Research On An Improved Algorithm For The Computational Accuracy Of Flat Shell Elements

The shell structure is one of the frequently used structural styles in civil engineering, hydraulic engineering, automotive engineering, aerospace engineering and other fields. Since the shell structure is a special three-dimensional structure with the properties of bending and membrane, its governing equations are a set of complex partial differential equations, which are difficult to be solved directly. Before the appearance of the computer, only some simple structures with regular-shapes can be solved. As the development of the Finite Element Method (FEM) based on the computer technology, many different types of shell elements have emerged through the efforts of numerous researchers. Among them, the flat shell element is one of the early developed shell elements. The essence of the flat shell element is to discretize a structure into some folded plates. The flat shell element is to simulate the bending and membrane state of the shell structure by the superposition of the membrane element and the plate bending element. Because of its advantages of simple expression, wide application, high computation efficiency and reliable results, the flat shell element is extensively applied to engineering practice problems and occupies an important position in the different types of shell elements. However, the flat shell element can not well simulate irregular shell structures, especially for curved shell structures, in which larger simulation errors may be resulted in. To achieve satisfactory accuracies, the meshes need to be refined, which would result in a large amount of computation.In order to improve the computational performance of the flat shell element, a new method to establish the local coordinate system over the element has been proposed in this thesis. In the new method, multiple local coordinate systems are established at all Gaussian points and they are located on the tangent planes to the element surface at the Gaussian points, thus the every transformation matrix of element stiffness can be adjusted in time to better fit the curved element surface. In order to obtain some parameters related to the element stiffness in the local coordinate systems created in the new method, a highly accurate algorithm is introduced in this thesis to calculate the shape functions with respect to the local coordinates and the determinant of the Jacobian matrix. The formulations of the highly accurate algorithm are deduced in the three-dimensional global coordinate system so that they can result in a high computational accuracy. The new method proposed for the flat shell element has achieved good results, both for the computation of the element stiffness matrices and the equivalent nodal loads. In addition, this thesis provides a new solver for the linear equations formed in FEM, and applies it to the flat shell element to improve the computing performance. The main contents of this thesis can be concluded as the following statements:(1) An improved algorithm to calculate the flat shell element stiffness matrices under the global coordinate system is proposed and applied to several different types of flat shell elements which are composed of different membrane and plate bending elements. The theoretical formulas of calculating the element stiffness matrices based on the improved algorithm are derived in detail, and the computational performance are discussed.(2) The applications of the improved algorithm to the flat shell elements are implemented in a developed code and the detailed design process and programming flow chart are presented. In order to validate the performance of the improved algorithm, this algorithm is applied to some typical flat shell elements. The correctness and accuracy are verified through several numerical examples.(3) The improved algorithm is extended further to the calculation of the equivalent nodal loads of the flat shell element. The theoretical formulas of the equivalent nodal loads under different load conditions are summarized and derived in detail. A code based on the new method to calculate the equivalent nodal loads is developed and carried out through some examples so that the computing performance of the new method can be verified.(4) A new linear equation solver, Simultaneous Elimination and Back-Substitution Method, is introduced to FEM, and a new technique to assemble the global stiffness matrix is proposed in the thesis. Then, the application of this solver is extended to the flat shell element. The detailed steps to use this new technique are given and implemented in coding. Some numerical examples are given to examine the performance of the solver.In conclusion, a new algorithm for improving the performance of the flat shell element is proposed in this thesis, which can provide a new tool for improving the performance of the flat shell element. This improved algorithm can also be applied to other flat shell elements and therefore it has a wide application prospect.