广义特征值问题中重特征值的特征向量导数

本文把重特征值的特征向量导数的计算方法，推广到非亏损矩阵的广义特征值问题，并给出了特征值导数也有重根时特征向量导数的计算式。本方法的优点是只需已知所考虑的重特征值的特征向量，因而计算量小，对于大型复杂结构更为适用。     

任意非亏损矩阵特征灵敏度分析的模态展开法

把特征向量的各阶导数表示成所有模态的线性组合，并利用左模态与右模态间的双正交性，首先导出了任意非亏损矩阵的重特征值的一阶导数所满足的特征值问题，然后根据此特征值问题无、看重根的情况，再导出了异导重特征值和等导重特征值对应的可微特征向量、特征值和特征向量各阶导数的一般计算公式。算例显示了方法的正确性。     

一般实矩阵特征值问题的多维灵敏度分析

应用Kronecker代数和矩阵微分理论系统发展了一般实矩阵特征值问题的矩阵灵敏度分析方法，结构系统特征值和特征向量的导数很方便地排列成二维矩阵，所得的结果很容易形成计算机程序。     

一种有效的广义特征值分析方法

提出了一种适合于自适应有限元分析中求解广义特征值问题的多重网格方法．这种方法充分利用了初始网格下的结果，通过插值或最小二乘拟合技术来得到网格变化后的新的近似特征向量，然后由多重网格迭代过程实现对结构广义特征值问题的求解．在多重网格迭代的光滑步中，选择了收敛梯度法以提高其收敛率；在粗网格校正步中，则导出了一种近似求解特征向量误差的方程．这种方法将网格离散过程和数值求解过程很好地相结合，建立了一个网格细分后广义特征值问题的快速重分析方法，与传统有限元方法相比较，具有计算简便、计算量少等特点，可以作为结构动力问题自适应有限元分析的一种十分有效的工具．     

求解对称矩阵特征值及特征向量的初等变换法

 在力学中有一类量的求解可归结为矩阵特征值和特征向量的求解，而求解 矩阵的特征值将要求解高次方程的根，这在数学上将遇到难以克服的困难. 应用初等变 换方法，将对称矩阵的特征矩阵对角化. 利用这种方法，同时可求出该矩阵所有的特征值和 正交的特征向量，避免了求解高次方程根的困难与把各特征向量正交化的麻烦.     

用动态缩聚法求解结构的特征值问题

基于逐级逼近思想提出了一种新的动态缩聚法。保留和压缩的自由度通过一个缩聚矩阵联系，最初的缩聚矩阵由系统子矩阵定义。用循环的方法逐级逼近特征值及特征向量。计算实例验证了上述方法的可行性。     

计算重根特征向量导数的消元法

本文提出了重根特征向量导数计算的消无法，它分为若干直接算法和一种迭代算法。这种方法概念简单、实施容易、精度良好。该方法可经简单移值用于非重根情况。     

随机结构系统的一般实矩阵特征值问题的概率分析

由于工程实际结构的复杂性和所用材料在统计上的离散性以及测量、加工、制造误差的存在，必然导致具有随机参数的随机结构振动系统，按结构参数的性质来划分，随机振动问题包括两方面内容：(1)确定结构问题；(2)随机结构问题。本文以现代数学理论为依托，研究了随机结构系统的一般实矩阵的特征值问题。根据Kronecker代数、向量值和矩阵值函数的灵敏度分析、一般二阶矩法和概率摄动技术给出了计算随机结构系统的一般实矩阵的特征值和特征向量的数值方法，可以有效地得出随机结构系统的一般实矩阵的特征向量的统计量，发展了2D矩阵值函数的随机结构系统的特征值问题概率分析理论。     

结构参数大修改时的特征值重分析方法

就结构参数发生大修改的情况提出了两种高精度的特征值重分析方法：Ｐａｄｅ　逼近法和推广的Ｋｉｒｓｃｈ混合法．利用这两种方法,计算了一个具有２０２个结点,３５７个梁单元的平面框架的近似特征值．计算结果表明,所提出的方法是结构参数修改时的特征值重分析的有效方法．     

边界元特征值分析及Burton-Miller法探究

运用围道积分方法将边界元非线性特征值问题转化为规模很小的广义特征值问题,从而构造出一种边界元特征值分析方法。数值算例验证了该方法的求解精度。针对外声场问题,通过对常规、法向导数和BurtonMiller边界积分方程的虚假特征频率的计算和比较,揭示了Burton-Miller法规避虚假特征频率的本质,并对其中的叠加常数的最优取值给出了一种新的解释。     

动态光弹性方法的主应力分离的研究

本文提出了动态光弹性、动态焦散线实验方法同边界元法结合的混合法，并用这种方法解决了动态光弹性主应力分离的问题，首先对现有的多火花高速摄影系统进行了改造，在动态实验过程中，成功地得到了不同瞬时的清晰的动态光弹性的等差线条纹和动态焦散线系列图像，这样，便可提取不同瞬时的边界应力值、全场主应力差值及边界上的外载。继而提出用Ｌａｐｌａｃｅ变换域上的边界元法来求解在冲击载荷作用下二维弹性模型全场的主应力和。最后，以受冲击载荷作用的圆盘为例，进行实验及边界元法计算，得到了分离的主应力场。     

An FMM for waveguide problems of 2-D Helmholtz’ equation and its application to eigenvalue problems

This paper discusses an FMM for solving waveguide problems and associated eigenvalue problems for Helmholtz’ equation in a two dimensional infinite strip with homogeneous Neumann boundary condition on the sides. Layer potentials with Green’s function for this problem are evaluated efficiently with the help of the method of images and FMM. We apply FMM to solve some boundary value problems in waveguides and associated resonance frequency problems using the Sakurai–Sugiura projection method after discussing the required analytic continuation of the solutions to complex frequencies. Some numerical examples show the accuracy and the efficiency of the proposed method.     

边界元特征值分析及Burton-Miller法探究BEM eigenvalue analysis and the investigation of the Burton-Miller method

运用围道积分方法将边界元非线性特征值问题转化为规模很小的广义特征值问题,从而构造出一种边界元特征值分析方法。数值算例验证了该方法的求解精度。针对外声场问题,通过对常规、法向导数和Burton‐Miller边界积分方程的虚假特征频率的计算和比较,揭示了Burton‐Miller法规避虚假特征频率的本质,并对其中的叠加常数的最优取值给出了一种新的解释。     

一种求解含振子及弹性支承圆板振动的新方法

将富里叶－贝塞尔级数引入积分方程[1],推导出一种研究含振子及弹性支承圆板振动特性的新方法,根据积分方程和富里叶－贝塞尔级数理论,首先用第一类贝塞尔函数构造圆板的格林函数,然后由叠加原理将圆板的自由振动问题转化为积分方程的特征值问题;进面将积分方程形式的特征值问题转化为无穷阶矩阵的标准特征值问题,计算时根据精度的要求,截取无穷阶矩阵的标准特征值为有限阶矩阵的标准特征值问题,采用Q－R算法,计算实践表明,本方法不仅具有运算简捷,精度高,适用性强的特点,而且能从整体上对系统的动态性加以研究,从而为这类系统的优化设计提供有;力的 工具。     

关于舰船横摇中的若干非线性问题

对船舶横摇运动中的一些非线性问题的若干研究现状进行了总结,讨论了所存在的一些问题及其发展趋势。对横摇非线性问题的产生以及导致非线性的因素作了简述,从大幅非线性横摇运动、横摇-垂荡耦合运动、和横摇-纵摇耦合运动三方面,详细综述了国内外关于舰船在波浪中的力学研究的发展趋势,提出了几个有待深入加强研究的问题。     

Wave propagation in a piezoelectric rod of 6 mm symmetry

The wave propagation in a piezoelectric rod of 6 mm symmetry is investigated by applying a 3-D piezoelectric elastic model. A self-adjoint method is introduced to solve this problem, this method avoids calculating the generalized eigenvalue equation, it completely draws the dispersion curves in the forms of Quasi-P wave, Quasi-SV wave and Quasi-SH wave under the self-adjoint boundary condition, and it can evaluate the dispersion curves of all kinds of boundary conditions. As an example, the dispersion curves of PLT-5H are completely drawn, we also found the Quasi-SV wave has standing wave phenomenon in the PLT-5H rod. In addition the relation of dispersion curves among different boundary conditions is discussed, and an experiment method is introduce to decide the dispersion curves for another boundary conditions.     

Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems

The aim of this paper is to show that the Jacobi–Davidson (JD) method is an accurate and robust method for solving large generalized algebraic eigenvalue problems with a singular second matrix. Such problems are routinely encountered in linear hydrodynamic stability analysis of flows that arise in various areas of continuum mechanics. As we use the Chebyshev collocation as a discretization method, the first matrix in the pencil is nonsymmetric, full rank, and ill‐conditioned. Because of the singularity of the second matrix, QZ and Arnoldi‐type algorithms may produce spurious eigenvalues. As a systematic remedy of this situation, we use two JD methods, corresponding to real and complex situations, to compute specific parts of the spectrum of the eigenvalue problems. Both methods overcome potentially severe problems associated with spurious unstable eigenvalues and are fairly stable with respect to the order of discretization. The real JD outperforms the shift‐and‐invert Arnoldi method with respect to the CPU time for large discretizations. Three specific flows are analyzed to advocate our statements, namely a multicomponent convection–diffusion in a porous medium, a thermal convection in a variable gravity field, and the so‐called Hadley flow. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlinear coupled dynamics of liquid-filled spherical container in microgravity

Nonlinear coupled dynamics of a liquid-filled spherical container in microgravity are investigated. The governing equations of the low-gravity liquid sloshing in a convex axisymmetrical container subjected to lateral excitation is obtained by the variational principle and solved with a modal analysis method. The variational formulas are transformed into a frequency equation in the form of a standard eigenvalue problem by the Galerkin method, in which admissible functions for the velocity potential and the liquid flee surface displacement are determined analytically in terms of the Gaussian hypergeometric series. The coupled dynamic equations of the liquid-filled container are derived using the Lagrange＇s method and are numerically solved. The time histories of the modal solutions are obtained in numerical simulations.     

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.