弹性力学中的一种非协调数值流形方法

AN INCOMPATIBLE NUMERICAL MANIFOLD METHOD FOR ELASTICITY PROBLEM

通过引入数学和物理双重网格,将插值域与积分域分别定义在不同的覆盖上,即在数学网格上进行插值函数的构造,物理网格上完成系统能量泛函积分运算,最后通过覆盖权函数将二者联结在一起．它的优点是单元网格划分随意,不受复杂边界形状和二相材料界面的限制,单元可以是任意形状,是较之于有限元方法更一般的数值模拟方法．在4节点四边形数值流形方法中,由于单元总体位移函数包含的完全多项式不完全,使得计算精度不够精确,为此,在单元总体位移函数上附加非协调位移基本项,使之趋于完全,提出了弹性力学问题的一种改进的数值流形方法——非协调数值流形方法．通过内部自由度静力凝聚处理,导出了消除内参后的单元应变矩阵和单元刚度矩阵,使得在不增加广义节点自由度的前提下,大大提高了数值流形方法的计算精度和计算效率．同时对非协调项进行了显式处理,可以对工程实践起到更切实的帮助．数值试验表明,它们能够保证收敛,有较高的精度,对畸变不敏感,从而证明了该方法的可行性．

The numerical manifold method （NMM） comes from the discontinuous deformation analysis. The method unifies mathematical and physical mesh into a system： the mathematical mesh provides the nodes for building a finite covering of the solution domain and the partition of unity functions; while the physical mesh provides the domain of integration. So the domains of the interpolation and the integration are defined on two different covers in NMM, respectively. Finally two meshes are connected by the weighted function in the method. The method has some advantages which are discreted by arbitrary meshes including circle, triangle and rectangle etc. Hence the method is a new universal numerical analysis method which is not restricted by complex geometrical shapes and different material interfaces of the analytical problem. In the NMM of the quadrangle with four nodes, the modeling results have some errors for the total element test functions in the polynomial space are not entire. For voiding the disadvantages, the incompatible displacement term is added in total element .test function which the test functions are made more complete. And the paper develops an incompatible numerical manifold method which is kind of the improved NMM. The method has high computing efficiency and accuracy under adding no generalized degrees of freedom. The paper are also derived the element strain matrix and the element stiffness matrix by eliminating the internal parameters. At the same time, an explicit equation for three-dimensional elasticity problem is derived to apply the method to practice engineering. Finally several numerical examples are analyzed to illustrate the stability and convergence of the present method. The results are shown that the method is highly validity and accuracy.