Discontinuous Galerkin Based Isogeometric Analysis And Applications

Since the time-consuming steps in mesh generation process and the difficulties encountered during the refinements, the traditional finite element analysis suffers a lot when dealing with complex geometries. Recently, isogeometric analysis was proposed to handle this important problem. In this papet, we turther propose a dis-continuous Galerkin based isogeometric analysis method to solve partial differential equations over complicated area. Complex geometries are composed of multiple non-overlapping NURBS patches joining with at least C0-continuity across the common interfaces. Since basis functions on different patches are completely independent, the discontinuous Galerkin method could be exactly used to glue the multiple patches together to get the right solution. Note that the discontinuous Galerkin ideology is adopted at patch level, i.e., we employ discontinuous Galerkin method only across the common interfaces between patches, and the standard isogeometric analysis is adopted within each patch. Our method enjoys the advantageous features of both isogeometric analysis method and the discontinuous Galerkin method, which en-ables us to design a superior numerical scheme. First, the time-consuming steps in mesh generation process are no longer necessary and the refinements can be easily performed by the knot insertion algorithm and order elevation algorithm of NURBS. Moreover, the geometrical error is eliminated or significantly reduced at the begin-ning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Additionally, this method can easily and nicely handle cases with non-conforming patches and different degrees across the patch interfaces due to the flexibility of the discontinuous Galerkin method.The main work of this paper is to solve partial differential equations on compli-cated surfaces using the proposed method. Chapter1briefly introduces the CAD, CAE, isogeometric analysis method and the discontinuous Galerkin method. Chap-ter2gives some necessary preliminaries. In Chapter3and Chapter4, we consider solving the elliptic equation and the Allen-Cahn equation on complex surfaces. After we derive the discontinuous Galerkin schemes, both theoretical and numerical re-sults are carried out. Theoretical results show that the proposed schemes are stable and could achieve the optimal convergence rates with respect to appropriate norms. Numerical results verify the theoretical results and show the good performance of our method. In Chapter5, we combine the proposed method with two geometric flows, including the mean curvature flow and qusi-surface diffusion flow, to handle some surface design problems, such as N-sided hole filling, surface blending and so on. After the numerical schemes are presented, we prove their stability under the (?)2norm. At last, we gauge the good performance of our method using numeri-cal results which show that we can achieve G1-continuity across not only the given boundaries but also the common interfaces between patches.