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 o...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.展开更多
The current work models a weak(soft) interface between two elastic materials as containing a periodic array of micro-crazes. The boundary conditions on the interfacial micro-crazes are formulated in terms of a system ...The current work models a weak(soft) interface between two elastic materials as containing a periodic array of micro-crazes. The boundary conditions on the interfacial micro-crazes are formulated in terms of a system of hypersingular integro-differential equations with unknown functions given by the displacement jumps across opposite faces of the micro-crazes. Once the displacement jumps are obtained by approximately solving the integro-differential equations, the effective stiffness of the micro-crazed interface can be readily computed. The effective stiffness is an important quantity needed for expressing the interfacial conditions in the spring-like macro-model of soft interfaces. Specific case studies are conducted to gain physical insights into how the effective stiffness of the interface may be influenced by the details of the interfacial micro-crazes.展开更多
The extraordinary transmission (ET) phenomenon is examined for waves propagating through gaps of vertical thin barriers in channels with a hypersingular boundary element method model on the linear potential theory, an...The extraordinary transmission (ET) phenomenon is examined for waves propagating through gaps of vertical thin barriers in channels with a hypersingular boundary element method model on the linear potential theory, and an estimate formula based on small gap approximation for predicting the number of ET frequencies is proposed. Numerical computations are carried out to examine the influences of barrier number, barrier interval, gap size, gap position and barrier arrangement on extraordinary transmission and wave height in the channel. It shows that all of those factors evidently affect the extraordinary transmission frequencies. The contours of wave amplitude show that very high waves can be excited in the basins between barriers at the extraordinary transmission frequencies. Proper arrangement of barriers in a channel can avoid the occurrence of ET phenomenon and reduce wave height in the channel.展开更多
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 hypersi...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.展开更多
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 app...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.展开更多
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 c...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.展开更多
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...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.展开更多
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 ce...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）.展开更多
In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate th...In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.展开更多
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 cu...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.展开更多
The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion...The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.展开更多
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 expan...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.展开更多
In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an exte...In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an extension of some known results.展开更多
With the rapidly increasing use of composite materials, a great deal of interest in the interface crack has been generated among the technicians, metal physicists and materials scientists. To the authors’ knowledge, ...With the rapidly increasing use of composite materials, a great deal of interest in the interface crack has been generated among the technicians, metal physicists and materials scientists. To the authors’ knowledge, many researches on this subject have been taken. However, they are usually applied to two-dimensional cases. For the three-dimensional ones, very few analyses have been done because of the great complexity on mathematics and mechanics. In this study, based on the point-force solutions for two perfectly展开更多
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
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; Inf...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.展开更多
文摘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.
文摘The current work models a weak(soft) interface between two elastic materials as containing a periodic array of micro-crazes. The boundary conditions on the interfacial micro-crazes are formulated in terms of a system of hypersingular integro-differential equations with unknown functions given by the displacement jumps across opposite faces of the micro-crazes. Once the displacement jumps are obtained by approximately solving the integro-differential equations, the effective stiffness of the micro-crazed interface can be readily computed. The effective stiffness is an important quantity needed for expressing the interfacial conditions in the spring-like macro-model of soft interfaces. Specific case studies are conducted to gain physical insights into how the effective stiffness of the interface may be influenced by the details of the interfacial micro-crazes.
基金financially supported by the National Natural Science Foundation of China(Grant Nos.51490672 and 51879039)
文摘The extraordinary transmission (ET) phenomenon is examined for waves propagating through gaps of vertical thin barriers in channels with a hypersingular boundary element method model on the linear potential theory, and an estimate formula based on small gap approximation for predicting the number of ET frequencies is proposed. Numerical computations are carried out to examine the influences of barrier number, barrier interval, gap size, gap position and barrier arrangement on extraordinary transmission and wave height in the channel. It shows that all of those factors evidently affect the extraordinary transmission frequencies. The contours of wave amplitude show that very high waves can be excited in the basins between barriers at the extraordinary transmission frequencies. Proper arrangement of barriers in a channel can avoid the occurrence of ET phenomenon and reduce wave height in the channel.
文摘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.
文摘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.
文摘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.
文摘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.
基金Supported by the National 973 Program of China(1999075105)National Natural Science Foundation of China(10271107)+1 种基金RFDP(20030335019)Natural Science Foundation of Zhejiang Proyince(RC97017)
文摘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）.
基金This research is supported in part by China NSF under grant 10071096NSF of Guangdong under grant 990228+1 种基金NSF of Hainan under grant 80525the One Hundred Distinguished Young Chinese Scientists Program of the Chinese Academy of Sciences from Yuesheng Xu.
文摘In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.
文摘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.
基金supported by National Natural Science Foundation of China(Grant Nos. 11101247 and 11201209)Shandong Provincial Natural Science Foundation of China (Grant No.ZR2011AQ020)+3 种基金a project of Shandong Province Higher Educational Science and Technology Program (GrantNo. J11LE08)supported by National Natural Science Foundation of China (GrantNo. 11101317)supported by National Basic Research Program of China (Grant No.2005CB321701)the Reward Fund of CAS for National Prize
文摘The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.
基金This work was supported by National Natural Science Foundation of China （Grant NO. 91430105 and Grant NO. 11771312）.
文摘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.
基金supported by National Natural Science Foundation of China (Grant Nos.10901017 and 10931001)Program for New Century Excellent Talents in University (Grant No. NCET-11-0574)Doctoral Fund of Ministry of Education of China (Grant No. 20090003110018)
文摘In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an extension of some known results.
基金National Natural Science Foundation of Chinathe National Ph.D. Foundation
文摘With the rapidly increasing use of composite materials, a great deal of interest in the interface crack has been generated among the technicians, metal physicists and materials scientists. To the authors’ knowledge, many researches on this subject have been taken. However, they are usually applied to two-dimensional cases. For the three-dimensional ones, very few analyses have been done because of the great complexity on mathematics and mechanics. In this study, based on the point-force solutions for two perfectly
基金supported by National Natural Science Foundation of China (Grant No. 10901017)Program for New Century Excellent Talents in University of China (Grant No. NCET-11-0574) +3 种基金the Fundamental Research Funds for the Central Universitiessupported by National Natural Science Foundation of China (Grant No. 10931001)the Research Fund for the Dectoral Program of Higher Education of China (Grant No. 20090003110018)Program for Changjiang Scholars and Innovative Research Team in University of China
文摘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
基金the National Natural Science Foundation of China, and the Foundation for Doctoralprogram of the State Education Commission of
文摘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.
