A New Method For The Finite Elements By The Area Coordinates Interpolation

The finite element method(FEM)plays a very important part in numerical anal-ysis.One common practice to construct compatible quadrilateral element is isopara-metric transformation. The Serendipity isoparametric elements are broadly used since their interpolate nodes are located on the boundary of the quadrilateral element.The Serendipity elements have a disadvantage of accuracy loss when elements are distorted. Because the Serendipity elements possess the low order completeness in the Cartesian coordinates though they have higher order terms of the isoparametric coordinates.The improvement of the Serendipity elements has attracted many researchers.Splines are piecewise polynomials satisfying certain continuity conditions, which are applied widely in the finite element method.The shape functions can be treated as splines. In this dissertation,several multivariate spline elements are presented to overcome the sensitiv-ity to mesh distortions based on triangular area coordinates and B-net method.Good results can be obtained by the proposed approaches.The main contributions of this dissertation are follows:1.In Chapter 3,we construct plane quadrilateral elements by using spline interpolation bases.The two diagonals are connected in a convex quadrangle,and the quadrangle is divided into four subtriangles.The triangular area coordinates and the B-net method are used in each subtriangle. Three quadrilateral spline elements with 4,12,17 nodes are proposed respectively. They compose a family of plane quadrilateral spline elements together with the known 8 nodes spline element, which possess complexness of orders 1-4 and also valid for non-convex quadrilateral element.Some numerical examples are given,in comparison with other known elements from the literatures,and the results show that the new spline family elements are competitive in both regular and distorted meshes.It is also demonstrated that the triangular are coordinates and the B-net method are efficient tools for developing quadrilateral elements.2.In Chapter 4,the way to construct quadrilateral elements is extend to n-sided polyg-onal spline element.The barycenter point of polygon is selected to divide the polygon into n subtriangles by linking the barycenter node and each corner node. The trian-gular area coordinates and B-net method are used in each subtriangle.The propose element can exactly model the quadratic fields.It is valid for both convex and non-convex polygonal element,and insensitive to mesh distortion.The numerical results show that this new polygon element has advantages of simpler formula and better accuracy. It is also accurate for the analysis of nearly incompressible problem.3.In Chapter 5,introduce and test two known 3D spline elements,21-node hexahedral element and 13-node pyramid element.Divide the hexahedron and pyramid elements into six and two tetrahedrons respectively. The 21 nodes hexahedral and 13 nodes pyramid spline elements are constructed based on the tetrahedron volume coordinates and the B-net method. These elements can exactly model the quadratic fields.The numerical results show that these new solid spline elements both have high accuracy. They are also accurate for the analysis of nearly incompressible problem.4.In Chapter 6,we discuss the construction of spline thin plate element.The discrete Kirchhoff element DKQ is a simple and efficient quadrilateral element for thin plate bending problems,which is formed based on the Kirchhoff hypothesis being imposed only at certain discrete points.Because of using the Serendipity Q8 element,the DKQ element is sensitive to mesh distortion.We establish refined spline element RD-KQS using the quadrilateral spline element and the refined non-conforming element method.The numerical results show that the new element is insensitive to mesh distortion and has high accuracy.