Systematic errors inherent in the current modeling of the reflected downward flux term used by remote sensing models

An underlying assumption of satellite data assimilation systems is that the radiative transfer model used to simulate observed satellite radiances has no errors. For practical reasons a fast-forward radiative transfer model is used instead of a highly accurate line-by-line model. The fast model usually replaces the spectral integration of spectral quantities with their monochromatic equivalents, and the errors due to these approximations are assumed to be negligible. The reflected downward flux term contains many approximations of this nature, which are shown to introduce systematic errors. In addition, many fast-forward radiative transfer models simulate the downward flux as the downward radiance along a path defined by the secant of the mean emergent angle, the diffusivity factor. The diffusivity factor is commonly set to 1.66 or to the secant of the satellite zenith angle. Neither case takes into account that the diffusivity factor varies with optical depth, which introduces further errors. I review the two most commonly used methods for simulating reflected downward flux by fast-forward radiative transfer models and point out their inadequacies and limitations. An alternate method of simulating the reflected downward flux is proposed. This method transforms the surface-to-satellite transmittance profile to a transmittance profile suitable for simulating the reflected downward flux by raising the former transmittance to the power of kappa, where kappa itself is a function of channel, surface pressure, and satellite zenith angle. It is demonstrated that this method reduces the fast-forward model error for low to moderate reflectivities.     

加窗插值FFT谐波分析算法的误差估计

非同步采样的情况下，快速傅里叶变换用于谐波分析存在较大的误差，很难获得准确的谐波参数。加窗插值FFT谐波分析算法能够减小谐波分析误差。针对该算法，借助计算机仿真研究了谐波分析时误差的变化规律以及最大的误差限。最后给出了谐波分析应用中的若干重要结论。     

Goal-oriented model reduction of parametrized nonlinear partial differential equations: Application to aerodynamics

We introduce a goal-oriented model reduction framework for rapid and reliable solution of parametrized nonlinear partial differential equations with applications in aerodynamics. Our goal is to provide quantitative and automatic control of various sources of errors in model reduction. Our framework builds on the following ingredients: a discontinuous Galerkin finite element (FE) method, which provides stability for convection-dominated problems; reduced basis (RB) spaces, which provide rapidly convergent approximations; the dual-weighted residual method, which provides effective output error estimates for both the FE and RB approximations; output-based adaptive RB snapshots; and the empirical quadrature procedure (EQP), which hyperreduces the primal residual, adjoint residual, and output forms to enable online-efficient evaluations while providing quantitative control of hyperreduction errors. The framework constructs a reduced model which provides, for parameter values in the training set, output predictions that meet the user-prescribed tolerance by controlling the FE, RB, and EQP errors; in addition, the reduced model equips, for any parameter value, the output prediction with an effective, online-efficient error estimate. We demonstrate the framework for parametrized aerodynamics problems modeled by the Reynolds-averaged Navier-Stokes equations; reduced models provide over two orders of magnitude online computational reduction and sharp error estimates for three-dimensional flows.     

Testing the effect of computer-generated hologram fabrication error in a cylindrical interferometry system

This paper presents a method of testing the effect of computer-generated hologram (CGH) fabrication error in a cylindrical interferometry system. An experimental system is developed for calibrating the effect of this error. In the calibrating system, a mirror with high surface accuracy is placed at the focal axis of the cylindrical wave. After transmitting through the CGH, the reflected cylindrical wave can be transformed into a plane wave again, and then the plane wave interferes with the reference plane wave. Finally, the double-pass transmitted wavefront of the CGH, representing the effect of the CGH fabrication error in the experimental system, is obtained by analyzing the interferogram. The mathematical model of misalignment aberration removal in the calibration system is described, and the feasibility is demonstrated via the simulation system established in Zemax. With the mathematical polynomial, most of the possible misalignment errors can be estimated with the least-squares fitting algorithm, and then the double-pass transmitted wavefront of the CGH can be obtained by subtracting the misalignment errors from the result extracted from the real experimental system. Compared to the standard double-pass transmitted wavefront given by Diffraction International Ltd., which manufactured the CGH used in the experimental system, the result is desirable. We conclude that the proposed method is effective in calibrating the effect of the CGH error in the cylindrical interferometry system for the measurement of cylindricity error.     

Tool path re-planning in free-form surface machining for compensation of process-related errors

Free-form surfaces are widely used in many applications in today’s industry. This paper presents a new approach to identify and compensate process-related errors in machining of free-form surfaces. The process-related errors are identified online by a newly developed in-process inspection technique. In this technique, the surface is first machined through an intermediate semi-finishing process that is specifically designed to machine different geometric shapes on the surface with different process parameters. An inspection method is developed to identify the process-related errors in the selected regions on the semi-finished surface. The relationship between the machining/surface parameters and process-related error is then achieved using a neural network. This relationship is used to predict the process-related errors in the finishing process. The process-related errors, together with the machine tool geometric errors identified using a method developed in our previous work, are compensated in the finishing tool paths through tool path re-planning. Experiment has been conducted to machine a part with a free-form surface to show the improvements in the machining accuracy.     

Novel interval theory‐based parameter identification method for engineering heat transfer systems with epistemic uncertainty

The parameter identification problem with epistemic uncertainty, where only a small amount of experimental information is available, is a challenging issue in engineering. To overcome the drawback of traditional probabilistic methods in dealing with limited data, this paper proposes a novel interval theory‐based inverse analysis method. First, the interval variables are introduced to represent the input uncertainties, whose lower and upper bounds are to be identified. Subsequently, an unbiased estimation method is presented to quantify the experimental response interval from limited measurements. Meanwhile, a quantitative metric is defined to characterize the relative errors between computational and experimental response intervals by which the interval parameter identification can be constructed as a nested‐loop optimization procedure. To improve the computational efficiency of response prediction with respect to various interval variables, a universal surrogate model is established in the support box via Legendre polynomial chaos expansion, where the expansion coefficients can be evaluated by a collocation method under Clenshaw‐Curtis points and Smolyak algorithm. Eventually, a heat conduction example is provided to verify the feasibility of proposed method, especially in the case with noise‐contaminated temperature measurements.     

Spatial coherence function of partially coherent Gaussian beams in atmospheric turbulence

We introduce a new method of estimating the coherence function of a Gaussian-Schell model beam in the inertial subrange of atmospheric turbulence. It is compared with the previously published methods based on either the quadratic approximation of the parabolic equation or an assumed independence between the source's randomness and the atmosphere using effective beam parameters. This new method, which combines the results of the previous two methods to account for any random source/atmospheric coupling, was shown to more accurately estimate both the coherence radius and coherence functional shape across much of the relevant parameter space. The regions of the parameter space where one method or another is the most accurate in estimating the coherence radius are identified along with the maximum absolute estimation error in each region. By selecting the appropriate estimation method for a given set of conditions, the absolute estimation error can generally be kept to less than 5%, with a maximum error of 7%. We also show that the true coherence function is more Gaussian than expected, with the exponential power tending toward 9/5 rather than the theoretical value of 5/3 in very strong turbulence regardless of the nature of the source coherence.     

Vector outcrossing probabilities by Monte Carlo

An exact method for calculating the probability that a vector-valued Gaussian stationary process will cross out of a safe set in a given time, is presented. It is based on representing the process by a random trigonometric polynomial and then calculating the probability by directional simulation. Exact bounds for the error can be calculated, giving a predetermined accuracy. A numerical example relating to the response of a structure to wind excitation is given.     

A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics

This paper presents an approach that extends the classical error bounding techniques to parametric models. It departs from appropriate pairs of complementary solutions of a linear elastic problem, obtained using a Proper Generalised Decomposition methodology, to determine approximations of selected local outputs and strict bounds of the error of these approximations. The paper starts by presenting the procedures used to obtain the complementary solutions. The properties, the convergence characteristics of the global error and the determination of an indicator of the error distribution are illustrated for a very simple example. The demonstration of the procedure used for determining local outputs and their bounds, also accompanied by illustrative examples, completes the paper.Copyright © 2013 John Wiley & Sons, Ltd.     

Error estimation for proper generalized decomposition solutions: A dual approach

The proper generalized decomposition is a well-established reduced order method, used to efficiently obtain approximate solutions of multi-dimensional problems in a procedure that controls the effects of the “curse of dimensionality.” The question of assessing the quality of the solutions obtained and adapting the approximations assumed, for example, the finite element meshes used, so that the best result is obtained at minimal cost, remains a relevant challenge. This article deals with finite element solutions for solid mechanics problems, using the error obtained from a dual analysis, the difference between complementary solutions, to bound the error in the solutions and to drive an optimal adaptivity process, which obtains meshes with errors significantly lower than those obtained using a uniform refinement.     

Guaranteed computable bounds on quantities of interest in finite element computations

We develop and compare a number of alternative approaches to obtain guaranteed and fully computable bounds on the error in quantities of interest of arbitrary order finite element approximations in the context of a linear second‐order elliptic problem. In each case, the bounds are fully computable and do not involve any unknown multiplicative factors. Guaranteed computable bounds are also obtained for the case when the Dirichlet boundary conditions are non‐homogeneous. This is achieved by taking account of the error incurred by the approximation of the Dirichlet data in the functional used to approximate the quantity of interest itself, which is found to generally give better results. Numerical examples are presented to show that the resulting estimators provide tight bounds with the effectivity index tending to unity from above. Copyright © 2012 John Wiley & Sons, Ltd.     

Generalized Least Squares With Ignored Errors in Variables

We present data, both real and simulated, that show generalized least squares (GLS) estimation, intended to account for correlated response error structure, can produce gross biasing in regression parameter estimates under misspecified models with ignored errors in explanatory-variable measurements. The bias, and its subsequent effect on mean squared error (MSE), can be much more severe than the apparently less appropriate ordinary least squares (OLS) estimator. This article provides a theoretical basis for these effects by deriving expressions for the bias and MSE for the general GLS estimator through Taylor-series expansions. The results are compared with simulations for two specific weight matrices and applied to a dataset relating atmospheric pollutant levels in Los Angeles with average recorded wind speed. We show that the bias (with subsequent implications for the MSE) is always worse for the exponential correlation model with equally spaced explanatory-variable observations and present a simple test to decide a preference for OLS or GLS in practice.     

五自由度测量中直线度现场标定方法研究

直线度现场标定是保证其在线测量精度的重要方法。在收发一体式激光五自由度测量结构的基础上,针对直线度现场标定中标定平台引入的阿贝误差和角锥棱镜成像误差,建立了直线度现场标定模型。根据该标定模型并结合五自由度测量装置的角度测量结果,提出一种直线度现场标定误差补偿方法。实验表明,该标定方法使标定系数误差减小到0.2%以内,有效提高了直线度现场标定精度。     

基于激光跟踪仪的协作机器人标定算法与实验研究

为了提高机器人末端绝对定位精度,提出了一种基于激光跟踪仪测量的机器人几何参数标定方法,对协作机器人的参数误差进行辨识和补偿.为了避免机械臂相邻两个轴平行时会出现奇异性的情况,采用了MDH参数法建立误差模型;为了把测量数据定义在同一坐标轴上,结合了机器人工具坐标系转换(靶球中心点相对于机器人末端坐标系的平移变换);进而用...     

基于响应面法的结构动力学模型修正

为了获得精确的结构动力学模型,提出了响应面和优化相结合的方法。利用参数化模型和优化拉丁方试验设计获取样本点构造多项式响应面模型,最小二乘法确定多项式系数并检验响应面的拟合精度。用响应面计算结果与实验结果的误差构造目标函数,自适应模拟退火算法来优化修正响应面参数,将修正后的参数值带入有限元模型得到修正模型。以欧洲航空科技组织的基准模型GARTEUR飞机模型为算例,对比修正前后模态频率,结果表明修正后的模型在测试频段和预测频段具有良好的复现和预测能力,进而验证了基于响应面法与优化方法相结合的结构动力学有限元模型修正的有效性。     

Geometric and algebraic errors in calculating a transducer'sspatial impulse response

Two types of errors resulting from the numerical evaluation of a transducer's spatial impulse response function have been identified, One is geometric and is due to the mismatch between the emulated piston surface (approximated using many small planar elements) and the true surface. The other error is algebraic and is due to the inexact formulation of the response of the emulated surface. The deviations from the true response were calculated for three different pistons. The results indicate that the shape and placement of the elements are important for the computational accuracy     

Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation

An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in Steffens and D'?ez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400). In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post‐processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in Steffens and Díez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400) is based in a polynomial least‐squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem. Copyright © 2011 John Wiley & Sons, Ltd.     

基于B样条曲面插值误差控制的内超环面齿轮齿面建模

针对内超环面齿轮齿面数控加工精度难以保证的问题,提出一种基于B样条曲面插值误差控制的内超环面齿轮齿面建模的方法.该方法以内超环面齿轮理论齿面原型为依据,运用B样条曲面构造插值曲面,计算插值曲面片与理论齿面之间的误差.通过插值误差分析,根据误差分布特点,将型值点网格不断细化,获得一组型值点阵,经插值重构后可得到满足精度要求的内超环面齿轮齿廓模型.最后,通过数控加工验证了建模的有效性.基于B样条曲面插值误差控制的内超环面齿轮齿面建模方法为获得高精度的内超环面齿轮实体模型奠定了理论基础.     

Optimal shape parameter for the solution of elastostatic problems with the RBF method

Radial basis functions (RBFs) have become a popular method for the solution of partial differential equations. In this paper we analyze the applicability of both the global and the local versions of the method for elastostatic problems. We use multiquadrics as RBFs and describe how to select an optimal value of the shape parameter to minimize approximation errors. The selection of the optimal shape parameter is based on analytical approximations to the local error using either the same shape parameter at all nodes or a node-dependent shape parameter. We show through several examples using both equispaced and nonequispaced nodes that significant gains in accuracy result from a proper selection of the shape parameter.     

The SPH equations for fluids

Existing smoothed particle hydrodynamics (SPH) formulations for simulating continuous fluids have errors that may be divergent and it has been known for some time that the SPH equations do not satisfy low‐order polynomial completeness conditions. Here SPH equations are derived that have convergent error terms and a correction method is presented for enforcing low‐order polynomial completeness irrespective of how many completeness conditions are required. Discretization is achieved through division of the model domain, in its initial state, into sub‐domains that have Lagrangian boundaries. It is shown that boundary integrals appearing in one derivation of the SPH equations may be treated as a convergent error. In simulations of basic fluid flows convergence and zeroth‐order completeness are demonstrated, but significant instabilities and a failure to conserve energy are observed. Copyright © 2009 John Wiley & Sons, Ltd.