Local Asymptotic Solution Research Articles

Solving Laplacian problems with boundary singularities: a comparison of a singular function boundary integral method with the p/ hp version of the finite element method

We solve a Laplacian problem over an L-shaped domain using a singular function boundary integral method as well as the p/ hp finite element method. In the former method, the solution is approximated by the leading terms of the local asymptotic solution expansion, and the unknown singular coefficients are calculated directly. In the latter method, these coefficients are computed by post-processing the finite element solution. The predictions of the two methods are discussed and compared with recent numerical results in the literature.

The Cauchy-Riemann conditions and localized asymptotic solutions of the linearized shallow-water equations

Singular solutions of the two-dimensional shallow-water equations with algebraic singularities of the “square root” type, which have been studied before [1–4], propagate along the trajectories of the external velocity field, over which this field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow on such a trajectory is proportional to an orthogonal operator. It turns out that in the linear approximation this situation is strongly linked with the “spreading” effect of solutions of the hydrodynamic equations (cf. [5,6]); namely, a localized asymptotic solution of the Cauchy problem for the linearized shallowwater equations maintains its form (i.e. does not spread) if and only if the Cauchy-Riemann conditions hold on the trajectory of the outer flow along which the disturbance is propagating.

Singular stress fields near contact boundaries in a composite bolted joint

Singular stresses arising in the neighborhood of contact surfaces introduced in laminated orthotropic plates by mechanical joining with clamp-up were investigated by using local asymptotic solutions and full-field numerical analysis. Three-dimensional B-spline approximation of the displacements and a penalty function-based contact solution was used in the numerical analysis. Recent work has shown that fracture in bolted composite joints may initiate near the outer edge of the bolt head or washer away from the hole edge, particularly if the joint is preloaded. Material and geometric discontinuities exist in these regions, resulting in singular stress behavior. Asymptotic stress analysis was performed to obtain the power of singularity in these regions as a function of the bolt-head (washer) stiffness. Frictionless contact conditions were assumed. It was found that the characteristics of the stress singularity for such practically important combinations as titanium bolt-head and carbon fiber composite plate are similar to a crack in terms of the power of singularity and uniqueness of the singular term. Coefficients of the singular terms of the asymptotic expansion were determined by comparison with the numerical solution in the close vicinity of the singular contour. Good agreement between the asymptotic and numerical solution in the transition regions was observed.

An experimental technique to determine critical stress intensity factors for delamination initiation

Interfacial delamination is an important reliability concern in multilayered microelectronic packages. Local asymptotic solutions, based on linear elastic assumptions, are available for estimating the free-edge stress intensity factor (SIF). However, to be able to predict delamination initiation, it is necessary to determine the critical free-edge SIF. In this paper, an experimental method has been proposed and conducted to determine the critical free-edge SIF. The test is based on the simple and standard shear strength test, which is well specified by ASTM standard for surface mount components, as opposed to specially designed or complicated test apparatus and fixture. The critical free-edge SIF for copper/ViaLux TM 81 photo-definable dry film PDDF interface has been determined with the proposed test method and calibrated using finite element analysis.

The solution of Laplacian problems over L‐shaped domains with a singular function boundary integral method

AbstractThe singular function boundary integral method is applied for the solution of a Laplace equation problem over an L‐shaped domain. The solution is approximated by the leading terms of the local asymptotic solution expansion, while the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. Estimates of great accuracy are obtained for the leading singular coefficients, as well as for the Lagrange multipliers. Comparisons are made with recent numerical results in the literature. Copyright © 2002 John Wiley & Sons, Ltd.

The 3D Stress Field at the Edge of a Circular Hole and the Interface of a Laminated Composite Plate

The author in this paper investigates the 3D stress field in the immediate vicinity of a bonded interface and the free edge of a hole in a laminated composite plate The laminates are assumed to be of homogeneous and isotropic materials, but of different elastic properties As to loading, a uniform tensile load is applied in the plane of the plate and at points far remote from the hole (see Figure 1) In constructing the local asymptotic solution, the author assumes the 3D field in a cer tain form which then permits a straightforward Williams approach for the determination of the stress singularities. The displacement and stress fields are recovered explicitly and a stress singularity is shown to exist for certain shear moduli ratios of G2/ G1. In general, the stress singularity is shown to be a function of the respective ratios of the shear moduli and Poisson's Moreover, the presence of a second singularity is observed which has sig nificant implications to the problem of adhesion An extension of the results to anisotropic layers is also discussed.

On the interlaminar stresses of a composite plate around the neighborhood of a hole

The author in this paper investigates the 3-D stress field in the immediate vicinity of a bonded interface and the free edge of a hole in a laminated composite plate. The laminates are assumed to be of homogeneous and Isotropic materials, but of different elastic properties. As to loading, a uniform tensile load is applied in the plane of the plate and at points far remote from the hole (shown in Fig. 1).In constructing the local asymptotic solution, the author assumes the 3-D field in a certain form which then permits a straightforward Williams approach for the determination of the stress singularities. The displacement and stress fields are recovered explicitly and a stress singularity is shown to exist for certain shear moduli ratios of G2/G1 In general, the stress singularity is shown to be a function of the respective ratios of the shear moduli and Poisson's ratio. Moreover, the presence of a second singularity is observed which has significant implications for the problem of adhesion. An extension of the results to anisotropic layers is also discussed.

The linear growth of the density wave in galaxies

The aim of this article is to study the linear growth of the density wave in galaxies by means of numerically resolving unsteady, two-dimensional hydrodynamic equations coupled with Poisson equation under the condition that the local asymptotic solution of linear density wave is given as an initial value. The results show that the perturbed peak density of linear density wave grows to the same order as the basic state density during merely tens of million years, the spiral pattern emerging which has barred structure in its inner region. The angular velocity of the spiral pattern and the growth rate of perturbed density vary gradually with changes in spatial place and time. The approximate property of quasistationary spiral structure hypothesis is discussed in this paper.

Influence of transverse slits on the hydrodynamic coefficients of a wing of finite span in stationary and nonstationary motion near a wall

The method of matched asymptotic expansions is used to construct an approximate solution to the problem of the influence of narrow transverse slits on the hydrodynamic coefficients of a thin rectangular wing moving near a wall. The flow in the neighborhood of a slit is described by a local asymptotic solution satisfying the condition of continuity of the pressure on the leading edge of the slit and matched to the main solution. Results of the calculations illustrate the influence of the slits on the hydrodynamic characteristics of the wing at different Strouhal numbers and aspect ratios.

The method of matched asymptotic expansions for the periodic solution of the van der pol equation

The purpose of this paper is to apply the method of intermediate matching for connecting the four local asymptotic solutions of the Van der Pol equation, given by Dorodnicyn [1]. It turns out that for the approximation of the periodic solution a fifth local solution is needed. The present approach results in a reduction of the computational work. The amplitude of the periodic solution is determined up to a higher order accuracy in v than has been done so far.