Matched Asymptotic Expansions Research Articles

Effective boundary conditions for second-order homogenization

Using matched asymptotic expansions, we derive an equivalent bar model for a periodic, one-dimensional lattice made up of linear elastic springs connecting both nearest and next-nearest neighbors. We obtain a strain-gradient model with effective boundary conditions accounting for the boundary layers forming at the endpoints. It is accurate to second order in the scale separation parameter ɛ≪1, as shown by a comparison with the solution to the discrete lattice problem. The homogenized modulus associated with the gradient effect (gradient stiffness) is found negative, as is often the case in second-order homogenization. Negative gradient stiffnesses are widely viewed as paradoxical as they can induce short-wavelength oscillations in the homogenized solution. In the one-dimensional lattice, the asymptotically correct boundary conditions are shown to suppress the oscillations, thereby restoring consistency. By contrast, most of the existing work on second-order homogenization makes use of postulated boundary conditions which, we argue, not only ruin the order of the approximation but are also the root cause of the undesirable oscillations.

Singular perturbation boundary and interior layers problems with multiple turning points

In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities.

Thermoelastic analysis of variable thickness truncated conical shell subjected to thermomechanical load with internal heat generation using perturbation technique

In this article, the behavior of the truncated conical shell subjected to thermomechanical loading in their inner and outer layers with internal heat generation source is investigated. The displacement field obeys kinematic of the first order shear deformation theory and the first-order temperature theory is used. Two-dimensional temperature analysis has been performed along the thickness and axis of the shell, which can be defined under various loading and thermomechanical boundary conditions. The set of governing equations is a system of differential equations with variable coefficients which are solved by using the analytical matched asymptotic expansion of the perturbations technique. The mentioned solution has little computational cost, therefore can be used well in parametric studies for optimization. It was shown, axial displacement is somehow independent of the radial axis and its maximum value occurs near the upper boundary. The conical shell has expanded in the radial direction. Also the maximum temperature happens approximately in the middle of the length of the conical shell. The parametric study showed, with the increase heat flux in the outer layer thereby expanding the area of heat application which would let more heat in than out. Also it was found, controlling the lifetime of the structure is the result of directing the cone angle. The results obtained from the analytical solution were compared with the Finite Element Method (FEM) analysis and the results of similar related articles, only to show a good agreement.

Modelling of glass matrix composites by the Coupled Criterion and the Matched Asymptotic Approach. The effect of residual stresses and volume fraction

The fracture toughness of glass can be improved by including ceramic reinforcing platelets. Among others, Al2O3 platelets are proposed as an environmentally friendly and low-cost solution. One of the critical points in the design of this composite is the generation of residual stresses after cooling in manufacture, due to the thermal expansion mismatch between the two materials. In this paper, the effect of these residual stresses and the role of the volume fraction are studied, following a methodology based on the Coupled Criterion together with the Matched Asymptotic Expansion. The model is validated through a comparison with experimental results found in the literature.

Research on hydrodynamic performance calculation of buoy-chain-sprocket wave energy collection device

Buoy-chain-sprocket wave energy collection device is a new type of oscillating buoy wave energy collection device, which has a good application prospect. The calculation method of the hydrodynamic performance of wave energy device (e.g. the first-level average collection efficiency of wave energy) is the core scientific problem of wave energy device, and it is also the theoretical basis for the optimization and control of wave energy device prototype. In this paper, on basis of linear potential flow theory, after the velocity potentials are calculated with eigenfunction expansion method and matching asymptotic expansion method, the frequency-domain wave loads in the heave, surge and pitch directions of the floating body are obtained. And then the motion coordination relationship of the floating body of buoy-chain-sprocket wave energy collection device is used to establish time-domain hydrodynamic motion response equations of the floating body of the device based on the Cummins method. After these equations are solved via the fourth-order Runge-Kutta method, the first-level average collection efficiency of the wave energy device is acquired. In order to further improve the calculation accuracy, the frequency-domain wave-exciting force is calculated by using variable draft depth when the hydrodynamic response of buoy-chain-sprocket wave energy collection device is calculated in this paper. Finally, the reliability of the above calculation method for the hydrodynamic performance of buoy-chain-sprocket wave energy collection device is verified through physical model experiment.

Homogenization of subwavelength free stratified edge of viscoelastic media including finite size effect

This paper proposes the homogenization for a stratified viscoelastic media with free edge. We consider the effect of two-dimensional periodically stratified slab over a semi-infinite viscoelastic ground on the propagation of shear waves hitting the interface. Within the harmonic regime, the second order homogenization and matched-asymptotic expansions method is employed to derive an equivalent anisotropic slab associated with effective boundary and jump conditions for the displacement and the normal stress across an interface. The reflection coefficients and the displacement fields are obtained in closed forms and their validity is inspected by comparison with direct numerics in the case of layers associated with Neumann boundary conditions.

Scattering of an Ostrovsky wave packet in a delaminated waveguide

We examine the scattering of an Ostrovsky wave packet, generated from a solitary wave, in a two layered waveguide with a delamination in the centre and soft (imperfect) bonding either side of the centre. The lower layer of the waveguide is assumed to be significantly denser than the upper layer, leading to a system of Boussinesq–Klein–Gordon (BKG) equations. Direct numerical modelling is difficult and so a semi-analytical approach consisting of several matched asymptotic multiple-scale expansions is used, which leads to Ostrovsky equations in soft bonded regions and Korteweg–de Vries equations in the delaminated region. The semi-analytical approach and direct numerical simulations are in good agreement with each other and theoretical estimates. The dispersion relations are used to estimate the wave speed and hence classify the length of the delamination, in addition to changes in the amplitude of the wave packet. We also show how to scale the non-dimensional results to material variables and an example for PMMA is presented. These results can provide a tool to control the integrity of layered structures.

Eigenvalue repulsions in the quasinormal spectra of the Kerr-Newman black hole

We study the gravito-electromagnetic perturbations of the Kerr-Newman (KN) black hole metric and identify the two $-$ photon sphere and near-horizon $-$ families of quasinormal modes (QNMs) of the KN black hole, computing the frequency spectra (for all the KN parameter space) of the modes with the slowest decay rate. We uncover a novel phenomenon for QNMs that is unique to the KN system, namely eigenvalue repulsion between QNM families. Such a feature is common in solid state physics where \eg it is responsible for energy bands/gaps in the spectra of electrons moving in certain Schr\"odinger potentials. Exploiting the enhanced symmetries of the near-horizon limit of the near-extremal KN geometry we also develop a matching asymptotic expansion that allows us to solve the perturbation problem using separation of variables and provides an excellent approximation to the KN QNM spectra near extremality. The KN QNM spectra here derived are required not only to account for the gravitational emission in astrophysical environments, such as the ones probed by LIGO, Virgo and LISA, but also allow to extract observational implications on several new physics scenarios, such as mini-charged dark-matter or certain modified theories of gravity, degenerate with the KN solution at the scales of binary mergers.

Spectral Analysis for Singularity Formation of the Two Dimensional Keller–Segel System

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller–Segel problem by the authors [8]. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [11]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

ASYMPTOTICS OF A MULTIZONAL INTERNAL LAYER SOLUTION TO A PIECEWISE-SMOOTH SINGULARLY PERTURBED EQUATION WITH A TRIPLE ROOT OF THE DEGENERATE EQUATION

A singularly perturbed boundary value problem for a stationary equation of reaction-diffusion type in the case when reactive term undergoes discontinuity along some curve that is independent of the small parameter is studied. This is a new class of problems with triple roots of the degenerate equation, which leads to the formation of complex multizonal internal layers in the neighborhood of the discontinuity curve. By the method of asymptotic differential inequalities and matching asymptotic expansion, the existence of a contrast structure solution is proved. Using a different modified boundary layer function method, the asymptotic representation of point itself and this solution are constructed.

Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions

This work, which can be considered as a small monograph, is devoted to the study of twoand three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

THEORETICAL METHOD OF THE OPTIMUM DESIGN OF A SUPERCAVITATING PROPELLER BLADE WITH SPOILER MOUNTED ON THE TRAILING EDGE

This paper presents theoretical design method to obtain 2-D optimum section with spoiler mounted on the trailing edge of a supercavitating propeller blade. Matched Asymptotic Expansions (MAE) is applied to determine the geometry of profile and cavity shape in the framework of potential flow theory. The blade section is of wedge-like shape and the opened cavity closure scheme is adopted. A typical section, on which the optimum blade design will be based, is singled out among the best individual sections from root to tip. The spoiler length of each hydrofoil section resulting from MAE method are finalized with CFD method so as to consider viscous effect under the same lift condition, others hydrofoil geometries being kept constant. The hydrodynamic performances of all blade sections being designed on the basis of the resulting typical section from linearized method are finally predicted with CFD method.

Matched asymptotic expansion approach to pulse dynamics for a three-component reaction–diffusion system

We study the existence and stability of standing pulse solutions to a singularly perturbed three-component reaction–diffusion system with one activator and two inhibitors. We apply the MAE (matched asymptotic expansion) method to construct solutions and the SLEP (singular limit eigenvalue problem) method to evaluate their stability. This approach is not just an alternative approach to geometric singular perturbation and the associated Evans function, it also yields two advantages: one is extendability to higher dimensional cases, and the other is that it allows us to obtain more precise information about the behaviors of critical eigenvalues. This implies the existence of codimension-two singularities of drift and Hopf bifurcations for the standing pulse solution. Moreover, it is numerically confirmed that stable standing and traveling breathers emerge around the singularity in a physically acceptable region.

Quasi-steady-state reduction of a model for cytoplasmic transport of secretory vesicles in stimulated chromaffin cells.

Neurosecretory cells spatially redistribute their pool of secretory vesicles upon stimulation. Recent observations suggest that in chromaffin cells vesicles move either freely or in a directed fashion by what appears to be a conveyor belt mechanism. We suggest that this observation reflects the transient active transport through molecular motors along cytoskeleton fibres and quantify this effect using a 1D mathematical model that couples a diffusion equation to advection equations. In agreement with recent observations the model predicts that random motion dominates towards the cell centre whereas directed motion prevails in the region abutting the cortical membrane. Furthermore the model explains the observed bias of directed transport towards the periphery upon stimulation. Our model suggests that even if vesicle transport is indifferent with respect to direction, stimulation creates a gradient of free vesicles at first and this triggers the bias of transport in forward direction. Using matched asymptotic expansion we derive an approximate drift-diffusion type model that is capable of quantifying this effect. Based on this model we compute the characteristic time for the system to adapt to stimulation and we identify a Michaelis-Menten-type law describing the flux of vesicles entering the pathway to exocytosis.

Microconfined electroosmotic flow of a complex fluid with asymmetric charges: Interplay of fluid rheology and physicochemical heterogeneity

Abstract We report a study on the micro-confined electroosmotic flow of an Oldroyd-B fluid over a surface having asymmetric charge modulation. We have employed a combination of regular and matched asymptotic expansion to obtain the analytical solution. We have also carried out a full numerical simulation of the physicochemical problem in order to assess the full parameter space of the problem. The analytical solution is shown to be in excellent agreement with the full numerical solution in the limiting conditions of thin electrical double layers and weakly viscoelastic fluids. We have used these solutions to describe the complex fluid flow pattern and the modified electroosmotic slip velocity. We have also highlighted the alteration in the net volumetric throughput and ionic current. We have subsequently presented the numerical solutions for the velocity field, flow rates, and streaming current by varying the physicochemical conditions at the surface and viscoelastic properties of the fluid. We report an augmentation in the electroosmotic slip velocity with the increase in relative strength. The symmetry breaking of flow and stress distributions are shown to affect the net-throughput and streaming current, which is indicative of the combined effect of phase angle, magnitude of charge distribution, and the fluid viscoelasticity. We have shown that the thickness of electrical double layer considerably affects the net-throughput and streaming current generation in the microchannel. Our results hold the key towards understanding the complex bio-fluid transport in a microchannel utilizing the electric field in the presence of non-uniform surface charge distribution.

Spot Dynamics of a Reaction-Diffusion System on the Surface of a Torus

Quasi-stationary states consisting of localized spots in a reaction-diffusion system are considered on the surface of a torus with major radius $R$ and minor radius $r$. Under the assumption that these localized spots persist stably, the evolution equation of the spot cores is derived analytically based on the higher-order matched asymptotic expansion with the analytic expression of the Green's function of the Laplace--Beltrami operator on the toroidal surface. Owing to the analytic representation, one can investigate the existence of equilibria with a single spot, two spots, and the ring configuration where $N$ localized spots are equally spaced along a latitudinal line with mathematical rigor. We show that localized spots at the innermost/outermost locations of the torus are equilibria for any aspect ratio $\alpha=\frac{R}{r}$. In addition, we find that there exists a range of the aspect ratio in which localized spots stay at a special location of the torus. The theoretical results and the linear stability of these spot equilibria are confirmed by solving the nonlinear evolution of the Brusselator reaction-diffusion model by numerical means. We also compare the spot dynamics with the point vortex dynamics, which is another model of spot structures.

Резонансы в двумерных квантовых волноводах, разделенных полупрозрачным барьером с малым отверстием

A pair of coupled quantum waveguides with a common semitransparent boundary having a small aperture is considered, which leads to resonance phenomena during the passage of an electron. Using the method of matching asymptotic expansions, formulas are obtained for the first terms in the expansions of resonances (quasi-eigenvalues) and resonance states. The influence of the geometry of the system and the transparency parameter of the boundary on the lifetime of resonance states is investigated

Effect of body shape on riblets performance

By adopting a proper slip length boundary condition in Large Eddy Simulations of airfoil flows at high Reynolds numbers it can be shown that riblets reduce both friction and form drag. We show that riblets induce small but significant modifications of the pressure distribution which tends towards inviscid with improved pressure recovery in the airfoil aft. A quantitative analysis using a classical matched asymptotic expansion reveals that riblets reduce the boundary layer displacement thickness. Reduced thickening of the equivalent body introduced by the boundary layer reduces form drag leading to the unexpected improvement in pressure gradient flows measured in various experiments.

Internal Layer for a Singularly Perturbed Equation with Discontinuous Right-Hand Side

We consider a boundary value problem for an ordinary singularly perturbed second-order differential equation whose right-hand side is a nonlinear function with a discontinuity along some curve that is independent of the small parameter. For this problem, we study the existence of a smooth solution with steep gradient in a neighborhood of some point lying on this curve. The point itself and an asymptotic representation for the solution are to be determined. The existence theorem is proved by the method of matching asymptotic expansions. To this end, we use theorems on existence of solutions of boundary value problems for singularly perturbed equations and methods for constructing asymptotic approximations to these solutions.

Domain-size effects on boundary layers of a nonlocal sinh–Gordon equation

This work investigates a nonlocal sinh–Gordon equation with a singularly perturbed parameter in a ball. Under the Robin boundary condition, the solution asymptotically forms a quite steep boundary layer in a thin annular region, and rapidly becomes a flat curve outside this region. Focusing more particularly on the structure of the thin annular layer in this region, the pointwise asymptotic expansion involving the domain-size is evaluated more sharply, where the domain-size exactly appears in the second term of the asymptotic expansion. It should be stressed that the standard argument of matching asymptotic expansions is limited because the model has a nonlocal coefficient depending on the unknown solution. A new approach relies on integrating ideas based on a Dirichlet-to-Neumann map in an asymptotic framework. The rigorous asymptotic expansions for the thin layer structure also matches well with the numerical results. Furthermore, various boundary concentration phenomena of the thin annular layer are precisely demonstrated.