The study of the convergence rate on B-spline finite element method in two dimensional plane stress problems of bi-material

• Main Authors: Bing-Chang Fang, 方炳昌
• Other Authors: Shi-Pin Ho
• Format: Others
• Language: zh-TW
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/09382212310149165356

碩士 === 國立成功大學 === 機械工程學系碩博士班 === 97 === We used the B-spline functions as the basis functions to solve the two dimensional plane stress problems of bi-material with irregular boundary shapes. Originally, we used the same basis function in different material. But stress was forced to be discontinues due to the continuity of strain and different modulus of elasticity. Finally, this problem induced by inaccuracy of stress. Therefore we used different basis functions in different material and applied compatibility condition in the border between two materials, producing the consistency of displacement in the border between two materials. We used B-spline functions from second order to sixth order simultaneously and different element size on the problems with different irregular boundary shapes. The first irregular boundary shape in these studies is a square plane with a circular shape. The h convergence of high order B-spline functions is excellent when the element size decreases, and the p convergence is also excellent when increases the order of B-spline functions in the suitable and the same element size. In order to use the B-spline finite element method widely on other irregular boundary shapes, we also study stress problems of bi-material on a square plate with a elliptic boundary shape, and a rectangle boundary shape with round edges. We also study the effect of the stiffness matrix with a large condition number, which is caused by small integration area.According to the study of a rectangle boundary shape with round edges in a square plate, for maintaining the property of Ck-2 continuity, the accuracy of the B-spline functions on areas around the maximum stress point is not as good as the results of the second-order finite element method, but the accuracy of maximum stress point is better than the results of the second-order finite element method. Specially, compatibility condition may affect convergence of B-spline function. We proposed a method of refinement for bi-material. With this mehod, we can add small B-spline functions on small elements at will, and we can use lesser degree of freedom to get a more accurate result in the analysis. The uses of refinement will increase the efficiency of the B-spline finite element method. In summary, the studies in this thesis show that the B-spline finite element method can be used widely in the analysis on the two dimensional plane stress problem of bi-material with irregular boundary shapes.