Wave computations in irregular layered media using a semidiscrete finite element perturbation method

A technique that takes into account nonparallel interfaces and lateral inhomogeneities as perturbations with respect to a reference, horizontally layered, laterally homogeneous medium and produces approximations of the perturbed wave motion with little additional computation effort is described. The formulation combines a perturbation method with a semidiscrete finite element technique, providing an approximate treatment of wave propagation in irregular layered media. Consistent transmitting boundaries and other semidiscrete hyperelements as well as Green's functions, already available for regular layered media, can be reformulated within the framework of the method. The method is relevant in problems of foundation dynamics, ground response to seismic waves and site characterization. Example problems are presented toward evaluation of the effectiveness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method with plane waves for sound‐absorbing materials

Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd.     

Mixed enrichment for the finite element method in heterogeneous media

Problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand, different enriched finite element methods such as the partition of unity, which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work, we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context, the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials. Copyright © 2014 John Wiley & Sons, Ltd.     

Coupled BEM/FEM approach for nonlinear soil/structure interaction

A general coupled boundary element/finite element formulation is presented for the investigation of dynamic soil/structure interaction including nonlinearities. It is applied to investigate the transient inelastic response of structures coupled with a halfspace. The structure itself and the surrounding soil in the near field are modeled with finite elements. In this part of the model inhomogeneities and an elastoplastic material behavior with hardening effects can be taken into account. The remaining soil region, i.e. the elastic halfspace, is discretized with the boundary elements. Thus wave radiation to infinity is included in the model. In representative examples it is shown that the methodology is computationally powerful and can be used efficiently for the nonlinear analyses of complex soil/structure interaction problems.     

The effect of crack face contact on the anisotropic effective moduli of microcrack damaged media

The generalized self-consistent method (GSCM) in conjunction with a computational finite element method is used to calculate the anisotropic effective moduli of a medium containing damage consisting of microcracks with an arbitrary degree of alignment. Since cracks respond differently under different external loads, the moduli of the medium subjected to tension, compression and an initially stress-free state are evaluated and shown to be significantly different, which will further affect the wave speed inside the damaged media. There are four independent material moduli for a 2-D plane stress orthotropic medium in tension or compression, and seven independent material moduli for a 2-D plane stress orthotropic cracked medium, which is initially stress free. When friction exists, it further changes the effective moduli. Numerical methods are used to take into account crack face contact and friction. The wave slowness profiles for microcrack damaged media are plotted using the predicted effective material moduli.     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element modeling of transient ultrasonic waves in linear viscoelastic media

Linear viscoelasticity offers a minimal framework within which to construct a causal model for wave propagation in absorptive media. Viscoelastic media are often described as media with `fading memory,' that is, the present state of stress is dependent on the present strain and the complete time history of strain convolved with appropriate time-dependent shear and bulk stress relaxation moduli. An axisymmetric, displacement-based finite element method for modeling pulsed ultrasonic waves in linear, homogeneous, and isotropic (LHI) viscoelastic media is developed that does not require storage of the complete time history of displacement at every node. This is accomplished by modeling stress relaxation moduli as discrete or continuous spectra of decaying exponentials and relaxation times. Details of the construction and computation of the time-dependent stiffness matrix are presented. As an application of the finite element method, a finite number of exponentials (amplitudes and relaxation times) are employed to represent a typical model for a continuous relaxation spectrum. It is demonstrated that a small number of discrete exponentials are required to model ultrasonic wave propagation of a typical band-limited pulse in a model material accurately. Previous work has shown this model to be consistent with other analytic models for wave propagation in viscoelastic media     

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

In this paper we present a new approach for finite element solution of time‐harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co‐ordinates. The system of FE equations is therefore reduced to a set of well‐known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.     

A wave finite element‐based formulation for computing the forced response of structures involving rectangular flat shells

The harmonic forced response of structures involving several noncoplanar rectangular flat shells is investigated by using the Wave Finite Element method. Such flat shells are connected along parallel edges where external excitation sources as well as mechanical impedances are likely to occur. Also, they can be connected to one or several coupling elements whose shapes and dynamics can be complex. The dynamic behavior of the connected shells is described by means of numerical wave modes traveling towards and away from the coupling interfaces. Also, the coupling elements are modeled by using the conventional finite element (FE) method. A FE mesh tying procedure between shells having incompatible meshes is considered, which uses Lagrange multipliers for expressing the coupling conditions in wave‐based form. A global wave‐based matrix formulation is proposed for computing the amplitudes of the wave modes traveling along the shells. The resulting displacement solutions are obtained by using a wave mode expansion procedure. The accuracy of the wave‐based matrix formulation is highlighted in comparison with the conventional FE method through three test cases of variable complexities. The relevance of the method for saving large CPU times is emphasized. Its efficiency is also highlighted in comparison with the component mode synthesis technique. Copyright © 2013 John Wiley & Sons, Ltd.     

层状正交各向异性材料弹性波导问题的数值计算

研究复合材料弹性波导问题的数值计算方法。将问题导向哈密顿体系,在哈密顿体系中以位移向量和应力向量综合成全状态向量,在杂交体系中建立动力-部分杂交元,导出一套哈密顿体系下新的半解析法。本文中给出了该方法在分层正交各向异性材料的弹性波导问题的数值算例。计算结果展现了该方法在弹性波导问题的应用前景。     

地下隧道工程地震动分析的有限元－人工透射边界方法

采用有限元－人工透射边界方法计算了含有地下隧道工程的地基在ＳＨ波和瑞利波作用下的地震动反应，讨论了各自不同的地震动特性。并通过与边界积分方法比较，验证了有限元—人工透射边界方法对解决此类问题的有效性。     

地下隧道工程地震动分析的有限元－人工透射边界方法

采用有限元－人工透射边界方法计算了含有地下隧道工程的地基在ＳＨ波和瑞利波作用下的地震动反应，讨论了各自不同的地震动特性。并通过与边界积分方法比较，验证了有限元—人工透射边界方法对解决此类问题的有效性。     

外源波动问题数值模拟的一种实现方式

将人工边界上的波动场分解为无局部场地效应影响的自由场与局部场地效应引起的散射场两部分,无限域地基中的波动传播对人工边界的影响通过将位移场、速度场转化为应力场施加到人工边界节点上来反映,其中散射波场的模拟采用了杜修力等提出的一种远场近似解。在此基础上,利用通用有限元软件实现了斜入射条件下瞬态平面地震波作用引起的局部场地效应的时域数值模拟,为计算二维、三维局部不规则、非均匀场地的地震局部场地效应以及土结动力相互作用等外源波动问题提供了一种有效、实用的实现方式。建议方法的优点是可直接借助通用有限元软件强大的求解器和前后处理功能来计算和分析近场波动反应,且实现简便。实际上,由于通用有限元软件强大的求解功能,可方便地用于局部场地存在不均匀、非线性的情况。     

Implementation of the transformation field analysis for inelastic composite materials

Conclusions The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (6₂); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.This work was supported by the Air Force Office of Scienctific Research, and by the Office of Naval Research     

LATIN: A new view and an extension to wave propagation in nonlinear media

The LATIN (acronym of LArge Time INcrement) method was originally devised as a non‐incremental procedure for the solution of quasi‐static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time‐dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.     

插值小波自适应求解层状介质波传问题

利用多尺度插值小波理论,提出一种适合于求解一般边界层状介质波传问题的快速自适应配点算法。将问题放在多尺度插值空间中进行,其插值小波系数与物理空间点一一对应,可简单快速地处理非线性、边界等。使非均匀变化的响应得到自适应压缩存储,改善了计算效率。地震勘探中数值实例显示了方法良好的潜力。     

Wave propagation analysis in inhomogeneous piezo‐composite layer by the thin‐layer method

The thin‐layer method (TLM) is used to study the propagation of waves in inhomogeneous piezo‐composite layered media caused by mechanical loading and electrical excitation. The element is formulated in the time‐wavenumber domain, which drastically reduces the cost of computation compared to the finite element (FE) method. Fourier series are used for the spatial representation of the unknown variables. The material properties are allowed to vary in the depthwise direction only. Both linear and exponential variations of elastic and electrical properties are considered. Several numerical examples are presented, which bring out the characteristics of wave propagation in anisotropic and inhomogeneous layered media. The element is useful for modelling ultrasonic transducers (UT) and one such example is given to show the effect of electric actuation in a composite material and the difference in the responses elicited for various ply‐angles. Further, an ultrasonic transducer composed of functionally graded piezoelectric materials (FGPM) is modelled and the effect of gradation on mechanical response is demonstrated. The effect of anisotropy and inhomogeneity is shown in the normal modes for both displacement and electric potential. The element is further utilized to estimate the piezoelectric properties from the measured response using non‐linear optimization, a strategy that is referred to as the pulse propagation technique (PPT). Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of the two-dimensional wave equation by using wave polynomials

The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.     

筒仓内散体静态屈服的研究

认为仓贮散体为服从Mohr—Coulomb屈服准则的理想弹塑性材料 ,散体与仓壁之间的摩擦属于Coulomb摩擦接触问题。就刚性筒仓讨论研究了静态时影响散体进入屈服状态的有关因素 ,提出了一种由散体临界内摩擦角来判断静态散体是否进入屈服状态的方法 ,并用有限元法做了数值模拟     

The method of fundamental solutions for acoustic wave scattering by a single and a periodic array of poroelastic scatterers

The method of fundamental solutions (MFS) is now a well-established technique that has proved to be reliable for a specific range of wave problems such as the scattering of acoustic and elastic waves by obstacles and inclusions of regular shapes. The goal of this study is to show that the technique can be extended to solve transmission problems whereby an incident acoustic pressure wave impinges on a poroelastic material of finite dimension. For homogeneous and isotropic materials, the wave equations for the fluid phase and solid phase displacements can be decoupled thanks to the Helmholtz decomposition. This allows for a simple and systematic way to construct fundamental solutions for describing the wave displacement field in the material. The efficiency of the technique relies on choosing an appropriate set of fundamental solutions as well as properly imposing the transmission conditions at the air–porous interface. In this paper, we address this issue showing results involving bidimensional scatterers of various shapes. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. Comparisons with results computed using a mixed pressure–displacement finite element formulation illustrate the great advantages of the MFS both in terms of computational resources and mesh preparation. The extension of the method for dealing with the scattering by an infinite array of periodic scatterers is also presented.