The Improvement And Application Of Meshless Local Petrov-Galerkin Method

Meshless local Pettrov-Galerkin (MLPG) method is a new numerical method. The implementation of integral form with weighted residual method is confined to a very small local sub-domain of the node, the trial and test functions are chosen from different function spaces. Meshless local Pettrov-Galerkin method requires integrations only over a localized quadrature domain, what we need now is only a background mesh of cells for the local quadrature, so it's also a truly meshless method. However, we need to improve meshless local Pettrov-Galerkin method as there're some drawbacks, such as there is an issue of imposition of essential boundary condition due to the lack of Kronecker-δfunction properties in the moving least square approximation (MLS) shape function, as well as the construction of trial function is complex, which limit the development and application of meshless local Pettrov-Galerkin method.First of all, the paper systematically introuduces development of meshless method and the basic theory of meshless local Pettrov-Galerkin method. This section mainly describes the moving least square approximation and the selection of test functions, as well as the imposition of essential boundary conditions. Secondly, the paper mainly describes the basic theory of meshless local Pettrov-Galerkin method, together with a classical mechanics example is presented to validate the method, which proved its accuracy, feasibility and efficiency.Then the paper studies on an improved meshless local Pettrov-Galerkin method. Unlike the general meshless local Pettrov-Galerkin method, natural neighbour interpolation is employed for constructing shape function, and the shape function has Kronecker-δproperties, so we can imposite essential boundary condition directly. Efforts are made to study its application in transient heat conduction, the rationality and feasibility of the method discussed in this paper is proved by several numerical examples.Lastly, a summary of the paper and the future research directions are forecasted.