基于改进Ritz法的耗能减震高层建筑结构地震响应分析

用改进Ritz法计算耗能减震高层建筑结构的地震响应。改进Ritz法采用基于外荷载空间分布的Ritz向量和基于外荷载频率的Ritz向量。其中,基于外荷载空间分布的Ritz向量用Lanczos法形成。基于外荷载频率的Ritz向量则用外荷载主频的平方进行特征值平移后再用Lanczos法形成。文中还给出上述两种Ritz向量的截断标准。之后用改进Ritz法的Ritz向量对结构动力方程进行线性变换。由于耗能减震高层建筑结构的阻尼是非比例阻尼,广义阻尼矩阵是非对角矩阵,所以文中采用拟力实模态法对线性变换后的耦联动力方程进行求解。最后的算例说明该方法用于耗能减震高层建筑结构的地震响应分析是有效可行的。     

陀螺特征值问题的广义SRR子空间迭代法及其加速法

本文基于非Ｈｅｒｍｉｔｉａｎ矩阵的Ｓｃｈｕｒ－Ｒａｙｌｅｉｇｈ－Ｒｉｔｚ加速的子空间迭代法，构造了状态空间表示的陀螺特征值问题广义ＳＲＲ子空间迭代法，这是与由质量阵和刚度阵构成的广义特征值问题的子空间迭代法平行的一种算法，在迭代中充分利用陀螺特征值问题的反对特征性，使得投影及ＳＲＲ特征值问题求解方法的本质是反对称Ｓｃｈｕｒ型的平方乃是对称Ｓｃｈｕｒ型。这一发现揭示了对称与反对称特征值问题之间的内在     

广义特征值问题求解的改进Ｒｉｔｚ向量法

从提高算法的稳定性和计算效率入手,采取迭代及防止漏根、多根的措施,对传统的Ｒｉｔｚ向量法进行改进,提出改进的Ｒｉｔｚ向量法。此算法仅需生成ｒ维的Ｋｒｙｌｏｖ空间,大大降低投影矩阵阶数,减少投影矩阵特征值计算时间。引入重正交方案和模态比较法,并给出Ｒｉｔｚ向量块宽ｑ与生成步数ｒ的建议取值。最后通过四参数的谱变换法,不     

计算特征向量及其导数的同步迭代法

以前的灵敏度分析方法都是先计算特征对然后再计算它们的导数，本文对特征对计算的矩阵迭代法及子空间迭代法进行改造，在迭代计算特征对的同时计算特征向量的导数。采用矩阵迭代法可以直接迭代计算特征向量导数，避免了对奇异灵敏度方程的求解。采用子空间迭代法可以将原来的大型特征方程和灵敏度方程缩阶为较小的方程。算例表明这两种算法精度较高，采用子空间迭代法计算多个特征向量对一个设计变量的导数计算效率较高。     

关于子空间迭代法初始向量选取问题的一点看法

本文叙述了三种形成子空间迭代法初始向量组的方法,选用不同的初始向量组作了算例,对于不同的结构可选取不同的初始向量,以便加速迭代收敛的速度。     

大型复杂结构系统陀螺模态综合技术的Lanczos法

本文建议了一种Lanczos减缩的陀螺模态综合技术,Lanczos矢量和三对角阵可在位形空间用实数运算获得,可以有效地求得大型复杂结构阻尼系统的复特征值和动力响应。研究表明,本方法适用于转子动力学,直升机旋翼-机身耦合系统以及具有流体流动的管道系统的动力分析。     

非亏损系统二次特征值问题的Ritz降价法

本文提出了一种利用Ritz降阶对含有重特征值及刚体模态的非亏损阻尼系统进行模态分析的方法，本方法利用保守系统的左、右截断模态作为Ritz基，综合出非保守系统二次特征值问题的待求左、右特征向量，文章证明了在大多数情况下，本方法可用较少的耗费，提取满足给定精度的待求二次特征值问题的模态解，文末的算例表明了本方法的有效性。     

结构特征向量导数计算的移位迭代模态法

提出了移位迭代模态法，可以大大提高特征向量对设计参数导数的计算效率。从理论上证明，经典模态法、修改模态法和迭代模态法只是移位迭代模态法的特殊情况。文中还给出以ＦＯＲＫ和３Ｄ－ＦＲＡＭＥ两种结构为实例的计算机仿真，并与已有的几种模态法进行了比较。理论分析与计算机仿真表明，本方法计算特征向量导数时，只需用很少几个模态即能保证精度，从而可以在保证精度的条件下大大提高计算速度。     

Ritz向量直接迭加法及其在微型机上的实现

利用 Ritz 向量直接迭加法进行动力分析通常比子空间迭代法平均要快十倍左右。尤其是 Ritz 向量法的步骤主要是一系列静力方程的求解。因此,本文应用分块外存求解器来求解大型结构的固有频率和振型是十分有效的,并且大大缩减了对内存容量的要求。对于内存很紧张的微型计算机,这一方法的优点则更为突出,以致使动力分折的容量提高了四倍左右。     

动力子结构法中Ritz基的一种选择

基于弹性动力学变分原理,本文导出了混合型动力子结构方法的一种Ritz基矢量,即采用正交化的块Lanczos基代替主模态,结出了相应的子结构模态变换的格式和综合后的系统方程式.文未给出了算例.     

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.     

Chebyshev filtering Lanczos' process in the subspace iteration method

Bathe's basic algorithm of subspace iteration for the solution of the symmetric eigenvalue problem is improved by including a Chebyshev filtering mechanism. To obtain satisfactory convergence for the largest eigenvalues, a shifting strategy is adopted. The shift factor is approximately computed by the Lanczos process.     

The accelerated power method

An algorithm is presented to accelerate the convergence of the inverse iteration method for the solution of algebraic symmetric eigensystems. The algorithm is based on the use of the Ritz analysis during inverse iteration to generate improved trial vectors at virtually no extra cost. Examples are shown to illustrate the computational advantages of the method. The results are compared with those obtained using the subspace iteration method, the determinant search method and the accelerated subspace iteration method.     

Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

When exact dynamic stiffness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally efficient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick–Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large‐scale structures. Copyright © 2003 John Wiley & Sons, Ltd.     

Comparison of the born iterative method and tarantola's method for an electromagnetic time-domain inverse problem

Two methods of solving the nonlinear two-dimensional electromagnetic inverse scattering problem in the time domain are considered. These are the Born iterative method and the method originally proposed by Tarantola for the seismic reflection inverse problems. The former is based on Born-type iterations on an integral equation, whereby at each iteration the problem is linearized, and its solution is found via a regularized optimization. The latter also uses an iterative method to solve the nonlinear system of equations. Although it linearizes the problem at each stage as well, no optimization is carried out at each iteration; rather the problem as a whole is posed as a (regularized) optimization. Each method is described briefly and its computational complexity is analyzed. Tarantola's method is shown to have a lower numerical complexity compared to the Born iterative method for each iteration, but in the examples considered, required more iterations to converge. Both methods perform well when inverting a smooth profile; however, the Born iterative method gave better results in resolving localized point scatterers.     

An accelerated iterative method for contact analysis

An iterative solution procedure for a frictionless contact problem is presented. Convergence to the exact solution is guaranteed by an error reduction method, and the rate of convergence is drastically improved by reduction of the ratio of extreme eigenvalues of the iteration matrix. For an ordinary linear equation, the present technique has the same theoretical value of convergence rate as the Chebyshev acceleration technique. The present acceleration technique can be directly extended for complicated frictional contact problems.     

Computer algorithms for calculating efficient initial vectors for subspace iteration method

In this paper, the effects of selecting initial vectors on computation efficiency for a subspace iteration method are investigated. Four algorithms are used for selecting the initial vectors. First, arbitrary starting iteration vectors are chosen according to Bathe and Wilson's algorithm.¹ In the other algorithms, the initial vectors are the retrieved eigenvectors from the Guyan and quadratic reduction methods. Improvement of the eigenvalue approximations of the subspace iteration method over reduction methods is presented. The computation effort is examined for the various algorithms used for initial iteration vectors.     

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。     

NUMERICAL ALGORITHMS FOR SOLUTIONS OF LARGE EIGENVALUE PROBLEMS IN PIEZOELECTRIC RESONATORS

Two algorithms for eigenvalue problems in piezoelectric finite element analyses are introduced. The first algorithm involves the use of Lanczos method with a new matrix storage scheme, while the second algorithm uses a Rayleigh quotient iteration scheme. In both solution methods, schemes are implemented to reduce storage requirements and solution time. Both solution methods also seek to preserve the sparsity structure of the stiffness matrix to realize major savings in memory. In the Lanczos method with the new storage scheme, the bandwidth of the stiffness matrix is optimized by mixing the electrical degree of freedom with the mechanical degrees of freedom. The unique structural pattern of the consistent mass matrix is exploited to reduce storage requirements. These major reductions in memory requirements for both the stiffness and mass matrices also provided large savings in computational time. In the Rayleigh quotient iteration method, an algorithm for generating good initial eigenpairs is employed to improve its overall convergence rate, and its convergence stability in the regions of closely spaced eigenvalues and repeated eigenvalues. The initial eigenvectors are obtained by interpolation from a coarse mesh. In order for this multi-mesh iterative method to be effective, an eigenvector of interest in the fine mesh must resemble an eigenvector in the coarse mesh. Hence, the method is effective for finding the set of eigenpairs in the low-frequency range, while the Lanczos method with a mixed electromechanical matrix can be used for any frequency range. Results of example problems are presented to show the savings in solution time and storage requirements of the proposed algorithms when compared with the existing algorithms in the literature.     

The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.