求解Cauchy型奇异积分方程的数值方法

1.引 言 断裂力学中许多裂纹问题的数学模型都可归结为奇异积分方程(SIE)[1,2].由于这些奇异积分方程的封闭解一般情况下都难以得到,因而数值方法受到广泛的注意.Muskhelishvili[3]对奇异积分方程的一般理论进行了深入的研究.这些研究成果为奇异积分方程的求解,不论     

A natural quadrature formula for the numerical evaluation of the Macgregor-Westergaard complex potentials es crack problems

The method of singular integral equations can be used for the numerical solution of crack problems in plane and antiplane elasticity. Here we consider the problem of the subsequent numerical evaluation of the stress components in the whole cracked medium by using the MacGregor-Westergaard complex potentials. To this end we use a modified quadrature formula for Cauchy type (but not principal value) integrals and their derivatives, where the poles of the integrands are properly taken into consideration. This is achieved by using a natural interpolation-extrapolation formula for singular integral equations and, for this reason, the new term ‘natural quadrature formula’ is proposed. Two simple applications to specific crack problems, based on the Gauss- and Lobatto-Chebyshev quadrature formulas, show the efficiency of the suggested quadrature formula.     

A frequency-domain BIEM combining DBIEs and TBIEs for 3-D crack-inclusion interaction analysis

The influence of a spherical elastic inclusion on a penny-shaped crack embedded in an infinite elastic matrix subjected to a time-harmonic crack-face or incident wave loading is investigated. A boundary integral equation method (BIEM) combining displacement boundary integral equations (DBIEs) on the matrix-inclusion interface and traction boundary integral equations (TBIEs) on the crack-surface is developed and applied for the numerical solution of the corresponding 3-D elastodynamic problem in the frequency domain. The singularity subtraction and mapping techniques in conjunction with a collocation scheme are implemented for the regularization and the discretization of the BIEs by taking into account the local structure of the solution at the crack-front. As numerical examples, the interaction of an elastic inclusion and a neighboring penny-shaped crack subjected to a tensile crack-surface loading or an incident plane longitudinal wave loading is investigated. The effects of the inclusion are assessed by the analysis of mixed-mode dynamic stress intensity factors (DSIFs) in dependence on the wave number, the material combination of the matrix and the inclusion, and the crack-inclusion orientation, size and distance.     

Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations

Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

Continuum shape sensitivity analysis of a mixed-mode fracture in functionally graded materials

This paper presents two new methods for conducting a continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic functionally graded material. These methods involve the material derivative concept from continuum mechanics, domain integral representation of interaction integrals, known as the M-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed to calculate the sensitivity of stress–intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. Three numerical examples are presented to calculate the first-order derivative of the stress–intensity factors. The results show that first-order sensitivities of stress intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

A numerical scheme for mixed boundary value problems in elasticity

A numerical scheme, based upon the Kobayashi-Tranter method with certain modifications, is given for axisymmetric punch and crack problems in elasticity. The problems are reduced to solving a system of linear algebraic equations instead of a Fredholm integral equation of the second kind. A standard program thus allows the treatment of a range of different cases.The indentation of a rigid punch on an elastic layer overlying an elastic foundation is formulated in this fashion and numerical results for various cases are presented.     

Boundary element analysis of three-dimensional mixed-mode cracks via the interaction integral

A three-dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed-mode three-dimensional cracks is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using their appropriate boundary integral equations. Special emphasis is put in the selection of the auxiliary function that represents the virtual crack advance in the domain integral. This is found to be a key feature to obtain reliable results at the intersection of crack fronts with free surfaces. Several examples are analysed to demonstrate the efficiency and accuracy of the implementation.     

On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems

A modification of the collocation method for the numerical solution of Cauchy type singular integral equations with generalized kernels is proposed. In accordance with this modification, although the abscissas and weights used in the numerical integration rule for the approximation of the integrals of the integral equation remain unaltered, yet the collocation points are selected in such a way that the poles of the integrands due not only to the Cauchy principal value part of the kernel, but also to the singularities of the generalized part of the kernel are taken into account. This modification assures the convergence of the method to the correct results since the error terms, usually neglected for the collocation points nearest to the end-points of the integration interval and generally tending to infinity, are now taken into consideration for the selection of the collocation points. The method was applied to the singular integral equations derived for the antiplane and plane elasticity problems of a crack terminating at a bimaterial interface.     

3D mode I crack analysis of piezoelectrics

The finite-part integral concept is first used to prove rigidly hypersingular integral equations for a mode I crack embedded an infinite transversely isotropic piezoelectric solid. With three-dimensional linear piezoelectricity theory, investigations on the singular electroelastic fields in the vicinity of the crack are thereafter made by the dominant-part analysis. Finally, numerical solutions of several typical planar cracks with hypersingular integral equation method are performed to give the electroelastic intensity factor K-fields and the energy release rate G. Accuracies of results are found to be very high.     

Finite difference solution of differential equations with stochastic inputs

This paper discusses application of two numerical methods (central difference and predictor corrector) for the solution of differential equations with deterministic as well as stochastic inputs. The methods are applied to a second order linear differential equation representing a series RLC netowrk with step function, sinusoidal and stochastic inputs. It is shown that both methods give correct answers for the step function and sinusoidal inputs. However, the central-difference method of solution is recommended for stochastic inputs. This statement is justified by comparing the auto-correlation and cross-correlation functions of the central-difference solution (with stochastic inputs) with the corresponding theoretical values of a continuous system. It is further shown that the more common predictor-corrector methods, although suitable for solution of differential equations with regular inputs, diverge for stochastic inputs. The reason is that these methods, by the application of several point integral formulas, use a high degree of smoothing on the variable and its derivatives. Inherent in the derivation of these integral formulas is the assumption of the continuity of the variable and its derivatives, a condition which is not satisfied in problems with stochastic inputs.Note that the second order differential equation chosen here for numerical experiments can be solved by classical methods for all of the given inputs, including the probabilistic inputs. The classical methods, however, unlike the numerical solutions, can not be extended to nonlinear differential equations which frequently arise in the digital simulation of engineering problems.     

Hybrid Bernstein Block–Pulse functions for solving system of fractional integro-differential equations

This paper deals with the numerical solution of system of fractional integro-differential equations. In this work, we approximate the unknown functions based on the hybrid Bernstein Block–Pulse functions, in conjunction with the collocation method. We introduce the Riemann–Liouville fractional integral operator for the hybrid Bernstein Block–Pulse functions. This operator will be approximated by the Gauss quadrature formula with respect to the Legendre weight function and then it is utilized to reduce the solution of the fractional integro-differential equations to a system of algebraic equations. This system can be easily solved by any usual numerical methods. The existence and uniqueness of the solution have been discussed. Moreover, the convergence analysis of this algorithm will be shown by preparing some theorems. Numerical experiments are presented to show the superiority and efficiency of proposed method in comparison with some other well-known methods.     

三维断裂弹性动力学的改进无单元Galerkin法

为提高断裂弹性动力学问题数值计算的精度,避免出现病态或奇异方程组,基于改进的移动最小二乘法建立三维弹性动力学问题的积分弱形式,采用罚函数法施加位移边界条件,引入隐式时间积分并且结合三维断裂力学的形函数考虑裂纹尖端的奇异性,探究将改进的无单元Galerkin(improved element-free Galerkin,IEFG)法用于断裂弹性动力学问题的数值计算.通过悬臂梁、柱和矩形板等3个算例,讨论节点分布、影响域比例参数、罚因子和时间步长等参数对计算精度的影响,证明IEFG法用于求解三维断裂弹性动力学问题的正确性和有效性.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

相关扰动下连续系统的连续时间最小二乘辨识的数值实现

基于数值积分和常微分方程数值解的欧拉法和龙格－－库塔法，给出随机连续系统的连续时间最小二乘辨识的两种数值实现方法，仿真结果表明了该方法的有效性。     

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

Sampled-data feedback practical semi-global controllability and stabilization for time-varying systems

In this paper it has been shown that—in the the general regulator problem—the methods of the calculus of variations, the principle of dynamic programming and Pontragi's maximum principle lead to the same equations for determining the optimal policy with respect to the general cost functional involving a positive definite function ot the terminal values of the variables plus an integral of a positive definite function. A numerical example has also been treated to illustrate the use of the theory in actual practtce.     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.