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One of the main problems of fracture mechanics is to describe the behavior of media with cracks. At the same time, more and more attention is paid to the study of large systems, including periodic structures. For several cracks, the coefficient of influence is considered to be an important parameter. It is the ratio of the stress intensity factor calculated in the problem of a system of cracks to the stress intensity factor for a single crack under the same load. The paper proposes a method that allows to accurately determine the coefficients of influence for large systems of cracks, including the case of periodic structures, in which the number of cracks is infinite. The method is based on a numerical algorithm developed by the authors that uses the method of discontinuous displacements of a high order of accuracy and an asymptotic representation of stress fields at the crack tip (M. Williams). To verify the method, the results of the implemented algorithm are compared with known analytical solutions both for single cracks and for an infinite doubly periodic system of cracks.

Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter ( e ) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O ( e 2 ) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number ( R e ). Furthermore, at subcritical R e with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

An analytical study on the propagation characteristics of shear wave in a spherical layered structure comprising of an isotropic sandy material medium covered by a concentric spherical transversely isotropic sandy material layer of finite thickness has been carried out in the present work. The source of excitation for wave propagation is considered in the outer layer and the analytical treatment with contour integration and Fourier-Legendre Expansion theorem is utilized to accomplish closed form of dispersion relation. Further, Debye Asymptotic Expansion analysis is used to deal the complication caused by the inclusion of Bessel's function, Hankel's function and their derivatives in the solution procedure. With help of Debye Asymptotic Expansion analysis, the obtained dispersion relation converts in the form of elementary functions and found to be in well-agreement with pre-established and classical results through the particular cases. Numerical computations are done to illustrate the influence of the affecting parameters (anisotropy, sandiness and radii ratio) on dispersion curves graphically. Moreover, to expose the impact of anisotropy, numerical data of three distinct transversely isotropic materials (Magnesium, Zinc and Beryl) are taken into the account to establish the comparative study with isotropic material which serves the major highlights of the present study.

The focus of this paper is to obtain explicit solutions to initial value problems, where numerical methods cannot provide one, and to verify the accuracy orders of the explicit solutions. One of available methods to obtain an explicit solution is the asymptotic (formal) expansion method. However, we must be sure with the accuracy order of the explicit solution. In this paper, an order verification method is proposed for truncated asymptotic formal expansion solutions to initial value problems. A least-squares fit of error data is used in the existing order verification method. The method that we propose does not involve any application of least-squares fit of error data, so is simpler, yet produces accurate expected accuracy orders of solutions of explicit truncated asymptotic formal expansions. With our proposed method, we are successful in verifying the accuracy orders of solutions of truncated asymptotic formal expansions to the linear and nonlinear initial value problems accurately.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Ito semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.

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.

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles.

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18] .

The Hurwitz zeta function ζ ( s , a ) admits a well-known (divergent) asymptotic expansion in powers of 1 / a involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Pade approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers ≥2. The latter can be viewed, in a certain sense, as analogues of Apery's celebrated sequences of rational approximations to ζ ( 2 ) and ζ ( 3 ) .

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method.

In this paper, we apply ∂ ‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equation u x t = u + 1 6 ( u 3 ) x x , u ( x , 0 ) = u 0 ( x ) ∈ H ( R ) , where H ( R ) = W 3 , 1 ( R ) ∩ H 2 , 2 ( R ) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale ( y , t ) . In any fixed space-time cone C ( y 1 , y 2 , v 1 , v 2 ) = { ( y , t ) ∈ R 2 : y = y 0 + v t , y 0 ∈ [ y 1 , y 2 ] , v ∈ [ v 1 , v 2 ] } , we compute the long time asymptotic expansion of the solution u ( x , t ) , which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N ( I ) -soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second | t | − 1 / 2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O ( | t | − 1 ) from a ∂ ‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable.

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.

For the two-dimensional elasticity problems, we give a uniqueness and existence analysis via the single-layer potential approach leading to a system of integral equation that contains a weakly singular operator. For its numerical solutions we describe a O ( h 3 ) order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power O ( h 3 ) is got and the accuracy of numerical approximations can be improved to the order of O ( h 5 ) by the Richardson extrapolation algorithm (EA) once. The efficiency of the method is illustrated by two numerical examples.

In this paper, we prove the global well-posedness for the incompressible magnetohydrodynamics (MHD) equations in the three-dimensional unbounded domain Ω : = R + × R 2 . More precisely, we construct global small Sobolev regularity solutions with the initial data near 0 for the three-dimensional MHD equations in Ω. The key point of the proof is to find the suitable initial approximation function such that the linearized equations around it admitted a partial dissipative structure when we carry out the weighted energy estimate. Meanwhile, the asymptotic expansion of Sobolev regularity solutions is given.

In this paper we study both the fast computation and the related error analysis of many different finite and infinite oscillatory Bessel transforms. By transforming the Bessel function into Confluent hypergeometric function and using its asymptotic expansion, we develop an efficient approach by the contours in the complex plane, corresponding to the paths of steepest descent. Furthermore, we conduct their error analyses in inverse powers of the frequency ω. The related error analyses show that the precision can be improved by either adding more Gauss-Laguerre nodes or increasing the frequency ω. In particular, the method exhibits the high error order. Meanwhile, we apply the proposed method to some other kinds of oscillatory integrals through some simple changes of variables. Numerical experiments can verify our theoretical analysis. It is also shown that the new algorithms are more efficient than the existing Filon-type method and can achieve the same level of accuracy with the method (the XMX method) proposed in Xu, Milovanovic and Xiang [46] by some numerical experiments under the given conditions. A combination of the n 1 -point Gauss-Laguerre quadrature rule and the n 2 -point generalized Gauss-Laguerre quadrature rule are used in the XMX method. However, we only use the Gauss-Laguerre quadrature rule for one time in the established method, which makes the theoretical and error analysis simpler than the XMX method.

An accurate evaluation of three-dimensional (3D) Time-Domain Green Function (TDGF) in infinite water depth is essential for ship’s hydrodynamic analysis. Various numerical algorithms based on the TDGF properties are considered, including the ascending series expansion at small time parameter, the asymptotic expansion at large time parameter and the Taylor series expansion combines with ordinary differential equation for the time domain analysis. An efficient method (referred as “Present Method”) for a better accuracy evaluation of TDGF has been proposed. The numerical results generated from precise integration method and analytical solution of Shan et al. (2019) revealed that the “Present method” provides a better solution in the computational domain. The comparison of the heave hydrodynamic coefficients in solving the radiation problem of a hemisphere at zero speed between the “Present method” and the analytical solutions proposed by Hulme (1982) showed that the difference of result is small, less than 3%.

Under investigation in this work is the general three-component nonlinear Schrodinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.

We study the critical periods of the Loud family of quadratic centers. We give the first terms of the asymptotic expansion of the period function and we conclude that no bifurcation occurs for certain values of the parameters. This result contributes to complete the bifurcation diagram of critical periods in the Loud family.

In this work the velocity of an isolated growth ledge located on an interphase boundary is solved by the method of matched asymptotic expansion where the “inner” solution is obtained by a conformal mapping procedure. In contrast to previous work, both a constant flux and local equilibrium is maintained at all points along the step face when formulating the solution. The matching procedure leads to a solvability condition for the problem, which in turn yields an analytic solution for the Peclet number as a function of supersaturation. The theoretical treatment is extended, in an approximate way, to the case of multiple steps. Predictions from the matched asymptotic expansion formulation are compared to experimental results for BCC precipitate growth in the Ni–Cr system.

We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of the rescaled Debye length, whose detailed derivation and analysis are carried out in our companion paper [8] . By establishing H m -norm, ( m ≥ 2 ) , estimate of the difference between the original and approximation solutions, provided that the well-prepared initial data is given, we show that the local-in-time solution exists in the time interval, uniform in the quasi-neutral limit, and we prove the difference converges to zero with a certain convergence rate validating the formal expansion order. Our results mathematically justify the quasi-neutrality of a plasma in the regime of plasma sheath, indicating that a plasma is electrically neutral in bulk, whereas the neutrality may break down in a scale of the Debye length.

In the steady state heat conduction problem, the heat flux could be infinite at the re-entrant corner, at the point where the boundary condition is discontinuous, or at the place where the material properties are changing abruptly. The conventional numerical methods, such as the finite element method (FEM) and the boundary element method (BEM), have difficulties in analyzing the singular heat flux field. Herein, a new singular element employed in the boundary integral equation is developed to interpolate the heat flux field near the singular point. The shape function of the singular element is set as an asymptotic expansion model with respect to the distance from the singular point. The singular order in the asymptotic expansion can be determined by solving the singularity eigen equation on the basis of the interpolating matrix method. The self-adaptive co-ordinate transformation method is introduced to deal with the weakly singular integral in the proposed method. Benefited from the singular element, more accurate temperature and heat flux results near the singular point are obtained with fewer elements. In addition, the proposed method is easy to be coupled with the conventional BEM without too much modifications.

Variable Angle Tow (VAT) composite panels, whereby fibre angle distributions vary continuously in-plane, have more scope for tuning structural properties than traditional straight fibre laminated composite materials. Currently, the geometrically non-linear response of such structures is typically modelled using path-following methods implemented in commercial finite element analysis. However the high computational cost given from both the incremental-iterative procedure and fine mesh required for accurate modelling of the variable trajectory of fibre direction, can be excessively time consuming. Driven by these shortcomings, we use a fast multi-modal Koiter asymptotic method for investigating the nonlinear buckling behaviour of VAT cylindrical panels in compression. In doing so, this paper provides new insight into the multi-modal description and nonlinear buckling mode interactions which are responsible for the highly nonlinear behaviour of cylindrical panels in compression with a particular emphasis on the nonlinear components in the mathematical asymptotic description. As such, through an extensive parametric virtual testing programme we construct, for each panel under investigation, an asymptotic solution projecting the equilibrium equations in the subspace of its first 30 buckling modes. Then, at several points of the resulting equilibrium path, the percentage contribution of each of these modes to the solution is measured. On the basis of the percentage obtained, those modes which give the largest contribution to the solution are identified. The main novelty of this work is that these results are used as a priori information for re-projecting the equilibrium equations into a new modal subspace to which only the buckling modes with the largest participation belong. Moreover, for the first time, we observe that in a multi-modal Koiter-inspired description the modal shapes giving the largest contribution to the asymptotic solution have the same degree of symmetry. In this way, for a generic panel under consideration, we show that it is possible to obtain a more computationally-efficient asymptotic description than current approaches. Finally, the new computed equilibrium paths are compared to benchmark results using the commercial finite element software ABAQUS and the computational advantages given by using a priori information in the asymptotic expansion are commented upon.

We study the mean exit time from a bounded multi-dimensional domain Ω of the stochastic process governed by the overdamped Langevin dynamics. This mean exit time solves the boundary value problem ( − e 2 Δ + ∇ V ⋅ ∇ ) u e = 1 in Ω , u e = 0 on ∂ Ω , e → 0 . The function V is smooth enough and has the only minimum at the origin contained in Ω; the minimum can be degenerate. At other points of Ω, the gradient of V is non-zero and the normal derivative of V at the boundary ∂Ω does not vanish. Our main result is a complete asymptotic expansion for u e . The asymptotics for u e involves an exponentially large term, which we find in a closed form. We also construct a power in e asymptotic expansion such that this expansion and a mentioned exponentially large term approximate u e up to arbitrary power of e.

We consider the problem of steady pumping of water from a line drain on the surface of a wet ground. Unlike the classical formulation, which regards the conductivity parameter K as uniformly distributed in the domain, the problem here is solved within a stochastic framework in order to account for the irregular (random), and more realistic, spatial vari- ability of K. Due to the linearity of the problem at stake, we focus on the derivation of the mean Green function G. This is computed by means of an asymptotic expansion. The fundamental result is an analytical (closed form) expression of G which general- izes the classical solution. Based on this, we develop an equivalent conductivity Keq which enables one to tackle the problem similarly to the classical one. In particular, it is shown that the equivalent conductivity grows monotonically with the radial distance r from the drain, and it lies within the range Keq (0) ≤ Keq (r) ≤ Keq (∞) < ∞.

Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen’s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation. The interactions between the SWCNT and its surrounding medium are modeled as a two-parameter Pasternak foundation model. After performing a series of mathematical processes, including nondimensionalization, asymptotic expansion, and successive integration, etc., the authors obtain recurrent sets of motion equations for various order problems. The nonlocal classical shell theory (CST) is derived as a first-order approximation of the current 3D nonlocal elasticity problem, and the equations of motion for higher-order problems retain the same differential operators as those of the nonlocal CST, although with different nonhomogeneous terms. The current asymptotic solutions for the natural frequency parameters of non-embedded or embedded SWCNTs and their corresponding through-thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

We consider a family of tronquee solutions of the Painleve II equation q ″ ( s ) = 2 q ( s ) 3 + s q ( s ) − ( 2 α + 1 2 ) , α > − 1 2 , which is characterized by the Stokes multipliers s 1 = − e − 2 α π i , s 2 = ω , s 3 = − e 2 α π i with ω being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if ω = 0 . We derive asymptotics of integrals of the tronquee solutions and the associated Hamiltonians over the real axis for α > − 1 / 2 and ω ≥ 0 , with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters α and ω are chosen to be special values. Some applications of our results in random matrix theory are also discussed.

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.

Theoretical study of superluminal sources of electromagnetic radiation boosted after the discovery of Cherenkov-Vavilov radiation. Later, the way to create fictitious sources moving superluminally was suggested. Different approaches have been proposed for the research of the distribution of the potential and the fields radiated by the superluminally moving charges. The simplest idealized cases of uniform rectilinear motion of the charge and of the charge rotating with constant angular speed open opportunities of a detailed analysis of the fields and potentials. We use Fourier series to calculate the potential distribution of point charge rotating with constant speed. An obvious advantage of this approach is that one no longer needs to calculate the retarded positions of the charge. The number of the retarded positions depends on the observation point and increases as the ratio {\omega}{R_0}/c rises, where c is the speed of light, {\omega} is the rotation frequency, and {R_0} is the radius of the circle. We demonstrate that equation of Mach surface can be obtained basing on the asymptotic expansion of the potential. We analyze some characteristics of the potential basing on this asymptotic expansion., Comment: 15 pages, 1 figure

This paper presents an asymptotic method-based analysis of composite laminates having interfacial imperfections. In general, imperfect interfaces are simply modeled by introducing a linear, spring-layer model, which empirically assumes that the displacement jumps that occur at the weakened interface are proportional to the transverse shear stresses of interface positions. In this study, we propose a composite plate model derived by using asymptotic expansion that does not make any assumptions, other than the scaling of coordinate systems. Within the framework of the asymptotic analysis, the spring-layer model is introduced to describe the effect of weakened interfaces, which is realized by the separation of domains in the through-the-thickness direction, and the integration of piecewise continuous warping functions. As a result, we newly define a spring parameter that is exactly the same as the stiffness of a spring. Therefore, a set of spring elements are added to the through-the-thickness modeling of the microscopic analysis, and the plate equations derived in the macroscopic problem are the same as those of the perfectly bonded laminates. As a consequence, we also derive the proposed plate model with the mathematical rigor that the previous asymptotic models contain. We provide some numerical results verifying that the proposed method shows good agreement with the elasticity and 3D FEM solutions.

A high-order three-scale approach developed in this work to analyze thermo-mechanical properties of porous materials with interior surface radiation is systematically studied. The microstructures of the porous structures are described by periodical layout of local cells on the microscopic domain and mesoscopic domain, and surface radiation effect at microscale and mesoscale is also investigated. At first, the three-scale formulas based on reiterated homogenization and high-order asymptotic expansion are established, and the local cell solutions in microscale and mesoscale are also defined. Then, two kinds of homogenized parameters are evaluated by upscaling methods, and the homogenized equations are derived on the whole structure. Further, heat flux and strain fields are constructed as the three-scale asymptotic solutions by assembling the higher-order unit cell solutions and homogenized solutions. The significant features of the proposed approach are an asymptotic high-order homogenization that does not require higher order continuity of the macroscale solutions and a new high-order three-scale formula derived for analyzing the coupled problems. Finally, some representative examples are proposed to verify the presented methods. They show that the three-scale asymptotic expansions introduced in this paper are efficient and valid for predicting the thermo-mechanical properties of the porous materials with multiple spatial scales.

This first-attempt study provided asymptotic expansion solutions of singular perturbation to disclose unsolved kinetics of bisphenol A (BPA) degradation via advanced oxidation processes (AOPs). Although different rate laws (e.g., first order kinetics) were applied to kinetic modeling via initial rate determination in AOPs, transient dynamics of pollutant degradation were still remained open to be unexplored, leading to problems for closed-form system optimization of AOPs. This study adopted “two-compartment open model” to decipher such fast reaction characteristics of BPA degradation. The area under the curve (AUC) was adopted to determine clearance of BPA degradation, revealing economically feasible treatment of AOPs. This performance index provided a more appropriate indicator than the percentage of degradation efficiency popularly used in literature. As the results indicated, k3-dominance should be avoided to guarantee the existence of “bi-exponential disposition” as first necessary condition for operation optimality. In addition, the phenomenon of “vanishing exponential” would lead to failure treatment of AOPs (i.e., divergent AUC value); thereby, AOPs should be away from such situation(s) to be taken place. Moreover, minimization of AUC could clearly exhibit the optimal performance index for maximal BPA degradation to be achieved.

This paper is concerned with the topological sensitivity analysis for a parabolic type operator. We consider the unsteady-state heat transfer equation as a model problem and we develop the topological asymptotic expansion valid for a large class of cost functions. The obtained theoretical results are based on a simplified mathematical analysis involving the elliptic part of the operator. In the numerical part, we proposed a simple and fast geometry reconstruction algorithm. The proposed algorithm is applied for recovering internal cooling holes in a turbine blade by minimizing various error functions measuring the difference between the current and actual temperature fields.

This paper is concerned with particle trajectories beneath solitary waves when a linear shear current exists. The fluid is assumed to be incompressible and inviscid, lying on a flat bed. Classical asymptotic expansion is used to obtain a Korteweg-de Vries (KdV) equation, then a forth-order Runge-Kutta method is applied to get the approximate particle trajectories. On the other hand, our particular attention is paid to the direct numerical simulation (DNS) to the original Euler equations. A conformal map is used to solve the nonlinear boundary value problem. High-accuracy numerical solutions are then obtained through the fast Fourier transform (FFT) and compared with the asymptotic solutions, which shows a good agreement when wave amplitude is small. Further, it also yields that there are different types of particle trajectories. Most surprisingly, periodic motion of particles could exist under solitary waves, which is due to the wave-current interaction. Keywords: Particle trajectories, Linear shear current, Solitary waves, Direct numerical simulation

In this study coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and are obtained by the use of finite element analysis. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The higher-order coefficients are extracted from finite element method calculations implemented in multi-purpose software package Simulia Abaqus. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy.

Thin-walled isogrid-stiffened structures have been widely used as highly efficient structural components in aerospace engineering. In this work, the stability behavior of isogrid-stiffened cylinder is analyzed using the smeared stiffener model based reduced-order modelling (SSM-ROM) method. The smeared stiffener method proposed based on mechanical characteristics of isogrid-stiffener cell is applied to facilitate the finite element(FE) modelling of structure. The reduced-order models constructed based on Koiter asymptotic expansion are extended to be applicable for the smeared stiffener model. Mode selection criteria for cylinders with measured geometric-shape imperfections are developed to lower the computational cost in construction of the reduced-order model. The proposed method is implemented into the finite element framework and achieved completely using the in-house codes. The isogrid-stiffened cylinders with two different configurations(strong or weak stiffeners) are considered to validate the performance of the proposed method. Linear and nonlinear buckling analyses are first applied to test the numerical accuracy of the smeared stiffener model. The sensitivity of the buckling load in terms of magnitude of initial imperfections is then carefully studied using three different geometric imperfection shapes. Numerical results demonstrate the good performance of the proposed method in both structural buckling analysis and imperfection sensitivity analyses.

In this study coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate are obtained by the use of the digital photoelasticity method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. The developed tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The higher-order coefficients are extracted from finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy.

We study different discretizations of the van Roosbroeck system for charge transport in bulk semiconductor devices that can handle nonlinear diffusion. Three common challenges corrupting the precision of numerical solutions will be discussed: boundary layers, discontinuities in the doping profile, and corner singularities in L -shaped domains. We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as advanced Scharfetter–Gummel type finite-volume discretization schemes. The most problematic of these challenges are boundary layers in the quasi-Fermi potentials near ohmic contacts, which can have a drastic impact on the convergence order. Using a novel formal asymptotic expansion, our theoretical analysis reveals that these boundary layers are logarithmic and significantly shorter than the Debye length.

An effective second-order reduced asymptotic expansion (SRAE) approach is proposed to analyze the thermo-mechanical coupling problems of nonlinear periodic heterogeneous materials. The first-order and second-order nonlinear unit cell solutions at a microscale obtained by calculating the different multiscale cell functions are given at first. Then, the homogenization coefficients are evaluated, and the nonlinear homogenized equations defined on whole structure are solved, successively. Also, the temperature and displacement fields are constructed as second-order multiscale approximate solutions by assembling the distinct local cell solutions and homogenized solutions. The main features of the proposed approach are: (i) an effective model reduction scheme for computing high-order nonlinear unit cell problems and (ii) an asymptotic high-order homogenization solution that does not need high-order continuity of the macroscale solutions. Further, the corresponding finite element-difference algorithms based on the SRAE approach are given in detail. Finally, by some typical numerical examples, the availability and correctness of the proposed algorithms are confirmed. The computational results clearly demonstrate that the SRAE approach reported in this work are efficient and valid to predict the macroscopic thermo-mechanical properties, and can catch the microscale information of the heterogenous materials accurately.

In this paper, we study efficient numerical methods for the computation of highly oscillatory Bessel transform ∫ a + ∞ f ( x ) J ν ( ω g ( x ) ) d x , g ( x ) ≠ 0 , g ′ ( x ) ≠ 0 for all x ∈ [ a , + ∞ ) . We first derive an asymptotic expansion formula for this integral by using integration by parts. Then we present an improved numerical steepest descent method, by applying complex integration theory to the remainder term of asymptotic expansion, which is also an oscillatory integral with Bessel kernel. In addition, the asymptotic orders about ω of the presented methods are given. The efficiency and accuracy of the proposed methods are investigated through theoretical results and numerical experiments.

A non-local dynamic homogenization technique for the analysis of wave propagation in viscoelastic heterogeneous materials with a periodic microstructure is herein proposed. The asymptotic expansion of the micro-displacement field in the transformed Laplace domain allows obtaining, from the expression of the micro-scale field equations, a set of recursive differential problems defined over the periodic unit cell. Consequently, the cell problems are derived in terms of perturbation functions depending on the geometrical and physical-mechanical properties of the material and its microstructural heterogeneities. A down-scaling relation is formulated in a consistent form, which correlates the microscopic to the macroscopic transformed displacement field and its gradients through the perturbation functions. Average field equations of infinite order are determined by substituting the down-scaling relation into the micro-field equation. Based on a variational approach, the macroscopic field equation of a non-local continuum is delivered and the local and non-local overall constitutive and inertial tensors of the homogenized continuum are determined. The problem of wave propagation is investigated in case of a bi-phase layered material with orthotropic phases and axis of orthotropy parallel to the direction of layers as a representative example. In such a case, the local and non-local overall constitutive and inertial tensors are determined analytically. Finally, in order to test the reliability of the proposed approach, the dispersion curves obtained from the non-local homogenized model are compared with the curves provided by the Floquet-Bloch theory.

In this paper, the extended boundary element method (XBEM) is employed to evaluate the singular stress fields of plane V-notched/cracked structures based on the linear elastic theory. Firstly, the V-notched/cracked structure is divided into two parts, which are a small region around the notch tip and the outer region without the tip region. Then, the expressions of the asymptotic series expansion of the singular stress fields in the notch tip region are assembled with the boundary integral equation of the outer region without stress singularity. Hence, the displacement and stress fields of both the tip region and outer structure are determined by the XBEM. Three benchmark examples are presented to show that XBEM can obtain the accurate displacement and stress fields from the tip region to the whole region for the V-notched/cracked structures.

The purpose of the paper is to provide an efficient method for the continuously monitored geometric Asian options under the double Heston model. By introducing two small parameters, we slightly modify the double Heston model. With singular and regular perturbation techniques, we derive the first-order asymptotic expansions for pricing geometric Asian options with fixed and floating strikes and provide the convergence of the asymptotic formulae. We also provide the Greeks of geometric Asian options. Numerical results verify the efficiency of the pricing method. We calibrate the modified model to real markets and examine the impacts of two-factor volatilities on geometric Asian option prices.

We are concerned in this paper with the numerical computation of the spectra of highly oscillatory integrals that arise in laser simulations. Discretised using the modified Fourier basis, the spectral problem for the integral equation is converted into two independent infinite systems of linear equations whose unknowns are the coefficients of the modified Fourier functions, namely the cosine and shifted sine functions, respectively. Each ( m , n ) entry of the resulting coefficient matrices can be represented exactly by expressions involving the error function with an argument that involves the oscillatory parameter ω and the numbers m and n . Moreover, considering the behaviour of the error function for a large argument, the asymptotics for each entry are analysed for large ω or for large m and n and this enables efficient truncation of the infinite systems. Numerical experiments are provided to illustrate the effectiveness of this method.

The Smith's approximation formula for the gamma function is given by Γ ( x + 1 2 ) = 2 π ( x e ) x ( 2 x tanh ⁡ 1 2 x ) x / 2 ( 1 + O ( 1 x 5 ) ) , as x → ∞ . In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions: Γ ( x + 1 / 2 ) 2 π ( x / e ) x ∼ ( 2 x tanh ⁡ 1 2 x ) x / 2 exp ⁡ ( ∑ n = 3 ∞ [ ( 2 n ) ! − ( 2 n − 1 ) 2 2 n − 1 ] ( 1 − 2 1 − 2 n ) 2 n ( 2 n − 1 ) ( 2 n ) ! B 2 n x 2 n − 1 ) , as x → ∞ , and give an estimate of remainder in the above asymptotic series, where B 2 n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ ( x + 1 / 2 ) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established.

We develop the asymptotic expansion theory for vector-valued sequences (F N) N ≥1 of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F N. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of F N and we illustrate our results by several examples. 2010 AMS Classification Numbers: 62M09, 60F05, 62H12

For a certain domain $\Omega$ in the Sierpinski gasket $\mathcal{SG}$ whose boundary is a line segment, a complete description of the eigenvalues of the Laplacian, with an exact count of dimensions of eigenspaces, under the Dirichlet and Neumann boundary conditions is presented. The method developed in this paper is a weak version of the spectral decimation method due to Fukushima and Shima, since for a lot of "bad" eigenvalues the spectral decimation method can not be used directly. Let $\rho^0(x)$, $\rho^\Omega(x)$ be the eigenvalue counting functions of the Laplacian associated to $\mathcal{SG}$ and $\Omega$ respectively. We prove a comparison between $\rho^0(x)$ and $\rho^\Omega(x)$ says that $ 0\leq \rho^{0}(x)-\rho^\Omega(x)\leq C x^{\log2/\log 5}\log x$ for sufficiently large $x$ for some positive constant $C$. As a consequence, $\rho^\Omega(x)=g(\log x)x^{\log 3/\log 5}+O(x^{\log2/\log5}\log x)$ as $x\rightarrow\infty$, for some (right-continuous discontinuous) $\log 5$-periodic function $g:\mathbb{R}\rightarrow \mathbb{R}$ with $0, Comment: 80 pages, 10 figures, 6 tables. arXiv admin note: text overlap with arXiv:0909.1066 by other authors

The goal of this paper is to propose new asymptotic models describing a viscous fluid flow through a pipe-like domain subjected to heating. The deformation of the pipe due to heat extension of its material is taken into account by considering a linear heat expansion law. The heat exchange between the fluid and the surrounding medium is prescribed by Newton's cooling law. An asymptotic analysis with respect to the small parameter $\varepsilon$ (the heat expansion coefficient of the material of the pipe) is performed for the governing coupled nonlinear problem. Motivated by applications, the special case of flow through a thin (or long) pipe is also addressed leading to an asymptotic approximation of the solution in the form of an explicit formula. In both settings, we rigorously justify the usage of the proposed effective models by proving the corresponding error estimates.

Let Ω ⊂ R N , N ≥ 2 , be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian u ↦ − Δ u in Ω with the Robin boundary condition ∂ n u = α u on ∂Ω with ∂ n being the outward normal derivative and α > 0 being a parameter. We show that for large α the associated eigenvalues E j ( α ) behave as E j ( α ) ∼ − ϵ j α ν , where ν > 2 and ϵ j > 0 depend on the dimension and the peak geometry. This is in contrast with the well-known estimate E j ( α ) = O ( α 2 ) for the Lipschitz domains.