| Numerical manifold method is a generalized numerical method with high-order accuracy. This method performs numerical computation with finite element cover system, which is composed of two independent cover grids:mathematic cover grid and physical cover grid. In NMM, the fixed mathematical grid in finite element cover system can avoid mesh distortion in large deformation; employing high order functions or analytic functions for physical cover functions can improve the field functions approximation. NMM has some distinct advantages over finite element methods and mesh-free methods for processing of boundary condition and improvement of accuracy of numerical solution. Recently, most studies of NMM are concentrated on the application of NMM and the enhancement of local cover function. Manifold method generally adopts traditional triangular elements and quadrilateral elements based on finite cover system as its manifold elements. They can be easily constructed, but both the number of elements and the amount of calculation are great. Traditional elements can generate self-locked shear problem. MMM based on these elements cannot give reliable numerical solution of complicated geometric shapes. With due consideration of these shortages of traditional elements, this paper provide a new conforming polygonal manifold element. In order to obtaining the form of weight function of polygonal manifold element and generating the polygonal grids, the main studies are made as follows:1. Different kinds of typical polygonal elements are introduced. Deliberate comparison and analysis of these elements are made for the selection of weight function. The factors of selection of weight function for NMM are proposed. The form of weight function is ascertained and the corresponding characteristics of polygonal elements are deduced.2. Deep analysis is made for the characteristics of selected weight function in this paper. Firstly, convergence, stability and continuity of general NMM is analyzed based on finite element interpolation scheme. Secondly, some relevant characteristic of weight function including nonnegativity, the quality of partition of unity and boundary behavior are given. Finally, based on analysis of convergence and stability of NMM, interpolation error in NMM is derived to conclude that the selected polygonal element have the same order accuracy with traditional triangular or quadrilateral elements.3. In order to achieve high-order continuity at the boundary of manifold elements which is composed of triangular elements and quadrilateral elements, further analysis is conducted to derive compatible equation and then process the equation.4. The overall process of generation of physical cover grid is given. Algorithm achievement of Delaunay triangulation coupled with process flow of generation of polygonal elements is also proposed in this paper.5. In view of distinctions of several typical numerical integration methods, Comparison of these methods is made and integration types of NMM are enumerated. Especially, the concrete selection of numerical integration methods for different types of integration in NMM is classified.6. On the basis of analysis of convergence and stability about NMM, further derivation is made to determinate some relevant factors as significant criteria to select optimal local cover function. An improved basis function of local cover function is provided in this paper.The analysis of convergence, stability and continuity about NMM consummates the fundamental theory of NMM. The efficiency of polygonal manifold element provided in this paper has been proved to be reliable. This dissertation introduces a new conception and method to solve complicated problems and is bound to be of great significance in theory and application.
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