Projection Operators And Error Analysis On Complex Physical Domains In Isogeometric Analysis

Isogeometric analysis is the development of finite element method,and has become a hotspot in the field of research and application.The essence of finite element analysis is a process of discretization and analysis of the solution domain.Therefore,when the traditional finite element(FEM)method performs mesh discretization on the CAD geometric model,it will face a large time for meshing,and the scale of the analytical model equation is huge.A series of unavoidable problems such as low calculation accuracy,and separation of analytical models from geometric models is also a major drawback.Correspondingly,the isogeometric analysis method emerges as the times require.It uses the NURBS spline to represent the boundary of the geometric model,and the mesh is refined by node insertion,thus avoiding the problems caused by finite elements.The use of spline representation is also a parametric process that achieves seamless stitching of CAD/CAE.This advantage has led to further extensions in physical simulation.However,due to the limitation expressed by the NURBS spline,the complex physical domains cannot be parameterized by a single patch of NURBS.Therefore,this dissertation has carried out research on related issues.Firstly,in the parameter domain,for smooth functions in multi-patch parameter domain,the bivariate Hermite interpolation function is used to approximate them,and its continuity and error order are discussed.Secondly,a projection mapping on complex physical domain is constructed based on multi-patch parameterization.Based on the projection mapping,the approximation error of the projection mapping for smooth functions in physical domain is discussed.Through theoretical analysis,the projection mapping can reach the optimal approximation order;the numerical results show that the approximation error order solved by this method can be optimal.Finally,a class of spline space which is suitable for solving second-order elliptic equations on complex physical domains based on isogeometric analysis is given.and the error between the numerical solution of the isogeometric analysis and the real solution of the equation is analyzed.