Iterative Solvers for Meshless Petrov Galerkin (MLPG) Method Applied to Large Scale Engineering Problems Challenges

Abstract

In recent years, significant research effort has been invested in development of mesh-free methods for different types of continuum problems. Prominent amongst these methods are element free Galerkin (EFG) method, RKPM, and mesh-less Petrov Galerkin (MLPG) method. Most of these methods employ a set of nodes for discretization of the problem domain, and use a moving least squares (MLS) approximation to generate shape functions. Of these methods, MLPG method is seen as a pure meshless method since it does not require any background mesh. Accuracy and flexibility of MLPG method is well established for a variety of continuum problems. However, most of the applications have been limited to small scale problems solvable on serial machines. Very few attempts have been made to apply it to large scale problems which typically involve many millions (or even billions) of nodes and would require use of parallel algorithms based on domain decomposition. Such parallel techniques are well established in context of mesh-based methods. Extension of these algorithms in conjunction with MLPG method requires considerable further research. Objective of this paper is to spell out these challenges which need urgent attention to enable the application of meshless methods to large scale problems. We specifically address the issue of the solution of large scale linear problems which would necessarily require use of iterative solvers. We focus on application of BiCGSTAB method and an appropriate set of preconditioners for the solution of the MLPG system.