Isogeometric Analysis Based On Polynomial Splines Over Hierarchical T-meshes And Its Application

Isogeometric analysis (IGA for short)introduced by Hughes et al. in2005is a kind of analysis framework, aimed at integrating CAD and CAE seamlessly. Within the framework of IGA, the smooth geometric basis is used as the basis for analysis. Comparing to traditional finite element method (FEM for short), mesh generation is avoided and the geometry is represented exactly in IGA. It was also demonstrat-ed that the smoothness of basis offers important computational advantages over standard finite elements. NonUniform Rational B-splines (NURBS) was first used as bases in IGA. However, NURBS is lack of local refinement, and is usually not suitable for adaptive analysis. T-splines was introduced by Sederberg et al. by per-mitting T-junctions in control mesh and can be locally refined. But T-splines are not always linearly independent and have excessive propagation of control points when refined. Polynomials splines over hierarchical T-meshes (PHT-splines)[邓建松2008] are a generalization of B-splines over hierarchical T-meshes. PHT-splines not only inherit the adaptivity and flexibility of T-splines, but also possess a very efficient local refinement algorithm. So it is expected to be suitable for isogeometric analysis.In Chapter1, we briefly reviewed the history of isogeometric analysis and the related knowledge of PHT-splines.In Chapter2, the frameworks of finite element method and NURBS-based iso-geometric analysis were outlined, which are the basic of IGA.In Chapter3, we introduced rational PHT-splines based isogeometric analy-sis. PHT-splines basis functions share some good properties with B-splines, such as nonnegativity, local support, linear independence and partition of unity. Most importantly, PHT-splines possess an efficient local refinement which is imperative in geometric modeling and adaptive analysis. However PHT-splines can not represent conic curves/surface which are used extensively in engineering. So we first intro-duced rational PHT-splines, and then applied it in IGA. Numerical examples are demonstrated to show the potential of rational PHT-splines as the basis for adaptive isogeoemtric analysis. In Chapter4, a residual-based posteriori error estimator using rational PHT-splines basis functions is derived to guide the local refinement in IGA adaptively. A Posteriori error estimation is the core of adaptive analysis. We also proved the convergence of adaptively numerically solving IGA. Several numerical examples are tested for the efficiency of the error estimator.We applied PHT-splines based IGA in convection-dominated problem based on the efficient local refinement and SUPG method in Chapter5, and in Chapter6, we applied PHT-splines based IGA in the fourth order of partial differential equation by means of global smoothness of PHT-splines.Finally in Chapter7, we briefly summarize the research and discuss some future work.