弹性力学的复变量数值流形方法  

COMPLEX VARIABLE NUMERICAL MANIFOLD METHOD FOR ELASTICITY

作　　者：高洪芬[1] 程玉民[1] 

机构地区：[1]上海大学上海市应用数学和力学研究所,上海200072

出　　处：《力学学报》2009年第4期480-488,共9页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(10871124);上海市教育委员会科研创新基金(09ZZ99)资助项目~~

摘　　要：数值流形方法通过引入数学和物理双重网格,将插值域和积分域分别定义在两个不同的覆盖上来完成系统能量泛函积分运算.当采用高阶函数构造位移函数时,广义节点自由度将大大增加.在求解系统的平衡方程中,运算量是与自由度的三次方成正比的,因此数值流形方法的计算量是较大的.为此,在复变量理论的基础上,采用一维基函数建立二维问题的逼近试函数,然后将其应用于弹性力学的数值流形方法,提出了复变量数值流形方法,推导了弹性力学的复变量数值流形方法的公式.与传统的数值流形方法相比,复变量数值流形方法具有计算量小、精度高的优点.The numerical manifold method （NMM） is a more general numerical method than finite element method with mathematical and physical meshes. The mathematical mesh provides the nodes to form a finite covering of the solution domain and the partition of unity functions, while the physical mesh provides the domain of integration. The numerical manifold method has some advantages, for example, the solution domain is discretized as an arbitrary mesh which is independent on the complex geometry of the boundary of the solution domain or the interface of bi-materials. However, the generalized degrees of freedom in the NMM will be enhanced to induce more computing cost when the higher polynomial interpolating function is used as the displacement function. In this paper, based on the complex variables theory, the approximation function of a two-dimensionai problem is developed with one-dimensional basis function, and the approximation function is applied to the NMM for two-dimensional elasticity. Then the complex variable numerical manifold method （CVNMM） for 2D elasticity is presented, and the formulae of the CVNMM are obtained. The influences of boundary conditions and initial stress on the final linear algebra equations system are discussed. In addition, two numerical cases were carried out. When the composite material structure is simulated using the CVNMM, the computing meshes can be easily generated along the interfaces of materials, then the CVNMM is more flexible than the finite element method. Furthermore, hi-material interface crack problem is analyzed using the CVNMM, and the stress intensity factor of the interface crack is obtained with the numerical extrapolation method. The CVNMM has greater computational efficiency and precision validated with the numerical cases.

关 键 词：数值流形方法 复变量数值流形方法 弹性力学 位移函数 广义节点自由度 

分 类 号：O343.2[理学—固体力学]