The Higher-order Polygonal Numerical Manifold Method And Its Application In Linear Elastics

The numerical manifold method(NMM), originally proposed by Dr. Genhua Shi,who also established the key block theory and discontinuous deformation analysis(DDA) for computational rock mechanics, is capable to solve both continuous and discontinuous problems. In the NMM, the physical covers and manifold elements are formed by mathematical cover system which is independent of the physical domain.Higher order NMM approximate can be easily achieved through the use of higher-order local functions on physical covers(named cover functions) on a fixed mathematical cover system(named higher-order NMM hereafter, for simplicity). In the present work,the high-order polygonal NMM is developed. The linear-dependence(LD) problem of the global stiffness matrix is carefully investigated. Meanwhile, the accuracy of the proposed approach is examined through some typical linear elastic cases. The major work is outlined as follows:(1)The research status of the NMM is comprehensively reviewed; then, the basic concepts of the NMM are presented; at the same time, the comparisons among the NMM and some other representative numerical methods, e.g., the finite element method(FEM) and the extended finite element method(XFEM) are made; further, the corresponding formulations of higher-order NMM are elaborated.(2) The linear dependence(LD) problem in the higher-order NMM has been analyzed. The associated codes for LD analysis are built using Fortran90 language.Detailed LD studies on the triangular, quadrilateral and hexagonal mathematical elements are carried out and quantitative conclusions are drawn.(3)The higher-order polygonal NMM for linear elastic problems is developed and the corresponding codes are constructed by C++ language. The accuracy of the proposed method on triangular, quadrilateral and polygonal mathematical elements is investigated through typical numerical examples.