Imposing Essential Boundary Conditions In Isogeometric Analysis

Recently,The isogeometric analysis becomes popular in numerical analysis field.With the increasing size of the computational problem and complex structure of the geometric models,the time-consuming steps in mesh generation process become a bottleneck restricting the application and development of finite element method.The isogeometric analysis was proposed to avoid above problems.The spline basis functions are employed directly in analysis process,which can represent the geometric shapes exactly.Design model and analysis models are described in a same form for bridging the gap between CAD and CAE.In the dissertation,the imposing method of essential boundary conditions is chosen as the breakthrough point for the further research.Due to the lack of interpolation of spline functions,the isogeometric analysis is difficult to deal with essential boundary conditions.If the conditions are enforced by assigning nodal value as by the finite element method,it will result the reduced accuracy and the degradation of convergence.To solve this problem,we propose two approaches.By considering the accuracy advantages of the exact-geometric to solve the PDEs on surface,the level set equation on the parametric surface is proposed.The main contents of the dissertation are presented as following:(1)Strongly imposed method based on spline fitting theoryStrongly imposed method means the conditions is solved directly on trail function space.Based on spline-fitting,we propose the three methods,including: boundary point collocation method,enhanced least-square method and the transformation matrix method.Based on spline-fitting,boundary point collocation method approximates the solution with an optimal set of collocation points on boundary parts.By similar ideas,enhanced least-square method approximates the solution by least square estimate of the residual error on boundary parts.For accuracy,numerical stability and applicability,least-square method is superior to the collocation method.The Laplace-Beltrami equations by using the rational Bezier elements which extracted directly from the NURBS surface.For the problem of lack of interpolation of Bernstein polynomials,a transformation matrix method is proposed to construct the interpolation functions form a set of Bernstein polynomials.The converted polynomials satisfy the interpolation property,avoiding the difficulties in essential boundary conditions.Because rational Bezier patches can represent exactly the quadratic surface shape contained in mostly industrial products,solving PDEs on surface with Bezier elements has higher finite elements.(2)Weakly imposed method based on Nitsche's variational theory.'Weak handling' method is a method of integral equivalent amendments.Chapter Nitsche method to linear elastic displacement boundary conditions of the problem,through Lagrange identification,functional structure without constraint.And according to the elastic problems proved Nitsche variational method of stability,stability coefficient calculation formulas are given.Compared with similar methods,Nitsche method with less degrees of freedom,numerical stability,punishment,easier to control advantages,the forth chapter is extended to two important problems of fluid mechanics.For the 2-dimensional Stokes flow in velocity-pressure form,the mixed FEM needs the element form satisfied the inf-sup stability condition.For this problem,the flow function is used to transform the Stokes flow to a forth order PDE by high-order continuous NURBS basis solving.At last,the Nitsche method and Least Squares method are mixed to handle two groups essential boundary condition.For incompressible Navier-Stokes equation,this paper proposed a new equivalent integral formula with Dirichlet boundary condition including whole tangential velocity and normal velocity.This new form can handle non-penetrates the wall in 'weak handling'.For nolinear equations,the tangential stiffness matrix is given and Newton-Raphson method is used for solving.(3)Level set equation and numerical anaysis on parametric surfaceBased on traditional level set equation in Euclidean space,a new level set equation is proposed on parametric surface which can be used in the dynamic interface tracking on surface.The geometric properties of level set equation is researched.The normal vector and curvature of constrains is deduced on parametric surface.At last,the spline-point collocation method is used to solve the level set equation on parametric surface.