

Experiments and simulations for large-amplitude vibrations of rectangular plates carrying concentrated masses

Large-amplitude (geometrically nonlinear) forced vibrations of a stainless-steel thin rectangular plate carrying different concentrated masses are experimentally studied. The experimental boundary conditions are close to those of a clamped plate. The plate is vertically and horizontally tested in order to investigate the gravity effect. Harmonic excitation is applied by using electrodynamic exciter and the plate vibration is measured by using a laser Doppler vibrometer with displacement decoder. The harmonic excitation is controlled in closed-loop in order to keep constant the desired force and is increased (or decreased) by very small discrete steps. Numerical simulations on reduced-order models, obtained by using Von Kármán nonlinear plate theory and global discretization, are also carried out and compared to experiments in order to better understand the system. Results show that concentrated masses have no effect on the trend of nonlinearity of the vertical plate, while they play a role in case of horizontal plate due to the static flexural deflection caused by gravity, which reduces the hardening-type nonlinearity. Initial geometric imperfection (deviation from flat surface in vertical position) of the plate is measured and taken into account; it plays a significant role.     

A note on transverse vibrations of a rectangular plate with a free,rectangular, corner cut-out

An approximate solution to the title problem is presented, obtained by using the Rayleigh-Ritz method. The analysis is presented for the case of simply supported and clamped plates. For the case of a rigidly clamped plate results are presented of numerical experiments on minimizing the calculated value of the fundamental frequency coefficient by using Schmidt's approach. An experimental investigation is described on a clamped square plate with a free square, corner cut-out, which has led to the conclusion that the fundamental frequency coeficient remains practically invariant with respect to size when compared with the frequency coefficient of the fully clamped plate. A similar conclusion is arrived at by means of the mathematical model. The problem under consideration is important from a practical viewpoint since cut-outs of the type considered here are quite common in engineering practice.     

Influence of acoustic loading on an effective single mass model of the vocal folds

Three-way interactions between sound waves in the subglottal and supraglottal tracts, the vibrations of the vocal folds, and laryngeal flow were investigated. Sound wave propagation was modeled using a wave reflection analog method. An effective single-degree-of-freedom model was designed to model vocal-fold vibrations. The effects of orifice geometry changes on the flow were considered by enforcing a time-varying discharge coefficient within a Bernoulli flow model. The resulting single-degree-of-freedom model allowed for energy transfer from flow to structural vibrations, an essential feature usually incorporated through the use of higher order models. The relative importance of acoustic loading and the time-varying flow resistance for fluid-structure energy transfer was established for various configurations. The results showed that acoustic loading contributed more significantly to the net energy transfer than the time-varying flow resistance, especially for less inertive supraglottal loads. The contribution of supraglottal loading was found to be more significant than that of subglottal loading. Subglottal loading was found to reduce the net energy transfer to the vocal-fold oscillation during phonation, balancing the effects of the supraglottal load.     

Enablers for robust POD models

This paper focuses on improving the stability as well as the approximation properties of reduced order models (ROMs) based on proper orthogonal decomposition (POD). The ROM is obtained by seeking a solution belonging to the POD subspace and that at the same time minimizes the Navier–Stokes residuals. We propose a modified ROM that directly incorporates the pressure term in the model. The ROM is then stabilized making use of a method based on the fine scale equations. An improvement of the POD solution subspace is performed, thanks to a hybrid method that couples direct numerical simulations and reduced order model simulations. The methods proposed are tested on the two-dimensional confined square cylinder wake flow in laminar regime.     

Data mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultrarelativistic particles

We propose a new approach that consists in using data mining techniques for scientific computing. Indeed, data mining has proved to be efficient in other contexts which deal with huge data like in biology, medicine, marketing, advertising and communications. Our aim, here, is to deal with the important problem of the exploitation of the results produced by any numerical method. Indeed, more and more data are created today by numerical simulations. Thus, it seems necessary to look at efficient tools to analyze them. In this work, we focus our presentation to a test case dedicated to an asymptotic paraxial approximation to model ultrarelativistic particles. Our method directly deals with numerical results of simulations and try to understand what each order of the asymptotic expansion brings to the simulation results over what could be obtained by other lower-order or less accurate means. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models.     

Finite correlation time effects in nonequilibrium phase transitions: I. Dynamic equation and steady state properties

We determine an approximate renormalized equation of evolution for an arbitrary nonlinear single-degree-of-freedom system externally driven by Gaussian parametric fluctuations of finite correlation time. The renormalization scheme used here gives a second order equation with a time-and-state-dependent “diffusion coefficient”. We are able to calculate the diffusion coefficient in closed form. The steady-state distribution can easily be obtained from the evolution equation. We are thus able to determine the parameter dependence of the steady-state distribution and, in particular, the influence of a correlation time of the fluctuations, which does not vanish, on the steady-state distribution.     

Nonergodicity of a Time Series Obeying Lévy Statistics

Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied. Such a process, called a Lévy walk, describes dynamical behaviors of many physical systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time. Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed, and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The considered situation is in full contrast to the usual assumptions of ergodicity and stationarity.     

Performance potential for simulating spin models on GPU

Graphics processing units (GPUs) are recently being used to an increasing degree for general computational purposes. This development is motivated by their theoretical peak performance, which significantly exceeds that of broadly available CPUs. For practical purposes, however, it is far from clear how much of this theoretical performance can be realized in actual scientific applications. As is discussed here for the case of studying classical spin models of statistical mechanics by Monte Carlo simulations, only an explicit tailoring of the involved algorithms to the specific architecture under consideration allows to harvest the computational power of GPU systems. A number of examples, ranging from Metropolis simulations of ferromagnetic Ising models, over continuous Heisenberg and disordered spin-glass systems to parallel-tempering simulations are discussed. Significant speed-ups by factors of up to 1000 compared to serial CPU code as well as previous GPU implementations are observed.     

Stationarity tests for financial time series

Much attention has been paid in recent years to the study of the order of integration of a time series, i.e. the number of differences that are necessary to transform it into a stationary series. The relevance of the subject arises because most of time series analysis in economics and finance are based on the stationarity hypothesis. In the paper we present the most common tests for the null hypothesis of stationarity, and apply them to study the order of integration in a Spanish financial series namely the IBEX-35, using unit root tests as well. We find empirical evidence of a unit root in the series.     

Active vibration control of beam structures using acceleration feedback control with piezoceramic actuators

In this study, the active vibration control of clamped–clamped beams using the acceleration feedback (AF) controller with a sensor/moment pair actuator configuration is investigated. The sensor/moment pair actuator is a non-collocated configuration, and it is the main source of instability in the direct velocity feedback control system. First, the AF controller with non-collocated sensor/moment pair actuator is numerically implemented for a clamped–clamped beam. Then, to characterize and solve the instability problem of the AF controller, a parametric study is conducted. The design parameters (gain and damping ratio) are found to have significant effects on the stability and performance of the AF controller. Next, based on the characteristics of AF controllers, a multimode controllable single-input single-output (SISO) AF controller is considered. Three AF controllers are connected in parallel with the SISO architecture. Each controller is tuned to a different mode (in this case, the second, third and fourth modes). The design parameters are determined on the basis of the parametric study. The multimode AF controller with the selected design parameters has good stability and a high gain margin. Moreover, it reduces the vibration significantly. The vibration levels at the tuned modes are reduced by about 12 dB. Finally, the performance of the AF controller is verified by conducting an experiment. The vibration level of each controlled mode can be reduced by about 12 dB and this value is almost same as the theoretical result.     

An efficient approach of unsteady flamelet modeling of a cross-flow-jet combustion system using LES

Steady flamelet models have been widely used in turbulent combustion simulations because of their simplicity, efficiency, yet physics-based nature. They are, however, unable to handle slow chemical and physical processes such as pollutant formation. Unsteady flamelet models have been shown to be able to provide accurate predictions especially for pollutants, but their implementations are usually not as straightforward as for the steady models, and additional assumptions are involved. One relatively straightforward approach of implementing the unsteady flamelet model is to tabulate the time history of unsteady flamelet solutions. This often leads to flamelet libraries of large sizes because of increased dimensions for the new physics. The purpose of this paper is to introduce a new and efficient approach of tabulating unsteady flamelet solutions in the LES of complex systems, here demonstrated in simulations of a cross-flow-jet combustion system. This approach employs Taylor series expansions to represent the time history of unsteady flamelet solutions. Compared with other approaches, the new approach retains the efficiency and simplicity benefits of steady flamelet models but possesses the accuracy of unsteady flamelet models. Various issues associated with the formulation and implementation of this approach are discussed, which include the selection of the base solution, the order of accuracy of the expansion, and the treatment of simultaneous wall heat losses and heat transfer through thermal radiation. This approach is validated in large eddy simulations of a cross-flow-jet combustion system. Good agreement with experiments is obtained for both temperature and NO concentration, as well as for major species.     

Defocusing role in femtosecond filamentation: Higher-order Kerr effect or plasma effect?

The femtosecond filamentation in the classical and high-order Kerr(HOK) models is numerically investigated by adopting multi-photon ionization(MPI) cross section with different values. It is found that in the case that the MPI cross section is relatively small, there exists a big difference between the electron density as well as clamped intensity calculated in the classical model and those calculated in the HOK one, while in the case that the MPI cross section is relatively large, the electron density and clamped intensity calculated in the two models are nearly in agreement with each other, and under this circumstance, even if the higher-order nonlinear terms do exist, the free-charge generation and the associated defocusing in a filament are enough to mask their effects. The different behaviors of the maximum intensity and on-axis electron density at the collapse position with the pulse duration provides an approach to determine which effect plays the dominant defocusing role. These results demonstrate that it is ionization that results in the difference between the two models.     

Dynamic stability of weakly damped oscillators with elastic impacts and wear

The dynamics of non-linear oscillators comprising of a single-degree-of-freedom system and beams with elastic two-sided amplitude constraints subject to harmonic loads is analyzed. The beams are clamped at one end, and constrained against unilateral contact sites near the other end. The structures are modelled by a Bernoulli-type beam supported by springs using the finite element method. Rayleigh damping is assumed. Symmetric and elastic double-impact motions, both harmonic and sub-harmonic, are studied by way of a Poincaré mapping that relates the states at subsequent impacts. Stability and bifurcation analyses are performed for these motions, and domains of instability are delineated. Impact work rate, which is the rate of energy dissipation to the impacting surfaces, is evaluated and discussed. In addition, an experiment conducted by Moon and Shaw on the vibration of a cantilevered beam with one-sided amplitude constraining stop is modelled. Bifurcation observed in the experiment could be captured.     

Chaos synchronization and parameter estimation of single-degree-of-freedom oscillators via adaptive control

This paper addresses the problem of synchronizing a class of single-degree-of-freedom oscillators with uncertain parameters. A modified adaptive control scheme is proposed to achieve globally asymptotic stable synchronization between the master and slave oscillators with arbitrary different initial conditions based on the Barbalat's Lemma. One of the advantages of the method is that it requires only a scalar driving signal which may be easily designed in practical applications. On the other hand, the method can ensure that the unknown parameters in the slave oscillator would be completely estimated with an adaptive updating law. Numerical simulations are performed on two examples to verify the analytical results.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

Effect of ACLD treatment configuration on damping performance of a flexible beam

A clamped–free beam with partial active constrained layer damping (ACLD) treatment is modelled by using the finite element method. The Golla–Hughes–McTavish (GEM) method is employed to account for the frequency-dependent characteristic of the viscoelastic material (VEM). As the resultant finite element model contains too many degrees of freedom due to the introduction of dissipative coordinates, a model reduction is performed to bring the system back to its original size. Finally, optimal output feedback gains are designed based on the reduced models. Numerical simulations are performed to study the effect of different ACLD treatment configurations, with various element numbers, spacing and locations, on the damping performance of a flexible beam. Results are presented for damping ratios of the first two vibration modes. It is found that to enhance the second mode damping, without deteriorating the first mode damping, splitting a single ACLD element into two and placing them at appropriate positions of the beam could be a possible solution.     

Fast and accurate X‐ray fluorescence computed tomography imaging with the ordered‐subsets expectation maximization algorithm

The ordered‐subsets expectation maximization algorithm (OSEM) is introduced to X‐ray fluorescence computed tomography (XFCT) and studied; here, simulations and experimental results are presented. The simulation results indicate that OSEM is more accurate than the filtered back‐projection algorithm, and it can efficiently suppress the deterioration of image quality within a large range of angular sampling intervals. Experimental results of both an artificial phantom and cirrhotic liver show that with a satisfying image quality the angular sampling interval could be improved to save on the data‐acquisition time when OSEM is employed. In addition, with an optimum number of subsets, the image reconstruction time of OSEM could be reduced to about half of the time required for one subset. Accordingly, it can be concluded that OSEM is a potential method for fast and accurate XFCT imaging.     

Dispersion management for solitons in a Korteweg-de Vries system

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.