Second-order sensitivity of eigenpairs in multiple parameter structures

This paper presents methods for computing a second-order sensitivity matrix and the Hessian matrix of eigenvalues and eigenvectors of multiple parameter structures. Second-order perturbations of eigenvalues and eigenvectors are transformed into multiple parameter forms,and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed.With these formulations,the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate changes of eigenvalues and eigenvectors when the design parameters are changed. The presented method avoids direct differential operation,and thus reduces difficulty for computing the second-order sensitivity matrices of eigenpairs.A numerical example is given to demonstrate application and accuracy of the proposed method.     

An analysis of perturbation based methods for the treatment of parameter uncertainty in numerical groundwater models

Monte Carlo methods are robust approaches for the estimation of prediction uncertainty in groundwater flow and transport modelling under uncertain model parameters. However Monte Carlo procedures estimate the prediction statistics by generating a population of solutions from random realisations of the model parameters which are consistent with the parameter statistics, and as a result are computationally demanding. Taylor series based procedures offer an alternative to Monte Carlo methods for calculating the prediction statistics. Two such approaches, the first-order second moment and McLaughlin and Wood's perturbation method, are based on using a Taylor series to derive approximate expressions for the model predictions first and second statistical moments. In this paper the perturbation method presented by McLaughlin and Wood is rederived using Vetter matrix notation. This is compared with the first-order second moment (FOSM) method and while the steady state expressions for these two approaches are shown to be equivalent, the transient forms are considerably different. A new form of the FOSM is derived, which is simpler and has a lower computational burden. However, the transient McLaughlin and Wood expression is found to have a significantly lower computational overhead than either of the FOSM methods presented.     

Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。     

有界不确定参数结构位移范围的区间摄动法

当结构参数具有误差或有界不确定性时,区间数学可以在不同知识不确定变量的概率分布的情况下定量地考察不确定参数对结构响应的影响。为计算出不确定结构参数对结构位移影响范围的上下界,文中提出了的两种区骚动国方法。     

Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis

The study was intended to evaluate the range of dynamic responses of structures with uncertain-but-bounded parameters by using the parameter perturbation method. The uncertain parameters were modeled as an interval vector. The first-order perturbation quantities of responses of the perturbed system were obtained through the parameter perturbation method, and then taking advantage of interval mathematics a new algorithm to estimate the response interval was presented. Comparisons between the parameter perturbation method and the probabilistic approach from mathematical proofs and numerical simulations were performed. The numerical results are in agreement with the mathematical proofs. The response range given by the parameter perturbation method encloses that obtained by the probabilistic approach. The results also show good robustness of the proposed method.     

An efficient 3D stochastic finite element method

Real life structural systems are characterized by their inherent or externally induced uncertainties in the design parameters. This study proposes a stochastic finite element tool efficient to take account of these uncertainties. Here uncertain structural parameter is modeled as homogeneous Gaussian stochastic field and commonly used two-dimensional (2D) local averaging technique is extended and generalized for 3D random field. This is followed by Cholesky decomposition of respective covariance matrix for digital simulation. By expanding uncertain stiffness matrix about its reference value, the Neumann expansion method is introduced blended with direct Monte Carlo simulation. This approach involves decomposition of stiffness matrix only once for the entire simulated structure. Thus substantial saving of CPU time and also the scope of tackling several stochastic fields simultaneously are the basic advantages of the proposed algorithm. Accuracy and efficiency of this method with reference to example problem is also studied here and numerical results validate its superiority over direct simulation method or first-order perturbation approach.     

基于二阶摄动法求解区间参数结构动力响应

在处理区间参数结构动力响应问题时,现有的分析方法大多局限于一阶区间分析方法. 如果参数的不确定量稍大,采用一阶区间分析方法对结构动力响应范围进行估计可能会失效,所以需要考虑二阶区间分析方法.但是采用基于区间运算的二阶区间分析方法得到的结果将会对动力响应范围过分高估. 为了克服以上缺点,首先基于二阶摄动法得到结构动力响应广义函数. 然后通过求解此动力响应函数的最大和最小值,将结构动力响应区间的问题转化为序列低维箱型约束下的二次规划问题. 最后采用DC 算法(di erence of convex functionsalgorithm) 对这些箱型约束下的二次规划问题进行求解. 这样可以在不引入过多计算量的情况下,避免了对动力响应范围的过分估计. 通过数值算例,将该方法和其他区间分析方法进行比较,验证了该方法的有效性与精确性.      

不确定非线性结构动力响应的区间分析方法

研究多自由度非线性不确定参数系统的动力响应问题. 以区间数学为基础,将不确定 性参数用区间进行定量化,借助一阶Taylor级数,给出了近似估计非线性振动系统动力响 应范围的区间分析方法. 从数学证明和数值算例两方面,将其与概率摄动有限元法进行了比 较,结果显示区间分析方法对不确定参数先验信息具有要求较少、精度较高的优点.     

具有随机参数的含裂纹板弯曲应力强度因子的统计分析

本文首次应用随机有限元法研究了具有随机参数的含裂纹板裂纹尖端弯曲应力强度因子的统计性质。文中首先给出了杂交模式的裂纹尖端奇异单元的刚度矩阵,然后基于随机场的局部平均理论和一阶泰勒展开得到了应力强度因子均值和方差的计算公式。作为数例,详细讨论了杨氏模量、泊松比及板厚度的不确定性对应力强度因子的影响。     

Stability of nonlinear time-varying perturbed differential equations

The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods.     

Finding vibrations of inclined cable structures by approximately solving governing equations for exterior matrix

In this paper, how to compute the eigenfrequencies of the structures composed of a series of inclined cables is shown. The physics of an inclined cable can be complicated, so solving the differential equations even approximately is difficult. However, rather than solving the system of 4 first-order equations governing the dynamics of each cable, the governing equations are instead converted to a set of equations that the exterior matrix satisfies. Therefore, the exterior matrix method (EMM) is used without solving the original governing equations. Even though this produces a system of 6 first-order equations, the simple asymptotic techniques to find the first three terms of the perturbative solution can be used. The solutions can then be assembled to produce a 6 × 6 exterior matrix for a cable section. The matrices for each cable in the structure are multiplied together, along with the exterior matrices for each joint. The roots of the product give us the eigenfrequencies of the system.     

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints(the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.     

Generalized-order perturbation with explicit coefficient for damage detection of modular beam

A general method is formulated to estimate damage location and extent from the explicit perturbation terms in specific set of eigenvectors and eigenvalues. At first, perturbed orthonormal equation is generated from the perturbation of eigenvectors and eigenvalues to obtain the k-th explicit perturbation coefficients. At second, perturbed eigenvalue equation is generated from the perturbation of eigenvector and eigenvalue, and first-order expansion of the stiffness matrix to obtain other explicit perturbation coefficients. Stiffness parameters are computed from these equations using an optimization method. The algorithm is iterative and terminates under certain criteria. A fixed–fixed modular beam with various numbers of elements is used as test structure to investigate the applicability of the developed approach. By comparison with the Euler–Bernoulli beam, discretization errors are analyzed. In six elements beam, first-order algorithm converges faster for small percentage damage. Second-order algorithm is more efficient for medium percentage damage. For large percentage damage, the second-order algorithm converges more effectively. Meanwhile, for eight elements large percentage damage and ten elements small percentage damage, second-order algorithm converges faster to the termination criterion.     

二维稳态辐射声场的光滑有限元-完美匹配层解法

针对外场声学有限元计算精度偏低的问题, 将光滑有限元技术引入到二维稳态辐射声场预测中, 提出了光滑有限元-完美匹配层解法. 该解法采用完美匹配层截断声场计算域, 并将其离散为等参四边形单元, 采用指数吸收函数实现完美匹配层内参数坐标和笛卡尔坐标的映射关系, 采用光滑声压梯度技术计算辐射声场刚度矩阵, 将形函数梯度的域内积分转换为形函数域边界积分. 某汽车二维声腔辐射声场的数值分析结果表明, 与标准有限元-完美匹配层相比, 光滑有限元-完美匹配层解法在完美匹配层内的声波吸收效果更好, 在计算域内的数值计算精度更高, 具有良好的工程应用前景.      

区间参数结构平稳随机载荷识别方法

针对贫信息下不确定性结构的随机载荷识别问题,使用基于Taylor展开的区间分析方法,提出了一种不确定性结构随机载荷识别的非概率区间方法。该识别方法在频域中进行,识别时使用区间变量描述工程结构中的不确定性参数。基于测点的响应谱密度函数,首先对不确定性参数的名义值点进行随机载荷识别,其次计算载荷关于不确定性参数的灵敏度,最后基于区间扩张理论获得识别载荷谱的上下界值。算例结果表明,使用区间方法得到的不确定性结构的载荷谱识别区间界值都能完全包含载荷真值,此方法能够在结构设计时给出更为可靠的载荷工况。     

Reflection and radiation of a wave system at the open end of a submerged elastic pipe

The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe’s interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.     

Higher order accuracy analysis of the second-order normal form method

The second-order normal form method has shown its intelligence in handling the weak nonlinear vibration problems, especially the lightly damped nonlinearities. The new technology can directly realize near identify transformation to the differential equation, while the first-order method has to change the differential equation to the first-order form at the very beginning. In order to get a more precise result, a lot of effort has been done to realize it through eliminating unnecessary approximations or reconsidering the influence of the nonlinearity in the subsequent processing. It is easy to conduct a simplified nonlinear transformation to get the first-order motion equation. Here in this paper, we focus on the higher-order accurate terms in the dynamic equation. The Taylor series expansion and the Poincaré expansion of the nonlinearity indicate that there are other resonance terms existing in final dynamic equation. A?general form of expression for higher-order resonance response function has been derived. The results show that the additional resonance terms cannot obviously increase the accuracy of the second-order normal form method; also, it cannot improve much of the predictions of sub and superharmonic responses.     

An improved analysis method for cross-wire signals obtained in supersonic flow

The use of crossed-wire probes to measure simultaneously the instantaneous stream-wise and normal velocities in supersonic turbulent flows has enabled researchers to investigate the characteristics of organized structures more fully. This paper examines both the practical aspects of using crossed-wire probes in supersonic flow and several methods of converting the resulting signals into useful quantities. Three small perturbation methods are compared in a Mach 2.9 boundary layer, and it is shown that the higher-order terms neglected in the traditional first-order perturbation analysis can alter the instantaneous velocity signals. This is particularly true for regions of intense streamwise mass flux fluctuations. A fourth method, which calculates the instantaneous flow angle directly from the inclined-wire formulation of King's Law, is introduced and discussed. While this method is potentially more accurate than the small perturbation techniques, it is more sensitive to parameter drift during the period between the wire calibration and actual testing.     

Interval finite difference method for steady-state temperature field prediction with interval parameters

A new numerical technique named interval finite difference method is proposed for the steady-state temperature field prediction with uncertainties in both physical parameters and boundary conditions. Interval variables are used to quantitatively describe the uncertain parameters with limited information. Based on different Taylor and Neumann series, two kinds of parameter perturbation methods are presented to approximately yield the ranges of the uncertain temperature field. By comparing the results with traditional Monte Carlo simulation, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed method for solving steady-state heat conduction problem with uncertain-but-bounded parameters.     

一种适用于强非线性结构力学问题数值求解的修正小波伽辽金方法

论文通过对有限区间上的任一连续函数在边界处采用基于泰勒展开的延拓处理,构造了一种与任意边界条件相协调的改进小波尺度基函数及在此基础上建立了小波逼近格式,由此可有效避免小波逼近在求解微分方程时在边界处的跳跃或抖动问题.在此基础上,结合论文后两位作者提出的广义小波高斯积分法,关于未知函数的任意非线性项的小波展开可以显式地用...