冲击载荷作用下动态应力强度因子的迭加积分法

介绍利用线性时不变系统求响应的迭加积分法(杜阿美尔积分)求解带裂纹构件的线弹性动态应力强度因子。即先求出结构的单位冲激响应,再与载荷做卷积求得总的响应;或者先计算出单位阶跃响应,再与载荷的微分做卷积来求得总的响应。理论和计算都证明了这两种迭加积分法的精度。在动态起裂韧度测试实验中,特别是用实验-数值法要处理同一组试样的大量实验数据,应用迭加积分法可以有效的避免数值求解的重复计算,减少计算量。     

A new integral identity for potential polygonal domain problems described by parametric linear functions

The paper presents a modification of the classical integral identity for two-dimensional potential boundary value problems with a linear segments boundary. The modification consists of describing integral contours in the integral identity by means of parametric linear functions. As a result of the modification, it was possible to obtain a new integral identity, which included the geometry of the boundary in its subinterval functions. The identity can be used for finding solutions in polygonal domains under the condition that the solutions previously obtained on the boundary are arrived at by the new parametric integral equation system. The proposed method makes it possible to obtain solutions of domain problems with no need for the discretization of the boundary geometry. The effectiveness of the method is illustrated by three testing examples.     

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。     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

Transient semianalytic analysis of a lossless multiconductor transmission line with nonlinear loads excited by a plane wave

A semianalytic method is used to calculate the transient responses of a lossless multiconductor transmission line with nonlinear loads. The Volterra integral equation for the current response is obtained by combining the linear transmission line equivalence and the voltage–current relations of the nonlinear terminations. The analytic expressions of parameters in the Volterra integral equation are obtained using the Baum–Liu–Tesche equation and analytic Fourier transformation, and then the Volterra integral equation can be simplified. Finally, a simulation example is illustrated to validate this method.     

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

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.     

A High‐order extended finite element method for extraction of mixed‐mode strain energy release rates in arbitrary crack settings based on Irwin's integral

An analytical formulation based on Irwin's integral and combined with the extended finite element method is proposed to extract mixed‐mode components of strain energy release rates in linear elastic fracture mechanics. The proposed formulation extends our previous work to cracks in arbitrary orientations and is therefore suited for crack propagation problems. In essence, the approach employs high‐order enrichment functions and evaluates Irwin's integral in closed form, once the linear system is solved and the algebraic degrees of freedom are determined. Several benchmark examples are investigated including off‐center cracks, inclined cracks, and two crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, because of its analytical nature, no special post‐processing is required. Thus, we conclude that this method may provide a good and simple alternative to the popular J‐integral method. In addition, it may circumvent some of the limitations of the J‐integral in 3D modeling and in problems involving branching and coalescence of cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

第二类Fredholm积分方程的小波快速算法

介绍了一种求解含有对数核的第二类Fredholm积分方程的有效的方法，该方法先用Nystrom法将积分方程离散，然后用小波矩阵变换方法稀疏系数矩阵，对系数矩阵预处理后再对线性方程组迭代求解。数值结果证明了该方法的有效性。     

Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem

An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.     

A direct integral equation method for the potential flow about arbitrary bodies

The velocity on the surface of an arbitrary three-dimensional body in the potential flow of an ideal fluid is determined by means of an integral equation method. The equations to be solved are a pair of singular integral equations of the second kind, the unknowns being the desired velocity components. The integral equations are approximated by a set of linear equations, which are solved numerically. For ellipsoids the results are compared with the analytic solutions.     

Cauchy-type integrals and integral equations with logarithmic singularities

Summary The Gauss-type quadrature methods with a logarithmic weight function can be extended to the evaluation of Cauchy-type integrals and to the solution of Cauchy-type integral equations by reduction of the latter to a linear system of algebraic equations. This system is obtained by applying the integral equation at properly selected collocation points. The poles of the integrand lying in the integration interval were treated as lying outside this interval. The efficiency of the method, both in evaluating integrals and solving integral equations, is exhibited by a numerical example. Finally, an application of the method to a crack problem of plane elasticity is made.     

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.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A SIMPLE PATH-INDEPENDENT INTEGRAL FOR CALCULATING MIXED-MODE STRESS INTENSITY FACTORS

A path-independent integral is introduced for calculating stress intensity factors. The derivation of the integral is based on the application of the known Bueckner's fundamental field solution for a crack in an infinite body and on the reciprocal theorem. The method was applied to two-dimensional linear elastic mixed-mode crack problems. The key advantage of the present path-independent integral is that the stress intensity factor components for any irregular cracked geometry under any kind of loading can be easily obtained by a contour integral around the crack tip. The method is simple to implement and only the far field displacements and tractions along the contour must be known. The required stress analysis can be made by using any analytical or numerical method, e.g. the finite element method, without special consideration of the modelling of crack tip singularity. The application of this integral is also independent of the crack type, that is, there is no difference between an edge crack and an embedded crack, provided that the crack tip asymptotic behaviour exists.     

Heat transfer during fluidized-bed coating

A fully implicit numerical method for linear parabolic free boundary problems with coupled and integral boundary conditions is described. The partial differential equation and the boundary conditions are time discretized with the method of lines. An auxiliary function is introduced to remove the coupled and integral boundary conditions from the resulting free boundary problem for ordinary differential equations. Once separated boundary conditions are obtained, invariant imbedding is used to solve the free boundary problem numerically. The method is illustrated by solving the heat transfer equations for the fluidized-bed coating of a thin-walled cylinder.     

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.     

速度面积法中不同特征点位置选择法之比较

介绍了用速度面积法测量流量时所用的三种基本方法,即等环面法[1]、切比雪夫积分法[7]和对数线性法[1],并用理论计算的方法讨论了这三种方法所得的数据和理论值的一致性.结果表明,在选取相同数目测点的情况下,对数线性法的一致性比其他两种方法要高,而等环面法一致性稍差.     

A boundary element approach for non-homogeneous potential problems

This paper reports an implementation of a Boundary Element Method dealing with two-dimensional inhomogeneous potential problems. This method avoids the tedious calculation of the domain integral contributions to the boundary integral equations. This is achieved by applying approximate particular solutions which are obtained by expressing the source distribution in terms of a linear combination of radial basis functions. Numerical examples show that the method is efficient and can produce accurate results.     

A weakly singular integral formulation for displacement prescribed problems of elasticity

Summary. This paper introduces a new integral formulation for displacement prescribed problems in linear elasticity. The formulation uses a weakly singular kernel and extends to the case of linear elasticity the integral formulation introduced by Mikhlin [1] to solve Dirichlet problems for Laplaces equation in multiply connected domains. A detailed proof of the proposed formulation is given for displacement prescribed problems in two-dimensional multiply-connected domains. The proposed method can also be readily extended to solve three-dimensional problems.