Thermo‐mechanical analysis of periodic multiphase materials by a multiscale asymptotic homogenization approach

A spatial and temporal multiscale asymptotic homogenization method used to simulate thermo‐dynamic wave propagation in periodic multiphase materials is systematically studied. A general field governing equation of thermo‐dynamic wave propagation is expressed in a unified form with both inertia and velocity terms. Amplified spatial and reduced temporal scales are, respectively, introduced to account for spatial and temporal fluctuations and non‐local effects in the homogenized solution due to material heterogeneity and diverse time scales. The model is derived from the higher‐order homogenization theory with multiple spatial and temporal scales. It is also shown that the modified higher‐order terms bring in a non‐local dispersion effect of the microstructure of multiphase materials. One‐dimensional non‐Fourier heat conduction and dynamic problems under a thermal shock are computed to demonstrate the efficiency and validity of the developed procedure. The results indicate the disadvantages of classical spatial homogenization. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

A high‐order time‐domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued‐fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued‐fraction approach for bounded domains is proposed, which yields numerically more robust time‐domain formulations. The coefficient matrices of the corresponding continued‐fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high‐order time‐domain formulation for bounded domains with a high‐order transmitting boundary suggested previously is also proposed. In the time‐domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency‐independent coefficient matrices, which can be solved efficiently using standard time‐integration schemes. Numerical examples for modal and time‐domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. Copyright © 2013 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.     

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Weight‐adjusted inner products are easily invertible approximations to weighted L² inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.     

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.     

Computational homogenization of coupled consolidation problems in micro‐heterogeneous porous media

Computational homogenization is exploited for the analysis of transient hydro‐mechanically coupled problems subjected to quasistatic loading (consolidation) in micro‐heterogeneous porous solids. The classical approach of first‐order homogenization is adopted in the spatial domain on representative volume elements (RVE), which are introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE ²‐algorithm is devised for the case of nonlinear permeability and storage coefficients, and it is applied to pore pressure changes in asphalt‐concrete (particle composite). Various parametric studies are carried out, in particular with respect to the influence of the characteristics (size, particle arrangement) of the RVE's. Copyright © 2011 John Wiley & Sons, Ltd.     

Three‐dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems

This paper discusses the matter of using higher order approximations in three‐dimensional problems through Coons macroelements. Recently, we have proposed a global functional set based on ‘Coons interpolation formula’ for the construction of large two‐dimensional macroelements with degrees of freedom appearing at the boundaries only of the domain. After successive application in many engineering problems, this paper extends the methodology to large three‐dimensional hexahedral macroelements with the degrees of freedom appearing at the 12 edges of the entire domain in case of smooth box‐like structures. Closed‐form expressions of the global shape functions are presented for the first time. It is shown that these global shape functions can be automatically constructed in a systematic way by arbitrarily choosing univariate approximations such as piecewise‐linear, cubic B‐splines, Lagrange polynomials, etc., along the control lines. Moreover, the mechanism of adding facial and internal nodes is presented. Relationships with higher order p‐methods are discussed. Following to excellent results previously derived for the solution of the Laplace equation as well as static and eigenvalue extraction analysis of structures, the paper investigates the performance of Coons macroelements in 3‐D eigenvalue and scalar wave propagation problems by implementing standard time‐integration schemes. Copyright © 2005 John Wiley & Sons, Ltd.     

Experimental validation for the diffraction effect in the ultrasonic field of piston transducers and its influence on absorption and dispersion measurements

The aim of this work was to study the diffraction effects in the ultrasonic field of piston source transducers and their importance for accurate measurements of attenuation and dispersion in viscoelastic materials. In laboratory measurements, the diffraction phenomena are mainly due to the beam spread of the ultrasonic wave propagating in viscoelastic materials. This effect is essentially related to the estimated attenuation and dispersion in the material. In this work, a frequency domain system identification approach, using the maximum likelihood estimator (MLE), was applied to the measured data in order to determine a function for correcting the diffraction losses in both normal and oblique incidences for a large frequency band (300 kHz to 3 MHz). The effective radius of the used transmitter was determined by the inverse problem when ultrasonic beam propagation was investigated in a water medium. Using the estimated radius, the propagation through viscoelastic materials was established, and the acoustic parameters of these materials were estimated. Attention was paid to the determination of the attenuation and dispersion in the materials. These quantities were compared to those obtained without diffraction correction in order to see the influence of introducing the diffraction correction into the propagation model     

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.     

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.     

Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

Non‐local dispersive model for wave propagation in heterogeneous media is derived from the higher‐order mathematical homogenization theory with multiple spatial and temporal scales. In addition to the usual space–time co‐ordinates, a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long‐term behaviour of the homogenized solution. By combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth‐order differential equation, also known as a ‘bad’ Boussinesq problem. Regularization procedures are then introduced to construct the so‐called ‘good’ Boussinesq problem, where the need for C¹ continuity is eliminated. Numerical examples are presented to validate the present formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

Local and global properties of the harmonic polynomial cell method: In‐depth analysis in two dimensions

A detailed and systematic analysis is performed on the local and global properties of the recently developed harmonic polynomial cell (HPC) method, a very accurate and efficient field solver for problems governed by the Laplace equation. At the local cell level, a simple rule is identified for the proper choice of harmonic polynomials in the local representation of the velocity potential in cells with symmetry properties. The local solution error, its convergence rate, its dependence on the cell topology, its distribution inside the cell, and its features across cells with different dimensions are carefully examined with relevant findings for HPC numerical implementations. At the global level, the error convergence rate is analytically estimated in terms of error contributions from the boundary conditions and from inside the liquid domain. In most cases, the error associated with boundary conditions dominates the global error. In order to minimize it, Quadtree grid strategies or high‐order local expressions of the velocity potential are proposed for cells near critical boundary portions. To model accurately the boundary conditions on rigid or deformable surfaces with generic geometries, 3 different grid strategies are proposed by adopting concepts of immersed boundary method and overlapping grids. They are comparatively studied for a circular rigid cylinder in infinite fluid and for the propagation of a free‐surface wave. Then, an immersed boundary strategy, using numerical choices suggested in this paper, is successfully compared against a fully nonlinear boundary element method for the case of a surface‐piercing circular cylinder heaving in otherwise calm water.     

On the influence of moisture absorption on Lamb wave propagation and measurements in viscoelastic CFRP using surface applied piezoelectric sensors

The understanding of the impact of environmental influence factors on propagation and damping of Lamb waves in composite materials is a topic of great interest for both design and utilization of structural health monitoring (SHM) systems. In this work, the influence of humidity absorption on the dispersive behavior of Lamb waves propagating in viscoelastic composite materials is investigated. Using a transversely isotropic material model and DMA measurements, the changes in the viscoelastic material properties due to water absorption are characterized. By means of a higher order plate theory and those mechanical properties, the dispersion curves for unconditioned and hot/wet-conditioned UD reinforced CFRP plates are then predicted. Both the changes in Lamb wave velocity and Lamb wave damping are investigated and compared with experimental values. Additionally, the changes of the sensor response, which are related to both the changes of the material properties and that of the adhesive layer, are investigated. The large impact of moisture absorption on Lamb wave excitation and propagation and its relevance for structural health monitoring (SHM) applications is shown and discussed.     

Computational aspects of a new multi‐scale dispersive gradient elasticity model with micro‐inertia

Computational aspects of a recently developed gradient elasticity model are discussed in this paper. This model includes the (Aifantis) strain gradient term along with two higher‐order acceleration terms (micro‐inertia contributions). It has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy. In this paper, the discretisation details of this model are thoroughly investigated, including both discretisation in time and in space. Firstly, the critical time step is derived that is relevant for conditionally stable time integrators. Secondly, recommendations on how to choose the numerical parameters, primarily the element size and time step, are given by comparing the dispersion behaviour of the original higher‐order continuum with that of the discretised medium. In so doing, the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length‐scales. A set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy. These guidelines are then verified numerically by examining the wave propagation in a one‐dimensional bar as well as in a two‐dimensional example. Copyright © 2016 John Wiley & Sons, Ltd.     

On attenuation and measurement of Lamb waves in viscoelastic composites

In this work, the influence of viscoelastic material properties, as featured by fibre reinforced plastics, on the measurement of Lamb waves with the aid of surface-applied piezoelectric sensors is examined. The focus points are frequency dependent material dampening and dispersion on the one hand and the impact of sensor size, wave excitation and measurement method on the other hand. The dependence of the measured wave propagation characteristics and the deviation from the actual characteristics is investigated to assess the relevance for Lamb wave based nondestructive testing and structural health monitoring methods. The sensor responses of piezoelectric sensors bonded to the surface of a viscoelastic composite are predicted by a comprehensive model including these influencing factors. The modelling approach is compared with experimentally measured values to evaluate both the methods and the relevance of the influencing factors.     

Numerical plate testing for linear two‐scale analyses of composite plates with in‐plane periodicity

A method of numerical plate testing (NPT) for composite plates with in‐plane periodic heterogeneity is proposed. In the two‐scale boundary value problem, a thick plate model is employed at macroscale, while three‐dimensional solids are assumed at microscale. The NPT, which is nothing more or less than the homogenization analysis, is in fact a series of microscopic analyses on a unit cell that evaluates the macroscopic plate stiffnesses. The specific functional forms of microscopic displacements are originally presented so that the relationship between the macroscopic resultant stresses/moments and strains/curvatures to be consistent with the microscopic equilibrated state. In order to perform NPT by using general‐purpose FEM programs, we introduce control nodes to facilitate the multiple‐point constraints for in‐plane periodicity. Numerical examples are presented to verify that the proposed method of NPT reproduces the plate stiffnesses in classical plate and laminate theories. We also perform a series of homogenization, macroscopic, and localization analyses for an in‐plane heterogeneous composite plate to demonstrate the performance of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two‐dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so‐called convolution quadrature method (CQM) by means of which the convolution integral, presented in time‐domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non‐homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo‐forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

On attenuation and measurement of Lamb waves in viscoelastic composites

In this work, the influence of viscoelastic material properties, as featured by fibre reinforced plastics, on the measurement of Lamb waves with the aid of surface-applied piezoelectric sensors is examined. The focus points are frequency dependent material dampening and dispersion on the one hand and the impact of sensor size, wave excitation and measurement method on the other hand. The dependence of the measured wave propagation characteristics and the deviation from the actual characteristics is investigated to assess the relevance for Lamb wave based nondestructive testing and structural health monitoring methods. The sensor responses of piezoelectric sensors bonded to the surface of a viscoelastic composite are predicted by a comprehensive model including these influencing factors. The modelling approach is compared with experimentally measured values to evaluate both the methods and the relevance of the influencing factors.     

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.