Study On Hierarchical Higher-order Quadrilateral Subparametric PCG Method And Its Applications

In the finite element analysis of two-dimensional problems,the unstructured quadrilateral mesh is needed for subdivision because the solution area is often irregular like gravity dam/arch-abdomen dam.This quadrilateral mesh can better reflect the displacement states and stress states in the deformed body.Quadrilateral high-order element is needed in the calculation for obtain high-precision finite element numerical solution?at present,the widely used finite element software such as ANSYS and ABAQUS only provide the option of secondary element?.However,compared with the low-order element,the high-order element needs more computer storage space,and it needs more calculation work to generate the corresponding element characteristic matrix and total stiffness matrix,how to improve the calculation efficiency is a very worthy of research.Morever,the discretized algebraic system corresponding to higher-order elements also has higher calculative complexity and morbidity,and the efficiency of the usual solution methods will be greatly reduced,need to design a fast solution for it?like pre-conditioned conjugate-gradient method?,in order to improve the overall calculation efficiency of finite element analysis.This paper focuses on the generation of unstructured quadrilateral mesh,the higher-order hierarchical subparametric finite element analysis of quadrilateral grids and the algebraic solution of the corresponding discrete system for the gravity dam/arch-abdomen dam problem.The main contents and conclusions are as follows:?1?The unstructured quadrilateral mesh generation of gravity dam/arch-abdomen dam is obtained by using the automatic quadrilateral mesh generation program: AUTOMESH-2D.The advantage of the program is that can get the arbitrary quadrilateral mesh with high quality and efficiency as long as setting the size of the element.Then,a quadrilateral hierarchical element and its corresponding hierarchical function are discussed by adding "virtual node" in the edge and "virtual node" in the plane,the corresponding higher-order hierarchical subparametric finite element method is designed and implemented by combining the sparse storage by three one-dimensional arrays?CSR format?.The validity and adaptability of the method are verified by applying the design method to the solution of gravity dam/arch-abdomen dam problem.?2?For the unstructured quadrilateral higher-order hierarchical element discrete system of gravity dam problem,a simple and effective preconditioner is designed by using the algebraic properties of the characteristics of hierarchical structure that its coefficient matrix is a diagonal block matrix,so that the efficiency of internal iterative calculation is greatly improved by PCG method.The basic idea of the method is to transform the quadrilateral hierarchical higher-order element discrete system into Q₄ element discretization system.The validity of the method is verified by the numerical test of the hierarchical Q₈ element and hierarchical Q₁₂ element discrete system of gravity dam/arch-abdomen dam.?3?A p-Version adaptive finite element method based on the local error control of "dam toe" and "dam heel" is designed by using ANSYS parametric design language?APDL?.It can keep the initial mesh unchanged and improve the accuracy of calculation by locally increasing the order of elements when the number of elements is small.By applying this method to the finite element calculation and simulation of gravity dam,the numerical results show that this p-Version adaptive finite element method can effectively improve the accuracy of the numerical solution near the "dam toe" and "dam heel",which is very effective for solving the gravity dam problem.