Modeling,Simulation And Analysis Of Two Classes Of Material Defects Problems

In this thesis,we investigate material defects of two major kinds of materials—the crystalline solids and the liquid crystals,based on the modeling,analysis,and simulation.The goal is to study the relation-ship between defects and their equilibrium states in those materials.Such a study can help us understand and predict the properties of vari-ous natural and artificial materials,and to develop new materials which possess desired properties.For crystalline solid defects,we use the atomic/continuum coupling method,which combines the accuracy of the fine scale model with the efficiency of the coarse scale model to achieve the optimal balance.We focus on the analysis of the error sources of the blended ghost force correction(BGFC)model and some related blending methods.We prove that the BGFC model using the piecewise quadratic finite element method has the optimal error when the Cauchy-Born model is adopted in the continuum region.At the same time,we construct a graded mesh,and implemented the atomistic/continuum coupling method for multi-body interaction potentials in three dimensions.The numerical experiment results are in good agreement with theoretical predictions.For nematic liquid crystal defects,Landau-de Gennes theory is used to study the configuration and defect structure of nematic liquid crystals at low temperatures.This theory resolves the problem of head-to-tail symmetry for rod-like liquid crystal molecules and can predict almost all kinds of liquid crystal defects.We focus on the reduced two-dimensional theory on a rectangular domain with tangential boundary conditions.When the reduced parameter tends to zero and infinity,we use the classical maximum principle and symmetry arguments to analyze the properties of the solutions of the equilibrium states;for ar-bitrary reduced parameters,we use numerical calculation and illustrate the bifurcation diagram to show the change of different equilibrium states with respect to the length-to-width ratio of the rectangular do-main and the reduced parameter.We compare the solutions on the gen-eral rectangular domains and those on the square domain and highlight the influence of the symmetry breaking.In addition,we numerically study the relaxation process from the nontrivial topological states to the trivial states,The observed defect splitting and motion patterns open an avenue to explore new transition mechanisms between trivial states.The two kinds of defects correspond to the breaking of the peri-odicity of space position and the singular points of the director field respectively.If the continuous model is regarded as a discrete model as the number of microscopic particles tends to infinity,then the logic behind these two kinds of defects is consistent.