Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

A family of effective equations for wave propagation in periodic media for arbitrary timescales O(epsilon-alpha), where epsilon MUCH LESS-THAN1 is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017, ; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, ). The effective solutions in the family are proved to be epsilon close to the original wave in a norm equivalent to the L infinity(0,epsilon-alpha T; L2(omega)) norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.

MATHICSE Technical Report : Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

Assyr Abdulle, Timothée Noé Pouchon

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain

\Omega \subset \mathbb{R}^d

over long time is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length

\varepsilon

, we prove that the solution of any member of our family of effective equations are

\varepsilon

-close in the

L^\infty (0; T^\varepsilon; L2(\Omega))

norm to the true oscillatory wave over a time interval of length

T^\varepsilon = O(\varepsilon^2)

.We show that the previously derived effective equation in [Dohnal, Lamacz, Schweizer, Multiscale Model. Simul., 2014] belongs to our family of effective equation. Moreover, while Bloch waves techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

MATHICSE2016

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

Timothée Noé Pouchon

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.

EPFL2017