Reduction method for the non-linear analysis of symmetric anisotropic panels

Reduction method and computational procedures are presented for reducing the size of the analysis model and the number of degrees of freedom used in predicting the non-linear response of symmetric anisotropic panels. The two key elements of the method are (a) operator splitting, or decomposition of the characteristic arrays of the finite element model into sums of orthotropic and non-orthotropic contributions, (b) application of a reduction method through the successive use of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate a small number of global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the classical Rayleigh-Ritz technique. The global approximation vectors are selected to be those commonly used in single (or multiple) parameter perturbation techniques, namely a non-linear solution corresponding to zero non-orthotropic arrays and a number of its derivatives with respect to an anisotropic tracing parameter (and possibly, to a load or arc-length parameter in the solution space). The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure. The effectiveness of the proposed reduction method is demonstrated by means of a numerical example, and its potential for solving quasi-symmetric non-linear problems of anisotropic panels is discussed.     

Nonlinear analysis via global–local mixed finite element approach

A computational algorithm, based on the combined use of mixed finite elements and classical Rayleigh–Ritz approximation, is presented for predicting the nonlinear static response of structures; The fundamental unknowns consist of nodal displacements and forces (or stresses) and the governing nonlinear finite element equations consist of both the constitutive relations and equilibrium equations of the discretized structure. The vector of nodal displacements and forces (or stresses) is expressed as a linear combination of a small number of global approximation functions (or basis vectors), and a Rayleigh–Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The global approximation functions (or basis vectors) are chosen to be those commonly used in static perturbation technique; namely a nonlinear solution and a number of its path derivatives. These global functions are generated by using the finite element equations of the discretized structure. The potential of the global–local mixed approach and its advantages over global–local displacement finite element methods are discussed. Also, the high accuracy and effectiveness of the proposed approach are demonstrated by means of numerical examples.     

Calculations of plate frequencies from complementary energy formulation

The complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self-equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non-zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.     

Incompressible and locking-free finite elements from Rayleigh mode vectors: quadratic polynomial displacement fields

Under pure bending, with an arbitrary patch of plane four-node finite elements, the exact analytical algebraic expressions of deformation, strain and stress fields are numerically captured by a computer algebra program for both compressible and incompressible continua. Linear combinations of Rayleigh displacement vectors yield the Ritz test functions. These coupled fields model pure bending of an Euler-Bernoulli beam with appropriate linearly varying axial strains devoid of shear. Such Courant admissible functions allow an undeformed straight side to curve in flexure. Since these displacement vectors satisfy equilibrium conditions, they are necessarily functions of the Poisson’s ratio. Applications in bio-, micro- and nano-mechanics motivated this formulation that blurs the frontier between the finite and the boundary element methods. Exact integration yields the element stiffness matrix of a compressible convex or concave quadrilateral, or a triangular element with a side node. For the generic energy density integral, the paper furnishes an analytical expression that can be incorporated in Fortran or C ⁺⁺. In isochoric plane strain problems, the Rayleigh kinematic mode of dilatation is replaced by a constant element pressure. The equivalent nodal loadings are calculated according to the Ritz variational statement. Subsequently, without assembling the global stiffness matrix, nodal compatibility and equilibrium equations are solved in terms of Rayleigh modal participation factors.     

大跨圆拱屋盖结构的风致响应分析

大跨屋盖特征值问题的求解是结构动力响应分析中最繁琐的一个环节，而且一些对结构响应贡献较大的高阶模态容易在传统的模态叠加法中被忽略。本文以典型的大跨圆拱屋盖为例，将里兹向量直接叠加法应用于屋盖系统特征值问题计算和风致响应分析，其特点是在误差逼近的基础上自动生成一组正交的里兹向量并用于缩减系统自由度数。与传统模态叠加法算得的结果相比，里兹向量直接叠加法只用很少数目的向量就可以得到较精确的结果，而且高阶模态的贡献不会被忽略。该方法不仅大幅度地减少了机时，而且提供了动力分析的误差估计。     

Hydroelastic vibrations of flexible rectangular tanks partially filled with liquid

In this paper, the three-dimensional vibratory characteristics of flexible rectangular tanks partially filled with liquid are studied. The surface waves of the liquid are taken into account in the analysis. Both the bulging modes of the tank-wall vibration and the sloshing modes of the liquid oscillation are investigated. The vibrating modes of the liquid–tank system are divided into four distinct categories: double symmetric modes (SS); antisymmetric–symmetric modes (AS); symmetric–antisymmetric modes (SA) and double antisymmetric modes (AA). Each of these categories is separately investigated. The velocity potential of the liquid is analytically deduced by using a combination of the superposition method and the method of separation of variables. According to the liquid–tank interface conditions and the orthogonality of trigonometric functions, the coefficients in the solution of liquid velocity potential are expressed in the integral forms including the tank–wall dynamic deflection. A set of reasonable static beam functions is constructed as the admissible functions of the tank-wall vibration. The eigenfrequency equation of the liquid–tank system is derived by using a combination of the Rayleigh–Ritz method and the Galerkin method. Convergence study demonstrates the high accuracy and small computational cost of the proposed approach. Finally, some numerical results are presented for the first time. Copyright © 2006 John Wiley & Sons, Ltd.     

Eigenproperties of large-scale structures by finite element partitioning and homotopy continuation

Finite element partitioning (or substructuring) is employed to estimate the eigenproperties of large-scale structural systems. A homotopy equation is constructed and its solutions are characterized by a number of curves which connect the eigensolutions of the partitions with those of the complete system. A step-by-step tracing procedure is developed to follow these curves. At each step, prediction and correction are performed. The Rayleigh–Ritz procedure and the conjugate gradient method are used as predictor and corrector, respectively. Compared with the sole use of either the Rayleigh–Ritz or gradient methods, the proposed method is more reliable and more efficient for large-scale problems. Numerical implementation is well suited for supercomputers.     

Isoparametric,finite element,variational solution of integral equations for three-dimensional fields

An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.     

Study of the effect of in-plane and interlaminar stresses on damping in fluid-filled laminated composite cylinders

The modal strain energy method is used in conjunction with a three-dimensional finite element analysis in the characterization of the effects of in-plane and interlaminar stresses in fluid-filled composite laminate cylindrical shells. A semi-analytical, 8-noded isoparametric finite element, which includes both the symmetric and antisymmetric modes in the circumferential direction, is used in the analysis. The effects of fiber angle, contained fluid height, size parameter of the shell, and stacking sequence on the contribution of in-plane and interlaminar stresses to the overall system damping in fluid-filled, composite laminate cylindrical shells are studied.     

Reduction methods for nonlinear steady-state thermal analysis

Hybrid analysis techniques based on the combined use of finite elements and the classical Bubnov–Galerkin approximation are presented for predicting nonlinear steady-state temperature distributions in structures and solids. In these hybrid techniques the modelling versatility of the finite element method is preserved and a substantial reduction in the number of degrees-of-freedom is achieved by expressing the vector of nodal temperatures as a linear combination of a small number of global-temperature modes, or basis vectors. The Bubnov–Galerkin technique is then used to compute the coefficients of the linear combination (i.e. the amplitudes of the global–temperature modes). The basis vectors chosen are the path derivatives commonly used in perturbation techniques, namely, the derivatives of the nodal–temperature vector with respect to a preselected control (or path) parameter(s). The vectors are generated by using the finite element model of the initial discretization. Also, the performance of alternate sets of basis vectors is investigated. In the alternate sets, only a few path derivatives are generated, and they are augmented by a constant vector representing a uniform temperature rise (or drop), and by reciprocal vectors with nonzero components equal to the reciprocals of the nonzero components of the path derivatives. A problem-adaptive computational algorithm is presented for efficient evaluation of global approximation vectors and generation of the reduced system of equations and for monitoring the accuracy of the reduced system of equations. The potential of the proposed reduction methods for the solution of large-scale, nonlinear steady-state thermal problems is also discussed. The effectiveness of these methods is demonstrated by means of four numerical examples, including conduction, convection and radiation modes of heat transfer. This study shows that the use of the uniform-temperature mode and the path derivatives as global approximation vectors significantly increases the accuracy of the solutions obtained by reduction methods, thereby enhancing the effectiveness of these methods for the solution of large-scale, nonlinear thermal problems.     

Reanalysis procedure for large structural systems

An efficient procedure is presented for repetitive analysis of structures, with large numbers of degrees of freedom and design variables, as they are progressively modified during the automated optimum design process. The three key elements of the procedure are: (a) lumping of the large number of design variables into a single tracing parameter; (b) operator splitting or additive decomposition of the different arrays in the governing finite element equations of the modified structure into the corresponding arrays of the original structure plus correction terms; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique. The reanalysis procedure is applied to the linear static and free vibration problems of framed structures. Changes in both the sizing and shape (configuration) design variables are considered. For static problems the similarities between the proposed procedure and the preconditioned conjugate gradient technique are identified and are exploited to provide a physical meaning for the preconditioned residual vectors. The effectiveness of the proposed procedure is demonstrated by means of numerical examples.     

Global/local analysis of composite plates with cutouts

 The study focuses on the development of a simple and accurate global/local method for calculating the static response of stepped, simply-supported, isotropic and composite plates with circular and elliptical cutouts. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function, which is accurate for uniform thickness plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cut-out boundary, nor does it accurately model the discontinuities caused by abrupt thickness changes. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. The methodology is then applied to the analysis of composite plates, and its usefulness successfully proved in such cases. The proposed approach resulted in accurate prediction of stresses, with considerable savings in CPU time and data storage for composite flat panels.     

Evaluation of fracture mechanics parameters for a general corner using a weight function method

Summary The weight function method (WFM) has been used recently as a reliable tool for evaluation of fracture mechanics parameters, where cracks are represented by zero opening traction free surfaces. The purpose of this paper is to extend this technique to general opening corner problem. The two dimensional singular fields for displacements and stresses are introduced in terms of generalized Bueckner's strength. By means of eigenvalue analysis the stress intensity factors (SIF) are then formulated after appropriate splitting the regular stress and displacement fields into symmetric and antisymmetric modes. Using Betti's reciprocal theorem, a new expression in a more general closed form is derived for Bueckner's strength consisten with the given nonzero opening case. The potentiality of the method is demonstrated by a numerical example for =/2 corner problem. The stress intensity factor for the symmetric mode is evaluated by WFM and by a simple collocation procedure using both boundary element (BE) and finite element (FE) discretization.     

Recycling of solution spaces in multipreconditioned FETI methods applied to structural dynamics

This article presents a new method to recycle the solution space of an adaptive multipreconditioned finite element tearing and interconnecting algorithm in the case where the same operator is solved for multiple right‐hand sides like in linear structural dynamics. It accelerates the computation from the second time step on by applying a coarse space that is generated from Ritz approximations of local eigenproblems, using the solution space of the first time step. These eigenproblems are known to provide very efficient coarse spaces but must usually be solved a priori at high computational cost. Their Ritz approximations are much smaller and less expensive to solve. Recycling methods based on Ritz approximations of global eigenproblems have been published for classical finite element tearing and interconnecting algorithms, but their efficient application to multipreconditioned variants is not possible. This article also presents the application of a simpler recycling procedure, which reuses plain solution spaces, to adaptive multipreconditioned finite element tearing and interconnecting. Numerical results of the application of the presented methods to four test cases are shown. The new Ritz approximation method leads to coarse spaces, which turn out to be as efficient as those obtained from solving the unreduced eigenproblems. It is the most efficient recycling method currently available for multipreconditioned dual domain decomposition techniques.     

基于Rayleigh法的独塔非对称悬索桥基频简化算法

为了便于计算独塔非对称悬索桥振动基频,采用Rayleigh法分别推导了一阶正、反对称竖弯及扭转振动基频估算公式,考虑了不对称跨径布置对振动基频的影响,并提出了非对称独塔悬索桥合理的跨径比例。将表征跨径关系的参数k取1即可得到独塔双跨对称悬索桥的基频估算公式,最后通过有限元法验证估算公式的有效性和可靠性。研究结果表明:独塔非对称悬索桥一阶正、反对称竖弯、扭转频率的有限元解和文中解的误差都在10%以内,表明推导的估算公式解与有限元解误差能满足设计阶段的要求,最后讨论了跨径相关系数变化对竖弯和扭转基频的影响并给出了合理边中跨比的建议,该公式可以方便指导独塔悬索桥方案设计和动力计算。     

高层建筑结构之非线性地震响应分析

将高层建筑结构近似地等效成非线性弹性支承梁，以悬臂梁，柱之本征函数作为Ｒｉｔｚ矢量，用能量法推导出结构的运动方程，其自由度数及计算量均比用有限元法时大为减小，用本文的方法可以计算具弹塑性铰或弹塑性耗能器之剪切墙及框架式高层建筑结构，计算结果表明，采用非线性耗能器并引起高阻尼，可以显著地降低结构之地震响应。     

Calculation of the elastic buckling loads of thin flat reinforced plates using triangular finite elements

Details are given of a finite element scheme for the determination of the elastic buckling loads of thin flat plates of arbitrary shape under general kinematic and traction boundary conditions. The buckling loads are calculated using a triangular finite element with corner connections which is based on a previously derived variational principle combining both stresses and displacements. Numerical solutions are presented to a number of example problems, including plates with reinforcing members, and comparisons are made, where possible, with results from the finite element analyses of other authors and with exact or classical Rayleigh–Ritz solutions.     

双轴受压反对称角铺设复合材料层合板在固支边界下的后屈曲和模态跃迁

为有效分析双轴受压反对称角铺设复合材料层压板在固支边界下的后屈曲性能, 由渐近修正几何非线性理论推导其双耦合四阶偏微分方程(即应变协调方程和稳定性控制方程), 通过双Fourier级数将耦合非线性控制偏微分方程转换为系列非线性常微分方程, 从而获得相对简单的求解方法。使用广义Galerkin方法求解与角交铺设复合层合板相关的边界值问题, 研究了模态跃迁前后不同复杂程度的后屈曲模式。对四层固支边界复合层合板的数值模拟结果表明: 该解析法与有限元方法在主后屈曲区域的线性屈曲荷载计算结果吻合良好; 有限元方法在解靠近二次分岔点时失去收敛性, 而解析方法可深入后屈曲区域, 准确捕捉模态跃迁现象; 对于反对称角铺设层合板, 可仅用纯对称模态来定性预测主后屈曲分支、二次分岔荷载及远程跃迁路径。      

Dispersion of Thermoelastic Waves in a Plate With and Without Energy Dissipation

In this paper, the dispersion and energy dissipation of thermoelastic plane harmonic waves in a thin plate bounded by insulated traction-free surfaces is studied on the basis of three generalized theories of thermoelasticity. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained. Some limiting and particular cases of the frequency equations are then discussed. Results obtained in three theories of generalized thermoelasticity are compared. The results for the coupled thermoelasticity can be obtained as particular cases of the results by setting thermal relaxations times equal to zero. Numerical evaluations relating to the lower modes of the symmetric and antisymmetric waves are presented for an aluminum alloy plate.     

Global-local finite element

Finite element procedures usually require more degrees of freedom for a specified accuracy than does a classical Ritz procedure if suitable coordinate functions are available. This paper develops a combined global and local dependent variable representation which couples the conventional and finite element Ritz methods. This hybrid method preserves much of the flexibility of the finite element method while increasing the solution accuracy for a specified system order. The method is illustrated by examination of a beam and a plate vibration problem.