【作者】 蔡永昌；

【导师】 张湘伟；

【作者基本信息】 重庆大学 ， 力学， 2001， 博士

【摘要】 在有限覆盖基础上新发展的流形方法统一解决了有限元、非连续变形分析和解析法的计算问题,是一种具有广泛应用前景的数值方法。本文的主要工作是分别采用标准的矩形网格、传统的有限单元网格和离散的扇形来构成流形方法的覆盖系统,并将其应用于二维连续体的弹性静力分析和张裂纹扩展过程的数值模拟。文中首先选取标准的矩形作为流形方法的基本覆盖,详细研究了流形方法里边界条件的施加,单元权函数的选取,连续的高精度应力场的计算,流形单元的数值积分方法以及各阶覆盖位移函数对流形方法精度的影响等关键问题。由于对各种复杂形状的结构体均可使用统一的规则矩形网格,本文方法的实施、使用及其与CAD 技术的一体化等都十分简单、方便。针对提出的理论和方法,论文首次把面向对象的思想引入流形方法的程序设计中,将流形方法的有限覆盖系统抽象为一些独立的数据类,给出了类的描述和它们的实现方法,并用树状结构对这些数据类进行管理。对任意形状的复杂结构体,用流形方法的数据类实现了流形单元有限覆盖系统的全自动生成。接着,文中采用围线积分法和流形方法的高阶位移函数来计算得到比较准确的混合型裂纹尖端的应力强度因子,然后基于线弹性断裂力学的裂纹扩展准则,用流形方法实现了平面Ⅰ-Ⅱ混合型张裂纹扩展的数值模拟,为数值方法研究裂纹的演化行为提供了一条简单而有效的途径。随后本文结合流形方法的覆盖理论和滑动最小二乘法,尝试提出了一种改进的无单元方法。它摒弃了单元和网格,采用圆形或扇形的覆盖图形并直接用结构域内的离散点来简化整体近似位移函数的构造,可以看作是流形方法实现的另一种特殊形式。该方法形式简单、计算时间少,比以往的一些无单元法具有更高的效率,并且更易于编程实现。更多还原

【Abstract】 The manifold method(MM),a prosperous numerical method based on the finite cover,is a very flexible numerical analysis method which can uniformly deal with calculating problems of the widely used finite element method(FEM), discontinuous deformation analysis(DDA) and analytic method.The main works of this dissertation are to adopt the standard rectangle,the traditional finite elements and the overlapping sectors to construct the cover system in MM,and apply MM to the linear analysis of two-dimensional continuous bodys and the simutation of open crack propagation. First of all,the rectangular meshes are chosen as the cover system of MM. Some pivotal techniques in MM such as the implemetion of essential boundary conditions, the selection of weight functions, the computation method of high precision stress,the numerical integration of manifold element and the influence of the high-order displacement functions are discussed in detail.Because the standard rectangular meshes are uniformly used for complex bodys,this method has the advantages of easy implementing,applying and incorporating with the CAD technology. For the proposed theory and method,the objected oriented programming is introduced into the program designing in MM.By abstracting the finite cover system of MM as independent data classes,the design and implemention of the objects about MM are studied,in which the objects are managed by techniques of tree structure.For complex area with arbitrary shape,the cover system of MM have been automatically generated by using these objects and trees. By applying contour integral method and high-order displacement functions of MM,we obtain the accurate stress intensity factors of mixed-mode crack.Then, the open crack propagation of the mixed mode I and II is simulated by means of theories of linear elastic fracture mechanics.This provides a simple and effective numerical way for crack evolvement studying. In addition,based on the theory of manifold cover and moving least squares (MLS),an improved formulation of the element-free method(IEFM) is introduced. The proposed method can avoid the element meshing and simplify the construction of approximations in the field by using the discrete circle and sector covers.IEFM can also be regarded as a special form of MM.By close comparison with other meshless methods,IEFM has advantages of low computation cost,simple form and easy programming.更多还原