Efficiency of a POD‐based reduced second‐order adjoint model in 4D‐Var data assimilation

Order reduction strategies aim to alleviate the computational burden of the four‐dimensional variational data assimilation by performing the optimization in a low‐order control space. The proper orthogonal decomposition (POD) approach to model reduction is used to identify a reduced‐order control space for a two‐dimensional global shallow water model. A reduced second‐order adjoint (SOA) model is developed and used to facilitate the implementation of a Hessian‐free truncated‐Newton (HFTN) minimization algorithm in the POD‐based space. The efficiency of the SOA/HFTN implementation is analysed by comparison with the quasi‐Newton BFGS and a nonlinear conjugate gradient algorithm. Several data assimilation experiments that differ only in the optimization algorithm employed are performed in the reduced control space. Numerical results indicate that first‐order derivative methods are effective during the initial stages of the assimilation; in the later stages, the use of second‐order derivative information is of benefit and HFTN provided significant CPU time savings when compared to the BFGS and CG algorithms. A comparison with data assimilation experiments in the full model space shows that with an appropriate selection of the basis functions the optimization in the POD space is able to provide accurate results at a reduced computational cost. The HFTN algorithm benefited most from the order reduction since computational savings were achieved both in the outer and inner iterations of the method. Further experiments are required to validate the approach for comprehensive global circulation models. Copyright © 2006 John Wiley & Sons, Ltd.     

Study of numerical errors in direct numerical simulation and large eddy simulation

By comparing the energy spectrum and total kinetic energy, the effects of numerical errors (which arise from aliasing and discretization errors), subgrid-scale (SGS) models, and their interactions on direct numerical simulation (DNS) and large eddy simulation (LES) are investigated. The decaying isotropic turbulence is chosen as the test case. To simulate complex geometries, both the spectral method and Pade compact difference schemes are studied. The truncated Navier-Stokes (TNS) equation model with Pade discrete filter is adopted as the SGS model. It is found that the discretization error plays a key role in DNS. Low order difference schemes may be unsuitable. However, for LES, it is found that the SGS model can represent the effect of small scales to large scales and dump the numerical errors. Therefore, reasonable results can also be obtained with a low order discretization scheme.     

A zonally symmetric model for the monsoon-Hadley circulation with stochastic convective forcing

Idealized models of reduced complexity are important tools to understand key processes underlying a complex system. In climate science in particular, they are important for helping the community improve our ability to predict the effect of climate change on the earth system. Climate models are large computer codes based on the discretization of the fluid dynamics equations on grids of horizontal resolution in the order of 100 km, whereas unresolved processes are handled by subgrid models. For instance, simple models are routinely used to help understand the interactions between small-scale processes due to atmospheric moist convection and large-scale circulation patterns. Here, a zonally symmetric model for the monsoon circulation is presented and solved numerically. The model is based on the Galerkin projection of the primitive equations of atmospheric synoptic dynamics onto the first modes of vertical structure to represent free tropospheric circulation and is coupled to a bulk atmospheric boundary layer (ABL) model. The model carries bulk equations for water vapor in both the free troposphere and the ABL, while the processes of convection and precipitation are represented through a stochastic model for clouds. The model equations are coupled through advective nonlinearities, and the resulting system is not conservative and not necessarily hyperbolic. This makes the design of a numerical method for the solution of this system particularly difficult. Here, we develop a numerical scheme based on the operator time-splitting strategy, which decomposes the system into three pieces: a conservative part and two purely advective parts, each of which is solved iteratively using an appropriate method. The conservative system is solved via a central scheme, which does not require hyperbolicity since it avoids the Riemann problem by design. One of the advective parts is a hyperbolic diagonal matrix, which is easily handled by classical methods for hyperbolic equations, while the other advective part is a nilpotent matrix, which is solved via the method of lines. Validation tests using a synthetic exact solution are presented, and formal second-order convergence under grid refinement is demonstrated. Moreover, the model is tested under realistic monsoon conditions, and the ability of the model to simulate key features of the monsoon circulation is illustrated in two distinct parameter regimes.     

Determination of correlation functions of turbulent velocity and sound speed fluctuations by means of ultrasonic technique

An experimental study of the propagation of high-frequency acoustic waves through grid-generated turbulence by means of an ultrasound technique is discussed. Experimental data were obtained for ultrasonic wave propagation downstream of heated and non-heated grids in a wind tunnel. A semi-analytical acoustic propagation model that allows the determination of the spatial correlation functions of the flow field is developed based on the classical flowmeter equation and the statistics of the travel time of acoustic waves traveling through the kinematic and thermal turbulence. The basic flowmeter equation is reconsidered in order to take into account sound speed fluctuations and turbulent velocity fluctuations. It allows deriving an integral equation that relates the correlation functions of travel time, sound speed fluctuations and turbulent velocity fluctuations. Experimentally measured travel time statistics of data with and without grid heating are approximated by an exponential function and used to analytically solve the integral equation. The reconstructed correlation functions of the turbulent velocity and sound speed fluctuations are presented. The power spectral density of the turbulent velocity and sound speed fluctuations are calculated.     

Wide-band measurement of frequency-dependent viscoelastic properties using an inverse nyquist plane parameter extraction methodology

This paper presents a new method for dynamically characterizing viscoelastics materials. A multi-degree of freedom vibration model is employed to extract frequency-dependent storage moduli and loss factors from experimental frequency-response functions of the elastomer specimen. The frequency range over which reliable data are available spans from 300 to 6,000 Hz. Hundreds of estimates can be obtained from a single test, Compared with conventional methods, it exhibits the remarkable advantages of covering a wider frequency range, of being more testwise efficient, and of providing more estimates within the frequency range of interest. Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics held in Cambridge, MA on May 28–June 1.     

A multi-entropy-level lattice Boltzmann model for the one-dimensional compressible Euler equations

In this paper, we propose a new lattice Boltzmann model for the one-dimensional compressible Euler equations. The new model is based on a three-entropy-level and three-speed lattice Boltzmann equation by using a method of higher-order moments of the equilibrium distribution functions. In order to obtain the second-order accuracy model, we employ the ghost field distribution functions to remove the non-physical dissipation terms in the Euler equations. We also use the conditions of the higher-order moments of the ghost field equilibrium distribution functions to obtain the equilibrium distribution functions. The numerical examples show that the numerical results can be compared with those classical methods.     

LOW-FREQUENCY RESPONSES OF NONLINEARLY MOORED VESSELS IN RANDOM WAVES: COUPLED SURGE,PITCH AND HEAVE MOTIONS

The responses of a multi-degree-of-freedom model of a moored vessel are analysed, accounting for the hydroelastic interaction between the nonlinear wave hydrodynamics and the nonlinear mooring stiffness. A two-scale perturbation method developed by Sarkar & Eatock Taylor to determine low-frequency hydrodynamic forces on a single-degree-of-freedom model of a nonlinearly moored vessel has been extended to analyse the nonlinear multi-degree-of-freedom dynamics of the system. Surge, heave and pitch motions are considered. The perturbation equations of successive orders are derived. To illustrate the approach, semi-analytical expressions for the higher-order hydrodynamic force components have been obtained for a truncated circular cylinder in finite water depth. In addition to conventional quadratic force transfer functions, a new type of higher-order force transfer function is introduced. This is used to characterize the hydrodynamic forces on the vessel which arise due to nonlinearity of the mooring stiffness. These are a type of radiation force, generated by the nonlinear interaction of the fluid–structure coupled system. Based on a Volterra series model, the power spectral densities of the new higher-order forces are then derived for the case of Gaussian random seas. It is shown that the additional response arising due to nonlinear dynamics of the mooring system can significantly contribute to low-frequency drift forces and responses of the vessel. Unlike conventional non-Gaussian second-order forces which are quadratic transformations of a Gaussian random process, the new higher-order forces arising due to the nonlinear mooring stiffness are polynomials of a Gaussian random process (up to fourth order for a Duffing oscillator model). This may significantly influence the extreme responses.     

Calculating response spectral moments by complex modal analysis

Response spectral moments are useful for system reliability analysis. Usually, spectral moments are calculated by the frequency domain method. Based on the time domain modal analysis of random vibrations, the authors present a new method for calculating response spectral moments through response correlation functions. The method can be applied to both classical and non-classical damping cases and to three kinds of random excitations, i.e., white noise, band-limited white noise, and filtered white noise. Project supported by the National Natural Science Foundation of China.     

High‐order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations

In this paper, we investigate the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under‐resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling. We take full advantage of the low numerical errors and associated superior scale resolving capabilities of high‐order spectral methods by using high‐order ansatz functions up to 12th order. We employ polynomial de‐aliasing techniques to prevent instabilities arising from inexact quadrature of nonlinearities. Without the need for any additional filtering, explicit or implicit modelling, or artificial dissipation, our high‐order schemes capture the turbulent flow at the considered Reynolds number range very well. Three classical large eddy simulation benchmark problems are considered: a circular cylinder flow at Re_D=3900, a confined periodic hill flow at Re_h=2800 and the transitional flow over a SD7003 airfoil at Re_c=60,000. For all computations, the total number of degrees of freedom used for the discontinuous Galerkin spectral method simulations is chosen to be equal or considerably less than the reported data in literature. In all three cases, we achieve an equal or better match to direct numerical simulation results, compared with other schemes of lower order with explicitly or implicitly added subgrid scale models. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient estimation of power spectral density from laser Doppler anemometer data

 A non-biased estimator of power spectral density (PSD) is introduced for data obtained from a zeroth order interpolated laser Doppler anemometer (LDA) data set. The systematic error, sometimes referred to as the “particle-rate filter” effect, is removed using an FIR filter parameterized using the mean particle rate. Independent from this, a procedure for estimating the measurement system noise is introduced and applied to the estimated spectra. The spectral estimation is performed in the domain of the autocorrelation function and assumes no further process parameters. The new technique is illustrated using simulated and measured data, in the latter case with direct comparison to simultaneously acquired hot-wire data. Received: 9 June 1997/Accepted: 14 October 1997     

Two-Dimensional Elastic Herglotz Functions and Their Applications in Inverse Scattering

In this work solutions of the spectral Navier equation that satisfy the Herglotz boundedness condition in two-dimensional linear elasticity are presented. Navier eigenvectors in polar coordinates are introduced and it is established that they form a linearly independent and complete set in the L ²-sense on every smooth curve. It is also proved that the classical solutions of the spectral Navier equation are expressed via Navier eigenvectors, and this expansion converges uniformly over compact subsets of R ². Two far-field patterns, the longitudinal and the transverse one corresponding to the displacement field are introduced, and the Herglotz norm is expressed as the sum of the L ²-norms of these patterns over the unit circle. It is also established that the space of elastic Herglotz functions is dense in the space of the classical solutions of the spectral Navier equation. Finally, connection to inverse elasticity scattering is established and reconstructions of rigid bodies are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

New truncated expansion method and soliton-like solution of variable coefficient KdV-MKdV equation with three arbitrary functions

The truncated expansion method for finding explicit and exact soliton-like solution of variable coefficient nonlinear evolution equation was described. The crucial idea of the method was first the assumption that coefficients of the truncated expansion formal solution are functions of time satisfying a set of algebraic equations, and then a set of ordinary different equations of undetermined functions that can be easily integrated were obtained. The simplicity and effectiveness of the method by application to a general variable coefficient KdV-MKdV equation with three arbitrary functions of time is illustrated.     

交错网格上应用有限谱QUICK法计算方腔流动

王健平教授提出的有限谱方法是一种局域化谱方法，具有精度高、无相位差、应用灵活等特点，在以往的实践中取得了很大成功。本文在交错网格上对二维驱动方腔流问题进行计算，求解了二维不可压缩流动的涡流流函数方程。其中微分部分采用有限谱法进行处理，对流项的处理则应用了QUICK格式。本文计算了雷诺数为1000、5000、10000、20000等多种情况，将所得的结果进行分析，并将中线上的速度分别同已有的文献数据进行对比，从而，验证有限谱微分的正确性和其在实际应用中的可行性。     

A model-free noise removal for the interpolation method of correlation and spectral estimation from laser Doppler data

A procedure to estimate and remove the contribution of data noise to the correlation function and the power spectral density calculated from laser Doppler data with the interpolation method is introduced. In comparison with earlier approaches, the new procedure is model free and, therefore, more objective. The new procedure is proven based on experimental data taken with a laser Doppler system, where the power spectral density obtained with the interpolation method is compared directly to data from a hot-wire reference measurement.     

基于仿射算法的深部隧道衬砌力学状态区间分析

根据深部地层隧道结构物理、力学参数信息资料缺乏及各类不确定性模型的特征,揭示了区间模型于深部地层隧道工程不确定性分析的优越性。将限制条件少、适用范围广的链杆模式的矩阵位移理论与响应面思路结合,提出了衬砌结构力学状态参量解析式的搭建方法,并进一步推导出了基于仿射运算的响应面函数矩阵表达形式及其区间上、下界计算公式,构筑起基于矩阵表达的响应面函数区间仿射运算模型。将区间离散逐步逼近规则引入,建立了该模型的优化求解方法。利用该模型及方法分析了某衬砌墙脚轴力及拱顶弯矩分布区间,与蒙特卡洛法计算结果比较,显示本文模型及方法有效地克服了经典区间方法在复杂状态函数中的扩张和溢出这一致命缺陷,提高了准确性和计算效率,分析结果精确度满足工程使用要求。     

Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model

Very recently, researchers dealing with constitutive law pertinent viscoelastic materials put forward the successful idea to introduce viscoelastic laws embedded with fractional calculus, relating the stress function to a real order derivative of the strain function. The latter consideration leads to represent both, relaxation and creep functions, through a power law function. In literature there are many papers in which the best fitting of the peculiar viscoelastic functions using a fractional model is performed. However there are not present studies about best fitting of relaxation function and/or creep function of materials that exhibit a non-linear viscoelastic behavior, as polymer melts, using a fractional model. In this paper the authors propose an advanced model for capturing the non-linear trend of the shear viscosity of polymer melts as function of the shear rate. Results obtained with the fractional model are compared with those obtained using a classical model which involves classical Maxwell elements. The comparison between experimental data and the theoretical model shows a good agreement, emphasizing that fractional model is proper for studying viscoelasticity, even if the material exhibits a non-linear behavior.     

Hybrid finite analytic solutions of shallow water circulation

IntroductionThepredictionoftidalandstormsurgesisessentialfornavigation ,fishingandprotectionofthecoast.Furtherthedeterminationofhydrodynamiccirculationbecomesnecessaryforthemodelingofdispersionofpollutants,sedimenttransportandsalinitydistribution .Recently ,somekindsofnumericalmodelingtechniqueshavebeenextensivelydevelopedinthelasttwodecades.Inordertoimprovethewaterqualityinabays/lagoons ,manymethodshavebeenproposedbyresearchers.Forexample ,Komatsuetal.(2 0 0 0 ) [1] proposedanewmethodtocontr…     

Operational modal filter and its applications

In the paper, the procedure for the estimation of modal filter coefficients from output-only data is presented. The basic concept of the procedure consists in frequency response functions synthesis based on the knowledge of an operational modal model. A method of operational mode shapes scaling is described. The method is then compared with the classical modal filter and with modal filtration of responses spectra, which is sometimes used as a solution for modal filtration based on the output-only data. Each solution is applied to load identification and damage detection. The study shows the method verification on data obtained from laboratory experiment.     

A spectral-analysis method for nonstationary field measurements

A method for dealing with the problem of spectral analysis of nonstationary field measurements is presented. The method hypothesizes that the nonstationary signal consists of two stationary signals which belong to different populations (environment and working plus environment) which occur consecutively. The analysis method entails segmenting the time history and estimating the population of each segment. Two estimators are presented (average absolute value and energy in a frequency band) and their frequency characteristics are described. Discrete-Fourier transforms of zero-padded segments are used for estimating the spectral-density functions. This method is simply implemented and treats the problem of smoothing the spectral estimates. Moreover, it simplifies the use of different window functions by using convolution in the frequency domain. This paper describes the computational facilities used as well as some electronic circuits that were developed. It also discusses the results of the analysis of typical data which were recorded by LBF (West Germany) personnel for the strain of the front axle of a private car running on three test roads. The detailed computational procedure is given and the results are presented. These results show the power of the suggested estimators in estimating the population of the segments of data and the suitability of the spectral-estimation method in extracting meaningful spectral estimates. In addition, the results identify the properties of the typical signal used. This analysis method can be effectively used in the simulation of field data for the study of random fatigue. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.