Research Of The Functionally Graded Materials By Complex Variable Meshless Local Petrov-galerkin Method

Functionally gradient material(FGM) is a heterogeneous material whose composition varies continuity in space. The analytical solutions of homogeneous materials and general composite materials are not suitable for FGMs. Numerical methods are usually used in engineering problems of FGMs.The meshless method is a new class of numerical approach for solving partial differential equations. The approximate functions of meshless method are based on a set of scattered nodes in the problem domain regardless of “element”or “mesh”. Meshless method has been demonstrated to be quite successful in dealing with problems of fracture mechanics?large deformation and simulation of manufacturing processes, and has become one of the hot topic in the field of computational mechanics.The meshless local Petrov-Galerkin method(MLPG), which is based on the moving least squares(MLS) method and local weak forms, is a truly meshless approach, since the method does not need a mesh or element either for the approximation of the trial function or for the integral of the local weak form. In the MLPG method, the MLS approximation is used for constructing the field variables, which leads to high computational cost ? ill-conditioned algebra equations and difficulties to impose the essential boundary conditions, thus reduce the computational efficiency of the MLPG method. The advantages of the complex variable moving least squares(CVMLS) approximation are that the trial function of a 2D problems is formed with 1D basis, which leads to the fewer coefficients in trial function, thus improve the computational efficiency.On the basis of the MLPG method, using complex variable theory, the complex variable meshless local Petrov-Galerkin(CVMLPG) method is proposed in this paper. Then the CVMLPG method is applied to solve the mechanical problems of functionally graded materials(FGMs). The outline of this paper is organized as follows:1. The basic theory of the CVMLS method is discussed in details. Some of the parameters, such as weight functions?the effects of radius of support domain,are analyzed. Numerical example, such as the fitting of the curve, is used to verify the effectiveness of the present approach.2. The CVMLPG method for elastic problems of FGMs is presented. The discrete equation is deduced and the effectiveness of the present method is testified by the several examples.3. The CVMLPG method for elastodynamic problems of FGMs is presented. And the corresponding formulae are deduced. Three numerical examples are used to show the availability of the CVMLPG method.4. The CVMLPG method for transient heat conduction problems of FGMs is established. The corresponding discrete equation is deduced. Some illustrative examples are employed to verify the effectiveness of the proposed method.On the basis of the above theory, the corresponding MATLAB codes are developed. All of the numerical examples in this paper are used to show the validity of the presented method.