Solutions of periodic notch problems with arbitrary configuration by using boundary integral equation and superposition method

This paper studies the periodic notch problem with arbitrary configuration. A complex variable boundary integral equation (CVBIE) is suggested to solve the problem. The periodic notch problem is considered as a superposition of infinite single notch problems. The influence on the domain point from the assumed boundary traction on the notch contour is reduced to formulate a matrix. In this paper, this matrix is formulated completely after the relevant BIE is solved in a matrix representation. The remainder estimation technique is suggested to evaluate the influence to the central notch from many (form N-th to infinity) neighboring notches. Many computed results for the stress concentration factor for the elliptic and square notches are carried out. The stacking effect is also studied.     

Green's function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material

Summary A Green's function approach based on the laminate theory is adopted for solving the two-dimensional unsteady temperature field (r, z) and the associated thermal stresses in an infinite hollow circular cylinder made of a functionally graded material (FGM) with radial-directionally dependent properties. The unsteady heat conduction equation is formulated as an eigenvalue problem by making use of the eigenfunction expansion theory and the laminate theory. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the unsteady temperature. The associated thermoelastic field is analyzed by making use of the thermoclastic displacement potential function and Michell's function. Numerical results are carried out and shown in figures.     

Degenerate scale problem for the Laplace equation in the multiply connected region with outer elliptic boundary

This paper investigates the degenerate scale problem for the Laplace equation in a multiply connected region with an outer elliptic boundary. Inside the elliptic boundary, there are many voids with arbitrary configurations. The problem is studied on the relevant homogenous boundary integral equation. The suggested solution is derived from a solution of a relevant problem. It is found that the degenerate scale and the eigenfunction along the elliptic boundary in the problem is the same as in the case of a single elliptic contour without voids, or the involved voids have no influence on the degenerate scale. The present study mainly depends on the integrations of two integrals, which can be integrated in closed form.     

Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation

Phase retrieval on an annular pupil and an annular sector pupil by using the eigenfunction method to solve the transport of intensity equation is proposed. The analytic expressions of Laplacian eigenfunctions with the Neumann boundary condition on an annular pupil and an annular sector pupil are given. The phase can be expanded as a set of eigenfunctions on the corresponding pupil, and the coefficients of the eigenfunctions can be obtained by the integral of the eigenfunction and the intensity derivative along the optical axis. Phase retrieval by the eigenfunction method on an annular pupil and an annular sector pupil is simulated, and accurate retrieved results are obtained.     

Degenerate scale problem in antiplane elasticity or Laplace equation for quadrilaterals with arbitrary configuration

This paper provides numerical solutions of the degenerate scale for shapes of quadrilaterals with arbitrary configuration in an exterior boundary value problem of antiplane elasticity or Laplace equation. The first step is to find the parameters in the Schwarz–Christoffel mapping. The first prevertex on the unit circle can be placed in a particular position, or at −1. From the single-valued condition of the mapping function, only one prevertex is independent. The real preverteces can be found from the condition that the computed ratio of two edges is equal to a ratio of two real edges assumed beforehand. An iteration is suggested to obtain the preverteces numerically. After those parameters are obtained, the degenerate sizes of four edges can be evaluated by a numerical integration. Several numerical examples and the computed results were provided.     

A Nonlinear Finite Element Eigenanalysis of Singular Plane Stress Fields in Bimaterial Wedges Including Complex Eigenvalues

A finite element formulation is developed for the analysis of variable-separable singular stress fields in power law hardening materials under conditions of plane stress. The displacement field within a sectorial element is assumed to be quadratic in the angular coordinate and of the power type in the radial direction as measured from the singular point. An iteration scheme that combines the Newton method and matrix singular value decomposition is used to solve the nonlinear homogeneous eigenvalue problem, where the eigenvalues and eigenfunctions are obtained simultaneously. The formulation and iteration scheme apply when the eigenvalue is complex. The examples considered include the single material crack and wedge to demonstrate convergence, and the bimaterial interface crack and the bimaterial wedge to demonstrate geometric versatility and the ability to handle complex eigenvalues. It is found that the real part of the complex eigenvalue for the interface crack agrees with the HRR value. In this case the associated complex eigenfunction is converted into an approximate real-valued eigenfunction that is valid for any mode-mix. In addition, the behavior of separable solutions near certain 'wedge paradox' geometries where non-separable solutions occur is investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Calculation of eigenfunction fluxes in nuclear systems

A new Monte Carlo method is being developed to calculate eigenfunction fluxes in critical or near-critical nuclear systems. The correct estimation of fluxes is essential for radiation protection and shielding near these systems, in addition to isotope production, isotope depletion, nuclear criticality and other applications. The proposed method applies to Monte Carlo criticality eigenvalue calculations in which the fission sites in one generation are used as fission sources in subsequent generations. The usual Monte Carlo power iteration method for such problems often calculates fluxes (eigenfunctions) that are inaccurate and very different in symmetric parts of a problem geometry. The proposed method calculates flux distributions by estimating an approximate fission matrix. The way the fission matrix is estimated and used differs from other recent works. Preliminary results are promising.     

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

In this paper, the dual boundary element method (BEM) and the null-field boundary integral equation method (BIEM) are both employed to solve two-dimensional eigenproblems. The positions of true and spurious eigenvalues for circular, elliptical, annular and confocal elliptical membranes are analytically examined in the continuous system and numerically studied in the discrete system. To analytically study eigenproblems, the polar and elliptical coordinates in conjunction with the Bessel functions, the Mathieu functions, the Fourier series and eigenfunction expansions are adopted. The fundamental solution is expanded into the degenerate kernel while the boundary densities of circular and elliptical boundaries are expanded by using the Fourier series and eigenfunction expansion, respectively. Dirichlet and Neumann eigenproblems are both considered as well as simply and doubly-connected domains are both addressed. By employing the singular value decomposition (SVD) technique in the discrete system, the common right unitary vectors corresponding to the true eigenvalues for the singular and hypersingular formulations are found while the common left unitary vectors corresponding to the spurious eigenvalues are obtained for the singular formulation or hypersingular formulation. True eigenvalues depend on the boundary condition while spurious eigenvalues depend on the approach, the singular formulation or hypersingular formulation of BEM/BIEM. Nonzero field in the domain are analytically derived and are numerically verified in case of the true eigenvalue while the interior null field and nonzero field for the complementary domain are obtained in case of the spurious eigenvalue. Four examples, circular, elliptical, annular and confocal elliptical membranes, are considered to demonstrate the finding of the present paper. After comparing with the analytical and numerical results, good agreements are made. The dual BEM displays the dual structure in the unitary vector and the null field.     

Application of two‐state M‐integral for analysis of adhesive lap joints

With the aid of the two‐state M‐integral and finite element analysis, the asymptotic solution in terms of the complete eigenfunction expansion is obtained for adhesive lap joints. The notch stress intensity is introduced to characterize the singular stress field near the notch vertex of adhesive lap joints. The proposed scheme enables us to extract the intensity of each eigenfunction term from the far field data without resort to special singular elements at the vertex. It turns out that a weak stress singularity is not negligible around the vertex when it exists in addition to the major singularity. For a thin adhesive layer, there exist two asymptotic solutions: one is the inner solution approaching the eigenfunction solution for the vertex at which the adherend meets with the adhesive and the other is intermediate solution represented by the eigenfunction series that would be obtained in the absence of the adhesive layer. An appropriate guideline for choosing the geometric parameters in designing the adhesive lap joints, particularly the overlap length or the size of the adhesive zone, is suggested from the viewpoint of minimizing the notch stress intensity. Copyright © 2001 John Wiley & Sons, Ltd.     

A POD reduced‐order model for eigenvalue problems with application to reactor physics

A reduced‐order model based on proper orthogonal decomposition (POD) has been presented and applied to solving eigenvalue problems. The model is constructed via the method of snapshots, which is based upon the singular value decomposition of a matrix containing the characteristics of a solution as it evolves through time. Part of the novelty of this work is in how this snapshot data are generated, and this is through the recasting of eigenvalue problem, which is time independent, into a time‐dependent form. Instances of time‐dependent eigenfunction solutions are therefore used to construct the snapshot matrix. The reduced order model's capabilities in efficiently resolving eigenvalue problems that typically become computationally expensive (using standard full model discretisations) has been demonstrated. Although the approach can be adapted to most general eigenvalue problems, the examples presented here are based on calculating dominant eigenvalues in reactor physics applications. The approach is shown to reconstruct both the eigenvalues and eigenfunctions accurately using a significantly reduced number of unknowns in comparison with ‘full’ models based on finite element discretisations. The novelty of this paper therefore includes a new approach to generating snapshots, POD's application to large‐scale eigenvalue calculations, and reduced‐order model's application in reactor physics.Copyright © 2013 John Wiley & Sons, Ltd.     

Wave scattering by a thin elastic plate floating on a two-layer fluid

The hydroelastic interaction between an incident gravity wave and a thin elastic plate floating on a two-layer fluid of finite depth is analyzed with the aid of the method of matched eigenfunction expansions. The fluid is assumed to be inviscid and incompressible. A two-dimensional problem is formulated within the framework of linear potential theory for small-amplitude waves. The fluid domain is divided into two and three regions for semi-infinite and finite plates, respectively, with the matching relations representing the continuities of the pressure and velocity. A new inner product involving two single integrals is proposed, in which the vertical eigenfunctions in the open water region of the two-layer fluid are orthogonal. Then the orthogonality of the eigenfunctions with respect to the newly defined inner product is used to obtain a set of simultaneous equations for the expansion coefficients of the velocity potentials, and the edge conditions are included as a part of the equation system. The effects of the fluid density ratio and the position of interface on the wave reflection and transmission are discussed. Numerical analysis shows that the method proposed herein is effective with a higher rate of convergence.     

Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems

The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.     

Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM

Degenerate scale of an ellipse is studied by using the dual boundary element method (BEM), degenerate kernel and unit logarithmic capacity. Degenerate scale stems from either the nonuniqueness of logarithmic kernel in the BIE or the conformal radius of unit logarithmic capacity in the complex variable. Numerical evidence of degenerate scale in BEM is given. Analytical formula for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that zero eigenvalue results in the degenerate scale. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale is imbedded in the left unitary vector for weakly singular and strongly singular integral operators. On the other hand, we obtain the common right unitary vector of a rigid body (physical) mode in the influence matrices of strongly singular and hypersingular operators after using the singular value decomposition. Null field for the exterior domain and interior nonzero fields are analytically derived and numerically verified in case of the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in case of the degenerate scale. No failure CHEEF point is confirmed in the nonzero exterior field to overcome the degenerate-scale problem. To deal with the nonuniqueness-solution problem, the constraint of boundary flux equilibrium instead of rigid body term, CHEEF and hypersingular BIE, is added to promote the rank of influence matrices to be full rank. Both analytical and numerical results agree well in the demonstrative example of an ellipse.     

Linking quasi-normal and natural modes of an open cavity

The present paper proposes a comparison between the extinction theorem and the Sturm–Liouville theory approaches for calculating the electromagnetic (e.m.) field inside an optical cavity. We discuss for the first time to the best of our knowledge, in the framework of classical electrodynamics, a simple link between the quasi normal modes (QNMs) and the natural modes (NMs) for one-dimensional (1D), two-sided, open cavities. The QNM eigenfrequencies and eigenfunctions are calculated for a linear Fabry–Pérot (FP) cavity. The first-order Born approximation is applied to the same cavity in order to compare the first-order Born approximated and the actual QNM eigenfunctions of the cavity. We demonstrate that the first-order Born approximation for an FP cavity introduces symmetry breaking: in fact, each Born approximated QNM eigenfunction produces values below or above the actual QNM eigenfunction value on the terminal surfaces of the same cavity. Consequently, the two error-functions for an approximated QNM are not equal in proximity to the two terminal surfaces of the cavity.     

Quantized electron states in nearly depleted hexagonal nanowires

The hexagonal quantum well (QW) is studied as a model for hexagonal nanowires, and the effects of donor impurities and geometrical deformations of the well are treated. By use of the Poisson equation the donor potential is calculated and the eigenspectrum of the hexagonal QW is shown to converge to that of a paraboloid quantum well with increasing donor density. Small deformations of the hexagon are shown to change the eigenspectrum significantly and give strong splittings of degenerate eigenvalues. Analytical approximations for the potential and eigenfunctions on the deformed hexagons are given.     

一个带m阶转向点的奇摄动特征值问题

研究了用以描述某热传导现象的一个带m阶转向点的特征值问题,采用Langer变换的方法,得到了方程的由Bessel函数表示的一致有效的渐近解,并给出了问题的特征值与特征函数,从而推广了已有的结果。     

Buckling of a rectangular plate under a self-equilibrating stress due to dislocations

Summary The buckling of thin elastic rectangular plates due to two arrays of edge dislocations symmetrically placed with respect to the longitudinal axis of the plate is studied for two different dislocation orientations. The analysis is based on the governing equation of plate buckling. Through a standard eigenvalue analysis the buckling criterion is analytically-numerically computed and numerical results are presented pertaining to cases of practical engineering interest such as welding.     

DELAMINATION: LAMINATE ANALYSIS AND FRACTURE MECHANICS

In the analysis of a multilayered composite laminate with an arbitrarily shaped delamination, accurate non-linear laminate analysis is required to obtain the kinematic and kinetic variables which characterize the local state of deformation at a generic point of the delamination front. These variables provide the strain/curvature parameters and the boundary tractions for the local analysis model of a multimaterial region containing an interfacial crack. The singular elasticity solution in the vicinity of the crack tip is obtained by an eigenfunction expansion, where the coefficients of the eigenfunctions are evaluated by boundary collocation. The accuracy of the present solution is indicated by the computed energy release rate, which is in close agreement with the result based on path-independent integrals.     

The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.     

The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains.