Research On A Boundary Collocation Meshless Method Based On Taylor Series

Based on Taylor Meshless Method(TMM),the aim of this thesis is to develop a simple,robust,efficient and accurate numerical method which is capable of solving large scale engineering problems and to provide a new idea for the follow-up study on meshless methods.To this end,the influence of the key factors in TMM has been stud-ied by solving three-dimensional and non-linear Partial Differential Equations(PDEs).The main idea of TMM is to use high order polynomials as shape functions which are approximated solutions of the PDE and the discretization concerns only the boundary.To solve the unknown coefficients,boundary conditions are accounted by collocation procedures associated with least-square method.TMM that needs only boundary col-location without integration process,is a true meshless method.The main contributions of the thesis are as following:1)TMM is used to solve two-dimensional PDEs,where the influence of the key factors on the accuracy of solu-tion has been studied through the obtained results.The efficiency and robustness of the algorithm have been verified;2)Based on TMM,a general and efficient algorithm has been developed for solving three-dimensional PDEs;3)Three coupling techniques in piecewise resolutions have been discussed and tested in cases of large-scale problems,including least-square collocation method and two coupling methods based on Lagrange multipliers;4)A general numerical method for solving non-linear PDEs has been pro-posed by combining Newton Method,TMM and Automatic Differentiation technique;5)To apply TMM for solving problems with singularties,the singular solutions sat-isfying the control equation are introduced as complementary shape functions,which provides a theoretical basis for solving singular problems.