Method Of Multiscale Asymptotic Expansions Research Articles

Vortex Dynamo in Rotating Media

The review highlights the main achievements in the theory of the vortex dynamo in rotating media. Various models of a vortex dynamo in rotating media are discussed: a homogeneous viscous fluid, a temperature-stratified fluid, a moist atmosphere, and a stratified nanofluid. The main analytical and numerical results for these models obtained by the method of multiscale asymptotic expansions are presented. The main attention is paid to models with a non-spiral external force. For a rotating moist atmosphere, characteristic estimates of the spatial and temporal scales of the generated vortex structures are obtained. New effects of the vortex dynamo in a rotating stratified nanofluid, which arise due to thermophoresis and Brownian motion of nanoparticles, are shown. The results of the analysis of the nonlinear equations of the vortex dynamo in the stationary regime are presented in the form of spiral kinks, periodic nonlinear waves, and solitons.

Topological optimization algorithm for mechanical-electrical coupling of periodic composite materials

<abstract><p>In this paper, a topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is studied. Firstly, the homogenization problem of the mechanical-electrical coupling topology optimization problem of periodic composite materials is established by the multi-scale asymptotic expansion method. Secondly, the topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is constructed by finite element method, solid isotropic material with penalisation method and homogenization method. Finally, numerical results show that the proposed algorithm is effective to calculate the optimal structure of the periodic composite cantilever beam under the influence of the mechanical-electrical coupling.</p></abstract>

Boundary layer problem and combined limits of the two-fluid Navier-Stokes-Poisson system

In this paper, we study the combined quasi-neutral and zero-viscosity limits of the two-fluid Navier-Stokes-Poisson system for a plasma composed of ions and electrons. By analyzing boundary layer under physically relevant condition and utilizing multi-scale asymptotic expansion method, we derive the boundary layer equations and construct the approximate solutions. Then, we use the energy estimate to give the linear stability of the approximate solutions.

Hydrodynamic α-effect in a rotating stratified moist atmosphere driven by small-scale non-helical force

In this paper we have studied the large-scale instability in a rotating stratified moist atmosphere with a small-scale turbulence. The axis of rotation of the medium is deviated from the vertical direction. The turbulence is excited by an external small-scale force with zero helicity and a low Reynolds number. The nonlinear equations of a vortex dynamo were obtained on the basis of the method of multiscale asymptotic expansions. The linear instability and stationary nonlinear modes were studied. The solutions in a form of localized vortex structures were obtained.

Nonlinear Vortex Structures Driven by Small-Scale Non-helical Forces in Obliquely Rotating Stratified Fluids

We study a new type of large-scale instability in obliquely rotating stratifi ed fl uids with small-scale nonhelical turbulence. The small-scale turbulence is generated by the external force with zero helicity and low Reynolds number. The theory uses the method of multiscale asymptotic expansions. The nonlinear equations for large-scale motions are obtained in the third order of perturbation theory. We consider a linear instability and stationary nonlinear modes. Solutions in the form of nonlinear Beltrami waves and localized vortex structures such as kinks of a new type are obtained.

Vortex Dynamo in an Obliquely Rotating Stratified Nanofluid by Small-Scale Non-Helical Forces

In this work, a large-scale instability of the hydrodynamic -effect in an obliquely rotating stratified nanofluid taking into account the effects of Brownian diffusion and particle flux under the influence of a temperature gradient (thermophoresis) is obtained. The instability is caused by the action of an external small-scale non-spiral force, which excites small-scale velocity oscillations with zero helicity and a low Reynolds number. Nonlinear equations for large-scale motions are obtained using the method of multiscale asymptotic expansions by a small parameter (Reynolds number). A linear large-scale instability of hydrodynamic -effect is investigated depending on the parameters of rotation , temperature stratification , and concentration of nanoparticles . A new effect of the generation of large-scale vortex structures in nanofluid at is associated with an increase in the concentration of nanoparticles is obtained. The maximum instability increment is reached at inclination angles for the Prandtl numbers , and for the Prandtl numbers at inclination angles . It has been found that the frequency changing of the parametric impact will make it possible to control and track the generation of large-scale vortex structures. It is shown that circularly polarized Beltrami vortices appear in nanofluid as the result of new large-scale instability development. In this paper, the saturation regime of large-scale instability in an obliquely rotating stratified nanofluid with an external small-scale non-spiral force is investigated. In the stationary regime was obtained a dynamic system of equations for large-scale perturbations of the velocity field. Numerical solutions of this system of equations are obtained, which show the existence of localized vortex structures in the form of nonlinear Beltrami waves and kinks. The velocity profile of kink tends to be constant at large Z values.

An assessment of multiscale asymptotic expansion method for linear static problems of periodic composite structures

The decoupled multiscale asymptotic expansion method (MsAEM) is applicable to the analysis of composite structures with periodic microstructures. When using the MsAEM to solve linear static problems, unit-cell influence functions must be solved from the microscale (or unit cell) problem with microscopic periodic boundary conditions, and homogenized displacements and their derivatives must be solved from the macroscale (or homogenized) problem with macrostructural boundary conditions. This paper compares the influence functions solved by using the normalization condition, the Dirichlet homogeneous boundary condition, and the oversampling technique or the super unit cell approach. It is concluded that the oversampling technique is a better way of dealing with periodic boundary conditions compared to the normalization condition. Another contribution of this work is to propose a combination method in which the MsAEM is used inside the structure and the multiscale eigenelement method (MEM) is used dealing with the boundary layer problem, and it is shown that this combination method can achieve accurate microscopic results for structures with any boundary conditions. In addition, the paper reveals the effects of different order expansion terms and gives some practical suggestions on how to determine the expansion order. Numerical comparisons validate the present combination method and all conclusions.

Nonlinear Dynamo in Obliquely Rotating Stratified Electroconductive Fluid in an Uniformly Magnetic Field

In this paper, we investigated a new large-scale instability that arises in an obliquely rotating convective electrically conducting fluid in an external uniform magnetic field with a small-scale external force with zero helicity. This force excites small-scale velocity oscillations with a small Reynolds number. Using the method of multiscale asymptotic expansions, we obtain the nonlinear equations for vortex and magnetic disturbances in the third order of the Reynolds number. It is shown that the combined effects of the Coriolis force and the small external forces in a rotating conducting fluid possible large-scale instability. The linear stage of the magneto-vortex dynamo arising as a result of instabilities of -effect type is investigated. The mechanism of amplification of large-scale vortex disturbances due to the development of the hydrodynamic - effect taking into account the temperature stratification of the medium is studied. It was shown that a «weak» external magnetic field contributes to the generation of large-scale vortex and magnetic perturbations, while a «strong» external magnetic field suppresses the generation of magnetic-vortex perturbations. Numerical methods have been used to find stationary solutions of the equations of a nonlinear magneto-vortex dynamo in the form of localized chaotic structures in two cases when there is no external uniform magnetic field and when it is present.

Second-Order Asymptotic Analysis and Computations of Axially and Spherically Symmetric Piezoelectric Problems for Composite Structures

The second-order coupled piezoelectric models based on the multi-scale asymptotic expansion method are developed for the axially and spherically symmetric composites. The governing piezoelectric equations are compactly formulated in cylindrical and spherical coordinates, and the composite domains are assumed to be periodically occupied by the representative cells. The multi-scale asymptotic expansions for the displacement and the electric potential are formally defined and the effective elastic, piezoelectric, and dielectric coefficients are expressed in terms of the microscopic functions defined on the cell domain. Particularly, the cell solutions and the homogenized solutions for the plane axisymmetric and spherically symmetric problem are derived analytically. The corresponding finite element procedure is proposed, in which the Newmark algorithm is applied to construct the computational scheme in the temporal domain. Numerical experiments are carried out to simulate both the static and dynamic asymptotic behavior of the space axisymmetric and the one-dimensional plane axisymmetric structures. It is validated from the numerical examples that the asymptotic models proposed in the current work are effective to capture the macroscopic performance of the piezoelectric structures and the second-order expansions of the solutions is essential for obtaining the correct distributions of the stress and electric displacement.

The multiscale asymptotic expansion method for three-dimensional static analyses of periodical composite structures

The decoupled multiscale asymptotic expansion method (MsAEM) is applied to the static analyses of three-dimensional (3D) periodical composite structures in this paper, and we focus on the attributes of the method in application. By comprehensive analyses and comparisons with finite element method, it is concluded that, 1) MsAEM can be viewed as the superposition method of different-order admissible deformation modes represented by influence functions; 2) the second-order expansion term is very necessary for the structure with considerable strain gradients; 3) the super unit cell approach can improve the computational accuracy of influence functions; 4) highly accurate differential quadrature finite element method can improve the computational efficiency of homogenized displacement and its derivatives; 5) the total potential energy is an effective measure to evaluate the computational accuracy.

Nonlinear vortex dynamo in a rotating stratified moist atmosphere

We have found a new type of large-scale instability in a rotating stratified moist atmosphere with small-scale turbulence. The turbulence is excited by an external small-scale force with a low Reynolds number. We have constructed the theory based on the method of multiscale asymptotic expansions. The nonlinear equations for large-scale motion have been derived in the third order of the perturbation theory. We have investigated the linear instability and stationary nonlinear regimes. Solutions in the form of localized vortex structures or kinks of a new type have been obtained.

NONLINEAR DYNAMO IN A ROTATING ELECTRICALLY CONDUCTING FLUID

We found a new large-scale instability, which arises in the rotating conductive fluid with small-scale turbulence. Turbulence is generated by small-scale external force with a low Reynolds number. The theory is built simply by the method of multiscale asymptotic expansions. Nonlinear equations for vortex and magnetic perturbations obtained in the third order for small Reynolds number. It is shown that the combined effects of the Coriolis force and the small external forces in a rotating conducting fluid possible large-scale instability. The large-scale increments of the instability, correspond to generation as the vortex and magnetic disturbances. This type of instability is classified as hydrodynamic and MHD alpha-effect. We studied the stationary regimes of nonlinear equations of magneto-vortex dynamo. In the limit of weakly conducting fluid found stationary solutions in the form of helical kinks. In the limit of high conductivity fluid was obtained stationary solutions in the form of nonlinear periodic waves and kinks.

Bragg Resonance around the Coast of Hardwall

Basically, when waves pass an uneven basis, then this wave will be split into transmission and reflection waves. First of all, it will be shown that a sinusoidal seabed can lead to the phenomenon of Bragg resonance. Bragg resonance occurs when the wave-length comes at twice the wave-length of a sinusoidal basis. The method used to obtain approximation solution is a multi-scale asymptotic expansion method. A research on the effect of Bragg resonance on sinusoidal basis had been studied. Sinusoidal basis can reduce the amplitude of the incoming wave so that the amplitude of the wave transmission is quite small. In these researcher, the coast is assumed ideal and can absorb all the energy of the wave transmission. If the beach can reflect waves, this indicates that the existence of sinusoidal basis is more harmful to the coast. This mechanism relies on the distance between the base sinusoidal and beaches. The present research will examined the influence of the base, when there was a beach of hard-wall on the right, which was perfectly capable of reflecting waves. Having regard to the phase difference, from super positioned waves when they hit the beach, so it can determine the safert and the most dangerous distance.

Solution methods for two key problems in multiscale asymptotic expansion method

The accuracy of decoupled multiscale asymptotic expansion method (MsAEM) depends completely on the accuracy of influence functions and the accuracy of homogenized displacement derivatives. Unit cell problem must be solved first for obtaining influence functions whose calculational accuracy depends largely on the boundary conditions of the unit cell problem; and homogenized problem must be solved for obtaining homogenized displacement derivatives whose accuracy depends largely on the order of used finite elements and meshes. In this work, as for two-dimensional (2D) periodical composite structures, super unit cell approach is proposed to solve for accurate influence functions, and quasi potential energy functional corresponding to influence functions (called quasi displacements in this paper), is constructed to evaluate the accuracy of influence functions or boundary conditions of unit cell problem, and it follows that clamp boundary condition is not always suitable for solving influence functions of different orders although it is exact for unit cell problem of one-dimensional (1D) rod. Finally, the differential quadrature finite element method is employed to improve the computational accuracy of homogenized displacement derivatives.

LARGE-SCALE MAGNETIC AND VORTEX STRUCTURES IN A TURBULENT MEDIUM

The survey covers a main achievements in the field of dynamo theory. Applications of the model of turbulent convective dynamo to the problem of generation of a large-scale vortexes and magnetic structures in the solar photosphere are discussed. Investigated are the new models of the vortex dynamo in a rotating fluid with small-scale external force. The main results in these models are obtained by the method of multiscale asymptotic expansions. The results of numerical analysis of nonlinear equations of a magnetovortex dynamo in a stationary mode are provided. Solutions are found in the form of spiral kinks, periodic nonlinear waves and solitons, which are also considered in the survey.

Thermo-mechanical analysis of periodic porous materials with microscale heat transfer by multiscale asymptotic expansion method

A novel multiscale asymptotic method used to simulate thermo-mechanical analysis of periodic porous materials with microscale heat transfer is systematically studied. In these materials, heat radiation and heat convection that account for the scale effect of unit cells have an important impact on the macroscopic temperature and stress fields, which is our particular interest in this study. The scale effect is thought to be the result of microscopic heat transfer, the amount of which depends on the microscale pore size of porous materials. The higher-order multiscale formulations for computing the dynamic thermo-mechanical coupling problem with the inertia term, coupling term, convection term and radiation term are given successively. Then, the corresponding numerical algorithm based on the finite element-difference method is brought forward in details. Finally, numerical examples are given to demonstrate the efficiency and validity of the proposed method. The results indicate the disadvantages of classical finite element method.

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.

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.

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.

Accuracy of multiscale asymptotic expansion method

The multiscale asymptotic expansion method (MsAEM) is generally implemented by finite element method. The calculating accuracy of MsAEM depends completely on the order of asymptotic expansion and the order of finite element. First, the necessary number of expansion term is decided in a mechanical view from the pseudo loads used for solving influence functions. Next for different order of load cases, the analytical solutions of the static problems of the periodical composite rod are obtained using different order of MsAEM and finite elements. In those solutions, the element order for solving analytical macro displacements depends on the external loads whereas the element orders for solving analytical influence functions are determined from the governing differential equations of influence functions. Then, two dimensional (2D) periodical composite are explored similarly. Finally, the potential energy functional is used to evaluate the accuracy of MsAEM, and numerical comparisons validate the conclusions.