Multiscale Asymptotic Expansion Research Articles

A multiscale asymptotic expansion for combustion system with composite materials

This paper discusses the multiscale asymptotic expansion for combustion system with composite materials or porous media in thermal protection system. The combustion system consists of two nonlinear equations with rapidly oscillating coefficients, which are posed on a time-varying domain and computationally expensive to solve with the finite element method. To efficiently solve this system, we propose the standard/novel multiscale asymptotic expansions by combining them with boundary layer solutions. Numerical simulation is performed to validate the accuracy and efficiency of the newly proposed methods, which show that the novel multiscale asymptotic expansion is more accurate than the standard one.

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.

Landau damping of electron-acoustic waves due to multi-plasmon resonances

The linear and nonlinear theories of electron-acoustic waves (EAWs) are studied in a partially degenerate quantum plasma with two-temperature electrons and stationary ions. The initial equilibrium of electrons is assumed to be given by the Fermi–Dirac distribution at finite temperature. By employing the multi-scale asymptotic expansion technique to the one-dimensional Wigner–Moyal and Poisson equations, it is shown that the effects of multi-plasmon resonances lead to a modified complex Korteweg–de Vries (KdV) equation with a new nonlocal nonlinearity. In addition to giving rise to a nonlocal nonlinear term, the wave–particle resonance also modifies the local nonlinear coupling coefficient of the KdV equation. The latter is shown to conserve the number of particles; however, the wave energy decays with time. A careful analysis shows that the two-plasmon resonance is the dominant mechanism for nonlinear Landau damping of EAWs. An approximate soliton solution of the KdV equation is also obtained, and it is shown that the nonlinear Landau damping causes the wave amplitude to decay slowly with time compared to the classical theory.

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.

Multiscale analysis and algorithm of transient electromagnetic scattering from heterogeneous materials

The paper is concerned with the multiscale analysis of the scattering problem for three-dimensional time-dependent Maxwell’s equations in heterogeneous materials. Firstly, an exact transparent boundary condition is developed to reduce the scattering problem into an initial–boundary value problem in heterogeneous materials. Secondly, the multiscale asymptotic expansions of the solution for the reduced problem and an explicit convergence rate for the approximate solutions are presented. Finally, a multiscale Crank–Nicolson mixed finite element method is proposed where the first-order approximation of the Silver–Müller radiation condition is utilized to truncate infinite domain problems. Numerical experiments are then carried out to validate the theoretical results.

Competition instabilities of spike patterns for the 1D Gierer–Meinhardt and Schnakenberg models are subcritical

Spatially localized 1D spike patterns occur for various two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. A competition instability of a steady-state spike pattern is a linear instability that locally preserves the sum of the heights of the spikes. This instability, which results from a zero-eigenvalue crossing of a nonlocal eigenvalue problem at a certain critical value of the inhibitor diffusivity, has been implicated from full PDE numerical simulations of various RD systems of triggering a nonlinear event leading to spike annihilation. As a result, this linear instability is believed to be a key mechanism for initiating a coarsening process of 1D spike patterns. As an extension of the linear theory, we develop and implement a weakly nonlinear theory to analyse competition instabilities associated with symmetric two-boundary spike equilibria on a finite 1D domain for the Gierer–Meinhardt and Schnakenberg RD models. Two symmetric boundary spikes interacting through a long-range bulk diffusion field is the simplest spatial configuration of interacting localized spikes that can undergo a competition instability. Within a neighborhood of the parameter value for the competition instability threshold, a multi-scale asymptotic expansion is used to derive an explicit amplitude equation for the heights of the boundary spikes. This amplitude equation confirms that the competition instability is subcritical and, moreover, it shows that the competition instability threshold corresponds to a symmetry-breaking bifurcation point where an unstable branch of asymmetric two-boundary spike equilibria emerges from the symmetric branch. Results from our weakly nonlinear analysis are confirmed from full numerical solutions of the steady-state problem using numerical bifurcation software.

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.

A novel second-order reduced homogenization approach for nonlinear thermo-mechanical problems of axisymmetric structures with periodic micro-configurations

A novel second-order reduced homogenization (SORH) approach is introduced for analyzing dynamic thermo-mechanical coupling problems in axisymmetric inelastic structures with periodic micro-configuration. The axisymmetric heterogeneous structures are periodically distributed in radial and axial directions and homogeneous distribution in circumferential directions. Firstly, the nonlinear coupled thermo-mechanical model is proposed, and the high-order nonlinear local problems, effective material parameters and the nonlinear homogenization equations are derived successively by the multiscale asymptotic expansion. Further, in order to reduce the large computational amount evaluated by the classical multiscale homogenization approach, the reduced-order nonlinear multiscale models and the corresponding finite-element algorithms are established in detail. The key features of the proposed approach are that an efficient reduced-model form based on transformation field analysis (TFA) to analyze nonlinear local cell problems is proposed and a nonlinear thermo-mechanical problem which considers the mutual coupling for the temperature and displacement fields is computed. In particular, a new SORH algorithm is proposed for investigating the axisymmetric inelastic structures. Finally, three typical numerical experiments are carried out, and the effectiveness and correctness of our presented algorithms in simulating and predicting the macroscopic behavior of the heterogeneous structures are confirmed.

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.

Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid

We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navier slip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We prove the small-time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body. The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls for which the “Navier-Stokes+rigid body” system behaves like a perturbation of the “Euler+rigid body” system. By the means of a multi-scale asymptotic expansion we construct a controlled solution to the “Navier-Stokes+rigid body” system thanks to some controlled solutions to “Euler+rigid body”-type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small.

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.

Weakly Nonlinear Theory for Oscillatory Dynamics in a One-Dimensional PDE-ODE Model of Membrane Dynamics Coupled by a Bulk Diffusion Field

We study the dynamics of systems consisting of two spatially segregated ODE compartments coupled through a one-dimensional bulk diffusion field. For this coupled PDE-ODE system, we first employ a multi-scale asymptotic expansion to derive amplitude equations near codimension-one Hopf bifurcation points for both in-phase and anti-phase synchronization modes. The resulting normal form equations pertain to any vector nonlinearity restricted to the ODE compartments. In our first example, we apply our weakly nonlinear theory to a coupled PDE-ODE system with Sel'kov membrane kinetics, and show that the symmetric steady state undergoes supercritical Hopf bifurcations as the coupling strength and the diffusivity vary. We then consider the PDE diffusive coupling of two Lorenz oscillators. It is shown that this coupling mechanism can have a stabilizing effect, characterized by a significant increase in the Rayleigh number required for a Hopf bifurcation. Within the chaotic regime, we can distinguish between synchronous chaos, where both the left and right oscillators are in-phase, and chaotic states characterized by the absence of synchrony. Finally, we compute the largest Lyapunov exponent associated with a linearization around the synchronous manifold that only considers odd perturbations. This allows us to predict the transition to synchronous chaos as the coupling strength and the diffusivity increase.

A high-order three-scale reduced asymptotic approach for thermo-mechanical problems of nonlinear heterogeneous materials with multiple spatial scales

An effective high-order three-scale reduced asymptotic (HTRA) approach is developed to simulate the coupled thermo-mechanical problems of nonlinear heterogeneous materials with multiple spatial scales. In this paper, the heterogeneous structures are constructed by periodical layout of unit cells on microscopic and mesoscopic scale. At first, the linear and nonlinear higher-order local cell functions defined at microscale and mesoscale are derived by multiscale asymptotic expansion approach. Further, two types of homogenized parameters are introduced, and related nonlinear homogenized solutions are obtained on the macroscopic domain. Then, the high-order three-scale temperature and displacement solutions are established by assembling the various multiscale local solutions and homogenization solutions. The key characteristic of the presented methods is an efficient reduced form for evaluating higher-order local cell problems, and hence greatly reducing the calculating amount in comparison to direct numerical simulations. Also, the new high-order nonlinear homogenization solutions that do not require higher-order continuity of the macroscale solutions. Finally, the effectiveness and accuracy of the multiscale algorithms are verified by some typical numerical examples.

Zero-viscosity limit of the incompressible Navier-Stokes equations with sharp vorticity gradient

It is well-known that the 3D incompressible Euler equations admit some local-in-time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time-dependent hypersurface which evolves with the flow. In this paper we prove that such a solution can be obtained as zero-viscosity limit of strong solutions to the Navier-Stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit. Indeed we exhibit some sequences of exact solutions to the Navier-Stokes equations with vanishing viscosity which are given by multi-scale asymptotic expansions involving some characteristic boundary layers given by some linear PDEs. The convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the Euler equations.

Multiscale computational method for nonlinear heat transmission problem in periodic porous media

This paper deals with multiscale analysis, by using the correctors, of a nonlinear heat transmission problem in a periodic microscopic structure with multiple components. The occurring nonlinearity is related to the flux condition at the interface between the two parts of the medium. In this work, we present first the multiscale asymptotic expansion of the solution for this kind of problems and the strong convergence of the asymptotic expansion to the solution of the multiscale model. Then, we introduce a numerical algorithm based on the multiscale method for solving this problem. Finally, in order to confirm the efficiency of the proposed algorithm, some numerical results obtained through finite element approximations are presented.

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.

Chemo-mechanical coupling analysis of composites by second-order multiscale asymptotic expansion

A novel second-order multiscale asymptotic expansion is developed in this work to analyze chemo-mechanical properties of composites with periodic microstructure. The chemo-mechanical coupling model which considers mutual interaction between the displacement and concentration fields of composites is introduced at first. Then, the second-order multiscale formulas based on homogenization methods and multiscale asymptotic expansion for evaluating the chemo-mechanical coupling problems are proposed, including the microscale cell functions, homogenized coefficients and homogenized equations. Further, the related numerical algorithms on the basis of the proposed multiscale models are brought forward. Finally, by some representative examples, the efficiency and accuracy of the presented algorithms are verified. The numerical results clearly illustrate that the second-order multiscale methods proposed in this work are effective and valid to predict the chemo-mechanical coupling properties, and can capture the microscale behavior of the composites accurately.