A multiscale Galerkin method for the hypersingular integral equation reduced by the harmonic equation

The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.

Geometric Conversion Approach for the Numerical Evaluation of Hypersin gular and Nearly Hypersingular Boundary Integrals over Curved Surface Boundary Elements

With the aid of the properties of the hypersingular kernels, a geometric conversion approach was presented in this paper. The conversion leads to a general approach for the accurate and reliable numerical evaluation of the hypersingular surface boundary integrals encountered in a variety of applications with boundary element method. Based on the conversion, the hypersingularity in the boundary integrals could be lowered by one order, resulting in the simplification of the computer code. Moreover, an integral transformation was introduced to damp out the nearly singular behavior of the kernels by the distance function defined in the local polar coordinate system for the nearly hypersingular case. The approach is simple to use, which can be inserted readily to computer code, thus getting rid of the dull routine deduction of formulae before the numerical implementations, as the expressions of these kernels are in general complicated. The numerical examples were given in three dimensional elasticity, verifying the effectiveness of the proposed approach, which makes it possible to observe numerically the behavior of the boundary integral values with hypersingular kernels across the boundary.

APPLICATION OF THE HYPERSINGULAR INTEGRAL EQUATIONS METHOD TO THE INTERACTION BETWEEN TWO PARALLEL PLANAR CRACKS IN 3-D ELASTICITY

By using the analytic theory of hypersingular integral equations in three- dimensional fracture mechanics, the interactions between two parallel planar cracks under arbitrary loads are investigated. According to the concepts and method of finite- part integrals, a set of hypersingular integral equations is derived, in which the unknown functions are the displacement discontinuities of the crack surfaces. Then its numerical method is proposed by combining the finite-part integral method with the boundary element method. Based on the above results, the method for calculating the stress intensity factors with the displacement discontinuities of the crack surfaces is presented. Finally, several typical examples are calculated and the numerical results are satisfactory.

Approximate Formulation and Numerical Solution for Hypersingular Boundary Integral Equations in Plane Elasticity

Based on the fact that the singular boundary integrals in the sense of Cauchy principal value can be represented approximately by the mean values of two companion nearly singular boundary integrals, a vary general approach was developed in the paper. In the approach, the approximate formulation before discretization was constructed to cope with the difficulties encountered in the corner treatment in the formulations of hypersingular boundary integral equations. This makes it possible to solve the hypersingular boundary integral equation numerically in a non regularized form and in a local manner by using conforming C 0 quadratic boundary elements and standard Gaussian quadratures similar to those employed in the conventional displacement BIE formulations. The approximate formulation is very convenient to use because the corner information is comprised naturally in the representations of those approximate integrals. Numerical examples in plane elasticity show that with the present approach, the compatible or better results can be achieved in comparison with those of the conventional BIE formulations.

A ROUGH HYPERSINGULAR INTEGRAL OPERATOR WITH AN OSCILLATING FACTOR ON FUNCTION SPACE

The singular integral operatorTα,βf（x）=p.v.∫R^n[e^i｜y｜^-βΩ（y’）]/[｜y｜^n＋α]f（x-y）dy,defined for all test functions f is studied, where Ω（y＇） is a distribution on the unit sphere S^n-1 satisfying certain cancellation condition. It is proved that Tα,β is a bounded operator from the Triebel-Lizorkin space Fp^s,q to the Triebel-Lizorkin space Fp^s＋γ,q, provided that Ω（y＇） is a distribution in the Hardy space H^r（S^n-1） with r = （n - 1）/（n - 1 ＋ γ）.

Antiplane shear crack in a prestressed elastic medium based on the couple stress theory

A prestressed elastic medium containing a mode-Ⅲcrack is studied by means of the couple stress theory(CST).Based on the CST under initial stresses,a governing differential equation along with a mixed boundary value problem is established.The singularities of the couple stress and force stress near the crack tips are analyzed through the asymptotic crack-tip fields resulting from the characteristic expansion method.To determine their intensity,a hypersingular integral equation is derived and numerically solved with the help of the Chebyshev polynomial.The obtained results show a strong size-dependence of the out-of-plane displacement on the crack and the couple stress intensity factor(CSIF)and the force stress intensity factor(FSIF)around the crack tips.The symmetric part of the shear stress has no singularity,and the skew-symmetric part related to the couple stress exhibits an r^(-3/2)singularity,in which r is the distance from the crack tip.The initial stresses also affect the crack tearing displacement and the CSIF and FSIF.

NEW TYPE OF DUAL BEM AND GREEN'S-FUNCTION-LIBRARY STRATEGY FOR FRACTURE ANALYSIS IN COMPLEX STRUCTURES

A new type of dual boundary integral equations(DBIE)is presented first,through which,a smaller system of equations needs to be solved in fracture analysis.Then a non-conforming crack tip element in two-dimensional problems is proposed.The exact formula for the hypersingular integral over the non-con- forming crack tip element is given next.By virtue of Green's-function-library strategy,a series of stress in- tensity factors(SIF)of different crack orientations,locations and/or sizes in a complicated structure can be obtained easily and efficiently.Finally,several examples of fracture analysis in two dimensions are given to demonstrate the accuracy and efficiency of the method proposed.

A NOVEL METHOD FOR EXACT SOLUTIONS OF ELLIPTICAL CRACKS IN PIEZOELECTRICS

According to the constitutive relationship in linear piezoceramics, elliptical crack problems in the impermeable case are reconsidered with the hypersingular integral equation method. Unknown displacement and electric potential jumps in the integral equations are approximated with a product of the fundamental density function and polynomials, in which the fundamental density function reflects the singular behavior of electroelastic fields near the crack front and the polynomials can be reduced to a real constant under uniform loading. Ellipsoidal coordinates are cleverly introduced to solve the unknown displacement and electric potential jumps in the integral equations under uniform loading. With the help of these solutions and definitions of electroelastic field intensity factors, exact expressions for mode Ⅰ, mode Ⅱ and mode Ⅲ stress intensity factors as well as the mode Ⅳ electric displacement intensity factor are obtained. The present results under uniform normal loading are the same as the available exact solutions, but those under uniform shear loading have not been found in the literature as yet.

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green＇s integral theorem in terms of difference of potential functicns across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.

FINITE-PART INTEGRAL AND BOUNDARY ELEMENT METHOD TO SOLVE FLAT CRACK PROBLEMS

Using the Somigliana formula and the concepts of finite-part integral, a set of hypersingular integral equations to solve the arbitrary flat crack in three-dimensional elasticity is derived and its numerical method is then proposed by combining the finite-part integral method with boundary element method. In order to verify the method, several numerical examples are carried out. The results of the displacement discontinuities of the crack surface and the stress intensity factors at the crack front are in good agrement with the theoretical solutions.

带振荡因子的粗双曲奇异积分算子(英文)

The singular integral operator FΩ.a, and the Marcinkiewicz integral operator μ^-Ω.a are studied. The kernels of the operators behave like ｜y｜^-n-a（a〉0） near the origin, and contain an oscillating factor e^i｜y｜^-β（β〉0） and a distribution Ω on the unit sphere S^n-1. It is proved that, if Ω is in the Hardy space H^r（S^n-1） with 0〈r=（n-1）/（n-1＋y）（r〉0）, and satisfies certain eancellation condition,then FΩ.a and μ^-Ω.a extend the bounded operator from Sobolev space L^pr to Lebesgue space L^p for some p. The result improves and extends some known results.

QUASI-INTERPOLATING OPERATORS AND THEIR APPLICATIONS IN HYPERSINGULAR INTEGRALS

The purpose of this paper is to propose and study a class of quasi-interpolating operators in multivariate spline space S-1/2(Delta(mn)(2*)) on non-uniform type-2 triangulation. Based on the operators, we construct cubature formula for two-dimensional hypersingular integrals. Some computing work have been done and the results are quite satisfactory.

APPROXIMATIONS OF HYPERSINGULAR INTEGRALS FOR NEGATIVE FRACTIONAL EXPONENT

This article presents approximations of the hypersingular integrals fa^b g（x）（x- t）^adx and fa^b g（x）｜x- t｜dx with arbitrary singular point t C （a, b） and negative fraction number 〈 -1. These general expansions are applicable to a large range of hypersingular inte- grals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.

Commutator of Hypersingular Integral with Rough Kernels on Sobolev Spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space Lpγ （Rn） to the Lebesgue space Lp（Rn） for 1

A SPLINE METHOD FOR SOLVING TWO-DIMENSIONAL FREDHOLM INTEGRAL EQUATION OF SECOND KIND WITH THE HYPERSINGULAR KERNEL

Provides information on a study which proposed a spline method for solving two-dimensional Fredholm Integral Equations of second kind space with hypersingular kernels. Details on the quasi-interpolating operators; Information on the cubature formulas; Formulas of the approximation method.

Stress Intensity Factor for the Interaction between a Straight Crack and a Curved Crack in Plane Elasticity

Formulation in terms of hypersingular integral equations for the interaction between straight and curved cracks in plane elasticity is obtained using the complex variable functions method. The curved length coordinate method and a suitable numerical scheme are used to solve such integrals numerically for the unknown function, which are later used to find the stress intensity factor, SIF.

NUMERICAL SOLUTIONS FOR A NEARLY CIRCULAR CRACK WITH DEVELOPING CUSPS UNDER SHEAR LOADING

In this paper, we study the behavior of the solution at the crack edges for a nearly circular crack with developing cusps subject to shear loading. The problem of finding the resulting force can be written in the form of a hypersingular integral equation. The equation is then trans-formed into a similar equation over a circular region using conformal mapping. The equation is solved numerically for the unknown coefficients, which will later be used in finding the stress intensity factors. The sliding and tearing mode stress intensity factors are evaluated for cracks and displayed graphically. Our results seem to agree with the existing asymptotic solution.

A NUMERICAL BOUNDARY EIGENVALUE PROBLEM FOR ELASTIC CRACKS IN FREE AND HALF SPACE

We present in this paper a numerical method for hypersingular boundary integral equations. This method was developed for planar crack problems： additional edge singularities are known to develop in that case. This paper includes a rigorous error analysis proving the convergence of our numerical scheme. Three types of examples are covered： the Laplace equation in free space, the linear elasticity equation in free space, and in half space.