Doubly spectral stochastic finite element method (DSSFEM) for random field problems

Uncertainties in complex dynamic systems play an important role in the prediction of a dynamic response in the mid- and high-frequency ranges. For distributed parameter systems, parametric uncertainties can be represented by random fields leading to stochastic partial differential equations. Over the past two decades, the spectral stochastic finite-element method has been developed to discretize the random fields and solve such problems. On the other hand, for deterministic distributed parameter linear dynamic systems, the spectral finite-element method has been developed to efficiently solve the problem in the frequency domain. In spite of the fact that both approaches use spectral decomposition (one for the random fields and the other for the dynamic displacement fields), very little overlap between them has been reported in literature. In this paper, these two spectral techniques are unified with the aim that the unified approach would outperform any of the spectral methods considered on their own. An exponential autocorrelation function for the random fields, a frequency-dependent stochastic element stiffness, and mass matrices are derived for the axial and bending vibration of rods. Closed-form exact expressions are derived by using the Karhunen-Loève expansion. Numerical examples are given to illustrate the unified spectral approach.

Guided-wave based structural health monitoring of built-up composite structures using spectral finite element method

The paper discusses basically a wave propagation based method for identifying the damage due to skin-stiffener debonding in a stiffened structure. First, a spectral finite element model (SFEM) is developed for modeling wave propagation in general built-up structures, using the concept of assembling 2D spectral plate elements and the model is then used in modeling wave propagation in a skin-stiffener type structure. The damage force indicator (DFI) technique, which is derived from the dynamic stiffness matrix of the healthy stiffened structure (obtained from the SFEM model) along with the nodal displacements of the debonded stiffened structure (obtained from 2D finite element model), is used to identify the damage due to the presence of debond in a stiffened structure.

Wave propagation analysis in laminated composite plates with transverse cracks using the wavelet spectral finite element method

This paper presents a newly developed wavelet spectral finite element (WFSE) model to analyze wave propagation in anisotropic composite laminate with a transverse surface crack penetrating part-through the thickness. The WSFE formulation of the composite laminate, which is based on the first-order shear deformation theory, produces accurate and computationally efficient results for high frequency wave motion. Transverse crack is modeled in wavenumber-frequency domain by introducing bending flexibility of the plate along crack edge. Results for tone burst and impulse excitations show excellent agreement with conventional finite element analysis in Abaqus (R). Problems with multiple cracks are modeled by assembling a number of spectral elements with cracks in frequency-wavenumber domain. Results show partial reflection of the excited wave due to crack at time instances consistent with crack locations. (C) 2014 Elsevier B.V. All rights reserved.

Wave propagation analysis in adhesively bonded composite joints using the wavelet spectral finite element method

A wavelet spectral finite element (WSFE) model is developed for studying transient dynamics and wave propagation in adhesively bonded composite joints. The adherands are formulated as shear deformable beams using the first order shear deformation theory (FSDT) to obtain accurate results for high frequency wave propagation. Equations of motion governing wave motion in the bonded beams are derived using Hamilton's principle. The adhesive layer is modeled as a line of continuously distributed tension/compression and shear springs. Daubechies compactly supported wavelet scaling functions are used to transform the governing partial differential equations from time domain to frequency domain. The dynamic stiffness matrix is derived under the spectral finite element framework relating the nodal forces and displacements in the transformed frequency domain. Time domain results for wave propagation in a lap joint are validated with conventional finite element simulations using Abaqus. Frequency domain spectrum and dispersion relation results are presented and discussed. The developed WSFE model yields efficient and accurate analysis of wave propagation in adhesively-bonded composite joints. (C) 2014 Elsevier Ltd. All rights reserved.

The partition of unity finite element method for elastic wave propagation in Reissner-Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four-noded element for the Reissner-indlin plate is derived in this paper, which is free of shear locking. Such a locking-free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency-dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright (C)006 John Wiley & Sons, Ltd.

Spectral element approach to wave propagation in uncertain beam structures

This paper presents a study on the uncertainty in material parameters of wave propagation responses in metallic beam structures. Special effort is made to quantify the effect of uncertainty in the wave propagation responses at high frequencies. Both the modulus of elasticity and the density are considered uncertain. The analysis is performed using a Monte Carlo simulation (MCS) under the spectral finite element method (SEM). The randomness in the material properties is characterized by three different distributions, the normal, Weibull and extreme value distributions. Their effect on wave propagation in beams is investigated. The numerical study shows that the CPU time taken for MCS under SEM is about 48 times less than for MCS under a conventional one-dimensional finite element environment for 50 kHz loading. The numerical results presented investigate effects of material uncertainties on high frequency modes. A study is performed on the usage of different beam theories and their uncertain responses due to dynamic impulse load. These studies show that even for a small coefficient of variation, significant changes in the above parameters are noticed. A number of interesting results are presented, showing the true effects of uncertainty response due to dynamic impulse load.

Spectral Element Approach to Wave Propagation in Uncertain Composite Beam Structures

This paper presents a study of the wave propagation responses in composite structures in an uncertain environment. Here, the main aim of the work is to quantify the effect of uncertainty in the wave propagation responses at high frequencies. The material properties are considered uncertain and the analysis is performed using Neumann expansion blended with Monte Carlo simulation under the environment of spectral finite element method. The material randomness is included in the conventional wave propagation analysis by different distributions (namely, the normal and the Weibul distribution) and their effect on wave propagation in a composite beam is analyzed. The numerical results presented investigates the effect of material uncertainties on different parameters, namely, wavenumber and group speed, which are relevant in the wave propagation analysis. The effect of the parameters, such as fiber orientation, lay-up sequence, number of layers, and the layer thickness on the uncertain responses due to dynamic impulse load, is thoroughly analyzed. Significant changes are observed in the high frequency responses with the variation in the above parameters, even for a small coefficient of variation. High frequency impact loads are applied and a number of interesting results are presented, which brings out the true effects of uncertainty in the high frequency responses. [DOI: 10.1115/1.4003945]

A spectral multiscale method for wave propagation analysis: Atomistic-continuum coupled simulation

In this paper, we present a new multiscale method which is capable of coupling atomistic and continuum domains for high frequency wave propagation analysis. The problem of non-physical wave reflection, which occurs due to the change in system description across the interface between two scales, can be satisfactorily overcome by the proposed method. We propose an efficient spectral domain decomposition of the total fine scale displacement along with a potent macroscale equation in the Laplace domain to eliminate the spurious interfacial reflection. We use Laplace transform based spectral finite element method to model the macroscale, which provides the optimum approximations for required dynamic responses of the outer atoms of the simulated microscale region very accurately. This new method shows excellent agreement between the proposed multiscale model and the full molecular dynamics (MD) results. Numerical experiments of wave propagation in a 1D harmonic lattice, a 1D lattice with Lennard-Jones potential, a 2D square Bravais lattice, and a 2D triangular lattice with microcrack demonstrate the accuracy and the robustness of the method. In addition, under certain conditions, this method can simulate complex dynamics of crystalline solids involving different spatial and/or temporal scales with sufficient accuracy and efficiency. (C) 2014 Elsevier B.V. All rights reserved.

A spectral element for wave propagation in honeycomb sandwich construction considering core flexibility

Spectral elements are found to be extremely resourceful to study the wave propagation characteristics of structures at high frequencies. Most of the aerospace structures use honeycomb sandwich constructions. The existing spectral elements use single layer theories for a sandwich construction wherein the two face sheets vibrate together and this model is sufficient for low frequency excitations. At high frequencies, the two face sheets vibrate independently. The Extended Higher order SAndwich Plate theory (EHSaPT) is suitable for representing the independent motion of the face sheets. A 1D spectral element based on EHSaPT is developed in this work. The wave number and the wave speed characteristics are obtained using the developed spectral element. It is shown that the developed spectral element is capable of representing independent wave motions of the face sheets. The propagation speeds of a high frequency modulated pulse in the face sheets and the core of a honeycomb sandwich are demonstrated. Responses of a typical honeycomb sandwich beam to high frequency shock loads are obtained using the developed spectral element and the response match very well with the finite element results. It is shown that the developed spectral element is able to represent the flexibility of the core resulting into independent wave motions in the face sheets, for which a finite element method needs huge degrees of freedom. (C) 2015 Elsevier Ltd. All rights reserved.

Multi-transform based spectral element to include first order shear deformation in plates

For obtaining dynamic response of structure to high frequency shock excitation spectral elements have several advantages over conventional methods. At higher frequencies transverse shear and rotary inertia have a predominant role. These are represented by the First order Shear Deformation Theory (FSDT). But not much work is reported on spectral elements with FSDT. This work presents a new spectral element based on the FSDT/Mindlin Plate Theory which is essential for wave propagation analysis of sandwich plates. Multi-transformation method is used to solve the coupled partial differential equations, i.e., Laplace transforms for temporal approximation and wavelet transforms for spatial approximation. The formulation takes into account the axial-flexure and shear coupling. The ability of the element to represent different modes of wave motion is demonstrated. Impact on the derived wave motion characteristics in the absence of the developed spectral element is discussed. The transient response using the formulated element is validated by the results obtained using Finite Element Method (FEM) which needs significant computational effort. Experimental results are provided which confirms the need to having the developed spectral element for the high frequency response of structures. (C) 2015 Elsevier Ltd. All rights reserved.

Estimation of gear tooth deflection by the finite element method

The finite element method (FEM) is used to determine for pitch-point, mid-point and tip loading, the deflection curve of a Image 1 diamentral pitch (DP) standard spur gear tooth corresponding to number of teeth of 14, 21, 26 and 34. In all these cases the deflection of the gear tooth at the point of loading obtained by FEM is in good agreement with the experimental value. The contraflexure in the deflection curve at the point of loading observed experimentally in the cases of pitch-point and mid-point loading, is predicted correctly by the FEM analysis.

Non-linear dynamic analysis of rotors by finite element method

Non-linear natural vibration characteristics and the dynamic response of hingeless and fully articulated rotors of rectangular cross-section are studied by using the finite element method. In the formulation of response problems, the global variables are augmented with appropriate additional variables, facilitating direct determination of sub-harmonic response. Numerical results are given showing the effect of the geometric non-linearity on the first three natural frequencies. Response analysis of typical rotors indicates a possibility of substantial sub-harmonic response especially in the fully articulated rotors widely adopted in helicopters.

A Legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids

A new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids. The formulation makes use of the Legendre polynomials in deriving the finite element interpolation shape functions in the Lagrangian frame of reference. The formulated element uses Gauss-Lobatto-Legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional Gauss-Legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes, which are normally associated with the Lagrangian FE formulation. The numerical performance of the spectral element formulated here is examined by doing the inf-sup test oil a standard rectangular rigid tank partially filled with liquid The eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h-type Lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence The efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements. and the obtained results are compared with available results in the published literature (C) 2009 Elsevier Inc All rights reserved

Modelling of solute transport in a mild heterogeneous porous medium using stochastic finite element method: Effects of random source conditions

Randomness in the source condition other than the heterogeneity in the system parameters can also be a major source of uncertainty in the concentration field. Hence, a more general form of the problem formulation is necessary to consider randomness in both source condition and system parameters. When the source varies with time, the unsteady problem, can be solved using the unit response function. In the case of random system parameters, the response function becomes a random function and depends on the randomness in the system parameters. In the present study, the source is modelled as a random discrete process with either a fixed interval or a random interval (the Poisson process). In this study, an attempt is made to assess the relative effects of various types of source uncertainties on the probabilistic behaviour of the concentration in a porous medium while the system parameters are also modelled as random fields. Analytical expressions of mean and covariance of concentration due to random discrete source are derived in terms of mean and covariance of unit response function. The probabilistic behaviour of the random response function is obtained by using a perturbation-based stochastic finite element method (SFEM), which performs well for mild heterogeneity. The proposed method is applied for analysing both the 1-D as well as the 3-D solute transport problems. The results obtained with SFEM are compared with the Monte Carlo simulation for 1-D problems.