Some Researches On The Spline Finite Element Method And The Scaled Boundary Finite Element Method

The finite element method is a powerful tool for numerical computation and an important method to deal with complex engineering problems.The spline finite element method and the scaled boundary finite element method are new numerical methods developed based on the finite element method.The spline finite element method takes advantage of the spline function satisfying certain continuity and having high approximation accuracy.The scaled boundary finite element method is a new semi-discrete and semi-analytical method proposed by Wolf and Song.This method combines the finite element method and the boundary element method,inherits the advantages of the two methods,and has its own characteristics,more flexible and more effective.At present,the spline finite element method and the scaled boundary finite element method are developing rapidly and widely used in numerical computation and engineering.This thesis studies the two methods and their combination in mathematical theory.The specific research works are the following three parts.1.The high-order completeness analysis of the scaled boundary finite element method for second-order elliptic problemsFor the scaled boundary finite element method,many rich research results have been obtained in engineering applications,but there is a few researches in mathematical theory so far.For the finite element method,the completeness analysis of the element is a very important part of the basic theory of the scaled boundary finite element.In Chapter 3,this thesis strictly gives a theoretical proof of the high-order completeness of the scaled boundary finite element method for solving the second-order problems.Starting from the expressions of the homogeneous solution(without force term)and non-homogeneous solution(with force term)of the scaled boundary finite element method,we introduce the circumferential interpolation operator and give the expression of the polynomials in the scaled boundary coordinates.Through analysis,it is concluded that the key to accurately reconstruct polynomials by the interpolation operator lies in the existence of a specific integer exponent r(i.e.,the degree of polynomials)in the expression form of polynomials.On the one hand,r in the homogeneous solution is determined by the eigenvalue of the matrix[Z]in the eigenvalue decomposition method.On the other hand,r in the particular solution of non-homogeneous solution is determined by the approximation precision of the force term.This chapter rigorously proves that for a closed S element,the specific integer exponent r can always be obtained in the calculation of the scaled boundary finite element method and is independent of the shape of the S element,that is,the scaled boundary finite element method has high-order completeness.Besides,in the completeness analysis,we also find some relevant theoretical problems in the process of solving the scaled boundary finite element equation and give the necessary proof.2.The spline scaled boundary finite element method for three-dimensional second-order elliptic problemsThe scaled boundary finite element method has good adaptability to mesh subdivision,especially,its equation derivation of polygonal or polyhedral elements is not different from that of triangular,quadrilateral,or hexahedral elements.In fact,the construction of polyhedral elements is quite difficult for three-dimensional problems.Considering the simple construction and good universality of the polyhedron element in the scaled boundary finite element method,in Chapter 4,we combine the spline elements with it.The quadratic and cubic spline scaled boundary finite element SBFEM-L8 and SBFEM-L12 are constructed,respectively.This element uses the quadrilateral spline element L8 or L12 as the surface element in the three-dimensional scaled boundary finite element method.According to the theoretical analysis in Chapter 3 and the numerical experiment in Chapter 4,SBFEM-L8 has the second-order completeness,and SBFEM-L12 has the third-order completeness.In addition,numerical experiments also show that SBFEM-L8 and SBFEM-L12 not only retain good adaptability of the scaled boundary finite element to meshes,but also exert the advantages of the spline element with few nodes,high accuracy,and insensitivity to meshes distortion.3.A superconvergent nonconforming quadrilateral spline element for forth-order elliptic problemsFor the spline finite element method,because the spline function has flexible continuity inside the element and strong adaptability to meshes,Chapter 5 constructs a 12-dof nonconforming quadrilateral spline element NCQS12 for the fourth-order elliptic problem.This element is based on the spline space S31(QT)on the ?-type triangulation.A subspace of S31(QT)containing a complete cubic polynomial is selected as the spline finite element space by the B-net method.The theoretical analysis shows that the interpolation error of the spline element NCQS12 is O(h2),and the consistent error is O(h1).In particular,for parallelogram mesh,the consistent error can reach O(h2),that is,the element has superconvergence.Numerical experiments verify our theoretical results.For two degenerate grids:bi-section meshes and asymptotically regular parallelogram meshes,numerical experiments show that element NCQS12 still has superconvergence.