Reaserch On Isogeometric Analysis Based On B-splines And NURBS

Finite element analysis is the most widely used numerical computation method inengineering analysis today. Due to its generality and effectiveness, it is received attention inthe engineering technological world. Along with the rapid development of computer scienceand technology, it has become an important part of computer-aided design (CAD) andcomputer-aided manufacturing (CAM). Isogeometric analysis (IGA) is a new finite elementmethod proposed by Professor Hughes in2005, and its purpose is to achieve unity betweenCAD and CAE, engineering design and analysis. In this thesis, IGA based on B-spline andNURBS functions is discussed; combined with the concept of isoparametric element, thebasis thought and theory of IGA are introduced; IGA-related package IGAtools is developed,and it’s applications are discussed on heat conduction, structural analysis, fluid mechanicsand so on.In this paper, basis functions generated from NURBS (Non-Uniform RationalB-Splines) are employed to construct an exact geometric model. For purposes of analysis,the basis is refined and/or its order elevated without changing the geometry or itsparameterization. Analogues of finite element h-and p-refinement schemes are presentedand a new, more efficient, higher-order concept, k-refinement, is introduced. Refinementsare easily implemented and exact geometry is maintained at all levels without the necessityof subsequent communication with a CAD (Computer Aided Design) description. In thecontext of structural mechanics, it means that all rigid body motions and constant strainstates are exactly represented. Standard patch tests are likewise satisfied. Applications arediscussed on heat conduction, structural analysis and so on. The effectiveness of IGA isvalidated through several numerical examples. The work of study is as follows.Chapter1introduces the background, significance and status quo of IGA. IGA is a newtype of finite element method, and it has become a hot spot today.Chapter2introduces the basic theory of B-spline and NURBS. Some important conceptssuch as knot vector are described; the relevant calculation formulas are listed; the important properties of basis function are analyzed; curve, curved surface and solid examples are given;finally the multiple pieces of geometry are introduced.Chapter3introduces the basic theory of IGA. Based on the concept of isoparametricelement in classic FEM, choosing the solution of Poisson equation and using Galerkinmethod, the detailed descriptions such as the idea and the process of IGA are introduced.Chapter4, based on the NURBS theory and thought of IGA, develops the IGA-relatedpackage IGAtools, and illustrates its structures and algorithms.Chapter5discussed the application of IGA in various fields, such as heat conduction,linear elasticity, fluid and electromagnetism, using IGAtools. The related formulas are given;2D and3D examples are verified to demonstrate the effectiveness and superiority of IGA.Chapter6is the conclusion and prospect of the paper. The chapter summarizes theresearch content of the thesis, and directed the future of the study.