数值流形方法中线性相关性问题的研究

Research on linear dependence problem of numerical manifold method

数值流形方法的形函数由覆盖函数和局部近似函数组成,形函数之间往往存在线性相关性。在现有研究成果的基础上对形函数线性相关性进行了分析,指出线性相关性的根源在于覆盖函数具有单位分解特性,并与单元形状有关。研究了线性相关性与整体刚度矩阵奇异性以及求解收敛性之间的关系,指出形函数线性相关不一定导致整体刚度矩阵奇异。对8结点六面体高阶流形单元的局部近似函数及单元形状与线性相关性之间的关系进行了分析,构造出一种完全线性独立的流形单元。通过算例分析了8结点六面体流形单元局部近似函数中一次完全多项式对求解精度和收敛性的影响,发现采用一次完全多项式局部近似函数的形函数虽然线性相关,但求解仍然收敛,且精度高于线性无关的单元。

The shape function of Numerical Manifold Method（NMM） is composed of cover functions and local approximate functions. The shape functions of NMM are prone to Linear Dependent （LD）. The LD problem was investigated. It is indicated that the origin of LD comes from partition of unity of local cover functions and LD is dependent of the shape of element. The relationship among LD of shape function, nullity of global stiffness matrices and convergence of solution was researched, and it revealed that the LD of shape functions is not inevitably null of global stiffness matrices. The relationship between local approximate polynomial function and LD of eight-node hexahedron manifold element was also analyzed and a linear independency element was devised. The effect of full first-order polynomial local approximate functions on calculation accuracy and convergence was investigated with case calculations. It indicated that the solution of full first-order polynomial local approximate functions is convergent although it is linear dependent, and it is more accurate than the linear independence element.