The global modal parameterization for non‐linear model‐order reduction in flexible multibody dynamics

In flexible multibody dynamics, advanced modelling methods lead to high‐order non‐linear differential‐algebraic equations (DAEs). The development of model reduction techniques is motivated by control design problems, for which compact ordinary differential equations (ODEs) in closed‐form are desirable. In a linear framework, reduction techniques classically rely on a projection of the dynamics onto a linear subspace. In flexible multibody dynamics, we propose to project the dynamics onto a submanifold of the configuration space, which allows to eliminate the non‐linear holonomic constraints and to preserve the Lagrangian structure. The construction of this submanifold follows from the definition of a global modal parameterization (GMP): the motion of the assembled mechanism is described in terms of rigid and flexible modes, which are configuration‐dependent. The numerical reduction procedure is presented, and an approximation strategy is also implemented in order to build a closed‐form expression of the reduced model in the configuration space. Numerical and experimental results illustrate the relevance of this approach. Copyright © 2006 John Wiley & Sons, Ltd.     

一种模态振型的直接扩阶方法

传统的模型缩阶及模态扩阶方法是以计算传递矩阵为手段.该文提出一种实测模态振型空间不完备问题的直接处理方法,即未测试自由度振型值的直接估计方法.通过引入一复合向量,该向量由对应于主自由度的实测振型值与对应于从自由度的待估算值组成.依靠实测模态及有限元模型信息对复合向量中未测试自由度的振型值进行重新估算,进而获得空间完备的模态振型.该文采用5 自由度的弹簧-质量体系验证方法的正确性;借助平面框架结构探讨新方法较传统方法在低阶模态振型估算精度上的优势.数值结果表明:所提出方法不需质量归一化条件以及实测模态与有限元振型的比例条件;同时,该方法较传统方法在低阶模态振型扩阶上具有更好的精度,并允许有限元模型具有一定的模型误差.     

Model reduction for LPV systems based on approximate modal decomposition

The paper presents a novel model order reduction technique for large‐scale linear parameter‐varying (LPV) systems. The approach is based on decoupling the original dynamics into smaller dimensional LPV subsystems that can be independently reduced by parameter‐varying reduction methods. The decomposition starts with the construction of a modal transformation that separates the modal subsystems. Hierarchical clustering is applied then to collect the dynamically similar modal subsystems into larger groups. The resulting parameter‐varying subsystems are then independently reduced. This approach substantially differs from most of the previously proposed LPV model reduction techniques, since it performs manipulations on the LPV model itself, instead of on a set of linear time‐invariant models defined at fixed scheduling parameter values. Therefore, the interpolation, which is often a challenging part in reduction techniques, is inherently solved. The applicability of the developed algorithm is thoroughly investigated and demonstrated by numerical case studies.     

自由度匹配技术在网壳结构损伤识别应用中的比较研究

测试自由度与有限元模型自由度不匹配是损伤识别技术应用于实际工程的最大障碍，动力模型缩聚和特征向量扩充技术是解决这一问题的主要手段。提出了基于剩余模态力和最小秩摄动理论的两阶段损伤识别算法，对几种模型缩聚技术和特征向量扩充技术在网壳结构损伤识别应用中的可行性、优缺点及适用范围进行了比较研究。通过单层短程线网壳结构模型的数值模拟，同时考虑测试噪声影响，实现了测试信息不完备情况下的网壳结构损伤识别。结果表明，本文所提出的损伤识别算法可以识别出网壳结构的单位置刚度损伤，而且利用Guyan静态缩聚法和Guyan扩充法进行自由度匹配后的损伤识别结果非常准确，它们是网壳结构损伤识别应用中模型自由度匹配的最佳选择。     

On the use of modal derivatives for nonlinear model order reduction

Modal derivative is an approach to compute a reduced basis for model order reduction of large‐scale nonlinear systems that typically stem from the discretization of partial differential equations. In this way, a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller, based on the resulting small‐scale state‐space model. We investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction, which has been missing so far. Concentrating on the application field of structural mechanics and structural dynamics, we show that the concept of modal derivatives can also be applied as nonlinear extension of the Craig–Bampton family of methods for substructuring. We furthermore generalize the approach from a pure projection scheme to a novel reduced‐order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables. This complexity reduction leads to a frequency‐preserving nonlinear quadratic state‐space model. Numerical examples with carefully chosen nonlinear model problems and three‐dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods, along with the current limitations. Copyright © 2016 John Wiley & Sons, Ltd.     

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

For the numerical solution of materially non‐linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point‐wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern‐day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel‐Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel‐Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non‐linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well‐known stress algorithms and show that the inclusion of a step‐size control based on local error estimations merely requires a small extra time‐investment. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparison of eigensolvers for large‐scale 3D modal analysis using AMG‐preconditioned iterative methods

The goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift‐invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd.     

仿真系统中DAE求解技术现状

目的 通过对仿真系统中微分代数方程（Differential Algebraic Equations,DAE）求解过程的描述,协助开发人员在仿真建模求解方面有更加深入的理解,便于分析和查找仿真建模中的问题。方法 利用指标约简技术将DAE方程的求解问题转化为ODE方程的求解问题,并利用数值求解和误差估计等现有技术进行求解。结果 通过对现有技术进行分析和归纳概括,总结了DAE系统现有技术在指标约简、数值求解和误差分析等关键环节中存在的优点和不足。结论 指出了仿真建模领域未来需要重点研究的方向。     

复杂变截面梁的轴向自由振动分析的近似方法

介绍了带有附加影响的变截面梁轴向自由振动问题的求解方法－模态摄动法，这一方法在由等截面均匀梁低阶主模态函数组成的模态子空间中，将复杂梁的变系数微分方程的求解化为线性代数方程组的求解，从而简化了计算过程，通过与其它方法的比较，说明了本文方法的优越性。     

Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis

A method of reducing the number of degrees of freedom and the overall computing times in finite element method (FEM) has been devised. The technique is valid for linear problems and arbitrary temporal variation of boundary conditions. At the first stage of the method standard FEM time stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. Simple matrix manipulations applied to the original FEM system produce a set of ordinary differential equations of a dimensionality greatly reduced when compared with the original FEM formulation. Using the concept of modal analysis the set is then solved analytically. Treatment of non‐homogeneous initial conditions, time‐dependent boundary conditions and controlling the error introduced by the reduction of the degrees of freedom are discussed. Several numerical examples are included for validation of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

模态测试设备系统校准方法研究

为了解决模态测试设备面向模态参数的校准问题,研究了模态分析与模态叠加理论,根据该理论得到一个闭环模态信号控制算法,由该闭环控制算法构建一个由5台激振器构成的闭环系统,该组激振器模拟模态测试过程中的一组信号,此组信号包含了标准模态参数的信号,最后用该组信号来校准模态测试设备。根据该方法进行了数值模拟和试验测试,结果显示该校准方法是可行的,并且可以脱离具体的模态结构,不同模态信号的模拟也可灵活实现。     

On Crack Detection in Tuned and Mistuned Repeating Structures Using the Modal Assurance Criterion

Repeating structures in the form of multiple‐bladed rotors are used widely in turbomachinery. Mistuning in turbomachinery is caused by small differences in individual blade properties because of inevitable material and manufacturing variations, resulting in the splitting of vibration modes of the tuned system. Modal characteristics of the blades are quite sensitive to the level of mistuning present inside the structure. In addition, the existence of damage also results in changed dynamics of the complete system. This paper introduces a modal assurance criterion (MAC)‐based approach for the investigation of small defects such as cracks in a repeating structure. In order to understand the key issues involved, initial work involved a numerical study of a simple comb‐like repeating structure, followed by a detailed numerical and experimental investigation of a tuned and mistuned bladed disc. Changes to the system mode shapes and mode order arising from damage are related to the location and severity of damage. Damage, in the form of small, open cracks, is modelled using different techniques such as follows: material removal, monotonic reduction in the modulus of elasticity of selected elements at the required location and mass modification. Damage indices based on differences in the MAC that give a measure of the change in the mode shapes are introduced. MAC matrices are obtained using a reduced number of data points. The damage index is obtained from the Frobenius norm of the MAC matrix subtracted from the AutoMAC of a reference tuned model without crack. A clear correlation between the damage indices and the crack depth/location is shown. Application of this approach to the limited data obtainable from developing techniques such as blade tip timing is also explained.     

Optimal modal reduction of vibrating substructures

A structure which consists of a main part and a number of attached substructures is considered. A ‘model reduction’ scheme is developed and applied to each of the discrete substructures. Linear undamped transient vibrational motion of the structure is assumed, with general external forcing and initial conditions. The goal is to replace each discrete substructure by another substructure with a much smaller number of degrees of freedom, while minimizing the effect this reduction has on the dynamic behaviour of the main structure. The approach taken here involves Ritz reduction and the Dirichlet‐to‐Neumann map as analysis tools. The resulting scheme is based on a special form of modal reduction, and is shown to be optimal in a certain sense, for long simulation times. The performance of the scheme is demonstrated via numerical examples, and is compared to that of standard modal reduction. Copyright © 2003 John Wiley & Sons, Ltd.     

Fast frequency response analysis of large‐scale structures with non‐proportional damping

Modal frequency response analysis is an economical approach for large and complex structural systems since there is an enormous reduction in dimension from the finite element frequency response problem. However, when non‐proportional damping exists, the modal frequency response problem is expensive to solve at many frequencies because modal damping matrices are fully populated. This paper presents a new algorithm to solve the modal frequency response problem for large and complex structural systems with structural and viscous damping. The newly developed algorithm, fast frequency response analysis (FFRA) algorithm, solves the damped modal frequency response problem with O(m²) operations at each frequency. Then the FFRA algorithm is extended for solving a system of equations in optimization application with the modal correction approach, in which the mass, stiffness and damping matrices of a modified configuration differ from the original configuration. Numerical results show that the FFRA algorithm dramatically improves the performance of the modal frequency response analysis compared to conventional methods in industry while obtaining the same accuracy. Copyright © 2006 John Wiley & Sons, Ltd.     

Advanced analysis of coupled buffeting response of bridges: a complex modal decomposition approach

This paper presents a framework based on a complex modal decomposition technique for predicting coupled buffeting response of bridges in both time and frequency domains. The coupled equations of motion in structural modal coordinates with frequency dependent aeroelastic self-excited terms are approximated by frequency independent state-space equations, without augmented aerodynamic states, which retain the complex modal properties of the original system. These equations are then decomposed into a set of uncoupled equations of motion for buffeting response analysis. The frequency dependent unsteady buffeting characteristics and their spanwise correlation are considered in both frequency and time domain analyses instead of invoking the customary quasi-steady assumption. This framework significantly enhances computational efficiency in the frequency domain analysis by avoiding system matrix inversion at each discretized frequency when evaluating the transfer function matrix. Furthermore, it also offers simulation of buffeting response in the time domain that includes frequency dependence of buffeting and self-excited forces. A detailed discussion concerning the complex modal frequencies, damping ratios, mode shapes, and the significance of structural modes on the multimode coupled buffeting response is provided. This helps to glean additional insight and to improve our understanding of the underlying physics of wind–structure interactions. Examples of long span suspension bridges are provided to illustrate the proposed scheme and to demonstrate its effectiveness.     

Coupling heat conduction and water–steam flow in a saturated porous medium

This paper is devoted to the simulation of water forced evaporation in a porous saturated medium in a 3D‐axisymmetric domain by resolution of partial differential algebraic equations (PDAE) that are encountered in different engineering applications. The goal of this paper is an attempt to present effective realizations, in order to determine the minimal duration of burning for prehistoric occupations. This multidisciplinary work includes scientists in Mathematics, Physics and Archaeology. The model proposed here couples the heat conduction in a water saturated soil with the water steam flow in the medium. We propose an efficient and robust global numerical method, based on a method of lines and differential algebraic equations (DAE) solvers, combined with a Newton method using a powerful sparse linear solver. After a brief overview of classes for numerical techniques applied for moving boundary problems, the Apparent Heat Capacity method (AHC) is used, and in order to validate our codes, a comparison with experiments is done. Copyright © 2010 John Wiley & Sons, Ltd.     

Generalized receptance‐based method for accurate and efficient modal synthesis

Modal synthesis plays an important role in efficient dynamic analyses of large structural assemblies. However, the numerical accuracy and the computational efficiency which existing modal synthesis methods have achieved to date are not quite satisfactory. This paper presents a new generalized receptance‐based modal synthesis method which is numerically very accurate and computationally very efficient. The method employs receptance data computed at just few frequency values to formulate a derived equivalent eigensys tem from which required eigenvalues and eigenvectors of interest can be solved. It provides mathematical generalization for existing mode‐based modal synthesis methods since modal data are essentially those receptance data at resonant frequencies. A normalization procedure is also presented which can be used to mass normalize computed mode shapes which are often required in practice. Compensation for the contribution of higher uncomputed modes to receptance data at lower frequency of interest is made to improve analysis accuracy by using lower calculated modes and known system matrices. Numerical results are presented to demonstrate the effectiveness of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

A three‐invariant hardening plasticity for numerical simulation of powder forming processes via the arbitrary Lagrangian–Eulerian FE model

In this paper, a three‐invariant cap plasticity model with an isotropic hardening rule is presented for numerical simulation of powder compaction processes. A general form is developed for single‐cap plasticity which can be compared with some common double‐surface plasticity models proposed for powders in literature. The constitutive elasto‐plastic matrix and its components are derived based on the definition of yield surface, hardening parameter and non‐linear elastic behaviour, as function of relative density of powder. Different aspects of the new single plasticity are illustrated by generating the classical plasticity models as special cases of the proposed model. The procedure for determination of powder parameters is described by fitting the model to reproduce data from triaxial compression and confining pressure experiments. The three‐invariant cap plasticity is performed within the framework of an arbitrary Lagrangian–Eulerian formulation, in order to predict the non‐uniform relative density distribution during large deformation of powder die pressing. In ALE formulation, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. This formulation introduces some convective terms in the finite element equations and consists of two phases. Each time step is analysed according to Lagrangian phase until required convergence is attained. Then, the Eulerian phase is applied to keep mesh configuration regular. Because of relative displacement between mesh and material, all dependent variables such as stress and strain are converted through the Eulerian phase. Finally, the numerical schemes are examined for efficiency and accuracy in the modelling of a rotational flanged component, an automotive component, a conical shaped‐charge liner and a connecting‐rod. Copyright © 2005 John Wiley & Sons, Ltd.     

Subsystem Global Modal Parameterization for efficient simulation of flexible multibody systems

This paper presents a new model order reduction strategy for flexible multibody simulation, namely the Subsystem Global Modal Parameterization. The proposed method is based on a system‐level reduction technique, named Global Modal Parameterization, but offers significant improvements for systems with many independent DOFs. The approach splits up the motion of a mechanism or part of a mechanism into a relative motion, in which the members move relatively with respect to each other, and a global motion of the system, in which the relative position of the members does not change. The relative motion is described by a local Global Modal Parameterization model expressed in a mechanism‐attached frame, and the global motion is described by the motion of the mechanism‐attached frame. In order to improve simulation efficiency, mass invariants are used, which are also introduced in this paper. Two numerical examples are presented, which show the good accuracy and the major simulation efficiency improvements this new approach offers. Copyright © 2011 John Wiley & Sons, Ltd.