A Study On The Meshless Local Petrov-Galerkin Method For The Kirchhoff Plate

Meshless methods possess many outstanding advantages over traditional numerical methods such as finite element method, boundary element method et al. A lot of the important pioneering effort has been done on meshless methods by scholars in a recent decade, and further investigations into many key problems of meshless methods should be made.The meshless local Petrov-Galerkin(MLPG) method is a new numerical technique presented in the recent years, for it doesn't need any finite element or boundary element meshes, no matter meshes for the use of energy integral or for the purpose of interpolation, it can analyze the problem flexibly and conveniently and is named as "a truly meshless method" with the great applied prospect. In the recent years, Atluri and Long SY et al have made a lot of investigations on the MLPG approach and its applications. On the basis of their work, applications of the MLPG method to the Kirchhoff plate are presented in this dissertation, which enriches theory of the MLPG and extends applications of the MLPG.At the beginning of the dissertation, recent developments of meshless methods are briefly reviewed. All kinds of meshless methods are reviewed and commented. Characteristics, advantages and disadvantages for a variety of meshless methods are pointed out. Among which, the research and development on meshless methods of plate and shell problems are reviewed in detail. Then, the basic equations of the plate based on the Kirchhoff hypotheses are formulated, and the moving least square (MLS) approximation for the solution variable of the Kirchhoff plate are presented.In the dissertation, the MLPG method to the static problem of the Kirchhoff plate is proposed. The symmetric weak forms of the equivalent integration equations to the governing differential equations for the isotropic, anisotropic plates and the plate on the elastic foundation in local subdomains are formulated by the weighted residual method (WRM). The deflection is interpolated by using the moving least squares (MLS) approximation. All integrals are carried out in subdomains and their boundaries with a regular shape. For the moving least square approximation does not possess properties of the Kronecker-delta function, the essential boundary condition can not be enforced directly, the penalty method is used to imposed the essential boundary condition. In the numerical implementation, the unsymmetrical linear system is solved by the iterative algorithm of the generalized minimal residual...