The Efficient Finite Element Methods And Fast Solvers For Thin-walled Structural Heat Conduction Problems In Three Dimensions

Because of some characteristics,thin-walled structures have been widely used in many kinds of engineering,such as coating,multi-layered structures and the heat-dissipation problem of electron device.It is difficult to analyze the thin-walled structures since its one dimension is different from two other dimensions obviously.Thus,numerical analysis method is necessary,such as the finite element method.When the finite element method is applied to analyze three-dimensional thin-walled structures,some thin hexahedral elements are usually used in order to reduce the number of elements,and the corresponding higher-order elements are preferred since they have some obvious advantages in the calculation accuracy,the degree of anti-distortion and so on.But compared with low-order elements,higher order elements need more computer storage space,the computational complexity of discrete linear systems is higher,and the coefficient matrix is also severely ill-conditioned.This leads to low efficiency for some commonly used methods.Firstly,an efficient and robust multi-level method is presented in this master thesis for the hierarchical quadratic discretizations of three-dimensional thin-walled structures by combining two special local block Gauss-Seidel smoothers and the DAMG algorithm based on the distance matrix.Since a hierarchical basis is used,we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types,and moreover the grid transfer operators are also trivial.This makes it easy to find the coarse level(linear element)matrix derived directly from the fine level matrix,and thus the overall efficiency is greatly improved.The numerical results verify the efficiency and robustness of the proposed method.For other higher-order elements as cube elements,the efficiency of the resulting multi-level method will be reduced due to the large computational complexity,and thus it is not suitable for large-scale practical problems.In the second part of this thesis,Wilson nonconforming elements are used to analyze thin-walled structural heat conduction problems in three-dimensions.The Wilson elements improve the degree of polynomial by setting extra degrees of freedom in the element considered,and have the advantages of few degrees of freedom,high accuracy and so on.Thus,they are widely used in practical computation.Two kinds of computational format of Wilson element are presented forthin-walled structural heat conduction problems in three dimensions with the general variable coefficients.Although the temperature is not consistent on the element boundary,the Wilson discrete system is transformed into the corresponding system of eight-node linear elements or twenty-node quadratic elements by using the internal condensation method.Compared with higher-order hexahedral elements,Wilson elements which do not have nodes on the edges or faces reduce computer storage space and the computational complexity largely when they are used in finite element methods.Then the DAMG method,which is suitable for the anisotropic mesh problems,is applied to the discrete system of eight-node Wilson elements.Compared to those commonly used methods,the DAMG method has better efficiency and robustness.This presents a fast solver for solving thin-walled structural heat conduction problems.Finally,some kinds of numerical examples are analyzed and computed,and the corresponding results verify the high efficiency of the proposed method.