Uncertainty propagation in modal analysis of viscoelastic sandwich structures using a stochastic collocation method

Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures. One challenge in solving this problem is enormous computation cost from the UP analysis. This paper aims to develop a novel computational effective method based on the combination of the adaptive sparse grid collocation (ASGC) method and the high dimensional model representation (HDMR) technique to evaluate the variability of the frequency response of viscoelastic damping structures. First, a well-validated layer-wise finite element method is employed to model the viscoelastic damping structures. The direct frequency response (DFR) method is utilized to calculate the response. Then, a modified adaptive strategy using expectation increments as the sampling indicator is proposed for cost-effective sparse grid construction. Lastly, high dimensional model representation (HDMR) technique is introduced to address the difficulty in moderate and high random dimensional situations. Two numerical examples are provided to assess the performances of the proposed method. Variations in constitutive parameters of viscoelastic material and thicknesses of the viscoelastic layer are considered. Numerical results show that, compared to the original high-dimensional ASGC method and Monte Carlo Simulation (MCS), the proposed method accurately predicts the variability of frequency responses and significantly improves the computational efficiency.

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

In vibration analysis of sandwich beam/plates, it is often assumed that there is shear deformation only, without extension or compression in the viscoelastic layer. Certainly, this assumption may have limitations, for example, with increase of the core thickness or frequency range of vibration. The purpose of this paper is to consider the normal, as well as shear strain of the core material for modal parameter estimation of the sandwich plates and to investigate how much error will be caused by neglecting the extension or compression in the core material. Natural frequencies and modal loss factors are estimated for a simply supported square plate by taking the normal as well as shear deformation into account for dynamic modeling. Nondimensional characteristic equations are formulated and solved numerically for various ratios of the base layer thickness to plate length, core layer thickness to base layer, constraining layer thickness to base layer, and shear modulus of core material to elastic modulus of base layer. The effects of the various parameters on the modal properties are shown to be intercorrelated to each other and hence difficult to summarize in one phrase. Normal deformation of the core material plays an important role when the thickness ratio of constraining layer to base layer is 0.5 and its Poisson’s ratio is smaller than 0.49, and hence need to be included in the dynamic modeling especially for estimation of modal damping when one of the following conditions are met; 1) the ratio of base layer thickness to plate length is greater than 0.02, 2) the thickness ratio of core layer to base layer is greater than 0.01, 3) the wavelength of a mode is less than one third of the plate length, 4) the ratio of shear modulus of core material to elastic modulus of base material is less than 10−5.

In this paper, the Stochastic Collocation Method is introduced into EMC uncertainty analysis. It is demonstrated to be accurate, with high computational efficiency, compared with the Method of Moments, the perturbation method and the Monte Carlo method. The Stochastic Collocation Method requires no modification to the original solver, and it is fit for being introduced into EMC simulation software.

Bending vibration of flat plates is controlled using patches of active constrained layer damping (ACLD) treatments. Each ACLD patch consists of a visco-elastic damping layer which is sandwiched between two piezo-electric layers. The first layer is directly bonded to the plate to sense its vibration and the second layer acts as an actuator to actively control the shear deformation of the visco-elastic damping layer according to the plate response. With such active/passive control capabilities the energy dissipation mechanism of the visco-elastic layer is enhanced and its damping characteristics of the plate vibration is improved. A finite element model is developed to analyze the dynamics and control of flat plates which are partially treated with multi-patches of ACLD treatments. The model is validated experimentally using an aluminum plate which is 0.05 cm thick, 25.0 cm long, and 12.5 cm wide. The plate is treated with two ACLD patches. Each of which is made of SOUNDCOAT (Dyad 606) visco- elastic layer sandwiched between two layers of AMP/polyvinylidene fluoride (PVDF) piezo- electric films. The piezo-electric axes of the patches are set at zero degrees relative to the plate longitudinal axis to control the bending mode. The effect of the gain of a proportional control action on the system performance is presented. Comparisons between the theoretical predictions and the experimental results suggest the validity of the developed finite element model. Also, comparisons with the performance of conventional passive constrained layer damping clearly demonstrate the merits of the ACLD as an effective means for suppressing the vibration of flat plates.

Fast and accurate evaluation of system failure rate is performed using stochastic collocation methods. First, we demonstrate that variability in I/O performance, such as driven voltage and slew rate, will impact failure probability of the link. For instance, higher slew rates will lead to increased levels of crosstalk between signals. Crosstalk above a certain threshold can be an indicator of system failure. Then, we demonstrate that by defining an upper bound for failure probability, an optimally relaxed set of I/O performance metrics can be generated corresponding to a stochastic interconnect model. This can be achieved by minimizing a cost function formulated with some performance metric, if these metrics are random variables. Lastly, we demonstrate that, if more information about the packaging design becomes available, the randomness of the interconnect model can be reduced, leading to a more relaxed set of I/O performance metrics. The method proposed in [1] is then be used to verify that the packaging design will meet the tolerated failure probability.

This paper concerns the dynamic analysis of composite beams containing elastic and viscoelastic (VE) layers. A method for determination of the dynamic characteristics of multilayered beams with VE layers (i.e., natural frequencies, non-dimensional damping ratios and modes of vibration) is presented. The Euler–Bernoulli beam theory and the Timoshenko theory are used to describe the elastic and VE layers, respectively. To describe the mechanical properties of a VE layer, the four-parameter rheological model with fractional derivative is applied. A number of particular rheological models are particular cases of this general model. The virtual work principle and the finite element method together with the Laplace transform are used to derive the equation of motion in the frequency domain. The dynamic characteristics of a beam with VE layers are obtained as the solution to a properly defined nonlinear eigenvalue problem. The continuation method is adopted for solving the nonlinear eigenproblem. Several conclusions concerning the accuracy of the method and variability of results are presented on the basis of numerical studies.

A computer model has been written to predict the interaction of a plane acoustic wave with an elastic or viscoelastic layered sphere. This model is based on the decomposition of the acoustic potentials in each layer in spherical harmonics. It can compute the scattered near‐ and farfields of spheres formed of up to ten layers, at frequencies up to ka = 120, a being the radius of the sphere and k the wavenumber of the incident wave. This model is used to study the influence of a constrained layer, consisting of a viscoelastic layer and a constraining outer shell, on the resonances of a metallic hollow sphere. Viscoelastic materials are modeled according to the equations of the Kelvin‐Voigt model, with complex elastic moduli. Plots of the acoustic pressure backscattered by the sphere, of the total pressure on its surface, of its surface displacements, and color‐coded maps of the pressure around the sphere are presented to show the influence of the thickness of the viscoelastic layer and of its absorption coefficients on the resonances of the metallic shell, and the effect of the resonances of the three layers on the scattered field.

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.

Uncertainty investigations in nonlinear aeroelastic systems

Viscoelastic layered surface treatments are widely used for passive control of vibration and noise, especially in passenger vehicles and buildings. When the viscoelastic layer is thick, the structural models must account for shear effects. In this work, a homogenised formulation for thick N-layered viscoelastic structures for finite element applications is presented, which allows for avoiding computationally expensive models based on solids. This is achieved by substituting the flexural stiffness in the governing thin beam or plate equation by a frequency dependent equivalent flexural stiffness that takes shear and the properties of the different layers into account. The formulation is applied to Free Layer Damping (FLD) and Constrained Layer Damping (CLD) beams and plates and its ability to accurately compute the eigenpairs and dynamic response is tested by implementing it in a finite element model and comparing the obtained results to those given by the standard for the application—Oberst for the FLD case and RKU for the CLD one—and to a solid model, which is used as reference. For the cases studied, the homogenised formulation is nearly as precise as the model based on solids, but requires less computational effort, and provides better results than the standard model.

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

In this study, we investigated the impact of various simulated skull bone geometries on the determination of skull speed of sound and acoustic attenuation values via optimization using transmitted pressure amplitudes beyond the bone. Using the hybrid angular spectrum method (HAS), we simulated ultrasound transmission through four model sets of different geometries involving sandwiched layers of diploë and cortical bone in addition to three models generated from CT images of ex-vivo human skull-bones. We characterized cost-function solution spaces for each model and, using optimization, found that when a model possessed appreciable variations in resolvable layer thickness, the predefined attenuation coefficients could be found with low error (RMSE < 0.01 Np/cm). However, we identified a spatial frequency cutoff in the models' geometry beyond which the accuracy of the property determination begins to fail, depending on the frequency of the ultrasound source. There was a large increase in error of the attenuation coefficients determined by the optimization when the variations in layer thickness were above the identified spatial frequency cutoffs, or when the lateral variations across the model were relatively low in amplitude. For our limited sample of three CT-image derived bone models, the attenuation coefficients were determined successfully. The speed of sound values were determined with low error for all models (including the CT-image derived models) that were tested (RMSE < 0.4 m/s). These results illustrate that it is possible to determine the acoustic properties of two-component models when the internal bone structure is taken into account and the structure satisfies the spatial frequency constraints discussed.

Viscoelastic material can significantly reduce the vibration energy and radiated noise of a structure, so it is widely used in lightweight sandwich structures. The accurate and efficient determination of the frequency-dependent complex modulus of viscoelastic material is the basis for the correct analysis of the vibro-acoustic behavior of sandwich structures. Based on the behavior of a sandwich beam whose core is a viscoelastic layer, a combined theoretical and experimental study is proposed to characterize the properties of the viscoelastic layer constituting the core. In this method, the viscoelastic layer is bonded between two constraining layers. Then, a genetic algorithm is used to fit the analytical solution of the frequency¬ response function of the free–free constrained beam to the measured result, and then the frequency-dependent complex modulus is estimated for the viscoelastic layer. Moreover, by varying the length of the beams, it is possible to characterize the frequency-dependent complex modulus of the viscoelastic material over a wide frequency range. Finally, the characterized frequency-dependent complex modulus is imported into a finite element model to compute the complex natural frequencies of a sandwich beam, and a comparison of the simulated and measured results displays that the errors in the real parts are within 2.33% and the errors in the imaginary parts are within 3.31%. It is confirmed that the proposed method is feasible, accurate, and reliable. This provides essential technical support for improving the acoustic vibration characteristics of sandwich panels by introducing viscoelastic materials.

In recent years, the demand to provide best-estimate predictions with confident bounds is increasing for the nuclear reactor performance and safety analysis. The Organisation for Economic Co-operation and Development Nuclear Energy Agency has been developing an international benchmark of the light water reactor (LWR) uncertainty analysis in modeling (UAM) for the examination of uncertainty quantification and propagation methodologies with various modeling and simulation code systems. The objective of the present work is to develop an uncertainty propagation mechanism based on the stochastic sampling method by taking into account the uncertainties of both basic nuclear data and fuel modeling parameters in the simulation of pressurized water reactors (PWRs) that can be incorporated in the conventional LWR simulation approach. More specifically, the Three Mile Island Unit 1–related exercises from the LWR-UAM benchmark have been modeled using the coupled TRACE/PARCS code system in the three-dimensional core representation. The input uncertainties of the neutronics simulation include few-group cross sections and kinetics parameters generated using the Sampler/Polaris sequence of SCALE 6.2.1. Several heat transfer–related variables for the fuel modeling were considered as sources of input uncertainty of the thermal-hydraulics simulations, including the thermal conductivity of fuel and cladding, fuel heat capacity, and the gap conductance. Dakota was used to sample input parameters of the coupled code system and to perform the uncertainty analysis. Two types of simulations were conducted: steady-state calculation at hot full-power condition and transient scenario initiated by the spatially asymmetric rod ejection accident. Quantities of interest for the steady-state calculation, including core multiplication factor and power peaking factors, were calculated with associated uncertainties. For transient calculations, best-estimate plus uncertainty results of the time evolution of core reactivity, core power, and peak fuel temperature were generated and analyzed. The Wilks’ formula was used to determine the necessary sample size to achieve a 95% confidence of the 95% limit of output quantities of interest. Although the uncertainty propagation and quantification method presented in this paper was developed for PWRs, it could be in general applicable to the multiphysics uncertainty quantification of other types of LWR cores.