Rigid body considerations for non-linear finite element analysis

The purpose of the paper is to demonstrate how the concept of rigid body motions can be employed to derive the external stiffness matrix for an initially stressed finite element. Such a matrix is as important as the elastic and geometric stiffness matrices. It can be used not only in an eigenvalue analysis for testing the zero energy modes of a finite element under initial loadings, but for calculating the element forces in a step-by-step non-linear analysis. The two-dimensional beam element presented in this paper serves as a vehicle to demonstrate the concept involved. The principle of virtual work in its updated Lagrangian form has been adopted as the method of formulation. Several examples are provided to illustrate the adequacy of the present approach.     

Scaled corrector and branch-switching in necking problems

A new branch-switching procedure for necking problems by finite element method is proposed. The scaled corrector method adopted in this paper for computing the critical eigenvector of the tangent stiffness matrix, which is indispensable to bifurcation analysis, is quite simple and effective for large scale problems because eigenvalue analysis is not required. The numerical background of the approximate critical eigenvector calculated during equilibrium iterations and the branch-switching procedure in elasto-plastic bifurcation problems are described in detail. As numerical examples, necking bifurcation problems of plane strain/stress states in finite strains are demonstrated to validate the proposed branch-switching procedure.     

用不完全模态测量数据修正有限元分析模型

提出一种基于不完全模态测量数据同时修正有限元质量矩阵与刚度矩阵的有效数值方法。运用代数特征值反问题的理论与方法，得到了满足正交关系及特征方程的最逼近有限元质量矩阵及刚度矩阵的唯一的修正质量矩阵与刚度矩阵（最优修正矩阵）。该方法有一个简洁的表达式，修正过程简单而且容易实现。数值算例表明，修正模型与模态试验数据具有非常好的一致性。     

A lanczos-based technique for exact vibration analysis of skeletal structures

This paper presents and discusses a Lanczos-based eigensolution technique for evaluating the natural frequencies and modes from frequency-dependent eigenproblems in structural dynamics. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency-dependent displacement fields in formulating the system matrices. The method is well suited to the solution of large-scale problems. The new solution methodology presented here is based on the ability to evaluate a specific set of parameterized non-linear eigenvalue curves at given values of the parameter through an implicitly restarted Lanczos technique. Numerical examples illustrate that the implicitly restarted Lanczos method with secant interpolation accurately evaluates the exact natural frequencies and modes of the non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models. In addition, the eigenvalue technique presented here is shown to be far more computationally efficient on large-scale problems than the determinant search techniques traditionally employed for solving exact vibration problems.     

模拟P波向无穷远域传播的一致边界

将人工边界设置在半无穷层单元和内部有限元区域的交界面上,建立了半无穷层单元的刚度矩阵后,得到了边界节点的动力平衡方程。任意给定激励圆频率,将边界节点系统的动力平衡方程转化为特征值方程。求解特征值方程得出边界节点系统的特征值和特征模态,利用模态叠加原理得到体现左半无穷层单元和右半无穷层单元对内部有限元区域作用的边界矩阵,这就是该文的一致边界。将其与内部有限元区域的刚度矩阵进行组装来模拟无穷远域介质对波的传播作用。最后用数值算例来说明一致边界的精确性和可行性。     

Design parameter variation and structural response

A general power series method is developed to aid in obtaining changes in the response variable due to changes in design parameters. The approach is considered for eigenvalue problems and systems of simultaneous equations which may represent static, dynamic and stability response problems. Particular emphasis is placed upon the eigenvalue problem and techniques for improving the calculations form the power series are suggested utilizing modified Rayleigh quotient expressions. Each of the methods proposed is treated computationally by the formulation of a finite element solution of column elastic stability. Investigations of solution accuracy versus percentage variation in element stiffness, whether the method underestimates or overestimates the changes, and the required number of eigenvalue solutions are presented.     

基于波有限元法的流固耦合结构波传导问题

本文采用波有限元方法研究流固耦合结构中的波传导问题。该方法以有限元法为基础,首先建立研究对象的有限元法模型,得到模型的动态刚度矩阵。通过对动态刚度矩阵的重新排列组合得到研究对象的传递矩阵,求解传递矩阵的特征值问题可以得到分别代表自由波传递的波数和波模。本研究首先分析独立流体结构和固体结构中的振动问题,并比较了采用波有限元法和理论方法求解得到的固体结构中波数分布情况,验证了模型的正确性。随后采用波有限元法分析流固耦合结构中的波传导问题。波有限元法的应用并不局限于本文所给出的均匀或周期性结构,还可将其应用于缓慢变化的非均匀结构。     

Large displacement analysis of thin-walled beams with generic open section

A finite element formulation for geometric non-linear analysis of elastic straight thin-walled beams is presented. The updated Lagrangian procedure is used for a stiffness matrix derivation-to analyse large displacements or buckling behaviour of space frames. The stiffness matrix is developed for a generic cross-section and is equally efficient for symmetric, monosymmetric or asymmetric open sections. Some non-linear strain terms usually neglected in the literature have been considered, resulting in a formulation which also includes the initial bimoment and torsion effect for all cross-sections analysed. The proposed method has been implemented and the results of sample problems are presented.     

A new finite element scheme for instability analysis of thin shells

This paper describes a new finite element scheme for the analysis of instability phenomena of arbitrary thin shells. A computationally efficient procedure is proposed for calculating the non-linear stiffness and tangential stiffness matrices for a doubly-curved quadrilateral element defined by co-ordinate lines. The essential feature is the explicit addition of the non-linear terms into the rigid-body motion of the element. Thus the non-linear and tangential element stiffness matrices can easily be generated by transforming the generalized element stiffness matrix for linear analysis, and the non-linear terms of these matrices are separated into a number of component terms multiplied by the rigid-body rotations. These component terms can be stored permanently and used to calculate efficiently the non-linear and tangential stiffness matrices at each iteration. Illustrative examples are presented which confirm the validity of the present approach in the analysis of instability phenomena of thin plates and shells.     

一种利用模态测量数据修正刚度矩阵的新方法

模型修正即为利用模态测量数据修正存在但不准确的有限元模型。本文在假定有限元模型的质量矩阵与刚度矩阵均为对称非负定矩阵,并且质量矩阵是精确的情况下,提出了一种修正刚度矩阵的新方法。该方法借助于矩阵的Kronecker积与拉直算子,把需修正的变量分离出来直接对其进行修正运算,得到了满足特征方程与正交性条件的最逼近有限元刚度矩阵的唯一修正矩阵。该方法不仅保证了修正矩阵带状稀疏的特点,而且修正过程简单易行。数值例子验证了该方法的有效性。     

Beam element matrices derived from Vlasov's theory of open thin-walled elastic beams

A uniform beam element of open thin-walled cross-section is studied under stationary harmonic end excitation. An exact dynamic (transcendentally frequency-dependent) 14 × 14 element stiffness matrix is derived from Vlasov's coupled differential equations. Special attention is paid to the computational problems arising when coefficients vanish in these equations because of symmetric cross-section, zero warping stiffness, etc. The dynamic element stiffness matrix is established via a generalized linear eigenvalue problem and a system of linear algebraic equations with complex matrices. A static stiffness matrix is also derived and the associated consistent mass and geometric stiffness matrices are given. Modal masses are evaluated. A FORTRAN program and a numerical example are included.     

计及二阶效应的梁杆系统动力响应分析方法研究

基于动力刚度法和有限元理论提出了一种考虑二阶效应计算梁杆动力响应的新方法。通过求解轴向力作用下Bernoulli-Euler 梁横向和轴向挠度自由振动微分方程,利用位移边界条件反解出待定系数,得到了动态精确形函数;使用经典有限元方法推导了考虑截面自身旋转惯量的质量阵和考虑二阶效应的刚度阵,该质量阵和刚度阵各元素均为轴力和圆频率的超越函数;建立了杆系结构瞬态动力学分析的动力平衡方程,给出了稳定和高效的求解方案。对几个典型的算例进行了计算分析,并与通用软件ANSYS 的计算结果进行了比较。计算结果表明:该分析梁杆系统动力响应的新方法具有较高的计算精度和效率,特别是能够准确地计入轴力对于梁杆动力响应的影响。     

力-变位关系全过程模拟的有限元位移控制新方法

材料、构件及结构的力-变位关系非线性全过程曲线有限元数值模拟中,极限强度和其后的软化下降段模拟一直是未得到较好解决的难题。位移增量控制法可以方便跨越力-变位关系中的极值点,因此它常被用于求解材料、构件及结构包含极限强度后软化下降段的力-变位关系非线性全过程曲线。但是,传统的位移增量控制法需要重新排列有限元方程的刚度矩阵,并存在求解非对称和非带状系数方程的问题,因而限制了其推广应用。该文提出了求解材料、构件及结构的力-变位关系非线性全过程曲线的一种新的位移增量控制方法,该方法通过修改刚度矩阵中相关对角系数的方式,将边界和力作用点的控制位移条件隐含到有限元方程中,从而可以采用有限元荷载控制方法同样的方式进行求解,保证了有限元方程系数矩阵在求解过程中的对称性和带状性,并可方便地在现有通用商业有限元软件中实现。算例分析表明了该文方法的有效性。     

Formulation and solution of the non-linear,damped eigenvalue problem for skeletal systems

This paper presents and discusses an Arnoldi-based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices. This modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems. The eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non-linear eigenvalue problem at given values of the parameter. Numerical examples illustrate that this method, used in conjunction with a secant interpolation, accurately evaluates the complex natural frequencies and modes of the quadratic non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models.     

On development of a finite dynamic element and solution of associated eigenproblem by a block Lanczos procedure

The paper first presents the details of the development of a new six-noded plane triangular finite dynamic element. A block Lanczos algorithm is developed next for the accurate and efficient solution of the quadratic matrix eigenvalue problem associated with the finite dynamic element formulation. The resulting computer program fully exploits matrix sparsity inherent in such a discretization and proves to be most efficient for the extraction of the usually required first few roots and vectors, including repeated ones. Most importantly, the present eigenproblem solution effort is shown to be comparable to that of the corresponding finite element analysis, thereby rendering the associated dynamic element method rather attractive owing to superior convergence characteristics of such elements, presented herein.     

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.     

A solid-shell layerwise finite element for non-linear geometric and material analysis

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios usually employs plate or facet-shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work an alternative approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This eight-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to solve several locking pathologies coming from the high aspect ratio of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, based on the finite element variables transformation matrix. The non-linear geometric and material capabilities are introduced into the finite element formulation, allowing for the representation of large displacements, large deformation and material non-linear behaviors. The developed formulation is numerically tested and benchmarked, being validated by using published experimental results obtained from sandwich specimens.     

动态刚度阵法的研究概况

动态刚度阵法被广泛的应用于工程结构振动分析中，尤其是在需要获得更高阶频率和更高精度的振动问题时。因为它不像传统的有限元方法和其它的近似方法，该方法通过极少的自由度就能较精确地计算出无数个固有频率和固有振型，所以也被称为精确方法。该方法的所有假设仅来自于建立该单元运动微分方程的经典理论，所形成的动态刚度阵是固有频率的超越函数，解决这类超越特征值问题的有效方法是Williams—Wittrick算法。     

MULTILEVEL AGGREGATION METHOD FOR SOLVING LARGE-SCALE GENERALIZED EIGENVALUE PROBLEMS IN STRUCTURAL DYNAMICS

In this paper a novel iterative method of multilevel type for solving large-scale generalized eigenvalue problems encountered in structural dynamics is presented. A preconditioned iterative technique, which can be viewed as a modification of the Subspace Iteration method, is used for simultaneous calculation of a group of lowest modes and frequencies. The paper demonstrates that a coarse aggregation model can be employed in the hierarchical structure of the preconditioner in order to provide a good resemblance of the latter to the stiffness matrix of the finite element approximation with respect to low-frequency modes. This leads to a fast convergent procedure of subspace iterations. As opposed to the coarse grid used in methods of multigrid type, this model allows for solving problems with different finite elements including reticulated structures in the framework of large comprehensive finite element software systems. Numerical experiments performed for three-dimensional truss, frame and solid structures demonstrate an excellent performance of the method. © 1997 by John Wiley & Sons, Ltd.     

Finite element method applied to the supersonic flutter of circular cylindrical shells

The application of the finite element method to the supersonic flutter of circular cylindrical shells subjected to internal pressure and axial compression is presented. A circular cylindrical shell element is used. The element stiffness, mass and initial stiffness matrices are given. The element aerodynamic matrix is derived based on a first order high Mach number approximation to the linear potential flow theory. The eigenvalue problem is solved by the QR algorithm. Numerical results are presented and these are compared with analytical solutions and experimental data.