求解夹杂问题的有限部积分──边界元法

利用Kelvin解及有限部积分的概念和方法,导出求解含夹杂二维有限弹性体的超奇异积分方程,继而使用有限部积分与边界元结合的方法,为其建立了数值求解方法,即有限部积分与边界元法.最后计算了若干典型数值例子夹杂端部的应力强度因子.      

Finite-part integral and boundary element method to solve embedded planar crack problems

Using the Somigliana formula and concepts of the finite-part integral, a set of hypersingular integral equations to solve the arbitrary flat crack in three-dimensional elasticity is derived, then its numerical method is proposed by combining the finite-part integral method with the 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 agreement with the theoretical solutions.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Three iterative methods for the numerical determination of stress intensity factors

Three iterative methods for the numerical determination of stress intensity factors at crack tips (by using the method of singular integral equations with Cauchy-type kernels) are proposed. These methods are based on the Neumann iterative method for the solution of Fredholm integral equations of the second kind. Two of these methods are essentially used for the solution of the system of linear algebraic equations to which the singular integral equation is reduced when the direct Lobatto-Chebyshev method is used for its approximate solution, whereas the third method is a generalization of the first two and is related directly to the singular integral equation to be solved. The proposed methods are useful for the determination of stress intensity factors at crack tips. Some numerical results obtained in a crack problem show the effectiveness of all three methods.     

A remark on the direct numerical determination of stress intensity factors at crack tips

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.     

Integral analysis of a magnetic field for an arbitrary geometry coil with rectangular cross section

The integral method can effectively analyze magnetic fields, but the traditional integral method can analyze only coils with regular geometries. Therefore, a new integral method was developed to calculate the three-dimensional (3-D) magnetic field created by an arbitrary geometry coil with a rectangular cross section using the local coordinate method and a 3-D coordinate transformation. However, when the field points are on the surface of the coil or the basic segment is the right angle trapezoidal prism, singularities occur that make the numerical analysis of the magnetic field more difficult. Thus, we present here some mathematical methods to eliminate the singularities to allow accurate numerical analysis of the magnetic field. We validate the integral method by comparing it with the analytical solutions for regular geometry coils.     

Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it.     

一强非线性随机振荡系统的路径积分解

应用数值路径积分解法计算了一强非线性随机动力系统的响应统计。该数值路径积分法是基于隐式的高斯-勒让德插值程序,而且概率密度的值是在子区间内的高斯积分点上获得的。查阅文献显示这是首次单独应用这种路径积分法来处理强非线性随机振荡系统问题。     

Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.     

Non-singular boundary integral representation of potential field gradients

In this paper we derive the non-singular boundary integral representation of the field gradients for two-dimensional problems of classical potential field theory. Numerical implementation of this representation is developed too. The proposed method eliminates the most inaccurate influence coefficients which arise when singular integral representations are used and the internal point approaches the boundary. Since the integrands in this new method are finite at any internal point, accurate numerical results are achieved even in that portion of a solid which is very close to a discretized boundary. Two test problems are analysed in which the numerical results computed by strongly singular, weakly singular and non-singular integral representations are compared mutually and with exact solutions.     

Non-singular boundary integral representation of stresses

This paper presents the derivation of the non-singular integral representation of stresses in two- and three-dimensional elastostatics. In contrast to the strongly singular and weakly singular integral representations, the numerical computation of the nearly singular integrals is eliminated because all the integrands are made finite in this new formulation even if the internal point approaches the boundary. Thus the method gives accurate numerical results even in that portion of a solid which is very close to a discretized boundary. Three test problems are analysed in which we present a comparison of the accuracies achieved by the numerical computations based on the use of strongly singular, weakly singular and non-singular integral representations of stresses.     

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

A boundary element-free method (BEFM) for three-dimensional elasticity problems

This study combines the boundary integral equation (BIE) method and improved moving least-squares (IMLS) approximation to present a direct meshless boundary integral equation method, the boundary element-free method (BEFM) for three-dimensional elasticity. Based on the improved moving least-squares approximation and the boundary integral equation for three-dimensional elasticity, the formulae of the boundary element-free method are given, and the numerical procedure is also shown. Unlike other meshless boundary integral equation methods, the BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus giving it a greater computational precision. Three selected numerical examples are presented to demonstrate the method.Aknowledgement The work in this project was fully supported by a grant from the Research Grants Council (RGC) of the Hong Kong Special Administrative Region, China (Project No. CityU 1011/02E).The work that is described in this paper was supported by Project No. CityU 1011/02E, which was awarded by the Research Grants Council of the Hong Kong Special Administrative Region, China. The authors are grateful for the financial support.     

基于小波包的数值积分误差分析及消除方法

在结构损伤识别的时域法研究中，数值积分的误差将直接影响反演参数结果的精度。积分初值的未知、消除数值积分引入的误差是结构损伤反演计算中信号处理的难题。针对积分初值、数值积分积累误差和数值积分引入的偏移误差进行了理论分析，得出此类误差均具有低频特性的性质，可作为信号低频噪声进行处理。根据数值计算误差的特点分析及小波包多尺度、高分辨的特性，设计了一种基于小波包的滤波器，较好地解决了消除数值积分计算引入误差的问题。通过四层剪切结构模型算例验证，结构刚度反演误差由73％-32．2％降低到0．6％-0．24％，获得了较理想的效果。该方法同样可以用于其它领域的数值处理。     

Extraction of stress intensity factors from generalized finite element solutions

The performance of several superconvergent techniques to extract stress intensity factors (SIFs) from numerical solutions computed with the generalized finite element method is investigated. The contour integral, the cutoff function and the J-integral methods are considered. An implementation of the extraction techniques based on a sequence of mappings that are independent of the underlying solution method or discretization is proposed. It is shown that this approach is suitable for virtually any mesh-free or mesh-based solution method. Several numerical examples demonstrating the convergence of the computed SIF and the flexibility of the proposed implementation are presented. The path independence of the extraction methods is also investigated. Numerical experiments demonstrate that the contour integral and the cutoff function methods are more robust than the J–integral method with the CFM being the most accurate.     

Numerical solution of the EFIE by the meshfree collocation method

This paper contains guidelines for numerical solution of the EFIE meshfree. Degrees of freedom including mathematical statement, the meshfree method, shape functions and their parameters are considered and proper choices are selected by logical deduction, experience or previous reports. The method is based on decomposing the differential and integral parts of the EFIE, which is an integro-differential equation. These two independent parts could be processed in parallel. The differential and the integral parts are expanded over interpolants and approximant meshless shape functions, respectively. The final arrangement is applied to various scattering problems. Even though we applied the method mostly to linear and rectangular structures, the approach is applicable to all Electromagnetic integral equations of arbitrary geometries. For simple geometries with equidistance node arrangements and considering only the central node for support of the approximants, suggestions are made for bypassing numerical integration. In this case, although the differential part is still meshless, the integral part cannot be regarded as a meshless method in a strict sense and it may be considered a high-order collocation method. The results are compared with the low-order method of moments, previous reports and the FEKO software.     

下限问题中基于四边形单元平衡方程的边界积分法

相对于三角形单元的下限分析,基于四边形单元的下限分析具有更高的精度和求解效率。该文利用格林公式把平衡方程的弱形式化为边界积分,从而得到简洁的线性方程,取代了以往的数值积分方案,克服了高斯积分中坐标变换等复杂的求解过程。此外还对应力连续性方程进行了简化。该积分方案不仅大大简化了计算,而且更易于编程实现。算例表明该文方法具有较高的精度。     

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.     

The numerical solutions of the periodic crack problems of anisotropic strips*

The numerical solutions of the periodic crack problems of anisotropic strip are investigated in this paper. Using the complex variable method the mathematical model, boundary value problems for analytic functions, is established. After constructing appropriate integral transformations the boundary value problems are transformed into integral equations. Employing Lobotto-Chebyshev quadrature formulas and Gauss quadrature formulas, the approximate analytical expressions of the stress intensity factors are obtained. And some numerical results are performed for several special cases.     

The boundary element modelling of the human body exposed to the ELF electromagnetic fields

In this paper a cylindrical model of human body exposed to the extremely low frequency (ELF) electromagnetic field is presented. The analysis is based on the solution of the simplified integral equation for thick wires. The numerical solution of the integral equations is performed by the Galerkin–Bubnov variant of the boundary element method. Several numerical results for the ELF exposures are presented.