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

Three non‐dispersive models in multi‐dimensions have been developed. The first model consists of a leading‐order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non‐local model, contains a fourth‐order spatial derivative and thus requires C¹ continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C⁰ continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd.     

Stabilized nonlocal model for dispersive wave propagation in heterogeneous media

In this paper, a general-purpose computational model for dispersive wave propagation in heterogeneous media is developed. The model is based on the higher-order homogenization with multiple spatial and temporal scales and the C⁰-continuous mixed finite element approximation of the resulting nonlocal equations of motion. The proposed nonlocal Hamilton principle leads to the stable discrete system of equations independent of the mesh size, unit cell domain and the excitation frequency. The method has been validated for plane harmonic analysis and for transient wave motion insemi-infinite domain with various microstructures.This work was supported by the Sandia National Laboratories under Contract DE-AL04–94AL8500, the Office of Naval Research through grant number N00014–97–1-0687, and the Japan Society for the Promotion of Science under contract number Heisei11-nendo 06542.     

Time‐domain hybrid formulations for wave simulations in three‐dimensional PML‐truncated heterogeneous media

We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly‐matched‐layer‐(PML)‐truncated media. We extend in three dimensions a recently developed two‐dimensional formulation, by treating the PML via an unsplit‐field, but mixed‐field, displacement‐stress formulation, which is then coupled to a standard displacement‐only formulation for the interior domain, thus leading to a computationally cost‐efficient hybrid scheme. The hybrid treatment leads to, at most, third‐order in time semi‐discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi‐axial PML. We discuss several time‐marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third‐order in time, either unsymmetric or fully symmetric semi‐discrete forms; (b) a standard implicit Newmark for the second‐order, unsymmetric semi‐discrete forms; and (c) an explicit Runge–Kutta scheme for a first‐order in time unsymmetric system. The latter is well‐suited for large‐scale problems on parallel architectures, while the second‐order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Non‐local damage model based on displacement averaging

Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress–strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behaviour, various generalized continuum theories have been proposed. Integral‐type non‐local damage models are often based on weighted spatial averaging of a strain‐like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the non‐local displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analysed body, both formulations differ and the non‐local displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non‐local weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

A method for computation of discontinuous wave propagation in heterogeneous solids: basic algorithm description and application to one‐dimensional problems

An explicit integration algorithm for computations of discontinuous wave propagation in heterogeneous solids is presented, which is aimed at minimizing spurious oscillations when the wave fronts pass through several zones of different wave speeds. The essence of the present method is a combination of two wave capturing characteristics: a new integration formula that is obtained by pushforward–pullback operations in time designed to filter post‐shock oscillations, and the central difference method that intrinsically filters front‐shock oscillations. It is shown that a judicious combination of these two characteristics substantially reduces both spurious front‐shock and post‐shock oscillations. The performance of the new method is demonstrated as applied to wave propagation through a uniform bar with varying courant numbers, then to heterogeneous bars. Copyright © 2012 John Wiley & Sons, Ltd.     

Non‐local boundary integral formulation for softening damage

A strongly non‐local boundary element method (BEM) for structures with strain‐softening damage treated by an integral‐type operator is developed. A plasticity model with yield limit degradation is implemented in a boundary element program using the initial‐stress boundary element method with iterations in each load increment. Regularized integral representations and boundary integral equations are used to avoid the difficulties associated with numerical computation of singular integrals. A numerical example is solved to verify the physical correctness and efficiency of the proposed formulation. The example consists of a softening strip perforated by a circular hole, subjected to tension. The strain‐softening damage is described by a plasticity model with a negative hardening parameter. The local formulation is shown to exhibit spurious sensitivity to cell mesh refinements, localization of softening damage into a band of single‐cell width, and excessive dependence of energy dissipation on the cell size. By contrast, the results for the non‐local theory are shown to be free of these physically incorrect features. Compared to the classical non‐local finite element approach, an additional advantage is that the internal cells need to be introduced only within the small zone (or band) in which the strain‐softening damage tends to localize within the structure. Copyright © 2003 John Wiley & Sons, Ltd.     

Equivalent continuum modeling for non‐linear wave propagation in jointed media

This paper describes a methodology to build equivalent continuum (ECC) models to represent jointed media for non‐linear wave propagation. In the present work, we study the response of randomly jointed rock media both to quasi‐static and dynamic loadings. We have found that, unlike in the quasi‐static case where no relaxation is observed, during dynamic loading, the deviatoric stress drops in the plastic wave after reaching its peak. Such stress relaxation phenomenon is calculated in jointed media, although the model used for intact rock blocks is rate‐independent. This is a strong indication that EC models for jointed rock should include some rate‐dependence when used in wave propagation simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Practical aspects of higher‐order numerical schemes for wave propagation phenomena

This paper examines practical issues related to the use of compact‐difference‐based fourth‐ and sixth‐order schemes for wave propagation phenomena with focus on Maxwell's equations of electromagnetics. An outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches: transformed plane method and physical space differencing. In the first technique, the truncation error expansion for the sixth‐order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available. Physical space‐differencing formulas are derived for the five‐point stencil by expressing the coefficients in terms of local spacing ratios. The order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes. To ensure stability for a broad range of problems, Fourier analysis is employed to develop a single‐parameter family of up to tenth‐order tridiagonal‐based spatial filters. The implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1‐D cavity where they are employed to suppress a late‐time instability. The paper concludes after demonstrating the application of the scheme to several 3‐D canonical problems utilizing Cartesian as well as curvilinear meshes. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. Government work and is in the public domain in the United States.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

Two‐scale diffusion–deformation coupling model for material deterioration involving micro‐crack propagation

In this paper, we introduce a two‐scale diffusion–deformation coupled model that represents the aging material deterioration of two‐phase materials involving micro‐crack propagations. The mathematical homogenization method is applied to relate the micro‐ and macroscopic field variables, and a weak coupling solution method is employed to solve the two‐way coupling phenomena between the diffusion of scalar fields and the deformation of quasi‐brittle solids. The macroscopic mechanical behavior represented by the derived two‐scale two‐way coupled model reveals material nonlinearity due to micro‐scale cracking induced by the scalar‐field‐induced deformation, which can be simulated by the finite cover method. After verifying the fundamental validity of the proposed model and the analysis method, we perform a simple numerical example to demonstrate their ability to predict aging material deterioration. Copyright © 2010 John Wiley & Sons, Ltd.     

Discrete and continuum methods for numerical simulations of non‐linear wave propagation in discontinuous media

The main objective of this paper is to develop a methodology to model dynamic loading of various discontinuous media. Two methods for modeling non‐linear waves in a media with multiple discontinuities are considered. The first one, a discrete method, is based on the Simple Common Plane contact algorithm. This method can be applied both to compliant contacts characterized by finite thickness and elastic moduli (such as joints in geomechanics) as well as to non‐compliant frictional contacts traditionally described by the slide lines in finite element/finite difference codes. The second one, a continuum method, assumes that the contacts are not compliant and can be modeled as one or several weakness planes cutting through the elements of the computational mesh. Both discrete and continuum methods described in the paper can be applied to derive equivalent continuum properties for media with multiple discontinuities. An example of such application for a randomly jointed media is given in the paper. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

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.     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

A meshless method with enriched weight functions for three‐dimensional crack propagation

The propagation of cracks in three dimensions is analysed by a meshless method. The cracks are modelled by a set of triangles that are added when the propagation occurs. Since the method is meshless, no remeshing of the domain is necessary during the propagation. To avoid using a large number of degrees of freedom, the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions. This enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper. Several algorithms for efficiently implementing the meshless method in three dimensions are detailed. The accuracy of the enrichment is first assessed on simple examples and some results of non‐planar propagation of multiple cracks are then presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Topology optimization using non‐conforming finite elements: three‐dimensional case

As in the case of two‐dimensional topology design optimization, numerical instability problems similar to the formation of two‐dimensional checkerboard patterns occur if the standard eight‐node conforming brick element is used. Motivated by the recent success of the two‐dimensional non‐conforming elements in completely eliminating checkerboard patterns, we aim at investigating the performance of three‐dimensional non‐conforming elements in controlling the patterns that are estimated overly stiff by the brick elements. To this end, we will investigate how accurately the non‐conforming elements estimate the stiffness of the patterns. The stiffness estimation is based on the homogenization method by assuming the periodicity of the patterns. To verify the superior performance of the elements, we consider three‐dimensional compliance minimization and compliant mechanism design problems and compare the results by the non‐conforming element and the standard 8‐node conforming brick element. Copyright © 2005 John Wiley & Sons, Ltd.     

The finite element method for the mechanically based model of non‐local continuum

In this paper the finite element method (FEM) for the mechanically based non‐local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter‐distance and proportional to the product of the interacting volume elements. The constitutive relations of the long‐range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance‐decaying function, which accounts for the decrement of the long‐range interactions as long as distance increases. In this study, the elastic problem involving long‐range central interactions for isotropic elastic continuum will be solved with the aid of the FEM. The accuracy of the solution obtained with the proposed FEM code is compared with other solutions obtained with Galerkins' approximation as well as with finite difference method. Moreover, a parametric study regarding the effect of the material length scale in the mechanically based model and in the Kr”oner–Eringen non‐local elasticity has been investigated for a plane elasticity problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of high‐order finite elements for convected wave propagation

In this paper, we examine the performance of high‐order finite element methods (FEM) for aeroacoustic propagation, based on the convected Helmholtz equation. A methodology is presented to measure the dispersion and amplitude errors of the p‐FEM, including non‐interpolating shape functions, such as ‘bubble’ shape functions. A series of simple test cases are also presented to support the results of the dispersion analysis. The main conclusion is that the properties of p‐FEM that make its strength for standard acoustics (e.g., exponential p‐convergence, low dispersion error) remain present for flow acoustics as well. However, the flow has a noticeable effect on the accuracy of the numerical solution, even when the change in wavelength due to the mean flow is accounted for, and an approximation of the dispersion error is proposed to describe the influence of the mean flow. Also discussed is the so‐called aliasing effect, which can reduce the accuracy of the solution in the case of downstream propagation. This can be avoided by an appropriate choice of mesh resolution. Copyright © 2013 John Wiley & Sons, Ltd.