广义扩展有限元法及其在裂纹扩展分析中的应用

结合广义有限元法(GFEM)和扩展有限元法(XFEM)的特点,提出了一种新的数值方法——广义扩展有限元法(GXFEM)。阐述了广义扩展有限元法的基本原理,对相关公式进行推导,探讨数值实施中需注意的重要问题,给出利用广义扩展有限元法进行断裂分析时应力强度因子的计算方法,编写了广义扩展有限元法程序。通过算例进行了应力强度因子的计算,模拟了结构裂纹的扩展过程。算例结果表明,利用广义扩展有限元法计算裂纹扩展问题,不需要进行过密的网格划分,且网格在裂纹扩展后无需重新剖分,具有相当高的计算精度。     

动载下裂纹应力强度因子计算的改进型扩展有限元法

相较于常规扩展有限元法(extended finite element method, XFEM), 改进型扩展有限元法(improved XFEM) 解决了现有方法线性相关与总体刚度矩阵高度病态问题, 在数量级上提升了总体方程的求解效率, 克服了现有方法在动力学问题中的能量正确传递、动态应力强度因子数值震荡、精度低下问题. 本文基于改进型XFEM, 采用Newmark 隐式时间积分算法, 重点研究了动载荷作用下扩展裂纹尖端应力强度因子的求解方法, 与静力学方法相比, 增加了裂纹扩展速度项与惯性项的贡献. 通过数值算例研究了网格单元尺寸、质量矩阵、时间步长、裂尖加强区域、惯性项、扩展速度项及相互作用积分区域J-domain的网格与单元尺寸对动态应力强度因子求解精度的影响, 验证了改进型XFEM计算动态裂纹应力强度因子方法的有效性. 针对文献中具有挑战性的 "I 型半无限长裂纹先稳定后扩展"问题, 改进型XFEM给出目前为止精度最好的动态应力强度因子数值解.      

分区混合有限元法计算应力强度因子

本文应用分区混合能量原理,提出分区混合有限元法,用以计算应力强度因子,方法的特点是:在裂纹尖端附近采用应力型奇异单元,在外部采用位移型常规单元。由于针对问题的受力特点,合理地把应力型与位移型、奇异元与常规元、解析解与数值解加以结合,各自发挥所长,从而能以较疏的网格取得较高的精度。 本文不仅为计算应力强度因子提供了一种有特点的有效解法,而且为分区混合有限元法的广泛应用提供了最初的例证。     

动载下裂纹应力强度因子计算的改进型扩展有限元法

相较于常规扩展有限元法(extended finite element method,XFEM),改进型扩展有限元法(improved XFEM)解决了现有方法线性相关与总体刚度矩阵高度病态问题,在数量级上提升了总体方程的求解效率,克服了现有方法在动力学问题中的能量正确传递、动态应力强度因子数值震荡、精度低下问题.本文基于改进型XFEM,采用Newmark隐式时间积分算法,重点研究了动载荷作用下扩展裂纹尖端应力强度因子的求解方法,与静力学方法相比,增加了裂纹扩展速度项与惯性项的贡献.通过数值算例研究了网格单元尺寸、质量矩阵、时间步长、裂尖加强区域、惯性项、扩展速度项及相互作用积分区域J-domain的网格与单元尺寸对动态应力强度因子求解精度的影响,验证了改进型XFEM计算动态裂纹应力强度因子方法的有效性.针对文献中具有挑战性的"I型半无限长裂纹先稳定后扩展"问题,改进型XFEM给出目前为止精度最好的动态应力强度因子数值解.     

复杂裂纹问题的多边形数值流形方法求解

数值流形方法是一种能统一处理连续和不连续问题的有效数值方法。该方法采用的数学覆盖系统可完全独立于物理域,能很好地求解各类裂纹问题,而n边形单元（n>4）则具有网格划分灵活,求解精度高等优点。本文基于数值流形方法,采用正六边形数学单元求解线弹性复杂裂纹问题。在导出相关方程的基础上对典型裂纹问题进行了分析,通过互能积分法得到了裂尖的应力强度因子,计算结果与参考解吻合得较好。除此之外,文中还对不同单元上的求解精度进行了比较,结果表明采用正六边形单元的求解精度较正四边形单元和正三角形单元上的精度均更高。     

围压对巴西圆盘应力强度因子影响的数值分析

为了验证巴西圆盘在围压作用下应力强度因子公式的正确性,论文使用有限元分析方法计算了不同相对裂纹长度下围压单独作用以及围压与集中力共同作用时巴西圆盘的应力强度因子,并与解析解进行了对比分析.计算结果表明:纯围压作用下巴西圆盘的应力强度因子的解析解与数值解结果非常接近,两者的相对误差最大仅为0.535%;围压与集中力共同作用时的I型应力强度因子解析解与数值解也非常吻合,两者计算误差很小,仅在纯II型裂纹临界加载角附近有较大误差,但最大相对误差仅为2%,从而证明了巴西圆盘在围压作用下应力强度因子公式的有效性和可靠性.计算结果亦表明:直接将试件放在液体中加压去研究围压对断裂韧度的影响,在实验方法上缺乏理论依据.     

求解混合型裂纹应力强度因子的围线积分法

本文用复变函数理论推导出裂纹的辅助场，并用Betti功互等定理给出求解混合型裂纹应力强度因子的远场围绕积分法．此方法与积分路径的选择无关，用有限元法计算出远离裂纹尖端的位移场和应力场，就可通过计算绕裂端的围线积分，精确地给出混合型裂纹的应力强度因子KⅠ和KⅡ的数值解．     

求解混合型裂纹应力强度因子的围绕积分法

本文用复变函数理论推导出裂纹的辅助场，并用Ｂｅｔｔｉ功互等定理给出求解混合型裂纹应力强度因子的远场围绕积分法，此方法与积分路径的选择无关，用有限元法计算出远离裂纹尖端的位移场和应力场，就可通过计算绕裂端的围线积分，精确地给出混合型裂纹的应力强度因子Ｋ１和Ｋ１的数值解。     

围压与径向荷载共同作用下巴西盘裂纹应力强度因子的解析解

利用权函数法推导了围压和径向荷载共同作用下,考虑裂纹面摩擦的预制裂纹巴西盘应力强度因子计算公式,从理论上分析了围压、径向荷载和裂纹面摩擦对巴西盘应力强度因子的影响。结果表明,围压对I型应力强度因子有很大影响,I型应力强度因子随围压的增大而减小。当裂纹面闭合后围压和摩擦系数对II型应力强度因子同样具有显著影响,考虑裂纹面有效剪应力的权函数法理论解与有限元数值解相吻合,表明理论分析的正确性。     

奇异边界法模拟二维弹性力学问题

奇异边界法是一种新的边界型无网格数值离散方法。该方法使用基本解作为插值基函数,在继承传统边界型方法优点的同时,不需要费时费力的网格划分和奇异积分,数学简单,编程容易,是一个真正的无网格方法。为避免配置点与插值源点重合时带来的基本解源点奇异性,该方法提出了源点强度因子的概念,从而将边界型强格式方法的核心归结为求解源点强度因子。本文首次将该方法应用于求解平面弹性力学问题。数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度。     

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method (SBFEM) is a recently developed numerical method combining advantages of both finite element methods (FEM) and boundary element methods (BEM) and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors (SIFs) directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.The project supported by the National Natural Science Foundation of China (50579081) and the Australian Research Council (DP0452681)The English text was polished by Keren Wang.     

A novel twice-interpolation finite element method for solid mechanics problems

Formulation and numerical evaluation of a novel twice-interpolation finite element method （TFEM） is presented for solid mechanics problems. In this method, the trial function for Galerkin weak form is constructed through two stages of consecutive interpolation. The primary interpolation follows exactly the same procedure of standard FEM and is further reproduced according to both nodal values and averaged nodal gradients obtained from primary interpolation. The trial functions thus constructed have continuous nodal gradients and contain higher order polynomial without increasing total freedoms. Several benchmark examples and a real dam problem are used to examine the TFEM in terms of accuracy and convergence. Compared with standard FEM, TFEM can achieve significantly better accuracy and higher convergence rate, and the continuous nodal stress can be obtained without any smoothing operation. It is also found that TFEM is insensitive to the quality of the elemental mesh. In addition, the present TFEM can treat the incompressible material without any modification.     

裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型

比例边界有限元法SBFEM(Scaled Boundary Finite Element Method)是一种半解析数值方法,在裂缝分析特别是强度因子计算上具有相当高的精度。本文提出了一种用于裂缝分析的基于虚拟结构面的SBFEM与常规FEM的耦合分析方法。首先选取裂缝周边一定范围的计算域,并将结构分成不含裂缝区域和含裂缝区域两部分。然后,对不含裂缝区域,采用FEM进行网格离散;对含裂缝区域,采用SBFEM进行网格离散;两者相互独立,在这两个域内,分别采用各自相应的位移模式。最后通过在SBFEM网格的外边界设置虚拟耦合结构面的模式,实现有限元网格和比例边界有限元网格的耦合。通过两个经典的含裂缝平板的算例研究,探讨了本文方法在I型开裂和混合型开裂分析中,影响应力强度因子精度的因素。算例表明,SBFEM具有的降维和半解析性质,使本文方法在裂缝分析中的前处理简单易行,且计算结果具有相当高的计算精度。     

求解弱不连续问题的p型自适应有限元方法

在实际工程计算中,存在大量的弱不连续问题,如含夹杂问题。利用通常的有限元方法,为确保界面上各点满足给定高精度,往往需要采用全域网格加密或全域提高单元阶次的方法,这将会导致计算机的物理内存和CPU时间的剧烈增长。p-型自适应有限元方法是一种能通过自适应分析逐步增加单元阶次以改善计算精度的数值方法。本文,我们针对弱不连续问题设计了相应的p-型自适应有限元方法,重点讨论了容许误差控制标准对界面上各点计算结果的影响,并对几类典型的弱不连续问题进行了数值计算与模拟。数值结果表明,本文设计的p-型自适应有限元方法对求解弱不连续问题是非常有效的,用较少的单元得到精度可靠的数值结果,可大大提高其有限元分析效率。     

Computations of modes I and II stress intensity factors of sharp notched plates under in-plane shear and bending loading by the fractal-like finite element method

The fractal-like finite element method (FFEM) is used to compute the stress intensity factors (SIFs) for different configurations of cracked/notched plates subject to in-plane shear and bending loading conditions. In the FFEM, the large number of unknown variables in the singular region around a notch tip is reduced to a small set of generalised co-ordinates by performing a fractal transformation using global interpolation functions. The use of exact analytical solutions of the displacement field around a notch tip as the global interpolation functions reduces the computational cost significantly and neither post-processing technique to extract SIFs nor special singular elements to model the singular region are required. The results of numerical examples of various configurations of cracked/notched plates are presented and validated via published data. Also, new results for cracked/notched plate problems are presented. These results demonstrate the accuracy and efficiency of the FFEM to compute the SIFs for notch problems under in-plane shear and bending loading conditions.     

Elastic wave scattering from a penny-shaped interface crack in coated materials

This paper is concerned with the elastic wave scattering induced by a penny-shaped interface crack in coated materials. Using the integral transform, the problem of wave scattering is reduced to a set of singular integral equations in matrix form. The singular integral equations are solved by the asymptotic analysis and contour integral technique, and the expressions for the stress and displacement as well as the dynamic stress intensity factors (SIFs) are obtained. Using numerical analysis, this approach is verified by the finite element method (FEM), and the numerical results agree well with the theoretical results. For various crack sizes and material combinations, the relations between the SIFs and the incident frequency are analyzed, and the amplitudes of the crack opening displacements (CODs) are plotted versus incident wavenumber. The investigation provides a theoretical basis for the dynamic failure analysis and nondestructive evaluation of coated materials.     

直接计算应力强度因子的扩展有限元法

系统地给出了直接计算应力强度因子的扩展有限元法。该方法以常规有限元法为基础,利用单位分解法思想,通过在近似位移表达式中增加能够反映裂纹面的不连续函数及反映裂尖局部特性的裂尖渐进位移场函数,间接体现裂纹面的存在,从而无需使裂纹面与有限元网格一致,无需在裂尖布置高密度网格,也不需要后处理就可以直接计算出应力强度因子,并且大大简化了前后处理工作。最后通过两个简单算例验证了该方法的精度,分析了影响计算结果的因素,并与采用J积分计算的应力强度因子作了对比,得出了两种方法计算精度相当的结论。     

网架结构拟夹层板法的有限元验证

用拟夹层板法和有限元法对网架结构进行分析,对三类屋面网架（正放四角锥网架、两向正交正放网架和正放抽空四角锥网架）进行了均布荷载、局部荷载（半跨均布荷载）作用下的静力分析以及固有振动分析,对三类竖向承重网架墙体进行了稳定性分析。通过与有限元法分析结果的对比,表明了拟夹层板法作为一种简化的计算方法,其精度是比较高的,绝大多数的误差都小于5％,是可以直接用于工程结构设计的一种有效方法。此外,拟夹层板法还可作为一种在宏观上检验有限元建模正确与否的实用方法。     

A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes

A robust finite volume method for viscoelastic flow analysis on general unstructured meshes is developed. It is built upon a general‐purpose stabilization framework for high Weissenberg number flows. The numerical framework provides full combinatorial flexibility between different kinds of rheological models on the one hand, and effective stabilization methods on the other hand. A special emphasis is put on the velocity‐stress‐coupling on colocated computational grids. Using special face interpolation techniques, a semi‐implicit stress interpolation correction is proposed to correct the cell‐face interpolation of the stress in the divergence operator of the momentum balance. Investigating the entry‐flow problem of the 4:1 contraction benchmark, we demonstrate that the numerical methods are robust over a wide range of Weissenberg numbers and significantly alleviate the high Weissenberg number problem. The accuracy of the results is evaluated in a detailed mesh convergence study.     

P-version hybrid analytical finite element method for plate bending problems

Two sets of trial functions with different variables are constructed for the admissible space of the finite element analysis. The trial functions satisfy the equilibrium differential equation inside elements, while the deflections and rotations on the edges of the elements are approximated by the Peano hierarchical interpolation functions. Then, a generalized variational principle is applied to set up the p-version hybrid analytical finite element method for plate bending problems. The accuracy of finite element computation can be improved by increasing the order of the interpolation polynomials with fixed mesh. In the finite element formulation, to obtain the stiffness matrices and the load vectors, it is only necessary to perform quadrature over the edges of the elements. These matrices and vectors possess an embedding structure. The conformability between the elements can be controlled automatically.This work is supported by the Natural Science Foundation of China and the Aeronautical Science Foundation of China.