Wave Propagation in Periodic Composites: Higher-Order Asymptotic Analysis Versus Plane-Wave Expansions Method

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
I. V. Andrianov ◽  
J. Awrejcewicz ◽  
V. V. Danishevs’kyy ◽  
D. Weichert
Keyword(s):  
Heterogeneous Media ◽  
Asymptotic Series ◽  
Stop Band ◽  
Fourier Method ◽  
Homogenization Method ◽  
Higher Order ◽  
Square Lattice ◽  
Infinite Matrix ◽  
Periodic Composite ◽  
Versus Plane

This work is devoted to a comparison of different methods determining stop-bands in 1D and 2D periodic heterogeneous media. For a 1D case, the well-known dispersion equation is studied via asymptotic approach. In particular, we show how homogenized solutions can be obtained by elementary series used up to any higher-order. We illustrate and discuss a possible application of asymptotic series regarding parameters other than wavelength and frequency. In addition, we study antiplane elastic shear waves propagating in the plane through a spatially infinite periodic composite material consisting of an infinite matrix and a square lattice of circular inclusions. In order to solve the problem, a homogenization method matched with asymptotic solution on the cell with inclusion of the large volume fracture is proposed and successfully applied. First and second approximation terms of the averaging method provide the estimation of the first stop-band. For validity and comparison with other approaches, we have also applied the Fourier method. The Fourier method is shown to work well for relatively small inclusions, i.e., when the inclusion-associated parameters and matrices slightly differ from each other. However, for evidently contrasting structures and for large inclusions, a higher-order homogenization method is advantageous. Therefore, a higher-order homogenization method and the Fourier analysis can be treated as mutually complementary.

Higher order asymptotic homogenization and wave propagation in periodic composite materials

2008 ◽  
Vol 464 (2093) ◽  
pp. 1181-1201 ◽  
Author(s):  
Igor V Andrianov ◽  
Vladimir I Bolshakov ◽  
Vladyslav V Danishevs'kyy ◽  
Dieter Weichert
Keyword(s):  
Composite Materials ◽  
Exact Analytical Solution ◽  
Higher Order ◽  
Square Lattice ◽  
Infinite Plane ◽  
Dimensional Problem ◽  
Asymptotic Homogenization ◽  
Plane Shear ◽  
Frequency Bands ◽  
Periodic Composite

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

scholarly journals Exponential asymptotics and Stokes lines in a partial differential equation

2005 ◽  
Vol 461 (2060) ◽  
pp. 2385-2421 ◽  
Author(s):  
S. Jonathan Chapman ◽  
David B Mortimer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Second Generation ◽  
Saddle Points ◽  
Asymptotic Series ◽  
Perturbation Expansion ◽  
Higher Order ◽  
Stokes Line ◽  
Partial Differential ◽  
Stokes Lines

A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al . (Berk et al . 1982 J. Math. Phys. 23 , 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki et al . (Aoki et al . 1998 In Microlocal analysis and complex Fourier analysis (ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line.

scholarly journals Study of wave dispersion in periodic composites by higher-order asymptotic homogenization method

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1090403-1090404
Author(s):  
Igor V. Andrianov ◽  
Vladimir I. Bolshakov ◽  
Vladyslav V. Danishevs'kyy ◽  
Dieter Weichert
Keyword(s):  
Wave Dispersion ◽  
Homogenization Method ◽  
Higher Order ◽  
Asymptotic Homogenization ◽  
Periodic Composites ◽  
Asymptotic Homogenization Method

A Dispersive Model for Wave Propagation in Periodic Heterogeneous Media Based on Homogenization With Multiple Spatial and Temporal Scales

2000 ◽  
Vol 68 (2) ◽  
pp. 153-161 ◽  
Author(s):  
W. Chen ◽  
J. Fish
Keyword(s):  
Wave Propagation ◽  
Heterogeneous Media ◽  
Initial Boundary ◽  
Multiple Scale ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Homogenization Theory ◽  
Temporal Scales ◽  
Spatial And Temporal Scales ◽  
Dispersive Model

A dispersive model is developed for wave propagation in periodic heterogeneous media. The model is based on the higher order mathematical homogenization theory with multiple spatial and temporal scales. A fast spatial scale and a slow temporal scale are introduced to account for the rapid spatial fluctuations as well as to capture the long-term behavior of the homogenized solution. By this approach the problem of secularity, which arises in the conventional multiple-scale higher order homogenization of wave equations with oscillatory coefficients, is successfully resolved. A model initial boundary value problem is analytically solved and the results have been found to be in good agreement with a numerical solution of the source problem in a heterogeneous medium.

scholarly journals The analysis of localized effects in composites with periodic microstructure

2013 ◽  
Vol 371 (1993) ◽  
pp. 20120373 ◽  
Author(s):  
Jacob Aboudi ◽  
Michael Ryvkin
Keyword(s):  
Wave Propagation ◽  
Fourier Transform ◽  
Single Cell ◽  
Order Theory ◽  
Higher Order ◽  
Thermomechanical Coupling ◽  
Transform Domain ◽  
Periodic Microstructure ◽  
Periodic Composite ◽  
Higher Order Theory

Several methods for the analysis of composite materials with periodic microstructure in which localized effects (such as concentrated loads, cracks and stationary/progressive damage) occur are resented. Owing to the loss of periodicity caused by these localized effects, it is no longer possible to identify and analyse a repeating unit cell that characterizes the periodic composite. For elastostatic problems, these methods are based on the combination of the representative cell method (RCM), the higher-order theory for functionally graded materials and often the high-fidelity generalized method of cells (HFGMC) micromechanical model. For elastodynamic problems, the combination of the dynamic RCM with a theory for wave propagation in heterogeneous media is used for the prediction of the time-dependent response of the periodic composite with localized effects. In the framework of the RCM, the problem for a periodic composite that is discretized into numerous identical cells is reduced to a problem of a single cell in the discrete Fourier transform domain. In the framework of the higher-order theory and the theory of wave propagation in composites, the resulting governing equations and interfacial conditions in the transform domain are solved by dividing the single cell into subcells and imposing the latter in an average (integral) sense. The HFGMC is often used for the prediction of the proper far-field boundary conditions based on the response of the unperturbed composite. The inverse of the Fourier transform provides the real elastic field at any point of a composite with localized effects. This research summarizes a series of investigations for the prediction of the behaviour of periodic composites with localized loading, fibre loss, damage and cracks subjected to static and dynamic loadings under isothermal and full thermomechanical coupling conditions.

Fem Based Simulation of Percolation Features of Disordered Heterogeneous Media Elastic Properties

1994 ◽  
Vol 367 ◽  
Author(s):  
S.A. Timan ◽  
V.G. Oshmian
Keyword(s):  
Mechanical Properties ◽  
Finite Element Method ◽  
Finite Element ◽  
Monte Carlo ◽  
Critical Exponents ◽  
Heterogeneous Media ◽  
Square Lattice ◽  
Monte Carlo Data ◽  
Disordered System ◽  
Phenomenological Renormalization

AbstractThe mechanical properties of the 2D elastic rigid-nonrigid disordered system in dependence on the concentrations of the rigid phase are studied. The system is constructed on the basis of the square lattice and finite element method (FEM) approximation. The elasticity threshold of the FE system and the critical exponents are detemined by the phenomenological renormalization (PR) of the Monte Carlo data.

Optimum split-step Fourier 3D depth migration: Developments and practical aspects

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S167-S175 ◽  
Author(s):  
Jianfeng Zhang ◽  
Linong Liu
Keyword(s):  
Finite Difference ◽  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Approximate Algorithm ◽  
Wide Angle ◽  
Data Set ◽  
Depth Migration ◽  
Varying Coefficients

We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.

Dispersion characteristics of periodic structural systems using higher order beam element dynamics

2019 ◽  
Vol 25 (2) ◽  
pp. 457-474
Author(s):  
M Ayad ◽  
N Karathanasopoulos ◽  
H Reda ◽  
JF Ganghoffer ◽  
H Lakiss
Keyword(s):  
Dynamic Equilibrium ◽  
Dimensional Space ◽  
Higher Order ◽  
Unit Cell ◽  
Square Lattice ◽  
Structural Systems ◽  
Discrete Dynamics ◽  
Two Dimensional ◽  
Dispersion Characteristics ◽  
Order Expansion

In the current work, we elaborate upon a beam mechanics-based discrete dynamics approach for the computation of the dispersion characteristics of periodic structures. Within that scope, we compute the higher order asymptotic expansion of the forces and moments developed within beam structural elements upon dynamic loads. Thereafter, we employ the obtained results to compute the dispersion characteristics of one- and two-dimensional periodic media. In the one-dimensional space, we demonstrate that single unit-cell equilibrium can provide the fundamental low-frequency band diagram structure, which can be approximated by non-dispersive Cauchy media formulations. However, we show that the discrete dynamics method can access the higher frequency modes by considering multiple unit-cell systems for the dynamic equilibrium, frequency ranges that cannot be accessed by simplified formulations. We extend the analysis into two-dimensional space computing with the dispersion attributes of square lattice structures. Thereupon, we demonstrate that the discrete dynamics dispersion results compare well with that obtained using Bloch theorem computations. We show that a high-order expansion of the inner element forces and moments of the structures is required for the higher wave propagation modes to be accurately represented, in contrast to the shear and the longitudinal mode, which can be captured using a lower, fourth-order expansion of its inner dynamic forces and moments. The provided results can serve as a reference analysis for the computation of the dispersion characteristics of periodic structural systems with the use of discrete element dynamics.

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