Free and forced vibrations of plates by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.     

The scaled boundary finite element method in structural dynamics

The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high‐order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high‐order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time‐domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity. Copyright © 2008 John Wiley & Sons, Ltd.     

开放系统悬臂类结构的动力特性计算方法

结构总是修建在一定的场地而形成土-结构相互作用的开放系统。为解决开放体系下悬臂类结构的自振频率、振型和考虑辐射阻尼下模态阻尼比的计算问题,提出了脉冲荷载响应模态分析法。该方法采用直接有限元法建立土-结构相互作用有限元模型,对结构施加脉冲荷载得到结构动力反应后,由模态识别方法计算结构的动力特性。随后,以一个悬臂类五层框架结构为例研究了计算动力特性随土体计算范围变化的规律和脉冲荷载激励点位置对计算结果的影响。在此基础上,讨论了土体材料阻尼对模态阻尼比的影响,并与集总参数模型和直接模态分析法进行对比,说明不同方法的计算精度。计算结果表明,随着土域计算范围的增加,脉冲荷载响应模态分析法所得的动力特性将逐渐收敛到精确解;当土体计算范围大于结构基频所对应的波长的2倍时,结构自振频率的误差小于1%,模态阻尼比的误差小于5%;以非模态节点作为激励点都可以得到比较精确的结果;三参数集总参数模型所得模态阻尼比存在显著误差,直接模态分析法所得模型的基频随土域范围增大而趋向于零;相比于辐射阻尼,土体材料阻尼对结构的各阶模态阻尼比的影响较小。     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 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.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Transient analysis of three-dimensional dynamic interaction between multilayered soil and rigid foundation

The study of dynamic soil-structure interaction is significant to civil engineering applications, such as machine foundation vibration, traffic-induced vibration, and seismic dynamic response. The scaled boundary finite element method (SBFEM) is a semi-analytical algorithm, which is used to solve the dynamic response of a three-dimensional infinite soil. It can automatically satisfy the radiation boundary condition at infinity. Based on the dynamic stiffness matrix equation obtained by the modified SBFEM, a continued fraction algorithm is proposed to solve the dynamic stiffness matrix of layered soil in the frequency-domain. Then, the SBFEM was coupled with the finite element method (FEM) at the interface to solve the dynamic stiffness matrices of the rigid surface/buried foundation. Finally, the mixed-variable algorithm was used to solve the three-dimensional transient dynamic response of the foundation in the time domain. Numerical examples were performed to verify the accuracy of the proposed algorithm in solving the dynamic stiffness matrix of the infinite domain in the frequency domain and the dynamic transient displacement response of the foundation in the time domain. Compared with the previous numerical integration technique, the dynamic stiffness matrix in the frequency domain calculated by using the proposed algorithm has higher accuracy and higher efficiency.     

Highly convergent dynamic models obtained by modal synthesis with application to short wave pulse propagation

A general approach for obtaining the matrices of a substructure ensuring minimum modal frequency errors of the whole structure is presented. The mass and stiffness matrices of a small component domain of selected dimension are obtained by applying the modal synthesis of a limited number of close‐to‐exact modes such that after assembling a larger joined domain model the modal convergence rate of the latter should be as high as possible. The goal is achieved by formulating the minimization problem for the penalty‐type target function representing the cumulative relative modal frequency error of the joined sample domain and by applying the gradient descent minimization method. After the optimum matrices of a component domain are obtained, they can be used in any structure as higher‐order elements or super‐elements. The well‐known generalized mass matrices obtained as a weighted sum of lumped and consistent components can be treated as a special case of the presented approach. The obtained dynamic models are used for modelling short transient waves and wave pulses propagating in elastic or acoustic environments by using a only a few nodal points per pulse length. Copyright © 2004 John Wiley & Sons, Ltd.     

Earthquake response analysis in the time domain for 2D soil–structure systems using analytical frequency‐dependent infinite elements

This paper presents a time domain method for soil–structure interaction analysis under seismic excitations. It is based on the finite element formulation incorporating analytical frequency‐dependent infinite elements for the far‐field soil region. Equivalent earthquake input forces are calculated based on the free‐field responses along the interface between the near‐ and far‐field soil regions using the fixed exterior boundary method in the frequency domain. Then, the input forces are transformed into the time domain by using inverse Fourier transform. The dynamic stiffness matrices of the far‐field soil region formulated using the analytical frequency‐dependent infinite elements in the frequency domain can be easily transformed into the corresponding matrices in the time domain. Hence, the response can be analytically computed in the time domain. A recursive procedure is proposed to compute the interaction forces along the interface and the responses of the soil–structure system in the time domain. Earthquake response analyses have been carried out on a multi‐layered half‐space and a tunnel embedded in a layered half‐space, and results are compared with those obtained by the conventional method in the frequency domain. Copyright © 2003 John Wiley & Sons, Ltd.     

基于弹性-黏弹性对应原理的复合材料层合板模态阻尼预测:理论及其有限元实现

提出一种利用通用有限元软件求解复合材料结构模态阻尼的有限元方法。该方法基于扩展弹性-黏弹性对应原理,定义出具有频率依存性的黏弹性复合材料复刚度矩阵,并借助ABAQUS提供的二次开发接口UMAT将其编入求解器中,结合复特征值法求解任意铺层层合板的模态阻尼。与已有的理论方法相比,本模型的计算结果更为接近实验数据。从而验证了本文提出的数值分析方法的有效性和精确性,为利用ABAQUS软件分析各向异性材料阻尼提供了一条有效途径。     

Solution of FE‐BE coupled eigenvalue problems for the prediction of the vibro‐acoustic behavior of ship‐like structures

To predict the vibro‐acoustic behavior of structures, both a structural problem and an acoustic problem have to be solved. For thin structures immersed in water, a strong interaction between the structural domain and fluid domain occurs. This significantly alters the resonance frequencies. In this work, the structure is modeled by the finite element method. The exterior acoustic problem is solved by a fast boundary element method employing hierarchical matrices. An FE‐BE formulation is presented, which allows the solution of the coupled eigenvalue problem and thus the prediction of the coupled eigenfrequencies and mode shapes. It is based on a Schur complement formulation of the FE‐BE system yielding a generalized eigenvalue problem. A Krylov–Schur solver is applied for its efficient solution. Hereby, the compressibility of the fluid is neglected. The coupled eigensolution is then used for a model reduction strategy allowing fast frequency sweep calculations. The efficiency of the proposed formulations is investigated with respect to memory consumption, accuracy, and computation time. Copyright © 2011 John Wiley & Sons, Ltd.     

Thin-layer method: Formulation in the time domain

The thin-layer method is a semi-discrete numerical technique that may be used for the dynamic analysis of laminated solids or fluids. In its classical implementation, the method is normally formulated in the frequency domain and requires the solution of a complex-valued quadratic eigenvalue problem; in this paper we present an alternative time-domain formulation which can offer advantages in some cases, such as avoiding the use of complex algebra. The proposed method entails expressing the governing equations in the frequency-wavenumber domain, solving a linear real-valued eigenvalue problem in the frequency variable, carrying out an analytical integration over frequencies, and performing a numerical transform over wavenumbers. This strategy allows obtaining the Green's functions for impulsive sources directly in the time domain, even when the system has little or no damping. We first develop the algorithm in its most general form, allowing fully anisotropic materials and arbitrary expansion orders; then we consider a restricted class of anisotropic materials for which the required linear eigenvalue problem involves only real, narrowly banded symmetric matrices and finally, we demonstrate the method by means of a simple problem involving a homogeneous stratum subjected to an antiplane impulsive source.     

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.     

Cyclic steady states of nonlinear electro‐mechanical devices excited at resonance

We present an efficient numerical method to solve for cyclic steady states of nonlinear electro‐mechanical devices excited at resonance. Many electro‐mechanical systems are designed to operate at resonance, where the ramp‐up simulation to steady state is computationally very expensive – especially when low damping is present. The proposed method relies on a Newton–Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time‐stepping from zero initial conditions. We use a recently developed high‐order Eulerian–Lagrangian finite element method in combination with an energy‐preserving dynamic contact algorithm in order to solve the coupled electro‐mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro‐electro‐mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed‐ups of the form S  = 0.6·ξ  ^? 0.8, where ξ is the linear damping ratio of the corresponding resonance frequency. Copyright © 2016 John Wiley & Sons, Ltd.     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.     

Computation of response moments of high dimensional MDOF structure excited by stationary non-Gaussian random fields

Output moments of a non-linear dynamical system excited by a non-Gaussian random field can be obtained in practice only by simulation techniques. When the dynamical system can be decomposed in a low dimension non-linear part acting on a high dimension linear part, the original problem reduces to calculate output moments of a high dimension linear system. The proposed method suggests that work should be directed in the frequency domain. Time trajectories are then obtained through Fourier transform. Such a procedure does not introduce any approximation errors due to the time integration numerical scheme nor does it introduce any transient state. Further quasi-static correction terms can be introduced when a truncated modal basis is utilized in order to describe the low frequency dynamic response.     

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

A high‐order time‐domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued‐fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued‐fraction approach for bounded domains is proposed, which yields numerically more robust time‐domain formulations. The coefficient matrices of the corresponding continued‐fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high‐order time‐domain formulation for bounded domains with a high‐order transmitting boundary suggested previously is also proposed. In the time‐domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency‐independent coefficient matrices, which can be solved efficiently using standard time‐integration schemes. Numerical examples for modal and time‐domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Transient vibration and sound radiation of a rectangular plate with viscoelastic boundary supports

This work considers transient vibration and sound radiation from an impact‐excited rectangular plate with viscoelastic boundary supports. The approach used is based on the modal strain energy (MSE) method. Vibration of the plate is approximated by a double infinite series in the spatial co‐ordinates. Each term of the series is constructed with vibration modes of beams having the same boundary conditions as the plate, multiplied by a time‐dependent function. Modal loss factor of each mode is obtained by the MSE method. The sound radiation pressure in the time and frequency domain is obtained by numerical integration of the Rayleigh integral. Effects of the viscoelastic boundary supports on the vibration response and the radiated sound pressure of the vibrating plate are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

基于复模态实验数据的粘性阻尼矩阵的修正

在实际工程中,由有限元模型得到的计算值与通过试验获得的测量值之间往往存在偏差,为了能够精确预测结构的动力响应,依据测量信息修正存在的动力模型是非常必要的。考虑用不完备复模态实验测量数据修正粘性阻尼矩阵的问题。在假定分析质量矩阵与刚度矩阵是精确的情况下,通过求解一个约束最优化问题,得到了满足特征方程的加权Frobenius范数意义下的最优对称修正矩阵。     

粘弹性分数阶导数模型的有限元法

本文给出了粘弹性分数阶导数模型的有限元格式,并用模态分析的方法进行了运动方程解耦。用Laplace变换及其反变换,解析地计算了解耦后单自由度系统的时域和频域响应。时域响应被分为极点部分和截断部分,它们分别代表短期内的衰减振动和长时间内的缓慢恢复效应。以一维杆件为例,对有限元算法和解析解做了对比,分析了响应精度与单元个数的关系。结果表明用有限元法计算的位移接近解析解,而要得到准确的高频加速度响应,则需要划分许多单元格。