Effective models and numerical homogenization methods for long time wave propagation in heterogeneous media

Modeling wave propagation in highly heterogeneous media is of prime importance in engineering applications of diverse nature such as seismic inversion, medical imaging or the design of composite materials. The numerical approximation of such multiscale physical models is a mathematical challenge. Indeed, to reach an acceptable accuracy, standard numerical methods require the discretization of the whole medium at the microscopic scale, which leads to a prohibitive computational cost. Homogenization theory ensures the existence of a homogenized wave equation, obtained from the original problem by a limiting process. As this equation does not depend on the microscopic scale, it is a good target for numerical methods. Unfortunately, for general media, the homogenized equation may not be unique and no formulas are available for its effective data. %Diverse numerical strategies have been developed to approximate a homogenized solution. Nevertheless, such formulas are known for media described by a locally periodic tensor. In that case, or more generally for problems with scale separation, methods such as the finite element heterogeneous multiscale method (FE-HMM) are proved to efficiently approximate the homogenized solution. For wave propagation in heterogeneous media, however, it is known that at large timescales the homogenized solution fails to describe the dispersive behavior of the original wave. Hence, a new equation that captures this dispersion is needed. In this thesis, we study such effective equations for long time wave propagation in heterogeneous media. The first result that we present holds in periodic media. Using the technique of asymptotic expansion, we obtain the characterization of a whole family of equations that describes the long time dispersive effects of the oscillating wave. The validity of our derivation is ensured by rigorous a priori error estimates. We also derive a numerical procedure for the computation of the tensors involved in the first order effective equations. This leads to a numerical homogenization method for long time wave propagation in periodic media. The second result that we present generalizes the procedure for deriving effective equations to arbitrary timescales. This generalization is also useful, for example, for the homogenization of the wave equation with high frequency initial data. We also provide a numerical procedure allowing to compute effective tensors of arbitrary order. The third result is the generalization of the family of first order effective equations from periodic to locally periodic media. A rigorous a priori error analysis is also derived in this situation. This constitutes the first analysis of effective models for the long time approximation of the wave equation in locally periodic media. In a second part of the thesis, we derive numerical homogenization methods for the long time approximation of the wave equation in locally periodic media. In one dimension, we analyze a modification of the FE-HMM called the FE-HMM-L. In higher dimensions, we design a spectral homogenization method. For both methods, we prove error estimates valid for large timescales and in arbitrarily large spatial domains. In particular, we show that these numerical homogenization methods converge to effective solutions that approximate the highly oscillatory wave equation over long time

The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms

An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. At certain frequencies strongly directional dynamic anisotropy is observed, and the asymptotic theory is shown to capture the effect, giving highly accurate qualitative and quantitative results as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour

High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice near the edges of the Brillouin zone. The leading order solution is expressed as a product between the standing wave solution and long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure

High-frequency homogenization of zero frequency stop band photonic and phononic crystals

We present an accurate methodology for representing the physics of waves, for periodic structures, through effective properties for a replacement bulk medium: This is valid even for media with zero frequency stop-bands and where high frequency phenomena dominate. Since the work of Lord Rayleigh in 1892, low frequency (or quasi-static) behaviour has been neatly encapsulated in effective anisotropic media. However such classical homogenization theories break down in the high-frequency or stop band regime. Higher frequency phenomena are of significant importance in photonics (transverse magnetic waves propagating in infinite conducting parallel fibers), phononics (anti-plane shear waves propagating in isotropic elastic materials with inclusions), and platonics (flexural waves propagating in thin-elastic plates with holes). Fortunately, the recently proposed high-frequency homogenization (HFH) theory is only constrained by the knowledge of standing waves in order to asymptotically reconstruct dispersion curves and associated Floquet-Bloch eigenfields: It is capable of accurately representing zero-frequency stop band structures. The homogenized equations are partial differential equations with a dispersive anisotropic homogenized tensor that characterizes the effective medium. We apply HFH to metamaterials, exploiting the subtle features of Bloch dispersion curves such as Dirac-like cones, as well as zero and negative group velocity near stop bands in order to achieve exciting physical phenomena such as cloaking, lensing and endoscope effects. These are simulated numerically using finite elements and compared to predictions from HFH. An extension of HFH to periodic supercells enabling complete reconstruction of dispersion curves through an unfolding technique is also introduced