Research On Meshless Manifold Method For Fracture Failure Of Underground Structure

In recent years Numerical Manifold Method (NMM) and Meshless Method (MM)have received worldwide attention in the fields of mechanics and engineering. MM is aclass of new numerical methods developed, only needs the information at nodes, anddoesn’t discretize the domain into a mesh. The advantages of MM are the simplerpre-processing and higher precision. MM is a hot topic of researches on scientific andengineering computation. Based on topological manifold methods and differentiablemanifold methods, NMM has two cover systems, namely the mathematical covers and thephysical covers, so NMM can integrate both continuous analysis and discontinuousanalysis in a united mathematical framework, it is very suitable for simulating the problemsof discontinuity and large deformation. However, several difficulties arise in the realizationand implementation of MM and NMM for dealing with very complex discontinuousdeformation problems such as multi-crack propagation. For example, MM usually uses thediffraction method, the transparency method or the see-through method to deal with thediscontinuity in the domain, which may make it difficult to construct the shape functions asa result of the insufficient interpolation points, thus leading to the instability of thenumerical solution. As for NMM, when dealing with complex discontinuous problems, thelimitation of its dual grids makes the forming algorithm of the cover system verycomplicated, thus seriously hinders the expansion of its application.In order to overcome the difficulties of MM and NMM when dealing with complexdiscontinuous problems, a new Meshless Shepard Interpolation Method (MSIM) based onthe partition of unity (PU) and the finite cover theory is employed to solve the problems ofcrack propagation and failure behavior of the underground structures. In the MSIM, Theshape functions are constructed by the partition of unity and the finite cover technology,and thus the shape functions are not affected by discontinuous domains and crack problemscan be more properly treated. The MSIM shape function possesses three distinguishedfeatures: the interpolation property, the arbitrarily high order consistency, and satisfyingKronecker delta property at any desired node. The essential boundary conditions in MSIMcan be treated as easily as they are in FEM. Compared with the conventional MM, theMSIM shape functions are not influenced by the discontinuities in the solution domainbecause of finite cover technology is used. Compared with the mesh-based NMM, the finitecovers and the partition of unity functions of the MSIM are constructed by using a series of nodes. The main works of this thesis are listed as follows:(1) Base on the partition of unity and the finite cover technology, a new MSIMinterpolation has been developed for the simulation of crack propagation and failurebehavior of the underground structures. Fundamental equations of MISM analysis arededuced. The approach of how to deal with discontinuity in MSIM method are alsointroduced, as well as the specific steps to achieve numerical method applied for crackexpansion analysis.(2) The J-integral method, the virtual crack extension method and the virtual crackclosure method are introduced into the MSIM method for the computation of the stressintensity factor. Furthermore, the selection of the corresponding crack propagation criterionand the corresponding crack step is investigated.(3) A simple but efficient algorithm for finite cover automatic generating is introducedinto the MSIM for dealing with complex discontinuous problems. Based on Matlabsoftware, the key implementation technology of finite cover system is proposed, and theMSIM program for concrete crack analysis is developed to implement the meshlessautomatic trace simulating of concrete structure crack expansion.(4) The accuracy and stability of the J-integral method, the virtual crack extensionmethod and the virtual crack closure method for several crack examples are studied byusing the MSIM method, then comparison of multi-crack propagation simulation betweenMSIM and experimental is carried out. Numerical examples indicate the advantages andthe accuracy of the MSIM for the analysis of crack propagation.(5) Combined with the experimental study on cracked tunnel lining reinforced withumbrella arch, the failure process analysis of cracked tunnel lining reinforced withumbrella arch is carried out by using MSIM, the property and feasibility of the MSIM isalso tested and verified.