Study On Linear Dependence Problem In High-order Numerical Manifold Method

The actual deformation and failure process of geo-materials and geo-structures is acomplex progressive evolution involving initially elastic deformation, crack propagation,large-scale displacement and even movement of the discrete system. However, the currentnumerical methods in geo-mechanics are suitable for either ideal continuum materials, orcompletely discontinuous media. On the one hand, such methods as FEM and BEM arebased on the hypothesis of total continuity or basal materials continuity. On the other hand,there are discontinuous deformation analysis methods such as DDA and DEM, which areon the basis of the blocky theory and the assumption that rock mass is discrete media. Thisleads to the cases that the actual deformation of rock mass must be represented to be eithercontinuous or completely discontinuous. A numerical method with ability to deal withcontinuous deformation problems, discontinuous deformation problems and crackpropagation problems seems to be a blue moon.In1991Shi Genhua proposed the numerical manifold method which can unify thecontinuous and discontinuous methods and can solve small deformation, large deformationand discontinuous deformation. Today the NMM has developed greatly. However, thebasic problem of NMM is that the whole stiffness matrix will be rank deficient if wechoose the high order (more than or equal to1) polynomials to approximate thedisplacement field. And we call this problem as linear dependence problem. In the Ma andAn’s invited reviews about NMM in2010, they pointed that the linear dependence problemof NMM is the principal problem which need to be solved. At the same time, the founderof NMM, Shi Genhua, calls this problem as a nail problem.This thesis firstly introduces the basic theory of NMM, summarizes the currentresearch states and develops codes of this method with the latest algorithm. Secondly thisthesis studies the linear dependence problem. The main works are stated as follows.Firstly, this thesis analyzed and proved the linear dependence problem.And it canpredict the rank deficiency due to introduction of polynomials of higher order.Secondly, with the system of linear equations with rank deficiency we develop twomethods. One is the modified LDLTmethod. The other is to construct a partition of unityfunction that is not polynomial shape function.Thirdly, using typical examples, we compare the proposed methods with othermethods, indicating that the algorithms of this article are convenient and efficient.