A Meshless Local Petrov-Galerkin Method Of Functionally Graded Materials

Functionally Graded Materials (FGMs) can be used in many fields such as aeronautical applications etc. due to their high heat resistance and good anti-corrosion properties over the traditional materials. The thermodynamics behaviors of FGMs have attracted many researchers'attention ever since. Owing to the nonhomogeneous characteristic of FGMs, analytical methods are generally difficult to apply to them, instead, numerical methods are needed for solving FGMs problems. Finite Element Method is the most widely used technique in engineering applications, but the assumption of homogeneous in each element makes it unsuitable for FGMs.The Meshless Method is a new numerical method developed recently based on the Finite Element Method. The meshless local Petrov-Galerkin (MLPG) method is a"truly meshless method"since it doesn't need any finite element or boundary element meshes, and it is suitable for the analysis of nonhomogeneous materials. The MLPG method uses local weak forms over a local sub-domain, and constructs shape functions by the moving least-squares (MLS) approximation. It is very flexible especially for the complex boundary problems. The MLPG method maitains the advantages of Element Free Galerkin method, but is more flexible than the latter. It is suitable for solving FGMs problems by less nodes.In this dissertation, the MLPG method is utilized to analyze the thermodynamics behaviors of FGMs. The history and advances in the research of meshless methods are reviewed comprehensively at first. The several main meshless methods and their characteristics have been commented, and the moving least square (MLS) approximation method is explored. The formats of three dimensional elasticity, steady heat transference, transient heat transference, thermodynamics problems in FGMs are derived. Furthermore, numerical examples of FGMs are worked out. It's found that the results from the MLPG agree well with the FEM results. The method and the program algorithms presented in the paper are confirmed to be effective, and have a good precision by using less nodes than FEM. But further study is needed for extending this method to more complicated problems such as crack growth, elastic-plastic deformation problems etc.