A tensor artificial viscosity using a finite element approach

We derive a tensor artificial viscosity suitable for use in a 2D or 3D unstructured arbitrary Lagrangian–Eulerian (ALE) hydrodynamics code. This work is similar in nature to that of Campbell and Shashkov [1]; however, our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework. The finite element point of view leads to novel insights as well as improved numerical results. We begin with a generalized tensor version of the Von Neumann–Richtmyer artificial viscosity, then convert it to a variational formulation and apply a Galerkin discretization process using high order Gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating (or shock entropy) term. This technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws; however, we motivate the use of such finite element approaches for discretizing each term in the Euler equations. We review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient. We also show, that under certain simplifying assumptions, the two-dimensional scheme from [1] can be viewed as an under-integrated version of our finite element method. This equivalence holds on general distorted quadrilateral grids. Finally, we present computational results on some standard shock hydro test problems, as well as some more challenging problems, indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids.     

A computational approach to calculating frequency-dependent structural operators using the spectral finite element method and a wavenumber-based gridding technique

Spectral finite element methods are used to compute exact vibration solutions of structural models at specific frequencies. The applicability of these methods to certain areas of structural dynamics is limited by two major factors: the lack of separate structural operators (mass, damping, and stiffness matrices), and the subsequent difficulty in computing mode shapes via eigenvalue decomposition. In the work presented in this article, a method is investigated to accurately calculate spectral finite elements while overcoming these limitations. The approach incorporates a two-dimensional, discrete solution utilizing a wavenumber-based gridding technique to compute frequency-dependent local mass, damping, and stiffness matrices which can be assembled into the global structural operators. Computed models are able to be used for precise vibration analysis as well as modal analysis via eigenvalue decomposition of the structural operators.     

The vibrations of generally orthotropic beams,a finite element approach

This paper presents two finite element models for the prediction of free vibrational natural frequencies of fixed-free beams of general orthotropy. The discrete models include the transverse shear deformation effect and the rotary inertia effect. Numerical studies show that the convergence rate of the approximations calculated from the finite element analysis is dependent on the fibre orientation.     

Assessment of rod behaviour theories used in spectral finite element modelling

In this work different theories of rods have been discussed and compared. The investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves. These are various single, two-mode and three-mode theories including the elementary, classical Love and Mindlin-Herrmann approaches as well as new two, three and four-mode theories proposed by the authors. Dispersion curves associated with each theory, obtained by the use of Hamilton's principle, have been presented and discussed in the paper. The investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application. Great attention has been paid on properties, limitations as well as difficulties associated with the use of the theories. The results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures.     

A finite element approach for predicting nozzle admittances

A finite element method is used to predict the admittances of axisymmetric nozzles. It is assumed that the flow in the nozzle is isentropic and irrotational, and the disturbances are small so that linear analyses apply. An approximate, two dimensional compressible model is used to describe the steady flow in the nozzle. The propagation of acoustic disturbances is governed by the complete linear wave equation. The differential form of the acoustic equation is transformed to an integral equation by using Galerkin's method, and Green's theorem is applied so that the acoustic boundary conditions can be introduced through the boundary residuals. The boundary conditions are described for both straight and curved sonic lines. A two dimensional FEM with linear elements is used to solve the acoustic equation. A one dimensional FEM is also used to solve the reduced equation of Crocco, and the solution verifies the sufficiency of the boundary residual formulation. Comparison between computed admittances and experimental data is shown to be quite good.     

基于有限元模拟的超声应力系数标定及分析*

应力系数的标定作为超声应力检测最为关键的环节,直接决定应力检测结果的准确性。传统的应力系数试验标定对于被测物的表面粗糙度、耦合剂厚度、声匹配块与被测物接触力等因素十分敏感,但缺少基本参照值。基于COMSOL建立多物理场耦合的超声应力检测模型,施加不同的拉伸载荷,计算临界折射纵波到达时间与不同应力值之间的关系,模拟标定45#钢的超声应力系数为13.7MPa/ns。单轴水平拉伸试验标定的45#钢应力系数为16.5MPa/ns。结果表明,通过两种方法标定的应力系数较为接近,试验标定的应力系数偏大,这是由于有限元方法能够消除试验过程中各种不确定因素对声时精确测量所造成的影响,能够更加纯粹的反映材料的声弹性效应,因此具有作为基础数据的参考价值。有限元方法作为传统试验方法的补充,可以减小试验标定数据的离散性,提高超声应力检测结果的可信度。     

A new multi-objective approach to finite element model updating

The single objective function (SOF) has been employed for the optimization process in the conventional finite element (FE) model updating. The SOF balances the residual of multiple properties (e.g., modal properties) using weighting factors, but the weighting factors are hard to determine before the run of model updating. Therefore, the trial-and-error strategy is taken to find the most preferred model among alternative updated models resulted from varying weighting factors. In this study, a new approach to the FE model updating using the multi-objective function (MOF) is proposed to get the most preferred model in a single run of updating without trial-and-error. For the optimization using the MOF, non-dominated sorting genetic algorithm-II (NSGA-II) is employed to find the Pareto optimal front. The bend angle related to the trade-off relationship of objective functions is used to select the most preferred model among the solutions on the Pareto optimal front. To validate the proposed approach, a highway bridge is selected as a test-bed and the modal properties of the bridge are obtained from the ambient vibration test. The initial FE model of the bridge is built using SAP2000. The model is updated using the identified modal properties by the SOF approach with varying the weighting factors and the proposed MOF approach. The most preferred model is selected using the bend angle of the Pareto optimal front, and compared with the results from the SOF approach using varying the weighting factors. The comparison shows that the proposed MOF approach is superior to the SOF approach using varying the weighting factors in getting smaller objective function values, estimating better updated parameters, and taking less computational time.     

Multiscale finite element methods for high-contrast problems using local spectral basis functions

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ₁)^1/2, where Λ₁ is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings.     

Analysis of nonlinear electric field of hvdc wall bushing with a finite element approach

The present research intends to establish a numerical model, on the basis of a theoretical analysis, for describing and analyzing the electric field of High Voltage Direct Current (HVDC) wall bushing that demonstrates highly nonlinear characteristics. The wall bushing is subjected high voltage with nonlinear electric field and the relationship between the electric field intensity and the resistance of the insulators of the wall bushing is highly nonlinear. With a parameter design language of a Finite Element Analysis software package for carrying out the numerical calculations, the effects of the nonlinearity on the electric field can be well taken into consideration in performing the numerical assessment. A technique utilizing the numerical iteration is developed for quantifying the electric intensity of the electric field. With the model and the iteration technique established, the nonlinear characteristics of the HVDC wall bushing can be investigated with efficiency.     

A finite element approach for scattering from inhomogeneous media with a rough interface

Electromagnetic scattering from an inhomogeneous medium with a one-dimensional rough interface is analysed. The proposed procedure combines the finite element method (FEM), to model the electromagnetic field in the inhomogeneous region, with a perturbative technique to account for the contributions due to the rough interface. Backscattering and bistatic scattering coefficients are computed and plotted for both plane wave and Gaussian beam incident fields in the case of TM_z polarization.     

A spectral finite element for analysis of wave propagation in uniform composite tubes

A spectral finite element model (SFEM) for analysis of coupled broadband wave propagation in composite tubular structure is presented. Wave motions in terms of three translational and three rotational degrees of freedom at tube cross-section are considered based on first order shear flexible cylindrical bending, torsion and secondary warping. Solutions are obtained in wavenumber space by solving the coupled wave equation in 3-D. An efficient and fully automated computational strategy is developed to obtain the wavenumbers of coupled wave modes, spectral element shape function, strain-displacement matrix and the exact dynamic stiffness matrix. The formulation emphasizes on a compact matrix methodology to handle large-scale computational model of built-up network of such cylindrical waveguides. Thickness and frequency limits for application of the element is discussed. Performance of the element is compared with analytical solution based on membrane shell kinematics. A map of the distribution of vibrational modes in wavelength and time scales is presented. Effect of fiber angle on natural frequencies, phase and group dispersions are also discussed. Numerical simulations show the ease with which dynamic responses can be obtained efficiently. Parametric studies on a clamped-free graphite-epoxy composite tube under short-impulse load are carried out to obtain the effect of various composite configurations and tube geometries on the response.     

Random response of periodic structures by a finite element technique

A finite element method is presented for analysing the response of periodic structures to convected random pressure fields. It is shown that the problem reduces to one of finding the response of a single periodic section to a harmonic pressure wave. In this case the inertia, stiffness and damping matrices become functions of the phase difference between the pressures at corresponding points in adjacent sections. The method is applied to a skin-rib type structure.     

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

A computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space using a truncated Karhunen–Loève expansion with several random variables. Due to the small correlation length of the covariance function, this often results in a high stochastic dimensionality. Therefore, a newly developed adaptive high dimensional stochastic model representation technique (HDMR) is used in the stochastic space. This results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method (ASGC). Numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method.     

Inversion of the classical second virial coefficient

It is shown that, on the basis of some weak assumptions regarding the nature of the intermolecular pair potential, the classical second virial coefficient determines the potential uniquely.Research supported by NSF Grant GP-19881.     

Inversion of acoustic mode coupling coefficient matrix

It is found that the normal mode amplitude time series consist of multi-frequency component by analyzing the structure of acoustical signal when internal wave propagation exists, and each frequency is the product of internal wave speed and the normal mode wave number difference between acoustical receivers and source. The amplitude of each component is proportional to the acoustic mode coupling coefficient. The structure of the normal mode coefficient time series is still complex even the internal waves do not reshape when they propagate from the acoustical receivers to the source. A method is presented to compute the AMCCM by the feature of IWs' motion and the relation between the AMCCM and the acoustical signal fluctuation amplitude. The IWs data measured in the 2001 Asia experiment (ASIAEX2001) is used to check the accuracy of this method by numerical simulation. It is show that the method is accurate to compute the AMCCM.     

A three-dimensional hybrid finite element/spectral analysis of noise radiation from turbofan inlets

This paper describes a new three-dimensional (3D) analysis of tonal noise radiated from non-axisymmetric turbofan inlets. The novelty of the method is in combining a standard finite element discretisation of the acoustic field in the axial and radial coordinates with a Fourier spectral representation in the circumferential direction. The boundary conditions at the farfield, fan face and acoustic liners are treated using the same spectral representation. The resulting set of discrete acoustic equations are solved employing the well-established BICGSTAB or QMR iterative algorithms and a very effective specialised preconditioner based on the axisymmetric mean geometry and flow field. Numerical examples demonstrate the suitability of the new method to engine configurations with realistic 3D features, such as relatively large degrees of asymmetry and spliced acoustic liners. The examples also illustrate the two advantages of the new method over a traditional 3D finite element approach. The new method requires a significantly smaller number of unknowns as relatively few circumferential Fourier modes in the spectral solution ensure an accurate field representation. Also, due to the effective preconditioner, the spectral linear solver benefits from stable iterations at a high rate of convergence.     

Free vibration analysis of a cracked beam by finite element method

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.