高阶数值流形方法中的线性相关问题研究  

STUDY ON LINEAR DEPENDENCE PROBLEM IN HIGH-ORDER NUMERICAL MANIFOLD METHOD

作　　者：郭朝旭[1] 郑宏[1,2] 

机构地区：[1]三峡大学土木与建筑学院,宜昌443002 [2]岩土力学与工程国家重点实验室,中国科学院武汉岩土力学研究所,武汉430071

出　　处：《工程力学》2012年第12期228-232,共5页Engineering Mechanics

基　　金：国家杰出青年基金项目(50925933);国家自然科学基金项目(90715028);国家“973”课题资助项目(2011CB013505)

摘　　要：数值流形方法(NMM)中整体逼近函数是通过单位分解将局部逼近函数进行"粘结"而形成的,当将局部函数取为阶数不低于一阶的多项式时便形成了所谓的高阶流形方法。然而高阶流形方法会导致刚度矩阵亏秩,这种亏秩即使在施加完整的位移约束后仍然存在,从而会导致NMM方程组的多解,但是每个解所对应的位移是唯一的,只要能稳定地求得任何一个特解即可。该文根据刚度矩阵的性质提出了改进的LDLT算法,可快速稳定地求得一个特解。结合典型算例,与摄动解法、最小二乘法和二次规划法进行了对比分析。In the numerical manifold method （NMM）, a global approximation function is built by ＂pasting together＂ the local approximation functions through the corresponding partition of unity. The high-order NMM is achieved by taking.high-order polynomials with an order no less than one as the local approximation functions. However, the high-order NMM will be associated with a stiffness matrix of rank deficiency. Even if the complete displacement constraints are enforced, such rank deficiency remains and causes a non-uniqueness solution to the system of linear equations of NMM. Nevertheless, the displacement corresponding to each solution is unique. Hence it suffices to develop a procedure that is able to find efficiently and stably out a particular solution. Based on the properties of the stiffness matrix, this study proposes an improved LDLT decomposition algorithm which can be used to find a particular solution for the high-order NMM system. Using a typical example, we make comparisons with the perturbation algorithm, the least squares method and the quadratic programming.

关 键 词：流形方法 刚度矩阵 线性相关 LDLT算法 最小二乘法 二次规划法 

分 类 号：TU452[建筑科学—岩土工程] TB115[建筑科学—土工工程]