任意非亏损系统特征灵敏度分析的直接摄动法

本文发展了一种任意非亏损系统特征灵敏度分析的二阶段摄动法，将未摄动的问题的解作为零阶近似，把摄动影响作为摄动后问题的高阶修正，经过严格的数学推导，得到了支配高阶修正量的完全方程组。本方法无损知道摄问题的全部特征向量，仅需知被摄模态的特征对。本方法可处理被摄问题具有重特征值，甚至具有等导重特征值这一高度退化极难处理的情况。算例显示了本方法的正确性。     

边界约束刚度不确定的结构振动特征值

利用摄动法 ,将随机的微分方程和边界条件化为一系列的确定性微分方程和边界条件。运用有限元离散方法 ,推导了统计特征值的二阶摄动近似表达 ,用算例对本文方法进行了说明并和 Monte-Carlo模拟法结果进行了比较     

渗流问题单参数摄动随机反演方法

在正演随机模拟方法的基础上,结合Taylor展开式和随机变量的摄动方程,讨论了随机参数反演问题,提出了摄动随机反演方法,给出了一阶均值反演准则和二阶均值反演准则,提出了单随机变量的均值和方差的表示方法,计算均值时采用改进的遗传算法,计算方差时采用统计的方法。给出了一个Thies模型反演导水系数的例子,计算表明该方法简单实用,效果良好。     

具有随机路径的振动传递路径系统的随机响应分析

基于一般概率摄动有限元法,解决了具有随机路径的振动传递路径系统的响应分析问题.应用Kronecker代数,矩阵微分理论,向量值和矩阵值函数的二阶矩技术,矩阵摄动理论和概率统计方法,提出了振动传递路径系统的随机响应分析方法,在考虑工程中的不确定因素以后,在时域内清晰地描述了振动传递路径的随机响应.     

复合材料层合板的摄动随机Cell-Based光滑有限元

随机性是实际工程结构的固有特性,如何更真实地描述含随机参数结构的随机响应及统计特性,对工程结构的可靠性设计具有非常重要的意义。本文基于Cell-Based光滑有限元,采用四边形单元,推导了基于一阶剪切变形理论的复合材料层合板的光滑有限元公式,降低了网格划分要求,适应不规则网格,并采用离散剪切间隙有效地消除了剪切自锁;结合摄动法和随机场理论,导出了复合材料层合板的摄动随机光滑有限元平衡方程,并给出了结构随机响应数字特征的计算公式,求解了材料属性含随机性的复合材料层合板的随机响应问题,数值算例结果表明了本方法的有效性和准确性。     

摄动有限差分方法研究进展

振动有限差分（ＰＦＤ）方法,既离散徽商项也离散非微商项（包括微商系数）,在微商用直接差分近似的前提下提高差分格式的精度和分辨率．ＰＦＤ方法包括局部线化微分方程的摄动精确数值解（ＰＥＮＳ）方法和摄动数值解（ＰＮＳ）方法以及考虑非线性近似的摄动高精度差分（ＰＨＤ）方法。论述了这些方法的基本思想、具体技巧、若干方程（对流扩散方程、对流扩散反应方程、双曲方程、抛物方程和ＫｄＶ方程）的ＰＥＮＳ、ＰＮＳ和ＰＨＤ格式,它们的性质及数值实验．并与有关的数值方法作了必要的比较．最后提出值得进一步研究的一些课题．      

局部损伤复合材料层合板的振动分析

本文采用了一种改进方法对局部损伤复合材料层合板进行了振动分析,将复合材料板中的损伤模拟为局部刚度的削减,并取三个损伤因子来刻画损伤的特性．利用高阶摄动法对其自由振动方程进行求解,主要计算了损伤板的自然振动频率和振动模态．相较于一阶摄动展开法,该方法在计算局部较大损伤问题中具有更高的准确度和敏感度．最后对损伤问题进行了参数研究,分析了不同的损伤因子（包括局部损伤程度、方向、面积大小）对板自由振动频率的影响．该方法为二维板局部损伤检测提供了有效精确的理论依据,并为损伤的定量评价提供了一种思路．     

轴对称粗糙圆杆的扭转变形

本文旨在研究轴对称表面粗糙圆杆的扭转变形。通过微元法,推导了轴对称表面粗糙圆杆的控制微分方程。结合摄动法和离散傅里叶变换,获得了粗糙圆杆的扭转角摄动解。对比典型粗糙圆杆的材料力学解析解或有限元解,验证了本文摄动解的有效性。通过系统计算,探究了粗糙面的统计参数对于扭转角的影响规律并建立了经验公式。本文的研究成果拓展了材料力学中扭转问题解的范围,并为数学物理方法提供了具体的力学实践。     

结构强度可靠性分析的模糊随机边界元法

利用模糊随机变量和模糊概率特征建立模糊随机边界元代数方程，对方程作λ水平截集，得到随机区间方程，将该方程中的系数矩阵，结点位移列阵和荷载列车在初始随机向量的均值处展开，利用区间数分解和小参数摄动理论导出求解应力统计特征、结构破坏概率指标和可靠度的计算公式，并给出算例。     

随机桁架结构几何非线性问题的混合摄动-伽辽金法求解

提出应用混合摄动-伽辽金法求解随机桁架结构的几何非线性问题.将含位移项的随机割线弹性模量以及随机响应表示为幂多项式展开,利用高阶摄动方法确定随机结构几何非线性响应的幂多项式展开的各项系数.将随机响应的各阶摄动项假定为伽辽金试函数,运用伽辽金投影对试函数系数进行求解,从而得到随机桁架结构几何非线性响应的显式表达式.同已有的随机伽辽金法相比,本文所给的试函数由摄动解的线性组合而成,在求解非线性问题时,试函数的获取具有自适应性.数值算例结果表明,对于具有不同概率分布的多随机变量问题,本文方法无需对随机变量的概率分布形式进行转换,避免了转换误差,因而比同阶的广义正交多项式方法 (generalized polynomial chaos, GPC)计算精度高.同时,在结果精度相当时,和GPC方法相比,本文方法得到的试函数系数的非线性方程维度不大,方程的求解工作量小且更易求解.当随机量涨落较大时,混合摄动-伽辽金法计算所得的结构响应的各阶统计矩比高阶摄动法所得结果更逼近于蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性.     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

High-order theory for arched structures and its application for the study of the electrostatically actuated MEMS devices

A high-order theory for arched rods and beams based on expansion of the two-dimensional (2D) equations of elasticity into Legendre’s polynomials series has been developed. The 2D equations of elasticity have been expanded into Legendre’s polynomials series in terms of a thickness coordinate. Thereby, all equations of elasticity including Hooke’s law have been transformed to corresponding equations for coefficients of Legendre’s polynomials expansion. Then system of differential equations in term of displacements and boundary conditions for the coefficients of Legendre’s polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in details. For obtained boundary-value problems, a finite element method has been used and numerical calculations have been done with COMSOL Multiphysics and MATLAB. Developed theory has been applied for study pull-in instability and stress–strain state of the electrostatically actuated micro-electro-mechanical Systems.     

Application of the random eigenvalue problem in forced response analysis of a linear stochastic structure

The random eigenvalue problem arises in frequency and mode shape determination for a linear system with uncertainties in structural properties. Among several methods of characterizing this random eigenvalue problem, one computationally fast method that gives good accuracy is a weak formulation using polynomial chaos expansion (PCE). In this method, the eigenvalues and eigenvectors are expanded in PCE, and the residual is minimized by a Galerkin projection. The goals of the current work are (i) to implement this PCE-characterized random eigenvalue problem in the dynamic response calculation under random loading and (ii) to explore the computational advantages and challenges. In the proposed method, the response quantities are also expressed in PCE followed by a Galerkin projection. A numerical comparison with a perturbation method and the Monte Carlo simulation shows that when the loading has a random amplitude but deterministic frequency content, the proposed method gives more accurate results than a first-order perturbation method and a comparable accuracy as the Monte Carlo simulation in a lower computational time. However, as the frequency content of the loading becomes random, or for general random process loadings, the method loses its accuracy and computational efficiency. Issues in implementation, limitations, and further challenges are also addressed.     

Free vibration analysis of elastic rods using global collocation

This paper investigates the performance of a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures when either basis or shape functions are used to approximate the displacement field. Although the methodology is generally applicable, numerical results are presented only for rods in which one-dimensional basis functions in the form of a power series, as well as equivalent Lagrange, Bernstein or Chebyshev polynomials are used. The new feature of the proposed methodology is that it can deal with any type of boundary conditions; therefore, the cases of two Dirichlet as well as one Dirichlet and one Neumann condition were successfully treated. The basic finding of this work is that all these polynomials lead to results identical to those obtained by the power series expansion; therefore, the solution depends on the position of the collocation points only.     

A high-order theory for functionally graded axially symmetric cylindrical shells

A high-order theory for functionally graded axially symmetric cylindrical shell based on expansion of the axially symmetric equations of elasticity for functionally graded materials into Legendre polynomials series has been developed. The axially symmetric equations of elasticity have been expanded into Legendre polynomials series in terms of a thickness coordinate. In the same way, functions that describe functionally graded relations has been also expanded. Thereby, all equations of elasticity including Hook’s law have been transformed to corresponding equations for coefficients of Legendre polynomials expansion. Then system of differential equations in terms of displacements and boundary conditions for the coefficients of Legendre polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in more details. For obtained boundary-value problems’ solution, a finite element has been used and numerical calculations have been done with COMSOL MULTIPHYSICS and MATLAB.     

随机场的投影展开法

本文基于伽辽金投影方法，提出了一种新的随机场的展开方法。它是通过构造一组标准正交基，并将随机场用其在这组基上的投影表示。这种展开方法避免了求解相关函数的特征函数的困难，最后通过算例说明了本文方法的特性。     

考虑场地介质随机特性的无限域波动分析

针对场地介质具有随机特性的无限域地震波动分析问题，在概率空间中将随机反应向量按随机介质场离散所得主导随机变量的正交多项式级数形式展开，使随机微分方程变换确定性的扩阶线性方程组，并在波动的元模拟技术的基础上，构造了扩阶透射人工边界公式，两者结合形成了求解无限域随机介质中波动传播问题的有限元分析方法，该方法不仅不受基于摄动思想各类方法的久期项的干扰，而且避免了采用模拟方法时人工边界区单元参数样本不均匀所引起的数值计算不稳定问题。     

随机杆系结构几何非线性分析的递推求解方法

建立了随机静力作用下考虑几何非线性的随机杆系结构的随机非线性平衡方程. 将和 位移耦合的随机割线弹性模量以及随机响应量表示为非正交多项式展开式，运用传统的摄动方法获 得了关于非正交多项式展式的待定系数的确定性的递推方程. 在求解了待定系数后，利用非 正交多项式展开式和正交多项式展开式的关系矩阵，可以很方便地得到未知响应量的二阶统计矩. 两杆结构和平面桁架拱的算例结果表明，当随机量涨落较大时，递推随机有限元方法比基于 二阶泰勒展开的摄动随机有限元方法更逼近蒙特卡洛模拟结果，显示了该方法对几何非线性 随机问题求解的有效性.     

Boundary Integral Equations Method for the Analysis of Acoustic Scattering from Line-2 Elastic Targets

The prediction of the acoustic scattering from elastic structures is a recurrent problem of practical importance as, for example, in underwater detection and target identification. We aim at setting out the diffraction problem of a transient acoustic wave by an axisymmetric shell composed of a cylinder bounded by hemispherical endcaps, called Line-2. Its time-dependent response is expanded in terms of the resonance modes of the fluid-loaded structure. The latter are well suited when the structure is submerged in a heavy fluid: it is an alternative to modal methods whose expansions as series of natural modes of the in vacuo shell are much better for describing the interaction between a structure and a light fluid. The resonance frequencies are defined as solutions of the nonlinear eigenvalue problem described by the set of homogeneous equations governing the structure displacement coupled to the acoustic radiated pressure. The resonance modes of the coupled system are the corresponding eigenvectors. Both hemisphere and cylinder equations are modeled by the approximation of Donnel and Mushtari which governs thin shells oscillations. The modeling of the sound pressure by a hybrid potential integral representation leads to a system of integro-differential equations defined on the surface of the structure only (boundary integral equations). The unknowns, the hybrid potential density as well as the shell displacement vector, are developed into Fourier series with respect to the natural cylindrical coordinate. Each angular component of the unknown functions is then expanded as series of Legendre polynomials, the coefficients of which are calculated thanks to a Galerkin method derived from the energetic form of the equations. The whole method can also be applied to predict the response of the coupled structure to a harmonic or a random excitation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Deformation of structure and spectrum of evolution equations

In this paper,we study the general structure of evolution equations of the AKNSeigenvalue problem q(x,t),r(x,t)with the spectrum varying asand A_1,B_1,C_1,are all positive or negative power polynomials of A where q.r are not limitedwith any additional conditions at infinity.