A spectrally formulated finite element for wave propagation analysis in functionally graded beams

In this paper, spectral finite element method is employed to analyse the wave propagation behavior in a functionally graded (FG) beam subjected to high frequency impulse loading, which can be either thermal or mechanical. A new spectrally formulated element that has three degrees of freedom per node (based upon the first order shear deformation theory) is developed, which has an exact dynamic stiffness matrix, obtained by exactly solving the homogeneous part of the governing equations in the frequency domain. The element takes into account the variation of thermal and mechanical properties along its depth, which can be modeled either by explicit distribution law like the power law and the exponential law or by rule of mixture as used in composite. Ability of the element in capturing the essential wave propagation behavior other than predicting the propagating shear mode (which appears only at high frequency and is present only in higher order beam theories), is demonstrated. Propagation of stress wave and smoothing of depthwise stress distribution with time is presented. Dependence of cut-off frequency and maximum stress gradient on material properties and FG material (FGM) content is studied. The results are compared with the 2D plane stress FE and 1D Beam FE formulation. The versatility of the method is further demonstrated through the response of FG beam due to short duration highly transient temperature loading.     

A finite wavelet domain method for wave propagation analysis in thick laminated composite and sandwich plates

An efficient numerical method is developed for the simulation of three dimensional transient dynamic response in thick laminated composite and sandwich plate structures involving very high frequencies and wave numbers. The proposed method incorporates Daubechies wavelet scaling functions for the interpolation of the in-plane displacements with a Galerkin formulation. It further explores the orthonormality and compact support of wavelet scaling functions to produce near diagonal consistent mass matrices and banded stiffness matrices. Hence, an uncoupled equivalent discrete spatial dynamic system is formulated, synthesized and rapidly solved in the wavelet domain using an explicit time integration scheme. The in-plane wavelet interpolation is further combined with an efficient high order layerwise laminate plate theory, that implements Hermite cubic splines for the through-the-thickness approximation of displacement fields. Numerical results are presented on the prediction of guided waves in laminated and thick sandwich composite plates and compared with respective solutions obtained by analytical, semi-analytical and time domain spectral element models. The method yielded higher convergence rates and substantial reductions in computational effort compared to respective time domain spectral finite elements.     

Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method

The work deals with the development of an effective numerical tool in the form of pseudospectral method for wave propagation analysis in anisotropic and inhomogeneous structures. Chebyshev polynomials are used as basis functions and Chebyshev–Gauss–Lobatto points are used as grid points. The formulation is implemented in the same way as conventional finite element method. The element is tested successfully on a variety of problems involving isotropic, orthotropic and functionally graded material (FGM) structures. The formulation is validated by performing static, free vibration and wave propagation analysis. The accuracy of the element in predicting stresses is compared with conventional finite elements. Free vibration analysis is carried out on composite and FGM beams and the computational resources saved in each case are presented. Wave propagation analysis is carried out using the element on anisotropic and inhomogeneous beams and layer structures. Wave propagation in thin double bounded media over long propagating distances is studied. Finally, a study on scattering of waves due to embedded horizontal and vertical cracks is carried out, where the effectiveness of modulated pulse in detecting small cracks in composites and FGMs has been demonstrated.     

Numerical modeling of PZT-induced Lamb wave-based crack detection in plate-like structures

This paper presents the numerical modeling and simulations of PZT-induced Lamb wave propagation in plate-like structures by using the spectral finite element method. A novel spectral plate finite element, which can efficiently model the three-dimensional (3D) behavior of Lamb waves, is proposed. In the formulation, linear displacement distributions in the thickness direction are assumed for both the PZT layer and the base plate. A way to avoid the thickness locking is proposed and used in the formulations. Two examples, one for the validation of the proposed two-dimensional (2D) spectral finite element and the other for the demonstration of crack detection in plates, are presented and discussed. The contact between the two faces of crack is considered. Numerical results show that (1) only the anti-symmetric mode is prone to thickness locking thus remedy should be made only on this part, (2) the proposed 2D spectral finite element can adequately model the Lamb wave propagation in plate-like structures and the complex scattering for the crack, and (3) crack location can be well determined by a PZT-induced Lamb wave-based diagnosis algorithm.     

Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements

This paper presents a formulation of an approximate spectral element for uniform and tapered rotating Euler–Bernoulli beams. The formulation takes into account the varying centrifugal force, mass and bending stiffness. The dynamic stiffness matrix is constructed using the weak form of the governing differential equation in the frequency domain, where two different interpolating functions for the transverse displacement are used for the element formulation. Both free vibration and wave propagation analysis is performed using the formulated elements. The studies show that the formulated element predicts results, that compare well with the solution available in the literature, at a fraction of the computational effort. In addition, for wave propagation analysis, the element shows superior convergence.     

PWAS激励Lamb波的谱有限元建模及参数研究

基于谱有限元法发展了一种由压电晶片主动传感器(PWAS)、胶层和主结构组成的三层模型,来模拟PWAS激励结构中Lamb波的传播。首先在各层使用不同的梁理论,推导PWAS-胶层-主结构三层模型的控制方程和力的边界条件,建立谱有限元模型。通过和传统的有限单元法进行比较,表明了在显著提高计算效率的同时,所发展谱有限元模型在分析结构中Lamb波传播上仍具有较高的精度。分析了激励频率、PWAS长度与厚度、胶层厚度等参数变化对输出电压信号的影响,可以为基于PWAS和Lamb波的主动健康监测技术提供参考。     

Propagation of surface (seismic) waves: Ordinary differential equations with strongly nonlinear damping

Second-order ordinary differential equations (ODEs) with strongly nonlinear damping (cubic nonlinearities) govern surface wave motions that entail nonlinear surface seismic motions. They apply to dynamic crack propagation and nonlinear oscillation problems in physics and nonlinear mechanics. It is shown that the nonlinear surface seismic wave equation (Rayleigh equation) admits several functional transformations and it is possible to reduce it to an equivalent first-order Abel ODE of the second kind in normal form. Based on a recently developed methodology concerning the construction of exact analytic solutions for the type of Abel equations under consideration, exact solutions are obtained for the nonlinear seismic wave (NLSW) equation for initial conditions of the physical problem. The method employed is general and can be applied to a large class of relevant ODEs in mathematical physics and nonlinear mechanics.     

Numerical simulation of flow‐induced noise using LES/SAS and Lighthill's acoustic analogy

We present a finite element (FE) formulation of Lighthill's acoustic analogy for the hybrid computation of noise generated by turbulent flows. In the present approach, the flow field is computed using large eddy simulation and scale adaptive simulation turbulence models. The acoustic propagation is obtained by solving the variational formulation of Lighthill's acoustic analogy with the FE method. In order to preserve the acoustic energy, we compute the inhomogeneous part of Lighthill's wave equation by applying the FE formulation on the fine flow grid. The resulting acoustic nodal loads are then conservatively interpolated to the coarser acoustic grid. Subsequently, the radiated acoustic field can be solved in both time and frequency domains. In the latter case, an enhanced perfectly matched layer technique is employed, allowing one to truncate the computational domain in the acoustic near field, without compromising the numerical solution. Our hybrid approach is validated by comparing the numerical results of the acoustic field induced by a corotating vortex pair with the corresponding analytical solution. To demonstrate the applicability of our scheme, we present full 3D numerical results for the computed acoustic field generated by the turbulent flow around square cylinder geometries. The sound pressure levels obtained compare well with measured values. Copyright © 2009 John Wiley & Sons, Ltd.     

Daubechies条件小波有限元法研究

为拓展小波理论在结构工程中的应用,提高结构计算精度,提出了以Daubechies条件小波Ritz法为基础的Daubechies条件小波有限元法。该法结合广义变分原理和拉格朗日乘子法构造修正泛函,根据修正泛函的驻值条件得到全域法求解方程矩阵。根据构件的边界条件,按左右边界对求解矩阵进行相应拆分,构建条件小波单元刚度矩阵,并依据公共节点位移相等原则形成总体刚度矩阵,由此解得各单元的小波基待定系数,即可进一步求解位移场函数、内力分布函数及荷载集度函数。以工程中常见的弹性拉压杆及平面弯曲梁为例,详细阐述了该方法的构造过程。并通过典型算例将Daubechies条件小波有限元法计算值与理论解进行了对比,结果表明:在弹性拉压杆算例中,位移、应力、载荷集度的相对误差均在1.22×10-3%以内;在平面弯曲梁算例中,挠度、弯矩、载荷集度的相对误差均在8.91×10-2%以内。     

Vibration modelling of structural networks using a hybrid finite element/wave and finite element approach

The vibration modelling of waveguide structures is considered. These structures comprise waveguides connected via joints. Traditionally, analytical models of the wave behaviour of such structures can be developed if they are simple (beams or rods connected at point joints, etc.). However, if the waveguides are of complicated constructions (truss-like, layered media, etc.) or the joints are complicated (e.g. of significant physical dimensions), obtaining the wave characteristics might be a formidable task. In this paper, such structures are modelled using a hybrid finite element/wave and finite element (FE/WFE) approach. The waveguides are modelled using the WFE method and thus their wave characteristics are obtained regardless of the complexity of their cross-section. The joints are modelled using standard FE, and the WFE and FE models are coupled to yield the scattering properties of the joints. The propagation and scattering models are assembled to describe the behaviour of the structure using relatively small models, while also providing information for other applications such as structure-borne sound, statistical energy analysis, etc. Numerical examples are presented to illustrate the approach.     

奇异Hamilton系统矩阵的精细积分法

常规位移有限元的结构振动方程是n个二阶常微分方程组.采用一般交分原理推导,将结构振动问题引入Hamiltoil体系,将得到2n个一阶常微分方程组.精细积分法宜于处理一阶方程,应用于线性定常结构动力问题求解,可以得到在数值上逼近精确解的结果.对于非齐次动力方程,当结构具有刚体位移时,系统矩阵将出现奇异.本文借鉴全元选大元高斯-约当法求解线性方程组的经验,提出全元选大元法求奇异矩阵零本征解的方法,该方法可以简便快速地寻求奇异矩阵零本征值对应的子空间.利用Hamiltoil体系已有研究成果及Hamilton系统的共轭辛正交归一关系,迅速将零本征值对应的子空间分离出来,通过投影排除奇异部分,然后用精细积分法求得问题的解.数值算例表明,该方法对Hamilton系统奇异问题,处理方便,计算量小,易于实现,同时保持了精细算法的优点.     

COMPACTLY SUPPORTED NON-TENSOR PRODUCT FORM TWO-DIMENSION WAVELET FINITE ELEMENT

Some theorems of compactly supported non-tensor product form two-dimension Daubechies wavelet were analysed carefully. Compactly supported non-tensor product form two-dimension wavelet was constructed, then non-tensor product form two dimension wavelet finite element was used to solve the deflection problem of elastic thin plate. The error order was researched. A numerical example was given at last.     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.     

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections

Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.     

Analysis of linear elastic wave propagation in piping systems by a combination of the boundary integral equations method and the finite element method

The combined methodology of boundary integral equations and finite elements is formulated and applied to study the wave propagation phenomena in compound piping systems consisting of straight and curved pipe segments with compact elastic supports. This methodology replicates the concept of hierarchical boundary integral equations method proposed by L. I. Slepyan to model the time-harmonic wave propagation in wave guides, which have components of different dimensions. However, the formulation presented in this article is tuned to match the finite element format, and therefore, it employs the dynamical stiffness matrix to describe wave guide properties of all components of the assembled structure. This matrix may readily be derived from the boundary integral equations, and such a derivation is superior over the conventional derivation from the transfer matrix. The proposed methodology is verified in several examples and applied for analysis of periodicity effects in compound piping systems of several alternative layouts.     

分层介质中Love波的扩展W-W算法

应用精细积分法(PIM)和扩展Wittrick-Williams(W-W)算法求解横观各向同性分层半空间中的Love波问题.Love波对应于波数-频率域线性常微分方程的本征值问题.精细积分法是求解线性常微分方程两端边值问题和初值问题的高精度算法.利用本征值计数技术,扩展W-W算法可以不遗漏地找到所有本征值.因此,文中使用的方法可以得到计算机精度意义下的精确解.     

Buckling analysis of laminated composite plates containing delaminations using the enhanced assumed strain solid element

This study investigates buckling behaviors of laminated composite structures with a delamination using the enhanced assumed strain (EAS) solid element. The EAS three-dimensional finite element (FE) formulation described in this paper, in comparison with the conventional approaches, is more attractive not only because it shows better accuracy but also it converges faster, especially for distorted element shapes. The developed FE model is used for studying cross-ply or angle-ply laminates containing an embedded delamination as well as through-the-width delamination. The numerical results obtained are in good agreement with those reported by other investigators. In particular, new results reported in this paper are focused on the significant effects of the local buckling for various parameters, such as size of delamination, aspect ratio, width-to-thickness ratio, stacking sequences, and location of delamination and multiple delaminations.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

Numerical modeling of damage detection in concrete beams using piezoelectric patches

Research and development of active monitoring systems for reinforced concrete structures should lead to improved structural safety and reliability. Numerical models of active monitoring and damage detection systems can help in the development and implementation of these systems. Modeling of damage detection process in a concrete beam with piezoelectric sensors/actuators based on wave propagation is investigated in this paper. Numerical modeling process is divided into two parts: (1) piezoelectric smart aggregates (SA), and (2) wave propagation models. Displacement obtained in the SA model is used as an input parameter for the modeling of wave propagation. Wavelet analysis is used as a signal processing tool and the damage index is calculated based on the wave energy. In this paper root-mean-square deviation (RMSD) damage index is used. Damage indices obtained by this numerical analysis are compared with experimental results. Very good fit between the finite element (FE) results and experimental results confirm a good FE approach of this problem.