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     

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。     

一种修正的电阻层析成像Landweber迭代算法

研究了灵敏度矩阵更新的Landweber迭代图像重建算法,以期提高重建图像精度。灵敏度矩阵更新时的初始图像由Landweber迭代法获得,对不同迭代次数的灵敏度矩阵更新间隔进行了比较,并且对灵敏度矩阵的更新次数进行了分析,仿真及实验结果表明,该方法能有效提高图像重建精度。     

非亏损动力学系统特征对导数的计算

提出了阻尼动力系统规范化特征向量的一个策略，分别给出了计算该系统单特征对和重特征对导数的方法。提出的方法不要求系统矩阵的对称性，直接由阻尼动力系统的特征值和特征向量计算特征对导数，从而减少了计算量。     

非保守系统特征灵敏度分析直接法

当结构动力学系统的阻尼矩阵不能同时通过质量和刚度矩阵对角化时,线性振动系统的特征值问题就转化为二次特征值方程,相应的特征值和特征值向量以及它们的导数都成为复空间内的量.针对非保守系统的二次特征值问题,通过求解非齐次线性方程组,直接导出非保守系统特征值和特征向量的一阶灵敏度.提出的方法不需要非保守系统的全部模态,因此,适用于大型复杂结构的特征灵敏度分析.     

压电结构系统辨识中的幂迭代子空间跟踪方法

针对子空间辨识压电结构系统模型,利用广义能观矩阵列空间与观测矢量相关矩阵信号子空间一致的特征,提出采用幂迭代子空间跟踪方法,保证全局且按指数收敛到主子空间。为避免跟踪过程中与主子空间偏离误差的传递,采用多级分解,形成多级幂迭代子空间跟踪方法。将其与投影逼近子空间方法比较,仿真结果表明,能观矩阵夹角小,输出均方根误差小,能提高跟踪精度。并运用于实际模型中,验证其有效性。     

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。     

计算特征向量灵敏度的Neumann级数展开法

在特征向量灵敏度模态法的基础上,提出,了计算特征向量灵敏度的Ｎｅｕｍａｎｎ级数展开法。该方法是胜矩阵组数的形式来等效未知模态对特征向量灵敏度的贡献,并证明它对低阶模态的特征向量灵敏度分析具有较高的计算效率。选用工程桁架结构作为算例说明本方法在计算特征向量灵敏度时,只需少数几阶模态,便能获得高精度的特征向量灵敏度。     

USSOR迭代法应用于相容次序矩阵的研究

本文主要研究 USSOR迭代法对于系数矩阵为相容次序矩阵的线性方程组的应用。在 Jacobi迭代矩阵的特征值是实数或纯度虚数这两种情况下 ,分别讨论 USSOR迭代法敛性及最优收敛性质 ,并且给出USSOR迭代矩阵谱半径的界     

一种有限元模型坐标动力缩聚技术

本文从修改后的系统特征方程出发,导出了反映系统主、副自由度上变形关系的动力缩聚矩阵的控制方程,并给出了求解该方程的迭代解法。与已有的动力缩聚迭代法相比,该方法的收敛速度最高。本文还提出了一种迭代收敛准则,该准则可大大减少计算效率很低的Rayleigh-Ritz操作的使用,从而使得该方法的迭代效率较高。     

基于分段悬链线理论的悬索分析矩阵迭代法

通过分析现有悬索计算的主要方法,在分段悬链线解析计算方法的基础上,结合求解非线性方程组的牛顿迭代法,提出了适用于悬索分析的矩阵迭代方法.根据解析推导给出了悬索矩阵迭代法的切线刚度矩阵构造方法,分析了刚度矩阵的特点,针对特定的工程问题列出了悬索找形的迭代计算流程.该方法计算量小,效率高,计算精度能够满足实际工程需要.通过与其他文献的数值计算结果进行对比,验证了所研究方法的可靠性.该方法适用于悬索桥、货运索道等各类悬索结构的分析计算.     

A Block Diagonal Preconditioner for Generalised Saddle Point Problems

A lopsided alternating direction iteration (LADI) method and an induced block diagonal preconditioner for solving block two-by-two generalised saddle point problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective.     

An Efficient Eigen-Solver and Some of Its Applications

This paper is mainly devoted to improve the efficiency of the subspace iteration strategy and propose practical new applications for the suggested scheme. In spite of utilizing subspace iteration method for many structures, it has not been mixed with the Sherman–Morrison–Woodbury process for problems with varying stiffness, yet. The authors' technique tries to achieve this goal. To corroborate the efficiency and capability of the present formulations, several numerical tests are performed. In order to assess the effects of various parameters on the eigenpairs of the problems, the required sensitivity analysis is also carried out.     

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。     

Algorithms for the calculation of limit and bifurcation points using Lanczos methods

The computation of limit and bifurcation points in structural mechanics using iterative preconditioned Lanczos solvers is studied. Contrary to classical implementations of algorithms for the calculation of limit and bifurcation points, which depend in general strongly on observing the diagonal elements of the decomposed matrix – obtained by a Gauß- or Cholesky decomposition – , we use an approach of determining limit and bifurcation points by examination of the subspace spanned by the iteration vectors of the Lanczos solver. Using a multilevel preconditioning with a coarse grid solver may result in a non positive definite preconditioning matrix if the coarse grid matrix is not positive definite in the post-critical solution branch. In that case the iteration has to be performed in the complex vector space. We prove by mathematical induction that all vectors and scalars are either purely real or purely imaginary. Therefore the generalized computation can be performed with about the same number of operations as in the case of a positive definite preconditioning matrix.     

复阻尼结构动力方程的增维精细积分法

为了避开求解复阻尼结构强迫激励动力学方程的积分运算,引入增维精细积分方法。根据复阻尼系统复化对偶原则,将动力学方程和激励波对偶复化为实部和虚部的形式,推导出增维矩阵的精细积分求解过程。结果表明,由于不用求解迭代矩阵H的逆矩阵,避免了矩阵奇异带来的计算解的不稳定性。在计算矩阵仅增加一维的情况下,化积分运算为代数运算,扩大了精细积分法的应用范围。通过对比增维精细积分法和频域法计算结果,二者结果保持较高的一致性。     

A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.