Order Asymptotic Solution Research Articles

Asymptotic analysis of the narrow escape problem in general shaped domain with several absorbing necks

This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several absorbing narrow necks. The narrow escape problem is to compute the mean first passage time (MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet–Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann–Robin type boundary value problem. By solving the new model, we obtain the three-term asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for narrow escape problem (NEP) in a general shaped domain with several absorbing neck windows. This work is motivated by Li (2014 J. Phys. A: Math. Theor. 47 505202), where the Neumann–Robin model was proposed to solve the NEP in a domain with a single absorbing neck.

Langevin dynamic simulations of magnetic hyperthermia in rotating fields

The effect of a rotating magnetic field on heat dissipation due to magnetic hyperthermia is investigated numerically. A robust and validated in-house code based on Langevin Dynamics is used. High order asymptotic solutions of the temperature rate are used to validate the numerical approach. Comparisons between a rotating and an oscillatory excitation indicate a nonlinear response of the magnetic suspension with respect to the amplitude and frequency of the field. The relaxation time and volume fraction of particles also play a relevant role in this dynamics. A complete analysis on the effect of these parameters on the thermal response of the system is presented. The timescales of the problem and geometric features of the suspension micro-structure are used to interpret the results. Hysteresis curves and the real and imaginary components of the complex susceptibility are also presented in order to provide a clear physical description of the problem.

A unified method to solve higher order asymptotic crack-tip fields of mode I, mode II and mixed mode I/II crack in power-law creeping solids

A higher order asymptotic analysis with separable variable form is presented in this paper for crack tip mechanics fields under mixed mode creep conditions. By defining a second order mode mixity factor, higher order term solutions are investigated by solving the boundary value problem. The solution procedure is validated first and then the characteristics of various kinds of second order asymptotic solutions for mixed I/II creep crack tip field are presented and discussed. The angular distribution functions of the second order stress and the stress exponent for various cases are given. Detailed relations between the amplitudes of the second order term and the first order term are also discussed. The method and solutions given in this paper encompass the higher order term solutions of pure mode I, pure mode II and mixed mode I/II.

Condensation of a Quiescent Vapor by a Stagnation-Point Liquid Flow

We analyze the condensation of a quiescent vapor, that is in equilibrium with its liquid, induced by a stagnation point flow in the liquid. The liquid flow brings subcooled liquid from far away to the interface. The ensuing heat transfer causes the vapor to condense. A similarity formulation for the liquid and vapor flow fields and the liquid temperature field is pursued, and a perturbation solution is performed when the ratio of the product of viscosity and density of the vapor to that of the liquid is small. A two-term higher order asymptotic solution is shown to be in excellent agreement with numerical results. The reduction in the rate of condensation due to the presence of a noncondensable gas in the vapor that is insoluble in the liquid is also analyzed.

一类Fermi气体光晶格非线性机制的轨线研究

A nonlinear disturbed model for a class of Fermi gases in optical lattices was investigated. Firstly, in the nondisturbed case, the exact solution of the model path curves of Fermi gases in optical lattices was given. Secondly, the generalized functional analysis homotopic mapping was introduced and an iterative system was constructed, the arbitrary order asymptotic solution to the nonlinear disturbed model for the path curves of Fermi gases in optical lattices was obtained. Finally, a nonlinear small disturbance system was studied. With the proposed method, the asymptotic expressions of the path curves can be conveniently formulated and further extended.

High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice. The leading order solution is expressed as a product between the standing wave solution and the long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones. A significant advantage of the method exposited herein is that it allows one to obtain analytical expressions for the leading order asymptotic solutions and effective material properties, even for complex systems; the methodology allows for greater physical understanding of the behaviour of the system than is usually the case for purely numerical homogenisation schemes.

Solution of the transfer models of femtosecond pulse laser for nano metal film

A class of transfer models for femtosecond pulse laser on nano metal film has been investigated. First, we solve the reduced solution. And then, the arbitrary order asymptotic solution of corresponding model is obtained by using the perturbation theory and method. Finally, the behavior of the solution is discussed.

The asymptotic analysis of interfacial stability with surface tension anisotropy for directional solidification of alloys

An asymptotic analysis of interfacial stability with the surface tension anisotropy is derived for directional solidification of alloys. The zero and first order solution for multi-scale and the zero order solution for single-scale are obtained. The dispersion relation adapted to all wave-length is presented by combining the short and the long wave-length solutions. It was concluded that the zero order asymptotic solution is equivalent to constitutional undercooling theory and the first order solution is consistent with M-S theory. The instability zone of control parameter enlarges with the surface tension anisotropy,especially for the high temperature gradient and high pulling velocitycases.

Comparison of dynamic light scattering measurements and mode-coupling theory for the tagged particle dynamics of a hard-sphere suspension

The mean-squared displacement, velocity autocorrelation function, and the non-Gaussian parameter, obtained by dynamic light scattering on suspensions of particles with hard-sphere interactions, are compared with the results of the idealized version of mode-coupling theory. Both leading order asymptotic and full numerical solutions of the mode-coupling equations are considered. Experiment and the full numerical results of the theory expose similar qualitative changes at the volume fraction of the first order freezing transition. In particular, the emergence of negative algebraic decays in the velocity autocorrelation function of the undercooled suspension suggest the emergence of clusters in which particles are trapped. Consistency of experiment, computer simulation, and theory in this regard suggests that, at particular strengths of the delayed, nonlinear feedback, contained in mode coupling theory, the latter predicts not only structural arrest which, as already established, is symptomatic of a glass transition, but also a more subtle change in dynamics that signals the onset of the first order transition.

Asymptotic solutions of the leaky modes and PML modes in a Pekeris waveguide

Leaky modes are useful to partially represent the continuous spectrum in open waveguides. The perfectly matched layer (PML) is a widely used technique for truncating unbounded domains in numerical simulations of wave propagation problems. When a Pekeris waveguide is terminated by a finite PML, it gives rise to three classes of modes corresponding to the trapped modes, the leaky modes and the modes that approximate the branch line integral along the Pekeris cut. In this paper, we derive high order asymptotic solutions for these modes in a Pekeris waveguide and a Pekeris waveguide terminated by a finite PML.

High order asymptotic solution of planar contraction flow with a slip boundary condition

The creeping flow of an incompressible viscous fluid in a planar contraction channel (Jeffery-Hamel flow) is investigated using slip boundary conditions. A high order asymptotic solution is presented to describe the velocity field accurately. Results indicate that the effect of slip is strong in the region close to the sink, while it is almost negligible in the far field, and in some cases, multi-extrema and reverse values appear in the velocity profile near the sink.

On the high order asymptotic solution of certain wave equations

On the high order asymptotic solution of certain wave equations

Higher Order Asymptotic Solution for an Interfacial Crack in Piezoelectric Bimaterial under Impact

This paper investigates the asymptotic transient structure of the near-tip field in an anisotropic piezoelectric bimaterial system under impact loading. By using an asymptotic methodology, the governing equation is reduced to a series of coupled partial differential equations. Using the extended version of Stroh's formulation and the method of analytical continuation, the problem is further recast into a Riemann-Hilbert problem of vector form. Upon solving the Riemann-Hilbert equation, the higher order near-tip transient asymptotic fields are obtained. For a bimaterial of PZT and PZT-4 piezoelectric ceramics, the direct and converse piezoelectric effects on the stress intensity factors and electric displacement intensity factors are clarified.

On the micromechanics theory of Reissner-Mindlin plates

A micromechanics model is developed for the Reissner-Mindlin plate. A generalized eigenstrain formulation, i.e., an eigencurvature/eigen-rotation formulation, is proposed, which is the analogue or counterpart of the eigenstrain formulation in linear elasticity. The micromechanics model of the Reissner-Mindlin plate is useful in the study of mechanical behavior of composite plates that contain randomly distributed inhomogeneities, whose sizes are close to the order of thickness of the plate; under those circumstances, the use of micromechanics of linear elasticity is not justified, and moreover, it is inconsistent with structural theories, such as the Reissner-Mindlin plate theory, that are actually used in engineering design. In this paper, the analytical solution of an elliptical inclusion embedded in an infinite thick plate is sought. In particular, the first order asymptotic (or approximated) solution of the elliptical inclusion problem is obtained in explicit form. Accordingly, the Eshelby tensors of the Reissner-Mindlin plate are derived, which relate eigencurvature and eigen-rotation to the induced curvature and shear deformation fields. Several variational inequalities of the Reissner-Mindlin plate are discussed and derived, including the comparison variational principles of Hashin-Shtrikman/Talbot-Willis, type. As an application, variational bounds are derived to estimate the effective elastic stiffness of Reissner-Mindlin plates, specifically, the flexural rigidity and transverse shear modulus. The newly derived bounds are congruous with the Reissner-Mindlin plate theory, and they provide an optimal estimation on effective rigidity as well as effective transverse shear modulus for unstructured composite thick plates.

Higher order fields for damaged nonlinear antiplane shear notch, crack and inclusion problems

Damaged nonlinear antiplane shear problems with a variety of singularities are studied analytically. A deformation plasticity theory coupled with damage is employed in analysis. The effect of microscopic damage is considered in terms of continuum damage mechanics approach. An exact solution for the general damaged nonlinear singular antiplane shear problem is derived in the stress plane by means of a hodograph transformation, then corresponding higher order asymptotic solutions are obtained by reversing the stress plane solution to the physical plane. As example, traction free sharp notch and crack, rigid sharp wedge and flat inclusion, and mixed boundary sharp notch problems are investigated, respectively. Consequently, higher order fields are obtained, in which analytical expressions of the dominant and second order singularity exponents and angular distribution functions of the near tip fields are derived. Effects of the damage and hardening exponents of materials and the geometric angle of notch/wedge on the near tip quantities are discussed in detail. It is found that damage leads to a weaker dominant singularity of stress, but to little stronger singularities of the dominant and second order terms of strain compared to that for undamaged material. It is also seen that damage has important effect on the angular distribution functions of the near tip stress and strain fields. As special cases, higher order analytical solutions of the crack and rigid flat inclusion tip fields are obtained, respectively, by reducing the notch/wedge tip solutions. Effects of damage and hardening exponents on the dominant and second order terms in the solutions of the crack and inclusion tip fields are discussed.

Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading

The higher order solutions of stress and deformation fields near the tip of a sharp V-notch in a power-law hardening material with continuous damage formation are analytically investigated under antiplane shear loading condition. The interaction between a macroscopic sharp notch and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable in the framework of damage mechanics. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the solution of damaged nonlinear notch problem in the stress plane. Then, inversion of the stress plane solution to the physical plane is performed. Consequently, higher order terms in the asymptotic solutions of the notch tip fields are obtained. Analytical expressions of the dominant and second order singularity exponents and associated angular distribution functions of notch tip stress and strain are presented. Effects of damage and strain hardening exponents and notch angle on the singular behavior of the notch tip quantities are discussed detailly. It is found that damage can lead to a weaker singularity of the dominant term of stress on one hand, but to stronger singularities of the second order term of stress and the dominant and second order terms of strain compared to that for undamaged case on the other. Also, both hardening exponent and notch angle have important effects on the notch tip quantities. Moreover, reduction of the notch tip solutions to a damaged nonlinear crack problem is carried out, and higher order solutions of the crack tip fields are obtained. Effects of damage and hardening exponents on the dominant and second order terms in the crack tip solutions are detailly discussed. Discussions on some other special cases are also presented, which shows that if damage exponent equals to zero, then the present solutions can be easily reduced to the solutions for undamaged cases.

Eigenspectra and Orders of Singularity at a Crack Tip for a Power-Law Creeping Medium

The objective of this paper is to study the eigenspectrum of a creeping body with a static mode I crack. The coordinate perturbation technique is employed to pose the required asymptotic equations. To attain eigensolutions a numerical scheme is worked out and the results obtained provide the information including the number of singularities, and their orders, as well as the angular distributions of stresses. In particular, additional eigenvalues of the HRR/RR problem that have not been reported previously are presented in the context, indicating the fact that other singular fields may exist for the problem. The discussion is also performed to the higher order asymptotic solutions that differ from those resulting from the eigen expansion treatment.

Buckling and postbuckling of shear-deformable rectangular plates of composite construction

Buckling and postbuckling behaviours of rectangular plates of composite construction are studied by means of the generalised von Karman equations counting for arbitrary lamination and material properties. In conjunction with four edges simply supported, higher order asymptotic solutions to the large deflection equations with incorporation of initial geometrical imperfection are constructed by the deflection type perturbation technique. Effects of initial imperfection, transverse shear deformation, shear-bending material interaction, and in-plane edge condition on buckling and post-buckling of the plates are examined through typical examples, and comparison of the present solution are made with other theoretical and numerical results. It is shown that the present predications are of fairly good accuracy.

One-dimensional bulk waves in a nonlinear magnetoelastic conductor of finite electric conductivity

In this paper, an attempt has been made to study one-dimensional (1-D) wave propagation in a magnetoelastic medium of finite electric conductivity subjected to an initial magnetic field-first longitudinal and then transverse to the direction of propagation—within the frame of a model including the first level of nonlinearity. The second order asymptotic solution is derived in terms of six arbitrary constants. Phase velocities and attenuation constants are obtained. A comparison of the first order solution with the existing literature reveals a complete agreement, except for the fact that the magnetic field now has non-vanishing fluctuations along the direction of propagation and this increases the velocity of the purely elastic waves of the usual state. An electric field is produced in the direction of propagation at the second order approximation if the initial magnetic field has a non-vanishing component in this same direction.

KBM METHOD OF ANALYZING LIMIT CYCLE FLUTTER OF A WING WITH AN EXTERNAL STORE AND COMPARISON WITH A WIND-TUNNEL TEST

The limit cycles flutters of two models of a wing/store with free play in the junction, one of two-dimensional wing, and the other a delta wing, are investigated. The equivalent linearization method based on the first order asymptotic solution of the KBM method and the method based on the second order asymptotic solution of the KBM method are used to analyze the limit cycle flutter of the wing with store system. The results are compared with those of numerical integration and a wind-tunnel test. It is shown that, in some cases, the features of the limit cycle flutter obtained by numerical integration can be predicted by the equivalent linearization method based on the second order asymptotic solution of the KBM method, whereas the conventional describing function method fails to do so. The results of the wind-tunnel test show that, for free play, the effect of high order harmonics on the limit cycle response to the aeroelasticity system is strong when the limit cycle amplitude is small.