On Bloch waves for the Stokes equations (original) (raw)
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The Bloch Approximation in Periodically Perforated Media
Applied Mathematics and Optimization, 2005
We consider a periodically heterogeneous and perforated medium filling an open domain of R N . Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in ε ( ε being minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where is R N and then localize the problem for a bounded domain , considering a homogeneous Dirichlet condition on the boundary of .
On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions
The Quarterly Journal of Mechanics and Applied Mathematics
Summary We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in mathbbRd\mathbb{R}^dmathbbRd, dgeqslant2d \geqslant 2dgeqslant2, and (b) exclusions, that is, ‘voids’ that are subject to homogeneous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch-wave expansion, we pursue this goal via asymptotic ansatz featuring the ‘spectral distance’ from a given wavenumber-eigenfrequency pair (situated anywhere within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion via projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the unit cell of periodicity, evaluated at the origin of a spectral neighborhood. For generality, we account for the presence of the source term in the wave equation and we consider—at a given wavenumber—generic cases of isolated, repeated, and nearby eigenvalues. In this way, we obtain ...
Physica D: Nonlinear Phenomena, 2010
In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multiperiodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.
Rayleigh–Bloch waves trapped by a periodic perturbation: exact solutions
Zeitschrift für angewandte Mathematik und Physik
Exact solutions describing the Rayleigh-Bloch waves for the two-dimensional Helmholtz equation are constructed in the case when the refractive index is a sum of a constant and a small amplitude function which is periodic in one direction and of finite support in the other. These solutions are quasiperiodic along the structure and exponentially decay in the orthogonal direction. A simple formula for the dispersion relation of these waves is obtained.
Applicable Analysis, 2021
We pursue a low-wavenumber, second-order homogenized solution of the timeharmonic wave equation in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium R d (d 1), we first use the Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term's projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result.
Bloch wave homogenization of a non-homogeneous Neumann problem
Zeitschrift für angewandte Mathematik und Physik, 2007
In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as "the strange term" in the literature.
Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator
Waves in Random and Complex Media, 2007
This paper is devoted to the asymptotic behaviour of the spectrum of the three-dimensional Maxwell operator in a bounded periodic heterogeneous dielectric medium T = [−T, T ] 3 , T > 0, as the structure period η, such that η −1 T is a positive integer, tends to 0. The domain T is extended periodically to the whole of R 3 , so that the original operator is understood as acting in a space of T-periodic functions. We use the so-called Bloch wave homogenisation technique which, unlike the classical homogenisation method, is capable of characterising a renormalised limit of the spectrum (called the Bloch spectrum). The related procedure is concerned with sequences of eigenvalues Λη of the order of the square of the medium period, which correspond to the oscillations of high-frequencies of order η −1. The Bloch-wave description is obtained via the notion of two-scale convergence for bounded self-adjoint operators, and a proof of the "completeness" of the limiting spectrum is provided.
Exact boundary conditions for wave propagation in periodic media containing a
2015
We present in this chapter a review of some recent research work about a new approach to the numerical simulation of time harmonic wave propagation in infinite periodic media including a local perturbation. The main difficulty lies in the reduction of the effective numerical computations to a bounded region enclosing the perturbation. Our objective is to extend the approach by Dirichlet-to-Neumann (DtN) operators, well known in the case of homogeneous media (as non local transparent boundary conditions). The new difficulty is that this DtN operator can no longer be determined explicitly and has to be computed numerically. We consider successively the case of a periodic waveguide and the more complicated case of the whole space. We show that the DtN operator can be characterized through the solution of local PDE cell problems, the use of the Floquet-Bloch transform and the solution of operator-valued quadratic or linear equations. In our text, we shall outline the main ideas without going into the rigorous mathematical details. The non standard aspects of this procedure will be emphasized and numerical results demonstrating the efficiency of the method will be presented.
The existence of Rayleigh–Bloch surface waves
Journal of Fluid Mechanics, 2002
Rayleigh–Bloch surface waves arise in many physical contexts including water waves and acoustics. They represent disturbances travelling along an infinite periodic structure. In the absence of any existence results, a number of authors have previously computed such modes for certain specific geometries. Here we prove that such waves can exist in the absence of any incident wave forcing for a wide class of structures.
Bloch waves in crystals and periodic high contrast media
ESAIM: Mathematical Modelling and Numerical Analysis, 2017
Analytic representation formulas and power series are developed describing the band structure inside periodic photonic and acoustic crystals made from high contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasi-periodic source free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. Convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation.