Update of corrected stiffness and mass matrices based on measured dynamic modal data

The physical parameters obtained from modal tests do not satisfy the dynamic constraints of eigenvalue function and orthogonality requirements due to modeling and measurement errors. The purpose of this study is to present the analytical equations on the updated stiffness and mass matrices in the satisfaction of such dynamic constraints. Minimizing the cost functions of the difference between analytical and desired physical parameter matrices, the corrected parameter matrices are straightforwardly derived by utilizing the Moore–Penrose inverse without using any multipliers. The cost functions given by a few researchers are utilized. From the comparison of the existing analytical results and the proposed equations, the validity of the proposed methods is evaluated in an application.     

Estimation of parameter matrices based on measured data

Finite element structural updating based on measured data may inherent significant errors due to uncertainties in the updated physical parameter matrices. This study presents analytical equations to estimate the change in the physical parameter matrices based on the measured modal data of dynamic systems and the measured displacement data of static systems. The equations for the parameter estimation are derived by minimizing cost functions in the satisfaction of the eigenvalue equation, the mode shape orthogonality requirements for the dynamic system, and the satisfaction of the measured displacement data for the static systems. The proposed method utilizes the Moore–Penrose inverse for the inverse of the rectangular matrices without using Lagrange multipliers. Comparing the analytical results with Berman & Nagy’s method and Yang & Chen’s method, this study demonstrates that the derived equations take simpler forms and produce more accurate results. The proposed method can be widely utilized in predicting static or dynamic parameter matrices for the design and analysis of any structure.     

New symmetry preserving method for optimal correction of damping and stiffness matrices using measured modes

Measured and analytical data are unlikely to be equal due to measured noise, model inadequacies, structural damage, etc. It is necessary to update the physical parameters of analytical models for proper simulation and design studies. Starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes, a new computationally efficient and symmetry preserving method and associated theories are presented in this paper to update the physical parameters of damping and stiffness matrices simultaneously for analytical modeling. A conjecture which is proposed in [Y.X. Yuan, H. Dai, A generalized inverse eigenvalue problem in structural dynamic model updating, J. Comput. Appl. Math. 226 (2009) 42-49] is solved. Two numerical examples are presented to show the efficiency and reliability of the proposed method. It is more important that, some numerical stability analysis on the model updating problem is given combining with numerical experiments.     

Eigenvalues of rank-one updated matrices with some applications

We prove a spectral perturbation theorem for rank-one updated matrices of special structure. Two applications of the result are given to illustrate the usefulness of the theorem. One is for the spectrum of the Google matrix and the other is for the algebraic simplicity of the maximal eigenvalue of a positive matrix.     

An iterative method for updating gyroscopic systems based on measured modal data

An efficient iterative method for updating the mass, gyroscopic and stiffness matrices simultaneously using a few of complex measured modal data is developed. By using the proposed iterative method, the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices. Numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters.     

矩阵最小二乘问题的迭代求解及其在机器翻译中的应用

研究一类线性矩阵方程最小二乘问题的迭代法求解,利用目标函数与矩阵迹之间的关系构造了矩阵形式的"梯度"下降法迭代格式,推广了向量形式的经典"梯度"下降法,并引入了两个矩阵之间的弱正交性来刻画迭代修正量的特点.作为本文算法的应用,给出了机器翻译优化问题的一种迭代求解格式.     

Scaled memoryless symmetric rank one method for large-scale optimization

This paper concerns the memoryless quasi-Newton method, that is precisely the quasi-Newton method for which the approximation to the inverse of Hessian, at each step, is updated from the identity matrix. Hence its search direction can be computed without the storage of matrices. In this paper, a scaled memoryless symmetric rank one (SR1) method for solving large-scale unconstrained optimization problems is developed. The basic idea is to incorporate the SR1 update within the framework of the memoryless quasi-Newton method. However, it is well-known that the SR1 update may not preserve positive definiteness even when updated from a positive definite matrix. Therefore we propose the memoryless SR1 method, which is updated from a positive scaled of the identity, where the scaling factor is derived in such a way that positive definiteness of the updating matrices are preserved and at the same time improves the condition of the scaled memoryless SR1 update. Under very mild conditions it is shown that, for strictly convex objective functions, the method is globally convergent with a linear rate of convergence. Numerical results show that the optimally scaled memoryless SR1 method is very encouraging.     

增长曲线模型中一致最小风险无偏估计的存在性

考虑协方差阵任意，或具有均匀协方差结构，或具有序列协方差结构的正态增长曲线模型本文将文[19]在设计矩阵满秩，且仅估计回归系数矩阵的情形获得的结果推广到设计矩阵不必列满秩，且同时估计回归系数矩阵的线性可估函数和协方差阵（或有关参数）的情形；在凸损失函数类和矩阵损失函数下，给出存在一致最小风险无偏估计的充分必要条件．     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.     

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive‐definite matrices with the mass matrix being diagonal and the damping and stiffness matrices being tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newton‐type algorithm for the optimization problem, which improves a pre‐existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method. Copyright © 2008 John Wiley & Sons, Ltd.     

On the convergence of nonstationary column-oriented version of algebraic iterative methods

Recently, Elfving, Hansen, and Nikazad introduced a successful nonstationary block-column iterative method for solving linear system of equations based on flagging idea (called BCI-F). Their numerical tests show that the column-action method provides a basis for saving computational work using flagging technique in BCI algorithm. However, they did not present a general convergence analysis. In this paper, we give a convergence analysis of BCI-F. Furthermore, we consider a fully flexible version of block-column iterative method (FBCI), in which the relaxation parameters and weight matrices can be updated in each iteration and the column partitioning of coefficient matrix is allowed to update in each cycle. We also provide the convergence analysis of algorithm FBCI under mild conditions.     

On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix–vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension.     

Damage detection approach based on the second derivative of flexibility estimated from incomplete mode shape data

The presence of damage in a structural member causes local and global deterioration of structural performance. The stiffness and flexibility can be predicted using the measured mode shapes. The stiffness and flexibility are estimated because it is not easy to collect the complete modal data corresponding to full modes and degrees of freedom (DOFs). The prediction using the lowest few modes provides a more accurate flexibility prediction compared with the stiffness. This work indicates that the updated flexibility matrix can extract more information concerning the state of the structural health compared with the stiffness matrix. Additionally, incomplete measurement data should be expanded to construct the flexibility matrix at damaged state. This study derives the analytical methods used to update the flexibility matrix based on an expanded full set of DOFs and to detect damage using the updated flexibility curvature. The validity of the proposed method is evaluated using a numerical experiment, and the method's effectiveness depending on the number of truncated modes is investigated.     

Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.     

关于反中心对称矩阵的某些性质探讨

利用反中心对称矩阵的定义以及翻转矩阵等技巧,给出了反中心对称矩阵的伴随矩阵、特征值及特征向量的一些新结论.     

On perturbations of matrix pencils with real spectra, a revisit

This paper continues earlier studies by Bhatia and Li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra. A unifying framework for creating crucial perturbation equations is developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained.

     

一类特征值反问题(IEP)的基于矩阵方程的Ulm型算法

应用求解算子方程的Ulm方法构造了求解一类矩阵特征值反问题（IEP）的新算法.所给算法避免了文献[Aishima K.，A quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems，Linear Algebra and its Applications，2018，542：310-33]中算法在每次迭代中要求解一个线性方程组的不足，证明了在给定谱数据互不相同的条件下所给算法具有根收敛意义下的二次收敛性.数值实验表明本文所给算法在矩阵阶数较大时计算效果优于上文所给算法.     

Differential equations over subalgebras of the full matrix algebra

We consider a general matrix differential equation whose parameters and solution are defined over subalgebras of the full matrix algebra. Some subalgebras are presented over which the equation splits into a set of equations over matrices of smaller orders. Multiplicative matrix equations of higher order in normal form are introduced and studied over some subalgebras of 2 × 2 matrices.     

Quasi‐optimal preconditioners for finite element approximations of diffusion dominated convection–diffusion equations on (nearly) equilateral triangle meshes

The paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a finite element approximation to diffusion‐dominated convection–diffusion equations. We consider a model setting in which the structured finite element partition is made by equilateral triangles. Under such assumptions, if the problem is coercive and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequences exhibit a strong eigenvalue clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and under the constant coefficients assumption, the eigenvector matrices have a mild conditioning. The obtained results allow to prove the conjugate gradient optimality and the generalized minimal residual quasi‐optimality in the case of structured uniform meshes. The interest of such a study relies on the observation that automatic grid generators tend to construct equilateral triangles when the mesh is fine enough. Numerical tests, both on the model setting and in the non‐structured case, show the effectiveness of the proposal and the correctness of the theoretical findings. Copyright © 2014 John Wiley & Sons, Ltd.