A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.     

A simple multi‐directional absorbing layer method to simulate elastic wave propagation in unbounded domains

The numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries. This point is critical for the analysis of wave propagation in heterogeneous or layered solids. Various techniques such as Absorbing Boundary Conditions, infinite elements or Absorbing Boundary Layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this paper, a simple absorbing layer method is proposed: it is based on a Rayleigh/Caughey damping formulation which is often already available in existing Finite Element softwares. The principle of the Caughey Absorbing Layer Method is first presented (including a rheological interpretation). The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences. A comparison with the PML method is first performed for pure P‐waves and the method is shown to be reliable in a more complex 2D case involving various wave types and incidences. It may thus be used for various types of problems involving elastic waves (e.g. machine vibrations, seismic waves, etc.). Copyright © 2010 John Wiley & Sons, Ltd.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Perfectly matched layers for transient elastodynamics of unbounded domains

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non‐tangential angles‐of‐incidence and of all non‐zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337–1375], the authors presented, inter alia, time‐harmonic governing equations of PMLs for anti‐plane and for plane‐strain motion of (visco‐) elastic media. This paper presents (a) corresponding time‐domain, displacement‐based governing equations of these PMLs and (b) displacement‐based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti‐plane PML is found to be symmetric, whereas that of the plane‐strain PML is not. Numerical results are presented for the anti‐plane motion of a semi‐infinite layer on a rigid base, and for the classical soil–structure interaction problems of a rigid strip‐footing on (i) a half‐plane, (ii) a layer on a half‐plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd.     

The performance of spheroidal infinite elements

A number of spheroidal and ellipsoidal infinite elements have been proposed for the solution of unbounded wave problems in the frequency domain, i.e solutions of the Helmholtz equation. These elements are widely believed to be more effective than conventional spherical infinite elements in cases where the radiating or scattering object is slender or flat and can therefore be closely enclosed by a spheroidal or an ellipsoidal surface. The validity of this statement is investigated in the current article. The radial order which is required for an accurate solution is shown to depend strongly not only upon the type of element that is used, but also on the aspect ratio of the bounding spheroid and the non‐dimensional wave number. The nature of this dependence can partially be explained by comparing the non‐oscillatory component of simple source solutions to the terms available in the trial solution of spheroidal elements. Numerical studies are also presented to demonstrate the rates at which convergence can be achieved, in practice, by unconjugated‐(‘Burnett’) and conjugated (‘Astley‐Leis’)‐type elements. It will be shown that neither formulation is entirely satisfactory at high frequencies and high aspect ratios. Copyright © 2001 John Wiley & Sons, Ltd.     

The Reflection of Guided Waves from Multiple Flaws in Pipes

Inspection of pipes and pipelines for corrosion is vital for their safe and cost effective operation. Ultrasonic guided wave inspection is a relatively new technique used for this. Recent research has advanced the technique so that it may be possible to characterise individual flaws, making guided waves a potentially viable alternative to direct examination where internal in-line inspection (ILI) is not feasible. However, corrosion often occurs in clusters and the presence of one flaw could affect the signals received from subsequent flaws. Therefore, the issue of multiple flaws must be dealt with if flaws are to be adequately characterised in the field. In order to address this, a semi-analytical modelling methodology has first been developed for the reflection and transmission of guided waves from a single flaw and then, this has been combined with an analytical model for wave propagation to extend it to simulate the reflections from two or more flaws. The semi-analytical model for two flaws has been validated by comparison with experimental data and verified against three-dimensional finite element simulations.     

模糊模式识别在轮箍超声横波探伤中的应用

轮箍是铁路机车运行的重要部件,在制造和使用过程中出现的各种危害性缺陷会严重威胁到列车的行驶安全.用超声无损检测缺陷,回波的识别易受轮箍标记、闸瓦、轮轨接触点及表面波等多种因素的干扰.在超声横波探伤基础上,可将模糊模式识别应用到机车轮箍的无损检测中.以内燃机车轮箍为实验检测对象,使用了多个标准人工伤模拟轮箍自然缺陷.通过提取缺陷回波频域相关特征建立典型缺陷的模糊子集,并运用基于贴近度的择近原则对未知缺陷进行分类识别.实验结果证明了该方法有效.对同一缺陷重复检测的正确识别率达92.5%.     

A multiscale flaw detection algorithm based on XFEM

We present a novel multiscale algorithm for nondestructive detection of multiple flaws in structures, within an inverse problem type setting. The key idea is to apply a two‐step optimization scheme, where first rough flaw locations are quickly determined, and then, fine tuning is applied in these localized subdomains to obtain global convergence to the true flaws. The two‐step framework combines the strengths of heuristic and gradient‐based optimization methods. The first phase employs a discrete‐type optimization in which the optimizer is limited to specific flaw locations and shapes, thus converting a continuous optimization problem in the entire domain into a coarse discrete optimization problem with limited number of choices. To this end, we develop a special algorithm called discrete artificial bee colony. The second phase employs a gradient‐based optimization of the Broyden–Fletcher–Goldfarb–Shanno type on local well‐defined and bounded subdomains determined in the previous phase. A semi‐analytical approach is developed to compute the stiffness derivative associated with the evaluation of objective function gradients. The eXtended FEM (XFEM), with both circular and elliptical void enrichment functions, is used to solve the forward problem and alleviate the costly remeshing of every candidate flaw, in both optimization steps. The multiscale algorithm is tested on several benchmark examples to identify various numbers and types of flaws with arbitrary shapes and sizes (e.g., cracks, voids, and their combination), without knowing the number of flaws beforehand. We study the size effect of the pseudo grids in the first optimization step and consider the effect of modeling error and measurement noise. The results are compared with the previous work that employed a single continuous optimization scheme (XFEM–genetic algorithm and XFEM–artificial bee colony methods). We illustrate that the proposed methodology is robust, yields accurate flaw detection results, and in particular leads to significant improvements in convergence rates compared with the previous work. Copyright © 2014 John Wiley & Sons, Ltd.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids

The dynamic responses of fluid-saturated semi-infinite porous continua to transient excitations such as seismic waves or ground vibrations are important in the design of soil-structure systems. Biot's theory of porous media governs the wave propagation in a porous elastic solid infiltrated with fluid. The significant difference to an elastic solid is the appearance of the so-called slow compressional wave. The most powerful methodology to tackle wave propagation in a semi-infinite homogeneous poroelastic domain is the boundary element method (BEM). To model the dynamic behavior of a poroelastic material in the time domain, the time domain fundamental solution is needed. Such solution however does not exist in closed form. The recently developed ‘convolution quadrature method’, proposed by Lubich, utilizes the existing Laplace transformed fundamental solution and makes it possible to work in the time domain. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time-stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method. Finally, two examples show both the accuracy of the proposed time-stepping procedure and the appearance of the slow compressional wave, additionally to the other waves known from elastodynamics.     

Explicit finite element perfectly matched layer for transient three‐dimensional elastic waves

The use of a perfectly matched layer (PML) model is an efficient approach toward the bounded‐domain modelling of wave propagation on unbounded domains. This paper formulates a three‐dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement‐based finite element setting. The novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration, thus making it practicable for use in large‐scale three‐dimensional dynamic analyses. An efficient method of calculating the strain terms in the PML is developed in order to take advantage of the lack of the overhead of solving equations at each time step. The PML formulation is studied and validated first for a semi‐infinite bar and then for the classical soil–structure interaction problems of a square flexible footing on a (i) half‐space, (ii) layer on a half‐space and (iii) layer on a rigid base. Numerical results for these problems demonstrate that the PML models produce highly accurate results with small bounded domains and at low computational cost and that these models are long‐time stable, with critical time step sizes similar to those of corresponding fully elastic models. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite–infinite element for dynamic analysis of axisymmetrically saturated composite foundations

The formulations of axisymmetrically infinite elements for dynamic analysis of vertical vibration problems in unbounded saturated composite foundations are presented. The theoretical basis as well as the implementation of the elements is discussed, and the element decay functions are derived using the analytical solutions of axially symmetric configurations. Using the proposed finite–infinite element method, the surface vertical displacements of air‐saturated soil (‘dry’ soil) and of water‐saturated soil with extremely low permeability subjected to a surface point excitation (called as the Lamb's problem) are calculated and the results agree very well with the existing theoretical solutions of single‐phase elastic media. As an application, the velocity admittances of a concrete block resting on cement mixing‐pile or gravel‐pile saturated composite foundations are calculated. The influence of soil permeability and pile rigidity on the dynamic response of the composite foundations is investigated. The method proposed by this paper is a simple and reliable numerical one that could be used to study axisymmetrically dynamic problems of layered saturated media and to get the mechanism of dynamic testing on single‐pile saturated composite foundations. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Mechanical properties of rock specimens containing pre‐existing flaws with 3D printed materials

Three‐dimensional printing (3DP) technology has undergone a rapid development in the last few years and become a useful tool in many research fields. This study applied 3DP technology to prepare solid specimens simulating rock‐type materials combined with computed tomography scanning and 3D image processing. 3DP specimens with pre‐existing flaws in different inclination angles were fabricated and then conducted a series of mechanical experiments to study the influence of number and inclination angle of pre‐existing flaw on strength and failure patterns under uniaxial compression. The experimental results indicated that 3DP specimens had similar mechanical properties with rock‐type materials. The 3DP specimens with 2 pre‐existing flaws had lower compressive strength with an average of 4.26 MPa, whereas compressive strength of specimens with one flaw was no less than 5.08 MPa. Different inclination angles led to various failure patterns and compressive strengths, which took on a V‐shaped curve with the increase of inclination angles. This study demonstrated that 3DP technology provided a new perspective for conducting laboratory experimental research of rock mechanics.     

Nondestructive identification of multiple flaws using XFEM and a topologically adapting artificial bee colony algorithm

We present a novel algorithm based on the extended finite element method (XFEM) and an enhanced artificial bee colony (EABC) algorithm to detect and quantify multiple flaws in structures. The concept is based on recent work that have shown the excellent synergy between XFEM, used to model the forward problem, and a genetic‐type algorithm to solve an inverse identification problem and converge to the ‘best’ flaw parameters. In this paper, an adaptive algorithm that can detect multiple flaws without any knowledge on the number of flaws beforehand is proposed. The algorithm is based on the introduction of topological variables into the search space, used to adaptively activate/deactivate flaws during run time until convergence is reached. The identification is based on a limited number of strain sensors assumed to be attached to the structure surface boundaries. Each flaw is approximated by a circular void with the following three variables: center coordinates (x_c, y_c) and radius (r_c), within the XFEM framework. In addition, the proposed EABC scheme is improved by a guided‐to‐best solution updating strategy and a local search (LS) operator of the Nelder–Mead simplex type that show fast convergence and superior global/LS abilities compared with the standard ABC or classic genetic algorithms. Several numerical examples, with increasing level of difficulty, are studied in order to evaluate the proposed algorithm. In particular, we consider identification of multiple flaws with unknown a priori information on the number of flaws (which makes the inverse problem harder), the proximity of flaws, flaws having irregular shapes (similar to artificial noise), and the effect of structured/unstructured meshes. The results show that the proposed XFEM–EABC algorithm is able to converge on all test problems and accurately identify flaws. Hence, this methodology is found to be robust and efficient for nondestructive detection and quantification of multiple flaws in structures. Copyright © 2013 John Wiley & Sons, Ltd.     

Experimental study on fracture behaviour of three‐flawed sandstone specimens after high‐temperature treatments

The crack coalescence of rocks significantly affects the stability of rock engineering, and extensive studies have been performed on preflawed rock specimens without thermal treatment. However, the fracturing behaviour of preflawed specimens after thermal treatment has not been investigated comprehensively. In this study, three‐flawed sandstone specimens with different flaw inclinations after high‐temperature treatments were tested under uniaxial compression. Photographic, acoustic emission and digital image correlation techniques were used to investigate the crack initiation, propagation and coalescence behaviour. Experimental results show that the peak strength, elastic modulus and peak strain of the three‐flawed specimens were lower than those of intact specimens and that they gradually recovered with increasing flaw angle. The peak strength and elastic modulus first increased and then decreased, whereas the peak strain increased with temperature. Noncoalescence, indirect coalescence and direct coalescence were three patterns observed between the two adjacent pre‐existing flaws. Finally, the mechanism of high temperature in alteration of the mechanical properties of sandstone was revealed through microobservations.     

An adaptive time domain deconvolution technique to characterize planar flaws in highly attenuative composites

To identify planar heterogeneities or flaws inside a structure using ultrasonic tests, it is necessary to characterize reflections originating from the boundaries of the heterogeneities. However, for composite materials with high, frequency-dependent wave attenuation, it is often not possible to clearly identify the reflections with either A-scan signals or conventional deconvolution techniques due to the combined effect of signal distortion and overlap. To enable deconvolution of a distorted propagating wave, a new time domain deconvolution technique which includes the frequencydependent acoustic properties of the host material has been developed. This approach is shown to be superior to conventional time domain deconvolution with L1 norm minimization in resolving flaw reflections in highly attenuative glass fiber reinforced polyurethane composites.