边界元法分析二维线弹性裂纹扩展

用边界元法研究裂纹扩展过程.首先将尖端区域Williams渐近展开的特征分析法与边界积分方程结合,解出切口尖端附近应力奇异性区域的各应力场渐近展开项系数,获得平面切口/裂纹结构完整的位移和应力场.再基于考虑非奇异应力项贡献的最大周向应力脆性断裂准则,运用边界元法分析边缘含裂纹半圆形弯曲试样在荷载作用下的启裂方向,对裂纹扩展过程给出自动跟踪方法,通过算例证明边界元法模拟裂纹扩展过程的正确性和有效性.     

Numerical solution for curved crack problem in elastic half-plane using hypersingular integral equation

A hypersingular integral equation for the curved crack problems of an elastic half-plane is introduced. Formulation of the equation is based on the usage of a modified complex potential. The potential is generally expressed in the form of a Cauchy-type integral. The modified complex potential is composed of the principal part and the complementary part. The principal part of the complex potential is actually equivalent to the original complex potential for the curved crack in an infinite plate. The role of the complementary part is to eliminate the boundary traction along the boundary of the half-plane caused by the principal part. From the assumed boundary traction condition, a hypersingular integral equation is obtained for the curved crack problems of an elastic half-plane. The curve length coordinate method is used to obtain a final solution. Several numerical examples are presented that prove the efficiency of the suggested method.     

A meshless Galerkin method with moving least square approximations for infinite elastic solids

Combining moving least square approximations and boundary integral equations, a meshless Galerkin method, which is the Galerkin boundary node method （GBNM）, for twoand three-dimensional infinite elastic solid mechanics problems with traction boundary conditions is discussed. In this numerical method, the resulting formulation inherits the symmetry and positive definiteness of variational problems, and boundary conditions can be applied directly and easily. A rigorous error analysis and convergence study for both displacement and stress is presented in Sobolev spaces. The capability of this method is illustrated and assessed by some numerical examples.     

T-matrix analysis of electromagnetic wave differaction from a Fourier grating

This paper is described for T-matrix analysis of the electromagnetic wave diffraction from a Fourier grating that the boundary value problem is treated by applying the extended boundary condition. The rigorous form of the expression of matrix elements is presented in the term of Bessel functions of the first kind. The error of power conservation versus the truncated number has been examined for mode number. Diffraction efficiencies versus groove depth and wavelength for a second or third harmonic wave of Fourier grating have been discussed. Numerical results are in good agreement with those obtained from other method and experimental values. Reasonable numerical results are presented for a groove depth per period of the Fourier grating less than 0.25.     

三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R²的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.     

任意复杂流-固边界的格子Boltzmann处理方法

本文提出了一种适用于流固耦合领域中任意复杂边界条件的lattice Boltzmann处理方法.该方法基于half-way反弹模型,在流固耦合处构建了一层虚拟边界,并结合有限差分的方法,获取虚拟边界上的变量值.改进后的方法确保了粒子反弹位置与宏观速度采集点的位置相同,计入了实际物理边界与网格线不重合时,偏移量对计算结果的准确影响,而且其适用范围被扩展到了任意静止或运动、平直或弯曲的复杂边界.文中研究了该方法在Poiseuille流、圆柱绕流和Couette流等经典条件下的边界处理能力,结果表明该方法与理论值符合良好,且当实际物理边界与网格线不重合时,与已发表文献中的结果相比,具有更高的精度.     

Applicability of locally reacting boundary conditions to porous material layer backed by rigid wall: Wave-based numerical study in non-diffuse sound field with unevenly distributed sound absorbing surfaces

To clarify the applicability of locally reacting boundary conditions in wave-based numerical analyses of sound fields in rooms, we numerically analyzed a non-diffuse sound field in a room with unevenly distributed sound absorbing surfaces and investigated the differences between the extended and local reactions. Each absorbing surface was a porous material layer backed by a rigid wall. Simulations were performed by the fast multipole boundary element method, a highly efficient boundary element method using the fast multipole method. At low frequencies, the extended and local reactions yielded similar reverberation decay curves because of the influence of the room. However, when the random incidence absorption coefficients were small at low frequencies or frequencies were high, the difference was greater than expected from the corresponding Eyring decay lines. We conclude at high frequencies, the locally reacting boundary conditions lead to a longer reverberation time than that expected from the absorption coefficient differences between the extended and local reactions. These differences were similar in sound-pressure-level and sound-intensity-level distributions, and in the oblique incidence absorption coefficient of the absorbing surfaces, but were increased at low frequencies.     

快速多极边界元法用于扩散光学断层成像研究

在求解扩散光学断层成像中的正向问题时, 目前普遍采用有限元法, 但是随着实际模型规模的增大, 有限元法的计算量问题日益显著, 而边界元法则由于可以降低计算维度使计算量减少而备受关注. 本文以均匀的高散射介质为模型, 研究了将快速多极边界元法用于扩散光学断层成像的正向问题. 快速多极边界元法利用核函数的多极展开, 将常规边界元法中系数矩阵和迭代矢量的乘积项等价为相应四叉树结构的一次递归, 再结合广义最小残量法进行迭代求解. 将计算结果和蒙特卡罗法的模拟结果进行了比较, 表明利用快速多极边界元法的模拟结果和蒙特卡罗法的结果有很好的一致性. 研究结果验证了快速多极边界元法可以用于扩散光学断层成像, 为其大规模和实时成像带来可观的前景. 关键词： 扩散光学断层成像 边界元法 快速多极边界元法     

DYNAMIC TRACTION VECTOR FIELD ESTIMATION IN A STRUCTURE USING HYBRID STRAIN ANALYSIS

The three-dimensional dynamic traction vector of a vibrating structure is obtained with the aid of the so-called hybrid strain analysis. Hybrid strain analysis is a method where vibration response measured at a limited number of points and numerically approximated continuous Hilbert space basis functions combined with spatial differentiation yields the frequency-dependent strain tensor field in a vibrating structure.Here, an extension and special application of hybrid strain analysis is proposed. In a special case with a built-up structure with unknown dynamic properties in the interfaces between the parts, the frequency- and spatial-dependent stress tensors, and thus also the traction vector, are obtained for the structural parts using the proposed technique.The method is validated using numerical simulation of measured vibration responses, with very good agreement between the calculated and the true traction vector. The calculated traction vector is shown to converge towards the true traction vector in an arbitrary small area on the boundary of the structure.The method is demonstrated for isotropic elastic material properties. This is, however, no limitation for the method; it can be also applied to a structure with anelastic material properties.     

A scaled boundary node method applied to two-dimensional crack problems

A boundary-type meshless method called the scaled boundary node method(SBNM) is developed to directly evaluate mixed mode stress intensity factors(SIFs) without extra post-processing.The SBNM combines the scaled boundary equations with the moving Kriging(MK) interpolation to retain the dimensionality advantage of the former and the meshless attribute of the latter.As a result,the SBNM requires only a set of scattered nodes on the boundary,and the displacement field is approximated by using the MK interpolation technique,which possesses the δ function property.This makes the developed method efficient and straightforward in imposing the essential boundary conditions,and no special treatment techniques are required.Besides,the SBNM works by weakening the governing differential equations in the circumferential direction and then solving the weakened equations analytically in the radial direction.Therefore,the SBNM permits an accurate representation of the singularities in the radial direction when the scaling center is located at the crack tip.Numerical examples using the SBNM for computing the SIFs are presented.Good agreements with available results in the literature are obtained.     

势问题的径向基函数——局部边界积分方程方法

将基于径向基函数构造的具有插值特性的近似函数和局部边界积分方程方法相结合，建立了求解势问题的径向基函数——局部边界积分方程方法，推导了相应离散方程.与其他边界积分方程的无网格方法相比，本文方法具有数值实现过程简单、计算量小、精度高的优点，并可直接施加边界条件.最后通过算例说明了该方法的有效性. 关键词： 径向基函数 无网格方法 局部边界积分方程 势问题     

Calculation of cutoff frequency of arbitrary weak guidance optical waveguides by extended boundary condition method

The extended boundary condition method is applied to both circular and elliptical regions for calculating cutoff frequencies V_c of weakly guiding optical waveguides, which consist of a uniform cladding and a core with arbitrary shape and index profile. The frequency shift formula method is explained by means of the extended boundary condition method on circular regions. Comparing the method with analytic and other rigorous methods, we show that it is indeed a powerful tool. Numerical results confirm that the mode designation and arrangement order in elliptical and rectangular waveguides are, in general, dependent on the aspect ratio.     

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.     

光滑粒子流体动力学方法固壁处理的一种新型排斥力模型

与传统网格法相比, 光滑粒子流体动力学方法不能直接施加壁面边界条件, 这就限制了该方法在工程中的应用.为此, 本文基于Galerkin加权余量法并结合传统排斥力方法, 推导出一种新的排斥力公式来施加壁面边界条件.该方法不含未知参数, 能在不减小边界粒子尺寸的情形下有效地防止流体粒子穿透壁面, 同时可避免邻近边界的流体粒子的速度及压力振荡. 分别通过静止液柱算例、液柱坍塌算例、容器中液体静止算例及溃坝算 例来验证本文方法的有效性, 并与传统边界处理方法进行对比, 结果表明: 本文方法克服了传统方法存在的缺陷, 是一种有效的固壁边界处理方法. 关键词： 光滑粒子流体动力学法 固壁边界 排斥力 加权余量法     

标量波传播问题的双互易精细积分法

为避免使用计算多种特征频率下的声场响应,采用双互易方法将边界积分方程中时间二次导数项的域积分转化为边界积分.首先,将计算场点配置在边界上并考虑边界条件,可以获得由内部节点上声压量线性表示的边界节点上的物理量;其次,将计算场点配置于域内离散节点上,将所得边界积分方程组中关于边界物理量用内部节点的声压量线性表示,获得关于声压量的二阶常微分方程组;第三,引入声压变化速度作为未知量,将二阶常微分方程组转化为一阶常微分方程组;最后,采用精细积分法精确求解常微分方程组.数值算例验证了双互易精细积分法的正确性和稳定性.     

Determination of the crack configuration in an anisotropic elastic medium

The problem of the identification of a single internal crack in an anisotropic elastic body is investigated. Using the dislocation theory approach, a system of boundary integral equations for the crack opening functions is constructed and studied by the boundary element method. A crack identification method is developed on the basis of the crack parametrization by a finite number of parameters, with their subsequent determination through the minimization of a certain nonquadratic residual functional. The problem of identifying a transverse tunnel crack in an orthotropic layer is solved for the cases of plane and antiplane deformations.     

An improved boundary element-free method （IBEFM） for two-dimensional potential problems

The interpolating moving least-squares (IMLS) method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method (BEFM), combining the boundary integral equation (BIE) method with the IMLS method, the improved boundary element-free method (IBEFM) for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker δ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker δ function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation 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 it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.     

Cohesive solutions of intersonic moving dislocations

A class of cohesive solutions of moving glide dislocations with intersonic speeds has been derived on the basis of the fundamental equation of a moving dislocation introduced by Weertman in conjunction with a proposed generalized Bilby–Cottrell–Swinden–Dugdale model. In this model we assume a straight weak path within an infinite elastic plate. Two length scales, namely the width (thickness) of the weak path and the material intrinsic length, which scales strain-gradient-induced hardening and energy dissipation, are taken into account by applying the traction–separation law for the decohesion of the weak path. Dislocations propagate along this weak path with a speed higher than the shear wave speed. The accumulation of these moving dislocations forms a macroscale crack growth with a cohesive zone ahead of the crack tip. Similar to the Bilby–Cottrell–Swinden–Dugdale model, the remote enforced stress and/or stress-rate boundary conditions are represented as an equivalent crack surface traction associated with the dislocation distribution. The involved Cauchy integral and corresponding eigenvalue problem are solved using the algorithms introduced by Muskhelishvili and by Weertman. The problems associated with three types of decohesion law are constant traction, traction linearly dependent on separation, and separation- and separation-rate-dependent traction. These problems are solved using three different solution strategies: the direct-integration method, the iteration method and the Jacobi polynomial expansion respectively. The derived solutions provide explicit relations between the remote load propagation speed, the material intrinsic length, the weak path thickness and the strain-rate-hardening parameter. The solutions demonstrate that the intersonic speed region can be divided into two subdomains; steady-state propagation occurs within the subdomain where the propagation speeds are equal to or greater than the Eshelby speed (c _s × 2^1/2, where c _s is the shear wave speed). For a weak path with a finite width and the corresponding decohesion law scaled by material intrinsic length, an intersonic crack propagation will not take place if only a constant remote stress is imposed. A ‘steady-state’ crack surface load and/or remote stress-rate boundary condition, which can be considered as a point force or a distributed force with a constant distance to the moving crack tip, is required to maintain steady-state intersonic crack propagation.     

Boundary element-free method for elastodynamics

The moving least-square approximation is discussed first. Sometimes the method can form an ill-conditioned equation system, and thus the solution cannot be obtained correctly. A Hilbert space is presented on which an orthogonal function system mixed a weight function is defined. Next the improved moving least-square approximation is discussed in detail. The improved method has higher computational efficiency and precision than the old method, and cannot form an ill-conditioned equation system. A boundary element-free method (BEFM) for elastodynamics problems is presented by combining the boundary integral equation method for elastodynamics and the improved moving least-square approximation. The boundary element-free method is a meshless method of boundary integral equation and is a direct numerical method compared with others, in which the basic unknowns are the real solutions of the nodal variables and the boundary conditions can be applied easily. The boundary element-free method has a higher computational efficiency and precision. In addition, the numerical procedure of the boundary element-free method for elastodynamics problems is presented in this paper. Finally, some numerical examples are given.     

DYNAMIC BEHAVIOUR OF A CRACKED SOLDERED JOINT

The two-dimensional “in-plane” time-harmonic elastodynamic problem for a multi-layered cracked soldered joint system is studied. This problem is solved by using a hybrid of both displacement and hyper-singular traction boundary integral equation method. The proposed method directly accounts for the effect of the outer boundary of a finite multi-layered body and the interaction between the internal and interface cracks. The open fracture model is used to present the interface crack. Numerical results are shown and discussed to reveal the effect of the existence and sizes of cracks, the crack interaction, the debonding effect, the influence of the wave frequency and the type of the material combination on the crack-tip fracture parameters and the displacement scattered far field.