有限元模型修正中的最佳矩阵逼近

1引言 在飞行器、船舶、桥梁等结构设计中,要定量、准确地进行结构动力学分析,解决飞行器、船舶、桥梁等工程结构中普遍存在的振动问题,首先必须建立结构的动力学模型.     

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.     

Correction of stiffness and mass matrices utilizing simulated measured modal data

Measured and analytical data are unlikely to be equal due to measured noise, model inadequacies and 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, this study presents the equations to update the physical parameters of stiffness and mass matrices simultaneously for analytical modelling by minimizing a cost function in the satisfaction of the dynamic constraints of orthogonality requirement and eigenvalue function. The proposed equations are straightforwardly derived by Moore–Penrose inverse matrix without using any multipliers. The cost function is expressed by the sum of the quadratic forms of both the difference between analytical and updated mass, and stiffness matrices. The results are compared with the updated mass matrix to consider the orthogonality requirement only and the updated stiffness matrix to consider the eigenvalue function only, respectively. Also, they are compared with Wei’s method which updates the mass and stiffness matrices simultaneously. The validity of the proposed method is illustrated in an application to correct the mass and stiffness matrices due to section loss of some members in a simple truss structure.     

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.     

Fast simplicial finite element algorithms using Bernstein polynomials

Fast algorithms for applying finite element mass and stiffness operators to the B-form of polynomials over d-dimensional simplices are derived. These rely on special properties of the Bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor-product techniques in rectangular domains. First, special structure leading to fast application of mass matrices is developed. Then, by factoring stiffness matrices into products of sparse derivative matrices with mass matrices, fast algorithms are also obtained for stiffness matrices.     

A new updating method for the damped mass-spring systems

In this paper, we concern the inverse problem of constructing a monic quadratic pencil which possesses the prescribed partial eigendata, and the damping matrix and stiffness matrix are symmetric tridiagonal. Furthermore, the stiffness matrix is positive semi-definite and weakly diagonally dominant, which has positive diagonal elements and negative off-diagonal elements. Based on the solution of the inverse eigenvalue problem, we apply the alternating direction method with multiplier to solve the finite element model updating problem for the serially linked mass-spring system. The positive semi-definiteness of stiffness matrix, nonnegativity of stiffness and the physical connectivity of the original model are preserved. Numerical results show that our proposed method works well.     

Effects of a small length scale on vibrations of an embedded double-walled carbon nanotube

Extensive continuum analyses are carried out to estimate the influence of matrix stiffness, a small length scale, and intertubular radial displacements on free vibrations of an individual double-walled carbon nanotybe. The analyses are based on both local and classical Euler–Bernoulli and Timoshenko elasticity theories with concentricity and nonconcentricity assumptions. The effect of a small length scale is incorporated in the formulations. New intertubular resonant frequencies are calculated based on these theories. Detailed results are demonstrated for the resonant frequencies as functions of matrix stiffness and the small length scale. The results indicate that the internal radial displacement and the stiffness of the surrounding matrix can greatly affect the resonant frequencies, especially at higher frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. More over, at high frequencies and small aspect ratios, the effect of the small length scale be comes more significant.     

动态环板元的动刚度矩阵

本文对于环形薄板单元取包含贝塞尔函数的谐振变形作为形状函数,解决了关于特殊函数的复杂积分问题.从而精确推导了环形单元的动刚度矩阵,并用直接刚度法进行了校核.接着,又着重于将封闭形式的动刚度矩阵,按频率平方的升幂式展开,得到了简洁完备的结果,以此作为结构动力特性分析和响应计算的基础.     

The tangent stiffness matrix in rigid multibody vehicle dynamics

In the development of the equations of motion of a rigid multibody system, particularly vehicles, it is quite common to linearize the equations after they are derived, or even to ignore the non-linear terms from the outset. When doing so, the tangent stiffness matrix, i.e., the stiffness term that results from preload of the system rather than physical flexibility, is often ignored. The motion analysis of preloaded mechanical systems, e.g., the ride quality analysis of vehicle suspensions, may be significantly altered by this omission. Explicit expressions for the tangent stiffness matrix for a few of the common constraint types, including the revolute joint and the rolling wheel, are derived in this article. These expressions are coded into software and included in an open-source linear equation of motion generator for rigid multibody systems. A sample automotive suspension system is analysed, comparing the results with and without the tangent stiffness matrix effects; additionally, a benchmark solution is developed using a commercial multibody dynamics code. The results provide confirmation of the significance of the tangent stiffness effect on motion analysis and correlate well with non-linear transient solutions.     

Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems

Two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied. Both techniques are based on spectral equivalence approach. The first technique consists in an incomplete factorization of the separate displacement component part of the stiffness matrix. The second technique uses an incomplete factorization of the isotropic approximation to the stiffness matrix. Results concerning existence, stability and efficiency of these preconditioning techniques are presented. The efficiency and robustness of the described techniques are illustrated by numerical experiments.     

约束矩阵方程的中心对称解及其在振动理论反问题中的应用

研究了中心主子矩阵约束下矩阵方程的中心对称解. 利用矩阵向量化、Kronecker乘积及奇异值分解方法,得到了有解的充分必要条件及解的一般表达形式．同时,考虑了与之相关的对任意给定矩阵的最佳逼近问题．进而,给出在振动理论反问题中的应用, 利用截断的主质量矩阵（或主刚度矩阵）、截断模态矩阵以及质量矩阵（或刚度矩阵）的中心主子阵,求系统的质量矩阵（或刚度矩阵）．最后用两个例子说明文中方法的有效性.     

Updating undamped structural models using displacement output feedback

A novel numerical method for updating stiffness matrix by displacement output feedback technique is presented. The required displacement output feedback gain matrix is determined by the QR-decompositions and the singular value decompositions of matrices, and thus the optimal approximation stiffness matrix is found and the large number of unmeasured and unknown eigeninformation of the original model is preserved. The proposed method is computationally efficient since it does not require iterations and the updated stiffness matrix is also symmetric and has the compact expression.     

Compatible wavelet-Galerkin method to solve stokes interior problem with circular boundary

This article addresses Neumann boundary value interior problem of Stokes equations with circular boundary. By using natural boundary element method, the Stokes interior problem is reduced into an equivalent natural integral equation with a hyper-singular kernel, which is viewed as Hadamard finite part. Based on trigonometric wavelet functions, the compatible wavelet space is constructed so that it can serve as Galerkin trial function space. In proposed compatible wavelet-Galerkin method, the simple and accurate computational formulae of the entries in stiffness matrix are obtained by singularity removal technique. It is also proved that the stiffness matrix is almost a block diagonal matrix, and its diagonal sub-blocks all are both symmetric and circulant submatrices. These good properties indicate that a 2 ^J+3 × 2 ^J+3 stiffness matrix can be determined only by its 2 ^J + 3J + 1 entries. It greatly decreases the computational complexity. Some error estimates for the compatible wavelet-Galerkin projection solutions are established. Finally, several numerical examples are given to demonstrate the validity of the proposed approach.     

Wavelet Collocation Methods for a First Kind Boundary Integral Equation in Acoustic Scattering

In this paper we consider a wavelet algorithm for the piecewise constant collocation method applied to the boundary element solution of a first kind integral equation arising in acoustic scattering. The conventional stiffness matrix is transformed into the corresponding matrix with respect to wavelet bases, and it is approximated by a compressed matrix. Finally, the stiffness matrix is multiplied by diagonal preconditioners such that the resulting matrix of the system of linear equations is well conditioned and sparse. Using this matrix, the boundary integral equation can be solved effectively.     

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.     

Estimating the manipulation errors in finite element analysis, Part I

In Part I of this two-part work, the relative errors in representing and processing real numbers in digital computers are reviewed, and the computation of the digits lost from the relative errors is shown. The upper bound estimate of the relative error in the solution of Kξ = p for the displacements ξ is given as a function of the condition number of the stiffness matrix K, and the relative errors in K and the load vector p. The computation of relative errors for p and K by equilibrium checks is outlined, and various estimates for the condition number of the stiffness matrix are given.     

有限元结构分析在MIMD计算机上的并行直接解法

§1.前言 并行计算是近十几年来随着并行计算机发展而发展起来的一门新兴学科,特别是对于多指令流多数据流(MIMD)的并行计算机,由于它是由多台普通计算机甚至是向量计算机相互以一定方式联结起来的新型计算机系统,因此无论是它的运算速度或存贮空     

Numerical solution of hypersingular equation using recursive wavelet on invariant set

In this paper, we construct the Chebyshev recursive wavelets on a unit interval of the first kind, the second kind and their corresponding weight functions. We apply wavelet collocation method to solve the natural boundary integral equation of the harmonic equation on the lower half-plane numerically. It is convenient and accurate to generate the stiffness matrix. Two numerical examples are presented. It is shown that the stiffness matrix is highly sparse when the order of the stiffness matrix becomes large. Current method allows choosing an appropriate weight function to increase the convergence rate and accuracy of the numerical results.     

Transient response of a transversely isotropic multilayered half-space due to a vertical loading

This paper investigates the transient response of a transversely isotropic multilayered half-space under vertical loadings. With the aid of a Laplace–Hankel transform, the global stiffness matrix for a multilayered half-space is acquired by assembling the analytical layer-element of each layer medium. The solutions for the displacements in the time domain are obtained by using the global stiffness matrix equations and a numerical inversion procedure. The accuracy of the proposed method is verified through comparisons with existing solutions for displacements induced by a step and rectangular pulse loading. In addition, selected numerical results for displacements induced by the buried loading are presented to illustrate the effect of transient loading type and material anisotropy on the transient response.     

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.