scholarly journals Bloch-Wave Homogenization on Large Time Scales and Dispersive Effective Wave Equations

2014 ◽  
Vol 12 (2) ◽  
pp. 488-513 ◽  
Author(s):  
T. Dohnal ◽  
A. Lamacz ◽  
B. Schweizer
Keyword(s):  
Time Scales ◽  
Wave Equations ◽  
Large Time ◽  
Bloch Wave ◽  
Effective Wave

scholarly journals A rational framework for dynamic homogenization at finite wavelengths and frequencies

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180547 ◽  
Author(s):  
Bojan B. Guzina ◽  
Shixu Meng ◽  
Othman Oudghiri-Idrissi
Keyword(s):  
Asymptotic Expansion ◽  
Brillouin Zone ◽  
Wave Motion ◽  
Scaling Law ◽  
Bloch Wave ◽  
Dispersion Branch ◽  
Effective Wave ◽  
Eigenfunction Basis ◽  
Point Of Departure ◽  
Effective Description

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency homogenization is pursued in R d via second-order asymptotic expansion about the apexes of ‘wavenumber quadrants’ comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and ‘dense’ clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as ‘blunted cones’. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the eigenfunction basis, (ii) the reference cell of medium periodicity, and (iii) the wavenumber–frequency scaling law underpinning the asymptotic expansion. We illustrate the analytical developments by several examples, including Green's function near the edge of a band gap and clusters of nearby dispersion surfaces.

Large time behavior of solutions to semilinear systems of wave equations

2006 ◽  
Vol 335 (2) ◽  
pp. 435-478 ◽  
Author(s):  
Hideo Kubo ◽  
Kôji Kubota ◽  
Hideaki Sunagawa
Keyword(s):  
Wave Equations ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Semilinear Systems

scholarly journals Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

2016 ◽  
Vol 26 (14) ◽  
pp. 2651-2684 ◽  
Author(s):  
Assyr Abdulle ◽  
Timothée Pouchon
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Multiscale Model ◽  
Bloch Wave ◽  
Time Interval ◽  
Numerical Homogenization ◽  
Effective Equation ◽  
Long Time ◽  
Effective Models

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 [Formula: see text] is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length [Formula: see text], we prove that the solution of any member of our family of effective equations is [Formula: see text]-close to the true oscillatory wave over a time interval of length [Formula: see text] in a norm equivalent to the [Formula: see text] norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave 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.

scholarly journals ASYMPTOTIC EXPANSION FOR DAMPED WAVE EQUATIONS WITH PERIODIC COEFFICIENTS

2001 ◽  
Vol 11 (07) ◽  
pp. 1285-1310 ◽  
Author(s):  
R. ORIVE ◽  
E. ZUAZUA ◽  
A. F. PAZOTO
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Heat Kernel ◽  
Wave Equations ◽  
Bloch Wave ◽  
Periodic Coefficients ◽  
First Approximation ◽  
Wave Decomposition ◽  
Dissipative Wave Equation

We consider a linear dissipative wave equation in ℝN with periodic coefficients. By means of Bloch wave decomposition, we obtain an expansion of solutions as t→∞ and conclude that, in a first approximation, the solutions behave as the homogenized heat kernel.

scholarly journals Universal spectral profile and dynamic evolution of muscle activation: a hallmark of muscle type and physiological state

2020 ◽  
Vol 129 (3) ◽  
pp. 419-441 ◽  
Author(s):  
Sergi Garcia-Retortillo ◽  
Rossella Rizzo ◽  
Jilin W. J. L. Wang ◽  
Carol Sitges ◽  
Plamen Ch. Ivanov
Keyword(s):  
Time Scales ◽  
Muscle Activation ◽  
Large Time ◽  
Maximal Exercise ◽  
Muscle Tone ◽  
Physiological State ◽  
Dynamic Evolution ◽  
Spectral Profile ◽  
Muscle Type ◽  
Evolution Path

To understand coordinated function of distinct fibers in a muscle, we investigated spectral dynamics of muscle activation during maximal exercise across a range of frequency bands and time scales of observation. We discovered a spectral profile that is specific for each muscle type, robust at short, intermediate, and large time scales, universal across subjects, and characterized by a muscle-specific evolution path with accumulation of fatigue and aging, indicating a previously unrecognized multiscale mechanism of muscle tone regulation.

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