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.     

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.     

Dimensional model reduction in non‐linear finite element dynamics of solids and structures

A general approach to the dimensional reduction of non‐linear finite element models of solid dynamics is presented. For the Newmark implicit time‐discretization, the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments. To deal with this, it is shown how the problem can be formulated in an approximation (Ritz) basis of much smaller dimension. Similarly, the explicit Newmark algorithm can be also written in a reduced‐dimension basis, and the computation time savings in that case follow from an increase in the stable time step length. In addition, the empirical eigenvectors are proposed as the basis in which to expand the incremental problem. This basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense. Because of this ‘global’ time viewpoint, the basis need not be updated as with reduced bases computed from a linearization of the full finite element model. If the dynamics of a finite element model is expressed in terms of a small number of basis vectors, the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved. At the same time, numerical experiments indicate that by using reduced models, substantial savings can be achieved even in the pre‐asymptotic range. Furthermore, the algorithm parallelizes very efficiently. The method we present is expected to become a useful tool in applications requiring a large number of repeated non‐linear solid dynamics simulations, such as convergence studies, design optimization, and design of controllers of mechanical systems. Copyright © 2001 John Wiley & Sons, Ltd.     

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。     

Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency

A rigorous computational framework for the dimensional reduction of discrete, high‐fidelity, nonlinear, finite element structural dynamics models is presented. It is based on the pre‐computation of solution snapshots, their compression into a reduced‐order basis, and the Galerkin projection of the given discrete high‐dimensional model onto this basis. To this effect, this framework distinguishes between vector‐valued displacements and manifold‐valued finite rotations. To minimize computational complexity, it also differentiates between the cases of constant and configuration‐dependent mass matrices. Like most projection‐based nonlinear model reduction methods, however, its computational efficiency hinges not only on the ability of the constructed reduced‐order basis to capture the dominant features of the solution of interest but also on the ability of this framework to compute fast and accurate approximations of the projection onto a subspace of tangent matrices and/or force vectors. The computation of the latter approximations is often referred to in the literature as hyper reduction. Hence, this paper also presents the energy‐conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi‐discrete), nonlinear, finite element structural dynamics models. Based on mesh sampling and the principle of virtual work, ECSW is natural for finite element computations and preserves an important energetic aspect of the high‐dimensional finite element model to be reduced. Equipped with this hyper reduction procedure, the aforementioned Galerkin projection framework is first demonstrated for several academic but challenging problems. Then, its potential for the effective solution of real problems is highlighted with the realistic simulation of the transient response of a vehicle to an underbody blast event. For this problem, the proposed nonlinear model reduction framework reduces the CPU time required by a typical high‐dimensional model by up to four orders of magnitude while maintaining a good level of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.     

Eigensolution reanalysis of modified structures using epsilon‐algorithm

Based on the Neumann series expansion and epsilon‐algorithm, a new eigensolution reanalysis method is developed. In the solution process, the basis vectors can be obtained using the matrix perturbation or the Neumann series expansion to construct the vector sequence, and then using the epsilon algorithm table to obtain the approximate eigenvectors. The approximate eigenvalues are computed from the Rayleigh quotients. The solution steps are straightforward and it is easy to implement with the general finite element analysis system. Two numerical examples, a 40‐storey frame and a chassis structure, are given to demonstrate the application of the present method. By comparing with the exact solutions and the Kirsch method solutions, it is shown that the excellent results are obtained for very large changes in the design, and that the accuracy of the epsilon‐algorithm is higher than that of the Kirsch method and the computation time is less than that of the Kirsch method. Copyright © 2005 John Wiley & Sons, Ltd.     

A method for modal reanalysis of topological modifications of structures

A method for structural modal reanalysis for three cases of topological modifications, the number of degrees of freedom (DOFs) is unchanged, decreased, and increased, is presented. In this method, the newly added DOFs are linked to the original DOFs of the modified structure by means of the dynamic reduction so as to obtain the condensed equation. The methods for forming the stiffness and mass increments, Δ K and Δ M , resulting from the three cases of topological modifications of structures are discussed. The extended Kirsch method is used to improve the accuracy of the starting solutions of the initial structure. And then, the eigenvectors of newly added DOFs resulting from topological modification can be recovered. At last, the Rayleigh–Ritz analysis is used to evaluate the eigenvalues and eigenvectors for the modified structure. Three numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural modal reanalysis of three cases of the topological modifications and it is easy to implement on a computer. By comparing with previous method, it can be seen that the present method can give good approximate eigenvalues and eigenvectors, even if the topological modifications are very large. Copyright © 2005 John Wiley & Sons, Ltd.     

Toward a multifrequency quasi‐static Ritz vector method for frequency‐dependent acoustic system application

Computational issues concerning the calculation of acoustic responses of a complex finite element (FE) model for various noise and vibration inputs have become prevalent. Such a model requires a significant amount of computation time because of repeated inversions of dynamic stiffness matrices. Thus, even state‐of‐the‐art computer hardware and software often face limitations where a model order reduction (MOR) scheme can help. The established MOR schemes such as Ritz vector or quasi‐static Ritz vector methods are efficient for general engineering systems, but these MOR methods become inaccurate for frequency response analyses in some acoustic systems with frequency‐dependent mass and stiffness matrices and force vectors (hereinafter frequency‐dependent acoustic systems). To cope with the inaccurate prediction by these methods for frequency‐dependent acoustic systems, this research presents and applies the multifrequency quasi‐static Ritz vector method. Unlike the Ritz vector or quasi‐static Ritz vector methods, the present multifrequency quasi‐static Ritz vector method employs direct Krylov subspace bases without an orthonormal procedure at multiple center frequencies. In comparison with the existing MOR scheme, a significant gain in computational efficiency is achieved, as well as enhanced accuracy. A comparison of these methods based on criteria such as efficiency, accuracy, and reliability was also conducted. Copyright © 2011 John Wiley & Sons, Ltd.     

A reduced basis approach to quantifying damage dependent dynamic response of laminated composite structures

A finite element (FE) based formulation is utilized to represent the damage-dependent response of laminated composite structures. An internal-state-variable (ISV) approach provides a definition of the stiffness reduction caused by intralaminar crack propagation at the ply level. These ISVs are combined with simple stress criteria to accommodate ply property changes caused by fiber fracture, fiber microbuckling and interior delaminations. A set of orthogonal Ritz vectors are chosen as basis vectors to transform the dynamical equations of motion to a reduced coordinate space. The reduced basis form of the equations provides significant numerical efficiencies, especially for large ordered systems. Furthermore, damping and its variation with damage can be generally represented in any number of vibratory modes. The Newmark integration operator is used to solve the dynamic equations of motion, and equilibrium iterations are performed in each incremental time step to assure convergence. Results are given for laminated beam and plate geometries subjected to dynamic loads.     

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.     

Model-size reduction for the analysis of symmetric structures with asymmetric boundary conditions

A simple computational procedure is presented for reducing the size of the analysis model for a symmetric structure with asymmetric boundary conditions to that of the corresponding structure with symmetric boundary conditions. The procedure is based on approximating the asymmetric response of the structure by a linear combination of symmetric and antisymmetric global approximation vectors (or modes). The key elements of the procedure are (a) restructuring the governing finite element equations to delineate the contributions to the symmetric and antisymmetric components of the asymmetric response, (b) successive application of the finite element method and the classical Rayleigh–Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh–Ritz technique. A tracing parameter is introduced which identifies all the contributions to the antisymmetric response. The global approximation vectors are selected to be the solution corresponding to a zero value of the tracing parameter and the various-order derivatives of the solution with respect to this parameter, evaluated at zero value of the parameter. The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions. The effectiveness of the computational procedure is demonstrated by means of numerical examples of linear static problems of shells, and its potential for solving non-linear problems is discussed.     

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。     

Optimum Design of Laminated Composite Plates Undergoing Large Amplitude Oscillations

The optimum design of composite laminated plates under going large amplitude free vibration is discussed. Von Karman's nonlinear strain displacement relations are considered to account for large amplitude. A higher order shear deformation theory with parabolic variation of transverse shear stresses through thickness is used in the finite element formulation. A nine-noded isoparametric element with 7 dof per node is adopted. Ritz formulation for nonlinear finite element analysis is implemented and the direct iteration method is used to solve the governing nonlinear equation. Optimization is carried out using genetic algorithm (GA) with tournament selection scheme.     

Application of load-dependent Ritz vectors to Bayesian probabilistic damage detection

This paper demonstrates the possibility of incorporating load-dependent Ritz vectors as an alternative to modal parameters into a Bayesian probabilistic framework for detecting damages in a structure. Recent research has shown that it is possible to extract load-dependent Ritz vectors from vibration tests. This paper shows that load-dependent Ritz vectors have the following potential advantages for damage detection over modal vectors: (1) in general, load-dependent Ritz vectors are more sensitive to damage than the corresponding modal vectors; and (2) substructures of interest can be made more observable using the load-dependent Ritz vectors generated from particular load patterns. An eight-bay truss example and a five-story frame example, explicitly considering both modeling error and measurement noise, are presented to illustrate the applicability of the proposed approach.     

An adaptive model order reduction with Quasi‐Newton method for nonlinear dynamical problems

Model Order Reduction (MOR) methods are extremely useful to reduce processing time, even nowadays, when parallel processing is possible in any personal computer. This work describes a method that combines Proper Orthogonal Decomposition (POD) and Ritz vectors to achieve an efficient Galerkin projection, which changes during nonlinear solving (online analysis). It is supported by a new adaptive strategy, which analyzes the error and the convergence rate for nonlinear dynamical problems. This model order reduction is assisted by a secant formulation which is updated by the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) formula to accelerate convergence in the reduced space, and a tangent formulation when correction of the reduced space is needed. Furthermore, this research shows that this adaptive strategy permits correction of the reduced model at low cost and small error. Copyright © 2015 John Wiley & Sons, Ltd.     

基于改进Ritz法的耗能减震高层建筑结构地震响应分析

用改进Ritz法计算耗能减震高层建筑结构的地震响应。改进Ritz法采用基于外荷载空间分布的Ritz向量和基于外荷载频率的Ritz向量。其中,基于外荷载空间分布的Ritz向量用Lanczos法形成。基于外荷载频率的Ritz向量则用外荷载主频的平方进行特征值平移后再用Lanczos法形成。文中还给出上述两种Ritz向量的截断标准。之后用改进Ritz法的Ritz向量对结构动力方程进行线性变换。由于耗能减震高层建筑结构的阻尼是非比例阻尼,广义阻尼矩阵是非对角矩阵,所以文中采用拟力实模态法对线性变换后的耦联动力方程进行求解。最后的算例说明该方法用于耗能减震高层建筑结构的地震响应分析是有效可行的。     

Nonlinear thermal analysis with a boundary element zone condensation technique

This paper presents a substantially more economical technique for the boundary element analysis (BEA) of a large class of nonlinear heat transfer problems including those with temperature dependent conductivity, temperature dependent convection coefficients, and radiation boundary conditions. The technique involves an exact static condensation of boundary element zones in a multi-zone boundary element model. The condensed boundary element zone contributions to be overall sparse blocked boundary element system matrices are formed once in the first step of the iterative nonlinear solution process and subsequently reused as the nonlinear parts of the overall problem are evolved to a convergent solution. Through a series of example problems it is demonstrated that the zone condensation technique facilitates the use of highly convergent iterative strategies for the solution of the nonlinear heat transfer problem involving modification and subsequent factorization of the overall boundary element system left had side matrix. For heat transfer problems with localized nonlinear effects, the condensation technique is shown to allow for the solution of nonlinear problems in less than half the CPU time required by methods that do not employ condensation.     

Generalized hybrid-combined methods for the singularity problems of homogeneous elliptic equations

Combinations of the Ritz–Galerkin and finite element methods are applied to solving singularity problems of homogeneous, elliptic equations. The Ritz-Galerkin method is used in the subdomains where there exist singular points; and the finite element method is still used in the rest of the solution domains. More general coupling techniques than those of Reference 6 along the common boundary of subdomains are discussed. Numerical experiments using these kinds of coupling techniques are provided for the first time. It is interesting that the calculated results of Motz's problem have shown the simplified hybrid strategy in Reference 6 to be optimal for both error bounds and stability of numerical solutions, among all general coupling techniques.     

Computer algorithms for calculating efficient initial vectors for subspace iteration method

In this paper, the effects of selecting initial vectors on computation efficiency for a subspace iteration method are investigated. Four algorithms are used for selecting the initial vectors. First, arbitrary starting iteration vectors are chosen according to Bathe and Wilson's algorithm.¹ In the other algorithms, the initial vectors are the retrieved eigenvectors from the Guyan and quadratic reduction methods. Improvement of the eigenvalue approximations of the subspace iteration method over reduction methods is presented. The computation effort is examined for the various algorithms used for initial iteration vectors.     

Prediction of Impact-Induced Fibre Damage in Circular Composite Plates

A simple analytical impact damage model for preliminary design analysis is developed on the basis of experimental findings observed from quasi-static lateral load and low velocity impact tests. The analytical model uses a non-linear approximation method (Rayleigh–Ritz) and the large deflection plate theory to predict the number of failed plies and damage area in a quasi-isotropic composite circular plate (axisymmetric problem) due to a point load at its centre. It is assumed that the deformation due to a static transverse load is similar to that occurred in a low velocity impact. It is found that the model, despite its simplicity, is in good agreement with finite element (FE) predictions and experimental data for the deflection of the composite plate and gives a good estimate of the number of failed plies due to fibre breakage. The predicted damage zone could be used with a fracture model developed by the second investigator to estimate the compression after impact strength of such laminates. This approach could save significant running time when compared to FE numerical solutions. Corresponding author.