New Plate And Shell Finite Elements Based On The Static-Geometric Analogy

It is well known that the construction of plate bending element is essentially more difficult than that of plane elasticity for the reason that the former has to meet the requirement of C1 continuity. Through years of researchers' persistent efforts, quite a number of plate and shell elements have been presented and many of them are excellent. However, some basic but crucial problems still remain unsettled: in the aspect of plate bending element, there exists the problem of imbalance between the development of plane elasticity elements and that of plate bending elements, which is not compatible with the similarity theory between plane elasticity and plate bending, for according to the theory the two systems, plane elasticity and plate bending, are isomorphic; in the aspect of thin shell element, the ultimate aim is to construct the shell element which can perform well in both membrane-dominated and bending-dominated situations, yet so far no widely accepted guiding theory or practicable method has been found.This paper focuses on the study of these basic problems. In the first place, with the similarity theory, a bridge is set up between plane elasticity and plate bending to transform those good plane elasticity elements into plate bending elements; then with the guiding theory of the static-geometric analogy, a new methodology of thin shell element is presented. Since the similarity between plane elasticity and plate bending is a special case of the static-geometric analogy, this paper, therefore, studies the new formulation based on the static-geometric analogy in a broad sense.A thorough study is exerted in this paper on the general method of transformation from plane elasticity displacement element, which is characterized by its simple formulation and sound reliability, to plate bending element. Firstly a flexible matrix of plate bending is obtained from a formulation which is the same as the one of a certain plane elasticity element except for the displacements replaced by stress functions. Secondly the flexible matrix is transformed into stiffness matrix by the use of Legendre transformation. Lastly the above stiffness matrix, the nodal variables of which are the dual of stress functions, is replaced by a new one with simple displacements vector regarded as unknown. Such finite element satisfies homogeneous equilibrium equations and can pass the patch test as long as the original plane elasticity element can pass the corresponding patch test.In order to avoid the difficulty in computing the general inverse of flexible matrix in Legendre transformation, this paper studies the mixed coordinates formulation for some quadrilateral plate bending elements instead of fully formulating in (e,n) plane for their original plane elasticity elements. On the one hand, the linear interpolation in (x , y) plane makes it easy to separate the three-dimensional null subspace corresponding to rigid body motions, hence what is left to do is just to compute the inverse of a symmetric definite submatrix numerically. In this way the numerical difficulty in computing general inverse can be avoided. On the other hand, the higher order interpolation in (e,n) plane can maintain the formulation of original plane elasticity element.The vector of nodal variables (called rolling vector) of the stiffness matrix obtained from Legendre transformation is not a simple vector with obvious geometrical meaning. Using the similarity between plane elasticity and plate bending, this paper successfully fulfils the identification of rolling vector and establishes a general method, the curvature lumping method, to represent the rolling vector by the use of simple displacements vector. The method removes thebottleneck of transformation from complementary energy element with stress functions vector to potential energy element with simple displacements vector. What is the most important about this method is that it never destroys the original convergence of the transformed plane elasticity element and that it can maintain the original p...