MESHLESS LOCAL PETROV- GALERKIN METHOD FOR PHASE CHANGE PROBLEMS

Abstract

The advent of faster and cheaper computational facility has made numerical methods very useful tool. It has helped engineers and scientists to study complicated physical phenomena which were earlier simplified drastically in the interest of solving them analytically. Since the invention of the finite difference method (FDM), many numerical methods such as the finite volume method (FVM), the finite element method (FEM) and the boundary element method (BEM) have been developed and applied to a wide range of engineering problems. Among all these methods, FEM has enjoyed superiority over others due to its greater flexibility and ease of application to complex geometry. Different formulations of FEM have been applied successfully to solve variety of practical problems. However, the method suffers from some drawbacks. Creation of a good quality mesh is a prerequisite for finite element analysis. This process is still distant from being completely automated. Hence, mesh generation is a time consuming process. Further, in the FEM analysis, the secondary variables such as stresses across the inter-element boundaries are discontinuous, which are required to be smoothened by using some special techniques. Element distortion in the cases of large deformation analysis is a severe problem associated with FEM. The ill-shaped elements resulting from element distortion perform poorly in the analysis. Re-meshing approaches are adopted to handle these types of problems. However, the remeshing and associated interpolation of the current solution onto the new mesh is a tedious process and also leads to degradation of accuracy in the successive stages of evolution. Hence, FEM due to its reliance on mesh, is less suitable for certain class of problems such as problems of large deformation, crack growth and phase change..