Meshless Galerkin Method For Solving Geometrically Nonlinear Problems

The element-free Galerkin method(EFGM) is a significant and promising method for its flexibility in solving a partial differential equation. In FEGM, the shape function is constructed by the moving least square(MLS) approximation, the weak form of the equivalent integral equation to the governing equation is employed and essential boundary conditions are imposed by the penalty function method. The advantages of EFGM are: only nodal data are necessary, high accuracy can be achieved and preprocess and postprocess is easy, etc.The mathematical basis of EFGM is the moving least squares approximation. To use MLS, it is only necessary to construct an array of nodes in the domain under consideration. Moving least square interpolants do not pass through the nodes because the interpolation functions are not equal to unity at the nodes unless the weight functions are singular. This is a disadvantage of EFGM as it suffers from problems in the imposition of essential boundary conditions and the application of point loads. However, these do not disadvantage EFGM significantly.In this thesis, an incremental-iterative solution procedure using the modified Newton-Raphson iteration is used to solve geometrically nonlinear problems. All measures are related back to the original configuration. This technique is named as total Lagrangian method. On the basis of nonlinear continuum mechanics, stress, strain and stress-strain relationship for large deformation are discussed, the expressions of geometrically nonlinear problems and matrix equations of the element free Galerkin method are derived. Then the corresponding computer program is developed and several examples are given to verified the correctness of the formulation and applicability of EFGM in the present thesis. Examples show that the element free Galerkin method still possesses some advantages such as good stability and high rate of convergence in solving geometrically nonlinear problems.