非连续问题中单元分割的模板方法

扩展有限元法(extended finite element method,XFEM)因具有裂纹几何独立于模拟网格、裂纹扩展时无需网格重分重映、计算精度高等优点,成为裂纹分析的主流数值方法之一.但该方法在工程实践中存在单元被裂纹分割的几何困难——现有精确几何分割方法实现复杂、计算量大、鲁棒性差.为克服这一困难,本文提出...     

Evolving crack patterns in thin films with the extended finite element method

This paper develops the extended finite element method (XFEM) to evolve patterns of multiple cracks, in a brittle thin film bonded to an elastic substrate, with a relatively coarse mesh, and without remeshing during evolution. A shear lag model describes the deformation in three dimensions with approximate field equations in two-dimensions. The film is susceptible to subcritical cracking, obeying a kinetic law that relates the velocity of each crack to its energy release rate. At a given time, the XFEM solves the field equations and calculates the energy release rate of every crack. For a small time step, each crack is extended in the direction of maximal hoop stress, and by a length set by the kinetic law. To confirm the accuracy of the XFEM, we compare our simulation to the exiting solutions for several simple crack patterns, such as a single crack and a set of parallel cracks. We then simulate the evolution of multiple cracks, initially in a small region of the film but of different lengths, showing curved crack propagation and crack tip shielding. Starting with multiple small cracks throughout the film, the XFEM can generate the well-known mud crack pattern.     

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.      

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。     

改进型扩展比例边界有限元法

结合了扩展有限元法(extended finite elementmethods,XFEM)和比例边界有限元法(scaled boundary finite elementmethods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improvedextended scaled boundary finite elementmethods,$i$XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的$i$XSBFEM的有效性,相比于常规XFEM,$i$XSBFEM的基于位移范数的相对误差收敛性较好;采用$i$XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合\较好.      

THE INVERSE ANALYSIS OF INTERNAL DEFECTS (INCLUSIONS) IN STRUCTURES USING A COMBINED XFEM AND GA METHOD

建立扩展有限元法与遗传算法相结合的结构缺陷反演分析模型.扩展有限元法通过引入不连续位移模式使得网格剖分无需依赖结构内部的不连续界面.通过改变水平集函数表征结构缺陷（夹杂）的位置和大小，遗传算法在每次迭代过程中具有全局和局部搜索能力，通过评估响应测点的响应量适应度值决定是否进一步迭代.对带有单个圆形缺陷（夹杂）和多个缺陷（夹杂）的结构进行了反演分析，并就响应测点的布置进行了讨论.结果表明，建立的反演分析模型能准确地探测结构存在的单个甚至多个缺陷（夹杂）.     

扩展有限元与遗传算法相结合的结构缺陷反演分析

建立扩展有限元法与遗传算法相结合的结构缺陷反演分析模型.扩展有限元法通过引入不连续位移模式使得网格剖分无需依赖结构内部的不连续界面.通过改变水平集函数表征结构缺陷（夹杂）的位置和大小,遗传算法在每次迭代过程中具有全局和局部搜索能力,通过评估响应测点的响应量适应度值决定是否进一步迭代.对带有单个圆形缺陷（夹杂）和多个缺陷（夹杂）的结构进行了反演分析,并就响应测点的布置进行了讨论.结果表明,建立的反演分析模型能准确地探测结构存在的单个甚至多个缺陷（夹杂）.     

Fatigue crack growth simulation in coated materials using X-FEM

In the present work, the eXtended Finite Element Method (XFEM) is used to study the effect of bi-material interfaces on fatigue life in galvanised panels. X-FEM and Paris law are implemented in ABAQUS software using Python code. The XFEM method proved to be an adequate method for stress intensity factor computation, and, furthermore, no remeshing is required for crack growth simulations. A study of fatigue crack growth is conducted for several substrate materials, and the influence of the initial crack angle is ascertained. This study also compares the crack growth rate between three types of bi-materials alloys zinc/steel, zinc/aluminium, and zinc/zinc. The interaction between two cracks and fatigue life, in the presence of bi-material interface, is investigated as well.     

准脆性材料中强不连续问题的数值方法若干应用及其研究进展

综述了模拟准脆性材料开裂过程的数值计算方法的研究进展和工程应用,比较了表征强不连续问题的显式非连续模型和隐式非连续模型的优缺点.结合混凝土粘结裂纹, 重点讨论了嵌入非连续模型,扩展有限元方法和富集有限元技术等非连续方法的构造特征和本质区别.从各种富集方法的理论完备性考察,以假定发展应变为基础的嵌入非连续方法虽然可以解决混凝土开裂过程中的应力锁死,满足内部边界的静力平衡条件以及反映开裂后的位移不连续问题,但嵌入非连续所采用的富集函数在开裂单元中并不能满足协调条件,使非连续两侧的应变不独立. 其局限性是由于富集自由度在单元的水平上引入,而以单位分解为基础的扩展有限元和富集有限元的富集函数以节点自由度的方式引入,除具有嵌入非连续的优点, 还可以有效消除嵌入非连续引起裂纹两侧应变的相互影响.文中同时指出了网格重构技术,弥散裂纹模型的局限性以及扩展有限元和富集有限元技术在构造方式上的细微差别.对于节点自由度方式引入的富集函数, 其操作困难性在文中也作了说明.      

考虑混凝土塑性耗散的CDM-XFEM裂缝计算方法

混凝土结构在服役期间受外界载荷的影响容易产生裂缝, 导致结构刚度降低、构件承载性能衰退, 而采用准确的计算方法预测混凝土裂缝的发展是治理裂缝的基本前提, 也是保障结构安全的重要手段. 连续损伤力学方法(continuou damage method, CDM)能够描述微裂缝的扩展过程, 但不能表示离散的开裂面, 且存在网格诱导偏差及虚假应力传递的弊端, 扩展有限单元法(mechanics-extended finite element method, XFEM)能够描述宏观裂纹的扩展过程, 但不能反映微裂缝的动态扩展, 两者计算出的裂纹分布与实际差异均较大. 现有的CDM-XFEM方法已经能够模拟混凝土微裂缝及宏观裂缝发展的整个过程, 但忽略了宏观裂缝出现时混凝土产生的塑性应变, CDM与XFEM的能量转化过程欠缺平衡性. 因此, 本文重点考虑能量转化时的塑性耗散, 选取指数型函数为粘结裂缝的牵引-分离模式, 基于能量及应力等效的条件重新构建了CDM与XFEM之间的能量转化方程. 采用广义逆最小二乘法求解能量转化系数, 确定能量转化时的临界位移, 并给出了裂缝面水平集的更新算法及整体计算方法的程序流程. 以双切口混凝土受剪拉开裂试验为例, 采用多种裂缝计算方法与试验进行了对比. 结果表明, 采用考虑混凝土塑性耗散的CDM-XFEM方法算出的裂缝分布及拉力-张开位移曲线与试验结果差异最小, 说明采用考虑混凝土塑性耗散的CDM-XFEM计算方法能够更好地计算混凝土裂缝.      

基于扩展有限元法的钢筋混凝土梁裂纹扩展的数值模拟

基于最大主应力准则,利用扩展有限元法对钢筋混凝土三点弯曲梁的裂纹扩展进行数值模拟。通过与已有模型试验结果的比对,验证了该方法的准确性与有效性。研究发现,带裂纹钢筋混凝土构件的裂纹扩展与力学性能不但受到单裂纹的尺寸、位置、倾角的影响,而且还受多裂纹间位置关系的影响;其中,裂纹横向分布位置和裂纹纵向分布位置是影响带裂纹钢筋混凝土梁裂缝扩展的主要因素。     

NUMERICAL SIMULATION OF CRACK GROWTH OF REINFORCED CONCRETE FLEXURAL MEMBERS BASED ON XFEM1)

基于最大主应力准则,利用扩展有限元法对钢筋混凝土三点弯曲梁的裂纹扩展进行数值模拟。通过与已有模型试验结果的比对,验证了该方法的准确性与有效性。研究发现,带裂纹钢筋混凝土构件的裂纹扩展与力学性能不但受到单裂纹的尺寸、位置、倾角的影响,而且还受多裂纹间位置关系的影响;其中,裂纹横向分布位置和裂纹纵向分布位置是影响带裂纹钢筋混凝土梁裂缝扩展的主要因素。     

磁电热弹耦合材料三维多裂纹超奇异积分法

用超奇异积分方程法将多场耦合载荷作用下磁电热弹耦合材料内含任意形状和位置三维多裂纹问题转化为求解一以广义位移间断为未知函数的超奇异积分方程组问题,退化得到内含任意形状平行三维多裂纹问题的超奇异积分方程组;推导出平行三维多裂纹问题的裂纹前沿广义奇异应力场解析表达式、定义了广义(应力、应变能)强度因子和广义能量释放率;应用有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以平行双裂纹为例,通过典型算例,研究了广义(应力、应变能)强度因子随裂纹位置、裂纹形状及材料参数变化规律,得到裂纹断裂评定准则. 最后,分析了裂纹间干扰、屏蔽作用及其在工程实际中的应用.      

基于XFEM与自适应网格的非均质材料裂纹扩展模拟

针对含有间断的非均匀材料的断裂问题,本文将虚节点多边形单元的形函数引入到扩展有限元（XFEM）中,提出了一种基于四叉树结构的动态网格细化方法,该方法可对间断面附近单元实现可调控的多层级细化,特别是对于裂纹扩展问题,可实现裂尖附近单元的动态网格细化与粗化。基于以上网格细化方法,本文提出了针对非均匀材质裂纹扩展问题的计算方法VP-XFEM。为验证算法的准确性与计算效率,针对含有孔洞及材料界面的断裂问题,本文给出了相应的算例。结果显示,与传统的一致性网格的XFEM相比,VP-XFEM能够明显改善计算精度与计算效率。     

A new integral equation formulation of two-dimensional inclusion–crack problems

A new integral equation formulation of two-dimensional infinite isotropic medium (matrix) with various inclusions and cracks is presented in this paper. The proposed integral formulation only contains the unknown displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks. In order to solve the inclusion–crack problems, the displacement integral equation is used when the source points are acting on the inclusion–matrix interfaces, whilst the stress integral equation is adopted when the source points are being on the crack surfaces. Thus, the resulting system of equations can be formulated so that the displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks can be obtained. Based on one point formulation, the stress intensity factors at the crack tips can be achieved. Numerical results from the present method are in excellent agreement with those from the conventional boundary element method.     

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采用有限元方法对表面含有两个不同大小半椭圆裂纹的有限厚矩形板在拉伸载荷作用下 进行参数化求解，得到两裂纹取不同尺寸和位置坐标时附属裂纹(尺寸较小裂纹)对主 导裂纹(尺寸较大裂纹)前沿参考点($\theta= 0, \pi/2, \pi $)处的应力强 度因子的影响系数$\beta$, 然后结合神经网络技术建立了相邻裂纹尺寸和位置参 数到主导裂纹前沿点处$\beta$的多变量非线性映射关系.     

A domain-independent interaction integral for magneto-electro-elastic materials

Magneto-electro-elastic (MEE) materials usually consist of piezoelectric (PE) and piezomagnetic (PM) phases. Between different constituent phases, there exist lots of interfaces with discontinuous MEE properties. Complex interface distribution brings a great difficulty to the fracture analysis of MEE materials since the present fracture mechanics methods can hardly solve the fracture parameters efficiently of a crack surrounded by complex interfaces. This paper develops a new domain formulation of the interaction integral for the computation of the fracture parameters including stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF) for linear MEE materials. The formulation derived here does not involve any derivatives of material properties and moreover, it can be proved that an arbitrary interface in the integral domain does not affect the validity and the value of the interaction integral. Namely, the interaction integral is domain-independent for material interfaces and thus, its application does not require material parameters to be continuous. Due to this advantage, the interaction integral becomes an effective approach for extracting the fracture parameters of MEE materials with complex interfaces. Combined with the extended finite element method (XFEM), the interaction integral is employed to solve several representative problems to verify its accuracy and domain-independence. Good results show the effectiveness of the present method in the fracture analysis of MEE materials with continuous and discontinuous properties. Finally, the particulate MEE composites composed of PE and PM phases are considered and four schemes of different property-homogenization level are proposed for comparing their effectiveness.     

一种XFEM断裂分析的裂尖单元新型改进函数

提出了一种适用于裂尖改进单元的新型改进函数, 基于三角变换的方法, 保留裂纹尖端场的应力奇异性和裂纹上、下表面的位移不连续性, 将常规扩展有限元法裂尖改进单元的4 项改进函数缩减为2 项, 裂尖改进单元的结点由常规的8 个改进自由度减少为4 个. 采用2 个正交的水平集函数表征材料内部裂纹面, 详细阐述了改进单元类型的判别方法, 给出一种改进单元的分区域积分方案. 最后, 若干断裂力学问题经典算例的数值计算结果表明:建议的裂尖改进函数具有较高的数值精度, 该方法是十分有效的.     

An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading

A plane crack problem of nonhomogeneous materials with interfaces subjected to static thermal loading is investigated. A modified interaction energy integral method (IEIM) is developed to obtain the mixed-mode thermal stress intensity factors (TSIFs). Compared with the previous IEIM, the original point of this paper is: the domain-independence of the modified IEIM still stands in nonhomogeneous materials with interfaces under thermal loading. Therefore, the modified IEIM can still be applied to obtain the TSIFs of nonhomogeneous material even if the integral domain includes interfaces. The modified IEIM is combined with the extended finite element method (XFEM) to solve several thermal fracture problems of nonhomogeneous materials. Good agreement can be obtained compared with the analytic solutions and the domain-independence of the IEIM is verified. Therefore, the present method is effective to study the TSIFs of nonhomogeneous materials even when the materials contain interfaces. The influence of the discontinuity of the material properties (thermal expansion coefficient, thermal conductivity and Young’s modulus) on the TSIFs is investigated. The results show that the discontinuity of both thermal expansion coefficient and Young’s modulus affects the TSIFs greatly, while the discontinuity of thermal conductivity does not arouse obvious change of the TSIFs.     

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。