High Precision Numerical Method And Its Application

Meshless method is a new kind of numerical method in recent score years. It can avoid the grid-dependent, and the shape function is of higher order continuity.So, meshless method has advantages in solving bending problems of the laminated plates. A group of meshfree methods have been proposed and developed. Among them,the element-free Galerkin(EFG) method is a very promising method and currently widely used in computational mechanics and other scientific and engineering areas.The high accuracy and stability of the EFG method has attracted a lot of researchers.A disadvantage of the MLS approximation is that the shape functions obtained are lack of Kronecker delta function property. As a consequence, the imposition of essential boundary conditions in the EFG method is quite awkward. Therefore, the meshless radial point interpolation method(RPIM) or RPIM Galerkin method meshless method has gradually received attention. A point interpolation meshless method based on radial basis functions is a method combines the Galerkin weak form and radial basis functions to form a meshfree radial point interpolation method(RPIM). In fact, it is a special case of the EFG method. In the RPIM method, it uses the radial point interpolation method to construct the shape functions. This is the major di?erence between the EFG method and the RPIM method. It has been studied that the RPIM method has more computational cost than the EFG method, due to the dimension of the moment matrix for every computational point is much bigger than that obtained by the MLS approximation in the EFG method. In this paper, a novel stratified interpolation meshless(SI) method is proposed. In this method, the shape function is divided into a plurality of approximation process with the last approximation has Kronecker property and the other have C0-consistency. Then, the shape functions obtained by the new scheme preserves the Kronecker delta function property in certain conditions. The new method has much more accuracy than the RPIM method, even it also has less time consuming than the RPIM method, since the number of nodes is generally much smaller than the number of integration points. Some numerical examples are illustrate the e?ectiveness of the proposed method.We also presents another scheme to reduce the CPU time of the RPIM meshless method: Reconstructing gauss domain meshless method(RGD). RGD is divided into two types: Reordering gauss point meshless method(RGP) and Reconstructing gauss domain meshless method(RGD). RGP is to find the gauss points with the same nodes,then the gauss points own one distance matrix and save time. However, it requires a search program to find the gauss points. This program consumes huge time in large scale problems, So we present the RGD meshless method. RGD specify the contact between the gauss points and the nodes to save the searching program. At the same time, the article gives the theoretical ratio of the method of time e?ciency, and some numerical examples verify this point.However, MLS and RPIM in the numerical process appears a series of singular problems(mainly concentrated in the rank deficient of the moment). So, we have to consider a new interpolation scheme and then expand a new conditionally positive definite function. Numerical experiments show that the reliability of this method. Based on the shape functions in meshless mainstream way are more or less defects, we have to look for a kind of interpolations compatible with the problem domain. Barycentric Lagrange interpolation gives us a hint. We can sacrifice freedom of node position and get the stability and high precision of the interpolation. Experiments show the trade-o?s in common areas(such as: square, parallelogram,and so on) is feasible and this is the most original spectral method. This paper study the in-plane sti?ness variable rectangular plate with the spectral collocation method, and obtain good results.At the same time, in order to better study the radial variable sti?ness of thin circular plate. We analyze from two aspects: One is through the polar coordinate theory,and through regularization conditions corresponding center conditions, Form a four order di?erential equation with variable coe?cient; Another is the use of original twodimensional theory. To overcome the singular case of thin circular plate stress boundary, we propose stratified spectral interpolation method. At the same time, this paper also used spectral-Galerkin method for structural analysis of composite laminates. Coupled higher-order shear and normal deformable plate theory(HOSNDPT) can well represent distribution of laminated plate force. The paper also presents the orthogonal higherorder shear and normal deformable plate theory(OHOSNDPT) to analyze the plate with arbitrarily variable thickness. The numerical results show that the thickness to span ratio is 1/1000, the shear stress of τxyalso won’t appear self-locking phenomenon.