Multiscale Expansion Research Articles

Resonance phenomena in a time-dependent, three-dimensional model of an idealized eddy

We analyze the geometry of Lagrangian motion and material barriers in a time-dependent, three-dimensional, Ekman-driven, rotating cylinder flow, which serves as an idealization for an isolated oceanic eddy and other overturning cells with cylindrical geometry in the ocean and atmosphere. The flow is forced at the top through an oscillating upper lid, and the response depends on the frequency and amplitude of lid oscillations. In particular, the Lagrangian geometry changes near the resonant tori of the unforced flow, whose frequencies are rationally related to the forcing frequencies. Multi-scale analytical expansions are used to simplify the flow in the vicinity of resonant trajectories and to investigate the resonant flow geometries. The resonance condition and scaling can be motivated by simple physical argument. The theoretically predicted flow geometries near resonant trajectories have then been confirmed through numerical simulations in a phenomenological model and in a full solution of the Navier-Stokes equations.

Transient heat conduction within periodic heterogeneous media: A space-time homogenization approach

Composite materials generally exhibit a highly anisotropic thermal behavior (due to the orientation of fibers), leading to strong difficulties to determine the thermal conductivity tensor. Two approaches can be developed for its evaluation. The first one is to carry out experimental measurements with one or several devices to get all the components of the tensor. The second one is to use predictive models based on homogenization theories from the properties and the arrangement at the microscopic scale of each component of the composite material.Following this second approach, a space-time homogenization based on the multi-scale asymptotic expansion method is first developed to model the transient heat conduction problem within periodic heterogeneous structures. The introduction of additional terms to correct the edge effects (i.e. close to the boundaries of the macroscopic domain) in the transient state is considered. We show how these transient correcting terms can be introduced and calculated, depending on the classical boundary conditions in conduction heat transfer problems. Moreover, we underline that correcting terms have also to be added to take into account “short time” effects.Furthermore, we propose to discuss numerical results of the heat transfer modeling in a Laser Flash experiment. We specifically show how the effective thermal diffusivity may be biased when edge effects are neglected in the homogenized model.

Acoustic solitons in waveguides with Helmholtz resonators: transmission line approach.

We report experimental results and study theoretically soliton formation and propagation in an air-filled acoustic waveguide side loaded with Helmholtz resonators. We propose a theoretical modeling of the system, which relies on a transmission-line approach, leading to a nonlinear dynamical lattice model. The latter allows for an analytical description of the various soliton solutions for the pressure, which are found by means of dynamical systems and multiscale expansion techniques. These solutions include Boussinesq-like and Korteweg-de Vries pulse-shaped solitons that are observed in the experiment, as well as nonlinear Schrödinger envelope solitons, that are predicted theoretically. The analytical predictions are in excellent agreement with direct numerical simulations and in qualitative agreement with the experimental observations.

The Curvelet Transform in the analysis of 2-D GPR data: Signal enhancement and extraction of orientation-and-scale-dependent information

The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR images of the subsurface frequently contain geometric information (constant or variable-dip reflections) from various structures such as bedding, cracks, fractures etc. Such features are frequently the target of the survey; however, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting their detectability. At the same time, the GPR method is very sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently blurs the reflections from such targets. The purpose of this paper is to investigate the Curvelet Transform (CT) as a means of S/N enhancement and information retrieval from 2-D GPR sections, with particular emphasis on the recovery of features associated with specific temporal or spatial scales and geometry (orientation/dip).The CT is a multiscale and multidirectional expansion that formulates an optimally sparse representation of bivariate functions with singularities on twice-differentiable (C2-continuous) curves (e.g. edges) and allows for the optimal, whole or partial reconstruction of such objects. The CT can be viewed as a higher dimensional extension of the wavelet transform: whereas discrete wavelets are isotropic and provide sparse representations of functions with point singularities, curvelets are highly anisotropic and provide sparse representations of functions with singularities on curves. A GPR section essentially comprises a spatio-temporal sampling of the transient wavefield which contains different arrivals that correspond to different interactions with wave scatterers in the subsurface (wavefronts). These are generally longitudinally piecewise smooth and transversely oscillatory, i.e. they comprise edges. Curvelets can detect wavefronts at different angles and scales because curvelets of a given angle and scale locally correlate with aligned wavefronts of the same scale.The utility of the CT in processing noisy GPR data is investigated with software based on the Fast Discrete CT and adapted for use with a set of interactive driver functions that compute and display the curvelet decomposition and then allow the manipulation of data (wavefront) components at different scales and angles via the corresponding manipulation (cancellation or restoration) of their associated curvelets. The method is demonstrated with data from archaeometric, geotechnical and hydrogeological surveys, contaminated by high levels of noise, or featuring straight and curved reflections in complex propagation media, or both. It is shown that the CT is very effective in enhancing the S/N ratio by isolating and cancelling directional noise wavefronts of any scale and angle of emergence, sometimes with surgical precision and with particular reference to clutter. It can as successfully be used to retrieve waveforms of specific scale and geometry for further scrutiny, also with surgical precision, as for instance distinguish signals from small and large aperture fractures and faults, different phases of fracturing and faulting, bedding etc. Moreover, it can be useful in investigating the characteristics of signal propagation (hence material properties), albeit indirectly. This is possible because signal attenuation and temporal localization are closely associated, so that scale and spatio-temporal localization are also closely related. Thus, interfaces embedded in low attenuation domains will tend to produce sharp reflections and fine-scale localization. Conversely, interfaces in high attenuation domains will tend to produce dull reflections with broad localization.

Symmetric solitonic excitations of the (1 + 1)-dimensional Abelian-Higgs classical vacuum.

We study the classical dynamics of the Abelian-Higgs model in (1 + 1) space-time dimensions for the case of strongly broken gauge symmetry. In this limit the wells of the potential are almost harmonic and sufficiently deep, presenting a scenario far from the associated critical point. Using a multiscale perturbation expansion, the equations of motion for the fields are reduced to a system of coupled nonlinear Schrödinger equations. Exact solutions of the latter are used to obtain approximate analytical solutions for the full dynamics of both the gauge and Higgs field in the form of oscillons and oscillating kinks. Numerical simulations of the exact dynamics verify the validity of these solutions. We explore their persistence for a wide range of the model's single parameter, which is the ratio of the Higgs mass (m(H)) to the gauge-field mass (m(A)). We show that only oscillons oscillating symmetrically with respect to the "classical vacuum," for both the gauge and the Higgs field, are long lived. Furthermore, plane waves and oscillating kinks are shown to decay into oscillon-like patterns, due to the modulation instability mechanism.

Stochastic Path Perturbation Approach Applied to Non–Local Non–Linear Equations in Population Dynamics

We described the first passage time distribution associated to the stochastic evolution from an unstable uniform state to a patterned one (attractor of the system), when the time evolution is given by an integro-differential equation describing a population model. In order to obtain analytical results we used the Stochastic Path Perturbation Approach introducing a minimum coupling approximation into the nonlinear dynamics, and a stochastic multiscale perturbation expansion. We show that the stochastic multiscale perturbation is a necessary tool to handle other problems like: nonlinear instabilities and multiplicative stochastic partial differential equations. A small noise parameter was introduced to define the random escape of the stochastic field. We carried out Monte Carlo simulations in a non-local Fisher like equation, to show the agreement with our theoretical predictions.

Nonlinear Vortex Structures in Obliquely Rotating Fluid

In this paper, we find a new large scale instability which appears in obliquely rotating flow with the small scale turbulence, generated by external force with small Reynolds number. The external force has no helicity. The theory is based on the rigorous method of multi-scale asymptotic expansion. Nonlinear equations for instability are obtained in the third order of the perturbation theory. In this article, we explain in detail the nonlinear stage of the instability and we find the nonlinear periodic vortices and the vortex kinks of Beltrami type.

Phase-field-crystal simulation of edge dislocation climbing and gliding under shear strain

Structural kinetics in crystalline solids is driven heterogeneously at an atomic level by localized defects, which in turn drive mesoscopic and macroscopic phenomena such as structural phase transformation, fracture, and other forms of plastic flows. A complete description of such processes therefore requires a multiscale approach. Existing modeling methods typically operate exclusively either on an atomic scale or on a mesoscopic scale and macroscopic scale. Phase-field-crystal model, on the other hand, provides a framework that combines atomic length scale and mesoacpoic/diffusive time scale, with the potential reaching a mesoacpoic length through systemic multiscale expansion method. In order to study the dislocation movement under shear strain, the free energy density functional including the exerting shear force term is constructed and also the phase field crystal model for system of shear stain is established. The climb and glide of single dislocation in two-grain system are simulated, and the glide velocity of dislocation and the Peierls potential for dislocation gliding are calculated. The results show that the energy curve changing with time are monotonically smooth under a greater shear strain rate, which corresponds to dislocation movement at a constant speed, which is of rigorous characteristic; while under less shear strain rate, the energy change curve of system presents a periodic wave feature and the dislocation movement in the style of periodic “jerky” for gliding with the stick-slip characteristic. There is a critical potential for dislocation starting movement. The Peierls potential wall for climbing movement is many times as high as that for gliding movement. The results in these simulations are in a good agreement with the experimental ones.

Limitation on the method of strained coordinates for vibrations with weak grazing unilateral contact

The nonlinear normal modes have recently been investigated with the method of strained coordinates for spring-mass models with some springs with piecewise linear stiffness. The N d.o.f. case and the mathematical validity of the method for large time were rigorously proved. The time validity is related to the nature of contact. For grazing contact, this method and also the multiscale expansion lose nonlinear features. For a small piecewise linear stiffness, we show that this method is less precise for a weak unilateral grazing contact. Thus, the validity of the asymptotic expansion for large time cannot be improved and the method has to be modified.

On a model of magnetization switching driven by a spin current: A multiscale approach

We study a model of magnetization switching driven by a spin current: the magnetization reversal can be induced without applying an external magnetic fi eld. We first write our one dimensional model in an adimensionalized form, using a small parameter $\epsilon$. We then explain the various time and space scales involved in the studied phenomena. Taking into account these scales, we fi rst construct an appropriate numerical scheme, that allows us to recover numerically various results of physical experiments. We then perform a formal asymptotic study as $\epsilon$ tends to 0, using a multiscale approach and asymptotic expansions. We thus obtain approximate limit models that we compare with the original model via numerical simulations.

Multiscale homogenization for fluid and drug transport in vascularized malignant tissues

A system of differential equations for coupled fluid and drug transport in vascularized (malignant) tissues is derived by a multiscale expansion. We start from mass and momentum balance equations, stated in the physical domain, geometrically characterized by the intercapillary distance (the microscale). The Kedem–Katchalsky equations are used to account for blood and drug exchange across the capillary walls. The multiscale technique (homogenization) is used to formulate continuum equations describing the coupling of fluid and drug transport on the tumor length scale (the macroscale), under the assumption of local periodicity; macroscale variations of the microstructure account for spatial heterogeneities of the angiogenic capillary network. A double porous medium model for the fluid dynamics in the tumor is obtained, where the drug dynamics is represented by a double advection–diffusion–reaction model. The homogenized equations are straightforward to approximate, as the role of the vascular geometry is recovered at an average level by solving standard cell differential problems. Fluid and drug fluxes now read as effective mass sources in the macroscale model, which upscale the interplay between blood and drug dynamics on the tissue scale. We aim to provide a theoretical setting for a better understanding of the design of effective anti-cancer therapies.

The Multiscale Loop Vertex Expansion

The loop vertex expansion (LVE) is a constructive technique which uses only canonical combinatorial tools and no space–time dependent lattices. It works for quantum field theories without renormalization. Renormalization requires scale analysis. In this paper, we provide an enlarged formalism which we call the multiscale loop vertex expansion (MLVE). We test it on what is probably the simplest quantum field theory which requires some kind of renormalization, namely a combinatorial model of the vector type with quartic interaction and a propagator which mimicks the power counting of \({\phi^4_2}\) . An ordinary LVE would fail to treat even this simplest superrenormalizable model, but we show how to perform the ultraviolet limit and prove its analyticity in the Borel summability domain of the model with the MLVE.

Characterization of spiraling patterns in spatial rock-paper-scissors games

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.

Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer

We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

A combination of downward continuation and local approximation for harmonic potentials

This paper presents a method for the approximation of harmonic potentials that combines downward continuation of globally available data on a sphere of radius R (e.g., a satelliteʼs orbit) with locally available data in a sub-region of the sphere of radius (e.g., the spherical Earthʼs surface). The approximation is based on a two-step algorithm motivated by spherical multiscale expansions: first, a convolution with a scaling kernel deals with the downward continuation from to , while in a second step, the result is locally refined by a convolution on with a wavelet kernel . The kernels and are optimized in such a way that the former behaves well for the downward continuation while the latter shows a good localization in . The concept is indicated for scalar as well as vector potentials.

Thin Layer Models for Electromagnetism

AbstractWe present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer. These models appear as first order approximations of the electromagnetic field. They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer, that makes possible to replace the thin layer by approximate conditions. We present the advantages and the drawbacks of several approximations together with numerical validations and simulations. The main motivation of this work concerns the computation of electromagnetic field in biological cells. The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane, which provides a specific behavior of the electromagnetic field at low frequencies.

Multiscale expansions in discrete world

In this paper, we show the attainability of KdV equation from some types of nonlinear Schrodinger equation by using multiscale expansions discretely. The power of this manageable method is confirmed by applying it to two selected nonlinear Schrodinger evolution equations. This approach can also be applied to other nonlinear discrete evolution equations. All the computations have been made with Maple computer packet program.

Physical interpretation of multiscale asymptotic expansion method

The multiscale asymptotic expansion method (MsAEM) up to any expansion order is formulated in a matrix form which is convenient for use as standard finite element method (FEM). Physical interpretations of the influence functions of different order are presented by analyzing the properties of self-balanced quasi load vectors used for solving the influence functions. The physical interpretation of MsAEM will lay the foundation for its applications. Numerical results validate the mathematical formulations and show that the second perturbation is necessary for micro analysis of periodical composite structures.

High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls

We derive a complete asymptotic expansion for the singularly perturbed problem of acoustic wave propagation inside gases with small viscosity. This derivation is for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity exhibits a boundary layer of size [Formula: see text] where η is the dynamic viscosity. The asymptotic expansion, which is based on the technique of multiscale expansion is expressed in powers of [Formula: see text] and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates.

Inner thermal resonance in thermoelastic geological structures

When investigating heterogeneous media such as composite materials or geological structures, it is convenient to replace them by macroscopic equivalent media, which simplifies computations a lot. In the paper, we look for the equivalent macroscopic model for describing seismic wave propagation and transient heat transfers in thermoelastic periodic geological structures made of rock or soil. We follow the route described in Auriault (2012), to investigating thermoelastic composite media. We use the method of multi-scale asymptotic expansions. By estimating the dimensionless numbers in the momentum and energy balances, we show that an equivalent macroscopic model exists for describing seismic waves at very low frequencies only. The model then shows a damping which is due to thermal resonance at the heterogeneity scale. At higher frequencies, such an equivalent macroscopic model does not exist. Macroscopic models for describing transient heat transfers do not exist.