Some Problems In Isogeometric Analysis

Finite element analysis (FEA) has developed rapidly in the last century and is widely applied in computer aided engineering such as the aircraft manufactory and ship manufactory. However there are some issues arouse in the application of FEA to CAD-based models. It is well known that mesh generation, which generates discrete geometry as computational domain from given CAD models, is a key and the most time-consuming step in FEA. Additionally, the given CAD-models are not used any-more in the analysis phase after being tesselated into meshes, leading to the geometry approximated error and no natural feedback loop between designer and analysts. These limitations call for a new kind of method to integrate CAD and CAE. In2005, Hughes et al. introduced the concept of isogeometric analysis (IGA) as a means of bridging the gap between computer aided engineering (CAE) and computer aided design (CAD). A simple explanation of IGA is using the basis functions represented the geometry as the basis in analysis phase.The main focus of the current thesis is to study some important issues in IGA. In chapter two, we introduce some splines which can be locally refined and give the frame of IGA. Then we apply PHT-splines in solving the4th order partial differential equa-tions and compare the results of mixed finite element method with our method using PHT-spline basis functions to approximate the trial function space H20. The smooth-ness of spline basis functions improves convergence result and is more effective when solving high order partial differential equation. In the last, we compare the condition number of stiffness matrix using different bases functions, such as B-splines, hierar-chical B-splines, AST-splines and PHT-splines. They behave quite differently in FEA, which is mainly the big difference of condition number of stiffness matrix. The large condition number induced by PHT-splines encourages us to find the reason and im-prove it.We analyze the decay phenomenon of PHT-spline bases during refinement in chapter three and proposed a method to construct new basis functions which can elim- inate the decay phenomenon. The new PHT-spline bases are used in solving elliptic equation, and the results show that convergence rate is optimal. The condition number of induced stiffness matrix is less than the one induced by original PHT-spline bases, which implies the new bases can bring more stable results than original PHT-spline bases.IGA avoid mesh generation which is a obstacle in traditional FEA, but another problem is proposed which is called parameterization. In chapter four, we propose a new framework of planar parametrization in IGA which is still an open problem. The core idea of our framework is firstly to partition the physical domain on which the partial equations are defined into subdomains, then use harmonic parametrization method to parameterize every subdomain. Compared to the existing work of pla-nar parametrization, our method can parameterize complicated domain with no self-intersection, and the numerical experiments demonstrate that the proposed method has a superior quality than the existing methods. This decomposition based parameteriza-tion method can be extended to high genus and high dimension domains. Our future work is given in chapter five.