离散振系的特征灵敏度分析（连载）

4.1 无阻尼陀螺特征值问题的解法及解的性质 陀螺特征值问题是位形空间中的一个二次特征值问题,它广泛出现在航空航天、旋转机械等许多具有旋转结构的动力学问题中。无阻尼陀螺特征值问题的解法可分两大类:一类是直接求解位形空间中的二次特征值问题的方法;另一类是把位形空间中N阶矩阵的二次特征值问题化为状态空间中2N阶矩阵的一次特征值问题后再求解的方法。后一类方法又可分为三种:一种是把所得一次特征值问题化为更易求解的矩阵特征值问题后再求解的方法;第二种是把一次特征值问题化为反对称阵的特征值问题后,对反对称阵进行一系列的二维或三维旋转相似变换,使之趋于对角块为二阶反对称阵的拟Jacobi方法;第三种是求解反对称阵的Lanzos方法。     

快速子空间迭代法、迭代Ritz向量法与迭代Lanczos法的比较

以高效的细胞稀疏直接快速解法为核心步骤，实现了快速的固有振动广义特征值问题解法。并在相同的允许模态误差的意义下检验了三种常用的大型矩阵特征模态算法——子空间迭代法、迭代Ritz向量法和迭代Lanczos法的计算效率。迭代Ritz向量法平均最快，子空间迭代法最慢，三种解法效率相差不是太大。与ANSYS的子空间迭代和Lanczos法相比。本文的子空间迭代比ANSYS的效率高很多，Lanczos法和ANSYS的效率差不多。大量较大规模的例题显示。本文对特征值算法的改进是十分有效的。算法的健壮性，通用性都达到了高水平。     

得分向量偏序集上Schur函数和奇异得分向量

所有ｎ维得分向量集合Ｌｎ在优超关系下是一个偏序集。Ｌ上的实函数ｇ（ｓ）称为（严格）Ｓｃｈｕｒ凸的，若对任意ｓ，ｓ′∈Ｌｎ，ｓ≠ｓ′，ｓ优超ｓ，恒有ｇ（ｓ）≥（〉）ｇ（ｓ）。本文证明了ｆ（ｘ）＝ｓ＾Ｔｓ和得分向量为ｓ的竞赛图Ｔｎ中３－圈个数ｃ３（ｓ）在Ｌｎ上分别是严格Ｓｃｈｕｒ凸和严格Ｓｃｈｕｒ凹的，称ｎ维得分向量ｓ为奇异的，若得分向量为ｓ的每个ｎ阶竞赛图Ｔｎ的邻接矩阵都是奇异的。最后，应用Ｌ上严格     

关于Schur补与逆M—矩阵问题的讨论

本文指出Ｌ．Ｎ．Ｉｍａｎ关于Ｓｃｈｕｒ补和逆Ｍ－矩阵问题的一些主要结果是错误的，我们修正了这些错误。     

关于Schur补与逆M─矩阵问题的讨论

本文指出Ｌ．Ｎ．Ｉｍａｎ关于Ｓｃｈｕｒ补和逆Ｍ－矩阵问题的一些主要结果是错误的，我们修正了这些错误。     

子空间旋转法分析及其仿真研究

本文对DOA估计的一种新方法－子空间旋转进行了全面的分析和详细的证明。指出了用子空间旋转法进行DOA估计的关键是求解奇异对的广义特征值，提供了一种求解的算法，并且利用此算法对子空间旋转法的一些性能进行了计算机仿真研究。     

基于SiPESC平台的声子晶体能带结构分析算法研究及软件实现EI北大核心CSCD

为了快速准确得分析声子晶体的能带结构,该文基于工程与科学计算仿真平台SiPESC,开发了一类高效三维声子晶体能带结构分析软件。软件针对能带结构分析过程中计算量庞大的Hermitian矩阵的广义特征值求解问题和边界约束节点匹配问题,提出了相关软件设计方案。针对Hermitian矩阵的广义特征值求解,在实对称矩阵子空间迭代法的基础上,发展了Hermitian矩阵子空间迭代法。针对边界约束节点匹配问题,该文将三维周期性条件划分为点、边、面约束分别处理。针对面约束,该文使用定位格匹配策略将单层的点-点匹配更改为2层的点-定位格-点加速匹配。开展了与多物理场分析软件COMSOL进行数值算例对比。使用三维局域共振声子晶体算例验证了软件在满足数值精度的前提下计算效率高于对比软件。通过大规模模型算例验证了软件具有高效的大规模计算能力。     

结构参数非对称变化特征值问题的一种摄动解法

本文提出结构动力这参数并非对称变化时，特征值问题的一种解法，把变化表达为对称和反对称两部分，依次对原系统作摄动求解，对称部分是实模态摄动，反对称部分是复模态摄动，由此可得出变化系统的特征值和特征向量的各阶渐近估计。     

圆弧曲梁面内自由振动的微分容积解法

用微分容积法求解圆弧曲梁在面内的自由振动问题。通过微分容积法将曲梁自由振动的控制微分方程和边界约束方程离散成为一组线性齐次代数方程组，这是一典型的特征值问题，求解这一特征值问题可以求得其自由振动的圆频率。中采用了考虑轴向变形、剪切变形和转动效应的理论，并采用子空间迭代法求解频率方程。数值算例表明，本方法稳定收敛、精度较高，对圆弧曲梁问题简单、有效。     

陀螺系统特征值问题的一些探讨

陀螺系统特征值问题是在转子动力学中提出的一个典型的数学问题。它的研究一直受到力学界的重视。研究它具有较大的理论意义和工程应用价值。作为本专题的第一部分，本文探讨三维旋转法本身存在的问题。指出三维旋转法不是对任何反对称矩阵都是合适的，而应有条件地应用它。     

陀螺系统辛子空间迭代法

转子系统的有限元分析可以导出陀螺系统的本征值问题．而陀螺本征值问题可在哈密顿体系下求解。基于辛子空间迭代法的思想，提出了一种求解陀螺系统本征值问题的算法。首先引入对偶变量，将陀螺动力系统导入哈密顿体系，将问题化为了哈密顿矩阵的本征值问题。由于稳定的陀螺系统其本征值必为纯虚数，利用这个特点。提出了对应陀螺系统的辛子空问迭代法，从而可以求出系统任意阶的本征值及其振型。算例证明了这种算法的有效性。     

求工程结构最低阶固有频率的一种算法

本文采用子空间迭代法将工程结构高阶动力系统减缩为低阶动力系统,然后用Ｃｏｌｌａｔｚ包含定理的推广求出该结构系统的最低阶固有频率。     

Multigrid solution procedures for structural dynamics eigenvalue problems

Three multigrid methods are described for solving the generalized symmetric eigenvalue problem encountered in structural dynamics. Two implicit algorithms are discussed that use a multigrid method to solve the linear matrix equations encountered in each iteration of the standard subspace and block Lanczos methods. An explicit method is also outlined which explicitly applies the basic multigrid philosophy of fine mesh relaxation and coarse mesh correction to the eigenvalue problem. All of these algorithms are capable of extracting the lower modes of the system, provided each required eigenvector can be represented on each coarse mesh. The behavior of the methods is studied by examining the selection of convergence tolerances and the solution of some ill-conditioned problems. A well-conditioned plate problem is solved to demonstrate the computational resources required by the algorithms. The explicit method is observed to be the most efficient method (in terms of storage and CPU time), whereas the implicit Lanczos method requires the most computational effort. A comparison between the multigrid algorithms and a commercially available implementation of the subspace iteration method is also presented.     

A Block–Stodola eigensolution technique for large algebraic systems with non-symmetrical matrices

A Block–Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = λ BU where A is real but non-symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vector as is the case in the classical Stodola–Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.     

Stability of Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme

The paper investigates the problem of numerical stability of the Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme. The instability is expressed as loss of positive definiteness of covariance matrix and is the result of modifications of standard exponential covariance functions that are commonly applied to increase the sparsity of the covariance matrix. The loss of positive definiteness of covariance matrix limits the use of efficient eigenvalue solvers that are needed for the solution of the resulting generalized eigenvalue problem. Two modifications of the shape of covariance function to avoid instability problems and at the same time to raise the numerical efficiency of Karhunen–Loève expansion by increasing the sparsity of the covariance matrix are proposed. The effects of the proposed modifications are demonstrated on numerical examples.     

Solution of FE‐BE coupled eigenvalue problems for the prediction of the vibro‐acoustic behavior of ship‐like structures

To predict the vibro‐acoustic behavior of structures, both a structural problem and an acoustic problem have to be solved. For thin structures immersed in water, a strong interaction between the structural domain and fluid domain occurs. This significantly alters the resonance frequencies. In this work, the structure is modeled by the finite element method. The exterior acoustic problem is solved by a fast boundary element method employing hierarchical matrices. An FE‐BE formulation is presented, which allows the solution of the coupled eigenvalue problem and thus the prediction of the coupled eigenfrequencies and mode shapes. It is based on a Schur complement formulation of the FE‐BE system yielding a generalized eigenvalue problem. A Krylov–Schur solver is applied for its efficient solution. Hereby, the compressibility of the fluid is neglected. The coupled eigensolution is then used for a model reduction strategy allowing fast frequency sweep calculations. The efficiency of the proposed formulations is investigated with respect to memory consumption, accuracy, and computation time. Copyright © 2011 John Wiley & Sons, Ltd.     

vibro‐Lanczos,a symmetric Lanczos solver for vibro‐acoustic simulations

An efficient symmetric Lanczos method for the solution of vibro‐acoustic eigenvalue problems is presented in this paper. Although finite element discretization results in real but nonsymmetric system matrices, we show that an efficient iteration scheme on a symmetric representation can be built up by using a transformation matrix. In order to decrease the numerical costs of the orthogonalizations performed, we propose to use a partial orthogonalization scheme for the symmetric case. The proposed method is tested on two large problems in order to demonstrate its efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

The generalized global basis (GGB) method

In this work, we present the generalized global basis (GGB) method aimed at enhancing performance of multilevel solvers for difficult systems such as those arising from indefinite and non‐symmetric matrices. The GGB method is based on the global basis (GB) method (Int J Numer Methods Eng 2000; 49 :439–460, 461–478), which constructs an auxiliary coarse model from the largest eigenvalues of the iteration matrix. The GGB method projects these modes which would cause slow convergence to a coarse problem which is then used to eliminate these modes. Numerical examples show that best performance is obtained when GGB is accelerated by GMRES and used for problems with multiple right‐hand sides. In addition, it is demonstrated that GGB method can enhance restarted GMRES strategies by retention of subspace information. Copyright © 2004 John Wiley & Sons, Ltd.     

Structured pseudospectra in structural engineering

This paper presents a new method for computing the pseudospectra of a matrix that respects a prescribed sparsity structure. The pseudospectrum is defined as the set of points in the complex plane to which an eigenvalue of the matrix can be shifted by a perturbation of a certain size. A canonical form for sparsity preserving perturbations is given and a computable formula for the corresponding structured pseudospectra is derived. This formula relates the computation of structured pseudospectra to the computation of the structured singular value (ssv) of an associated matrix. Although the computation of the ssv in general is an NP‐hard problem, algorithms for its approximation are available and demonstrate good performance when applied to the computation of structured pseudospectra of medium‐sized or highly sparse matrices. The method is applied to a wing vibration problem, where it is compared with the matrix polynomial approach, and to the stability analysis of truss structures. New measures for the vulnerability of a truss structure are proposed, which are related to the ‘distance to singularity’ of the associated stiffness matrix. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of fracture mechanics parameters for a general corner using a weight function method

Summary The weight function method (WFM) has been used recently as a reliable tool for evaluation of fracture mechanics parameters, where cracks are represented by zero opening traction free surfaces. The purpose of this paper is to extend this technique to general opening corner problem. The two dimensional singular fields for displacements and stresses are introduced in terms of generalized Bueckner's strength. By means of eigenvalue analysis the stress intensity factors (SIF) are then formulated after appropriate splitting the regular stress and displacement fields into symmetric and antisymmetric modes. Using Betti's reciprocal theorem, a new expression in a more general closed form is derived for Bueckner's strength consisten with the given nonzero opening case. The potentiality of the method is demonstrated by a numerical example for =/2 corner problem. The stress intensity factor for the symmetric mode is evaluated by WFM and by a simple collocation procedure using both boundary element (BE) and finite element (FE) discretization.